From 5f4fb67faefa155b3d212629bf2c46e3861bae96 Mon Sep 17 00:00:00 2001 From: martinholters Date: Wed, 4 Dec 2024 08:41:18 +0000 Subject: [PATCH] Delete preview of PR#604 --- previews/PR604/.documenter-siteinfo.json | 1 - previews/PR604/appendix/index.html | 2 - previews/PR604/assets/documenter.js | 1082 ----------------- .../PR604/assets/themes/catppuccin-frappe.css | 1 - .../PR604/assets/themes/catppuccin-latte.css | 1 - .../assets/themes/catppuccin-macchiato.css | 1 - .../PR604/assets/themes/catppuccin-mocha.css | 1 - .../PR604/assets/themes/documenter-dark.css | 7 - .../PR604/assets/themes/documenter-light.css | 9 - previews/PR604/assets/themeswap.js | 84 -- previews/PR604/assets/warner.js | 52 - previews/PR604/convolutions/index.html | 9 - previews/PR604/estimation/index.html | 15 - previews/PR604/filters/index.html | 68 -- previews/PR604/index.html | 2 - previews/PR604/internals/index.html | 10 - previews/PR604/lpc/index.html | 8 - previews/PR604/objects.inv | Bin 1959 -> 0 bytes previews/PR604/periodograms/index.html | 201 --- previews/PR604/search_index.js | 3 - previews/PR604/siteinfo.js | 1 - previews/PR604/util/index.html | 31 - previews/PR604/windows/index.html | 294 ----- 23 files changed, 1883 deletions(-) delete mode 100644 previews/PR604/.documenter-siteinfo.json delete mode 100644 previews/PR604/appendix/index.html delete mode 100644 previews/PR604/assets/documenter.js delete mode 100644 previews/PR604/assets/themes/catppuccin-frappe.css delete mode 100644 previews/PR604/assets/themes/catppuccin-latte.css delete mode 100644 previews/PR604/assets/themes/catppuccin-macchiato.css delete mode 100644 previews/PR604/assets/themes/catppuccin-mocha.css delete mode 100644 previews/PR604/assets/themes/documenter-dark.css delete mode 100644 previews/PR604/assets/themes/documenter-light.css delete mode 100644 previews/PR604/assets/themeswap.js delete mode 100644 previews/PR604/assets/warner.js delete mode 100644 previews/PR604/convolutions/index.html delete mode 100644 previews/PR604/estimation/index.html delete mode 100644 previews/PR604/filters/index.html delete mode 100644 previews/PR604/index.html delete mode 100644 previews/PR604/internals/index.html delete mode 100644 previews/PR604/lpc/index.html delete mode 100644 previews/PR604/objects.inv delete mode 100644 previews/PR604/periodograms/index.html delete mode 100644 previews/PR604/search_index.js delete mode 100644 previews/PR604/siteinfo.js delete mode 100644 previews/PR604/util/index.html delete mode 100644 previews/PR604/windows/index.html diff --git a/previews/PR604/.documenter-siteinfo.json b/previews/PR604/.documenter-siteinfo.json deleted file mode 100644 index 6b35610f9..000000000 --- a/previews/PR604/.documenter-siteinfo.json +++ /dev/null @@ -1 +0,0 @@ -{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-04T08:23:37","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/previews/PR604/appendix/index.html b/previews/PR604/appendix/index.html deleted file mode 100644 index 2e33a1e93..000000000 --- a/previews/PR604/appendix/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Index · DSP.jl
GitHub

Index

diff --git a/previews/PR604/assets/documenter.js b/previews/PR604/assets/documenter.js deleted file mode 100644 index 7d68cd808..000000000 --- a/previews/PR604/assets/documenter.js +++ /dev/null @@ -1,1082 +0,0 @@ -// Generated by Documenter.jl -requirejs.config({ - paths: { - 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', - 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', - 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', - 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', - 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', - 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', - 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', - 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', - 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', - }, - shim: { - "highlight-julia": { - "deps": [ - "highlight" - ] - }, - "katex-auto-render": { - "deps": [ - "katex" - ] - }, - "headroom-jquery": { - "deps": [ - "jquery", - "headroom" - ] - }, - "highlight-julia-repl": { - "deps": [ - "highlight" - ] - } -} -}); -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { -$(document).ready(function() { - renderMathInElement( - document.body, - { - "delimiters": [ - { - "left": "$", - "right": "$", - "display": false - }, - { - "left": "$$", - "right": "$$", - "display": true - }, - { - "left": "\\[", - "right": "\\]", - "display": true - } - ] -} - - ); -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { -$(document).ready(function() { - hljs.highlightAll(); -}) - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -let timer = 0; -var isExpanded = true; - -$(document).on( - "click", - ".docstring .docstring-article-toggle-button", - function () { - let articleToggleTitle = "Expand docstring"; - const parent = $(this).parent(); - - debounce(() => { - if (parent.siblings("section").is(":visible")) { - parent - .find("a.docstring-article-toggle-button") - .removeClass("fa-chevron-down") - .addClass("fa-chevron-right"); - } else { - parent - .find("a.docstring-article-toggle-button") - .removeClass("fa-chevron-right") - .addClass("fa-chevron-down"); - - articleToggleTitle = "Collapse docstring"; - } - - parent - .children(".docstring-article-toggle-button") - .prop("title", articleToggleTitle); - parent.siblings("section").slideToggle(); - }); - } -); - -$(document).on("click", ".docs-article-toggle-button", function (event) { - let articleToggleTitle = "Expand docstring"; - let navArticleToggleTitle = "Expand all docstrings"; - let animationSpeed = event.noToggleAnimation ? 0 : 400; - - debounce(() => { - if (isExpanded) { - $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); - $("a.docstring-article-toggle-button") - .removeClass("fa-chevron-down") - .addClass("fa-chevron-right"); - - isExpanded = false; - - $(".docstring section").slideUp(animationSpeed); - } else { - $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); - $("a.docstring-article-toggle-button") - .removeClass("fa-chevron-right") - .addClass("fa-chevron-down"); - - isExpanded = true; - articleToggleTitle = "Collapse docstring"; - navArticleToggleTitle = "Collapse all docstrings"; - - $(".docstring section").slideDown(animationSpeed); - } - - $(this).prop("title", navArticleToggleTitle); - $(".docstring-article-toggle-button").prop("title", articleToggleTitle); - }); -}); - -function debounce(callback, timeout = 300) { - if (Date.now() - timer > timeout) { - callback(); - } - - clearTimeout(timer); - - timer = Date.now(); -} - -}) -//////////////////////////////////////////////////////////////////////////////// -require([], function() { -function addCopyButtonCallbacks() { - for (const el of document.getElementsByTagName("pre")) { - const button = document.createElement("button"); - button.classList.add("copy-button", "fa-solid", "fa-copy"); - button.setAttribute("aria-label", "Copy this code block"); - button.setAttribute("title", "Copy"); - - el.appendChild(button); - - const success = function () { - button.classList.add("success", "fa-check"); - button.classList.remove("fa-copy"); - }; - - const failure = function () { - button.classList.add("error", "fa-xmark"); - button.classList.remove("fa-copy"); - }; - - button.addEventListener("click", function () { - copyToClipboard(el.innerText).then(success, failure); - - setTimeout(function () { - button.classList.add("fa-copy"); - button.classList.remove("success", "fa-check", "fa-xmark"); - }, 5000); - }); - } -} - -function copyToClipboard(text) { - // clipboard API is only available in secure contexts - if (window.navigator && window.navigator.clipboard) { - return window.navigator.clipboard.writeText(text); - } else { - return new Promise(function (resolve, reject) { - try { - const el = document.createElement("textarea"); - el.textContent = text; - el.style.position = "fixed"; - el.style.opacity = 0; - document.body.appendChild(el); - el.select(); - document.execCommand("copy"); - - resolve(); - } catch (err) { - reject(err); - } finally { - document.body.removeChild(el); - } - }); - } -} - -if (document.readyState === "loading") { - document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); -} else { - addCopyButtonCallbacks(); -} - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { - -// Manages the top navigation bar (hides it when the user starts scrolling down on the -// mobile). -window.Headroom = Headroom; // work around buggy module loading? -$(document).ready(function () { - $("#documenter .docs-navbar").headroom({ - tolerance: { up: 10, down: 10 }, - }); -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -$(document).ready(function () { - let meta = $("div[data-docstringscollapsed]").data(); - - if (meta?.docstringscollapsed) { - $("#documenter-article-toggle-button").trigger({ - type: "click", - noToggleAnimation: true, - }); - } -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -/* -To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc - -PSEUDOCODE: - -Searching happens automatically as the user types or adjusts the selected filters. -To preserve responsiveness, as much as possible of the slow parts of the search are done -in a web worker. Searching and result generation are done in the worker, and filtering and -DOM updates are done in the main thread. The filters are in the main thread as they should -be very quick to apply. This lets filters be changed without re-searching with minisearch -(which is possible even if filtering is on the worker thread) and also lets filters be -changed _while_ the worker is searching and without message passing (neither of which are -possible if filtering is on the worker thread) - -SEARCH WORKER: - -Import minisearch - -Build index - -On message from main thread - run search - find the first 200 unique results from each category, and compute their divs for display - note that this is necessary and sufficient information for the main thread to find the - first 200 unique results from any given filter set - post results to main thread - -MAIN: - -Launch worker - -Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) - -On text update - if worker is not running, launch_search() - -launch_search - set worker_is_running to true, set last_search_text to the search text - post the search query to worker - -on message from worker - if last_search_text is not the same as the text in the search field, - the latest search result is not reflective of the latest search query, so update again - launch_search() - otherwise - set worker_is_running to false - - regardless, display the new search results to the user - save the unfiltered_results as a global - update_search() - -on filter click - adjust the filter selection - update_search() - -update_search - apply search filters by looping through the unfiltered_results and finding the first 200 - unique results that match the filters - - Update the DOM -*/ - -/////// SEARCH WORKER /////// - -function worker_function(documenterSearchIndex, documenterBaseURL, filters) { - importScripts( - "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" - ); - - let data = documenterSearchIndex.map((x, key) => { - x["id"] = key; // minisearch requires a unique for each object - return x; - }); - - // list below is the lunr 2.1.3 list minus the intersect with names(Base) - // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) - // ideally we'd just filter the original list but it's not available as a variable - const stopWords = new Set([ - "a", - "able", - "about", - "across", - "after", - "almost", - "also", - "am", - "among", - "an", - "and", - "are", - "as", - "at", - "be", - "because", - "been", - "but", - "by", - "can", - "cannot", - "could", - "dear", - "did", - "does", - "either", - "ever", - "every", - "from", - "got", - "had", - "has", - "have", - "he", - "her", - "hers", - "him", - "his", - "how", - "however", - "i", - "if", - "into", - "it", - "its", - "just", - "least", - "like", - "likely", - "may", - "me", - "might", - "most", - "must", - "my", - "neither", - "no", - "nor", - "not", - "of", - "off", - "often", - "on", - "or", - "other", - "our", - "own", - "rather", - "said", - "say", - "says", - "she", - "should", - "since", - "so", - "some", - "than", - "that", - "the", - "their", - "them", - "then", - "there", - "these", - "they", - "this", - "tis", - "to", - "too", - "twas", - "us", - "wants", - "was", - "we", - "were", - "what", - "when", - "who", - "whom", - "why", - "will", - "would", - "yet", - "you", - "your", - ]); - - let index = new MiniSearch({ - fields: ["title", "text"], // fields to index for full-text search - storeFields: ["location", "title", "text", "category", "page"], // fields to return with results - processTerm: (term) => { - let word = stopWords.has(term) ? null : term; - if (word) { - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - word = word - .replace(/^[^a-zA-Z0-9@!]+/, "") - .replace(/[^a-zA-Z0-9@!]+$/, ""); - - word = word.toLowerCase(); - } - - return word ?? null; - }, - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not - // find anything if searching for "add!", only for the entire qualification - tokenize: (string) => string.split(/[\s\-\.]+/), - // options which will be applied during the search - searchOptions: { - prefix: true, - boost: { title: 100 }, - fuzzy: 2, - }, - }); - - index.addAll(data); - - /** - * Used to map characters to HTML entities. - * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts - */ - const htmlEscapes = { - "&": "&", - "<": "<", - ">": ">", - '"': """, - "'": "'", - }; - - /** - * Used to match HTML entities and HTML characters. - * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts - */ - const reUnescapedHtml = /[&<>"']/g; - const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); - - /** - * Escape function from lodash - * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts - */ - function escape(string) { - return string && reHasUnescapedHtml.test(string) - ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) - : string || ""; - } - - /** - * RegX escape function from MDN - * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping - */ - function escapeRegExp(string) { - return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - } - - /** - * Make the result component given a minisearch result data object and the value - * of the search input as queryString. To view the result object structure, refer: - * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult - * - * @param {object} result - * @param {string} querystring - * @returns string - */ - function make_search_result(result, querystring) { - let search_divider = `
`; - let display_link = - result.location.slice(Math.max(0), Math.min(50, result.location.length)) + - (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div - - if (result.page !== "") { - display_link += ` (${result.page})`; - } - searchstring = escapeRegExp(querystring); - let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); - let text = - textindex !== null - ? result.text.slice( - Math.max(textindex.index - 100, 0), - Math.min( - textindex.index + querystring.length + 100, - result.text.length - ) - ) - : ""; // cut-off text before and after from the match - - text = text.length ? escape(text) : ""; - - let display_result = text.length - ? "..." + - text.replace( - new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence - '$&' - ) + - "..." - : ""; // highlights the match - - let in_code = false; - if (!["page", "section"].includes(result.category.toLowerCase())) { - in_code = true; - } - - // We encode the full url to escape some special characters which can lead to broken links - let result_div = ` - -
-
${escape(result.title)}
-
${result.category}
-
-

- ${display_result} -

-
- ${display_link} -
- - ${search_divider} - `; - - return result_div; - } - - self.onmessage = function (e) { - let query = e.data; - let results = index.search(query, { - filter: (result) => { - // Only return relevant results - return result.score >= 1; - }, - combineWith: "AND", - }); - - // Pre-filter to deduplicate and limit to 200 per category to the extent - // possible without knowing what the filters are. - let filtered_results = []; - let counts = {}; - for (let filter of filters) { - counts[filter] = 0; - } - let present = {}; - - for (let result of results) { - cat = result.category; - cnt = counts[cat]; - if (cnt < 200) { - id = cat + "---" + result.location; - if (present[id]) { - continue; - } - present[id] = true; - filtered_results.push({ - location: result.location, - category: cat, - div: make_search_result(result, query), - }); - } - } - - postMessage(filtered_results); - }; -} - -/////// SEARCH MAIN /////// - -function runSearchMainCode() { - // `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! - const filters = [ - ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), - ]; - const worker_str = - "(" + - worker_function.toString() + - ")(" + - JSON.stringify(documenterSearchIndex["docs"]) + - "," + - JSON.stringify(documenterBaseURL) + - "," + - JSON.stringify(filters) + - ")"; - const worker_blob = new Blob([worker_str], { type: "text/javascript" }); - const worker = new Worker(URL.createObjectURL(worker_blob)); - - // Whether the worker is currently handling a search. This is a boolean - // as the worker only ever handles 1 or 0 searches at a time. - var worker_is_running = false; - - // The last search text that was sent to the worker. This is used to determine - // if the worker should be launched again when it reports back results. - var last_search_text = ""; - - // The results of the last search. This, in combination with the state of the filters - // in the DOM, is used compute the results to display on calls to update_search. - var unfiltered_results = []; - - // Which filter is currently selected - var selected_filter = ""; - - $(document).on("input", ".documenter-search-input", function (event) { - if (!worker_is_running) { - launch_search(); - } - }); - - function launch_search() { - worker_is_running = true; - last_search_text = $(".documenter-search-input").val(); - worker.postMessage(last_search_text); - } - - worker.onmessage = function (e) { - if (last_search_text !== $(".documenter-search-input").val()) { - launch_search(); - } else { - worker_is_running = false; - } - - unfiltered_results = e.data; - update_search(); - }; - - $(document).on("click", ".search-filter", function () { - if ($(this).hasClass("search-filter-selected")) { - selected_filter = ""; - } else { - selected_filter = $(this).text().toLowerCase(); - } - - // This updates search results and toggles classes for UI: - update_search(); - }); - - /** - * Make/Update the search component - */ - function update_search() { - let querystring = $(".documenter-search-input").val(); - - if (querystring.trim()) { - if (selected_filter == "") { - results = unfiltered_results; - } else { - results = unfiltered_results.filter((result) => { - return selected_filter == result.category.toLowerCase(); - }); - } - - let search_result_container = ``; - let modal_filters = make_modal_body_filters(); - let search_divider = `
`; - - if (results.length) { - let links = []; - let count = 0; - let search_results = ""; - - for (var i = 0, n = results.length; i < n && count < 200; ++i) { - let result = results[i]; - if (result.location && !links.includes(result.location)) { - search_results += result.div; - count++; - links.push(result.location); - } - } - - if (count == 1) { - count_str = "1 result"; - } else if (count == 200) { - count_str = "200+ results"; - } else { - count_str = count + " results"; - } - let result_count = `
${count_str}
`; - - search_result_container = ` -
- ${modal_filters} - ${search_divider} - ${result_count} -
- ${search_results} -
-
- `; - } else { - search_result_container = ` -
- ${modal_filters} - ${search_divider} -
0 result(s)
-
-
No result found!
- `; - } - - if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { - $(".search-modal-card-body").removeClass("is-justify-content-center"); - } - - $(".search-modal-card-body").html(search_result_container); - } else { - if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { - $(".search-modal-card-body").addClass("is-justify-content-center"); - } - - $(".search-modal-card-body").html(` -
Type something to get started!
- `); - } - } - - /** - * Make the modal filter html - * - * @returns string - */ - function make_modal_body_filters() { - let str = filters - .map((val) => { - if (selected_filter == val.toLowerCase()) { - return `${val}`; - } else { - return `${val}`; - } - }) - .join(""); - - return ` -
- Filters: - ${str} -
`; - } -} - -function waitUntilSearchIndexAvailable() { - // It is possible that the documenter.js script runs before the page - // has finished loading and documenterSearchIndex gets defined. - // So we need to wait until the search index actually loads before setting - // up all the search-related stuff. - if (typeof documenterSearchIndex !== "undefined") { - runSearchMainCode(); - } else { - console.warn("Search Index not available, waiting"); - setTimeout(waitUntilSearchIndexAvailable, 1000); - } -} - -// The actual entry point to the search code -waitUntilSearchIndexAvailable(); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// Modal settings dialog -$(document).ready(function () { - var settings = $("#documenter-settings"); - $("#documenter-settings-button").click(function () { - settings.toggleClass("is-active"); - }); - // Close the dialog if X is clicked - $("#documenter-settings button.delete").click(function () { - settings.removeClass("is-active"); - }); - // Close dialog if ESC is pressed - $(document).keyup(function (e) { - if (e.keyCode == 27) settings.removeClass("is-active"); - }); -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -$(document).ready(function () { - let search_modal_header = ` -
-
-

- - - - -

-
-
- -
-
- `; - - let initial_search_body = ` -
Type something to get started!
- `; - - let search_modal_footer = ` - - `; - - $(document.body).append( - ` - - ` - ); - - document.querySelector(".docs-search-query").addEventListener("click", () => { - openModal(); - }); - - document - .querySelector(".close-search-modal") - .addEventListener("click", () => { - closeModal(); - }); - - $(document).on("click", ".search-result-link", function () { - closeModal(); - }); - - document.addEventListener("keydown", (event) => { - if ((event.ctrlKey || event.metaKey) && event.key === "/") { - openModal(); - } else if (event.key === "Escape") { - closeModal(); - } - - return false; - }); - - // Functions to open and close a modal - function openModal() { - let searchModal = document.querySelector("#search-modal"); - - searchModal.classList.add("is-active"); - document.querySelector(".documenter-search-input").focus(); - } - - function closeModal() { - let searchModal = document.querySelector("#search-modal"); - let initial_search_body = ` -
Type something to get started!
- `; - - searchModal.classList.remove("is-active"); - document.querySelector(".documenter-search-input").blur(); - - if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { - $(".search-modal-card-body").addClass("is-justify-content-center"); - } - - $(".documenter-search-input").val(""); - $(".search-modal-card-body").html(initial_search_body); - } - - document - .querySelector("#search-modal .modal-background") - .addEventListener("click", () => { - closeModal(); - }); -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// Manages the showing and hiding of the sidebar. -$(document).ready(function () { - var sidebar = $("#documenter > .docs-sidebar"); - var sidebar_button = $("#documenter-sidebar-button"); - sidebar_button.click(function (ev) { - ev.preventDefault(); - sidebar.toggleClass("visible"); - if (sidebar.hasClass("visible")) { - // Makes sure that the current menu item is visible in the sidebar. - $("#documenter .docs-menu a.is-active").focus(); - } - }); - $("#documenter > .docs-main").bind("click", function (ev) { - if ($(ev.target).is(sidebar_button)) { - return; - } - if (sidebar.hasClass("visible")) { - sidebar.removeClass("visible"); - } - }); -}); - -// Resizes the package name / sitename in the sidebar if it is too wide. -// Inspired by: https://github.com/davatron5000/FitText.js -$(document).ready(function () { - e = $("#documenter .docs-autofit"); - function resize() { - var L = parseInt(e.css("max-width"), 10); - var L0 = e.width(); - if (L0 > L) { - var h0 = parseInt(e.css("font-size"), 10); - e.css("font-size", (L * h0) / L0); - // TODO: make sure it survives resizes? - } - } - // call once and then register events - resize(); - $(window).resize(resize); - $(window).on("orientationchange", resize); -}); - -// Scroll the navigation bar to the currently selected menu item -$(document).ready(function () { - var sidebar = $("#documenter .docs-menu").get(0); - var active = $("#documenter .docs-menu .is-active").get(0); - if (typeof active !== "undefined") { - sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; - } -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// Theme picker setup -$(document).ready(function () { - // onchange callback - $("#documenter-themepicker").change(function themepick_callback(ev) { - var themename = $("#documenter-themepicker option:selected").attr("value"); - if (themename === "auto") { - // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); - window.localStorage.removeItem("documenter-theme"); - } else { - // set_theme(themename); - window.localStorage.setItem("documenter-theme", themename); - } - // We re-use the global function from themeswap.js to actually do the swapping. - set_theme_from_local_storage(); - }); - - // Make sure that the themepicker displays the correct theme when the theme is retrieved - // from localStorage - if (typeof window.localStorage !== "undefined") { - var theme = window.localStorage.getItem("documenter-theme"); - if (theme !== null) { - $("#documenter-themepicker option").each(function (i, e) { - e.selected = e.value === theme; - }); - } - } -}); - -}) -//////////////////////////////////////////////////////////////////////////////// -require(['jquery'], function($) { - -// update the version selector with info from the siteinfo.js and ../versions.js files -$(document).ready(function () { - // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the - // siteinfo.js file, we just return immediately and not display the version selector. - if ( - typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && - DOCUMENTER_VERSION_SELECTOR_DISABLED - ) { - return; - } - - var version_selector = $("#documenter .docs-version-selector"); - var version_selector_select = $("#documenter .docs-version-selector select"); - - version_selector_select.change(function (x) { - target_href = version_selector_select - .children("option:selected") - .get(0).value; - window.location.href = target_href; - }); - - // add the current version to the selector based on siteinfo.js, but only if the selector is empty - if ( - typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && - $("#version-selector > option").length == 0 - ) { - var option = $( - "" - ); - version_selector_select.append(option); - } - - if (typeof DOC_VERSIONS !== "undefined") { - var existing_versions = version_selector_select.children("option"); - var existing_versions_texts = existing_versions.map(function (i, x) { - return x.text; - }); - DOC_VERSIONS.forEach(function (each) { - var version_url = documenterBaseURL + "/../" + each + "/"; - var existing_id = $.inArray(each, existing_versions_texts); - // if not already in the version selector, add it as a new option, - // otherwise update the old option with the URL and enable it - if (existing_id == -1) { - var option = $( - "" - ); - version_selector_select.append(option); - } else { - var option = existing_versions[existing_id]; - option.value = version_url; - option.disabled = false; - } - }); - } - - // only show the version selector if the selector has been populated - if (version_selector_select.children("option").length > 0) { - version_selector.toggleClass("visible"); - } -}); - -}) diff --git a/previews/PR604/assets/themes/catppuccin-frappe.css b/previews/PR604/assets/themes/catppuccin-frappe.css deleted file mode 100644 index 32e3f0082..000000000 --- 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.button.is-primary.is-light.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-primary.is-light:active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-frappe .button.is-primary.is-light.is-active,html.theme--catppuccin-frappe .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:hover,html.theme--catppuccin-frappe .button.is-link.is-hovered{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-link:focus,html.theme--catppuccin-frappe 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe 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.button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe 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.button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text 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a.navbar-item:focus,html.theme--catppuccin-frappe a.navbar-item:focus-within,html.theme--catppuccin-frappe a.navbar-item:hover,html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link:focus,html.theme--catppuccin-frappe .navbar-link:focus-within,html.theme--catppuccin-frappe .navbar-link:hover,html.theme--catppuccin-frappe .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#8caaee}html.theme--catppuccin-frappe .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .navbar-item img{max-height:1.75rem}html.theme--catppuccin-frappe .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-frappe .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-frappe .navbar-item.is-tab:focus,html.theme--catppuccin-frappe .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee}html.theme--catppuccin-frappe .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#8caaee;border-bottom-style:solid;border-bottom-width:3px;color:#8caaee;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-frappe .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-frappe .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-frappe .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-frappe .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-frappe .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-frappe 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a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#838ba7}html.theme--catppuccin-frappe .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#8caaee}.navbar.is-spaced html.theme--catppuccin-frappe .navbar-dropdown,html.theme--catppuccin-frappe .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-frappe .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-frappe .navbar-divider{display:block}html.theme--catppuccin-frappe .navbar>.container .navbar-brand,html.theme--catppuccin-frappe .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-frappe .navbar>.container .navbar-menu,html.theme--catppuccin-frappe .container>.navbar 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body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-frappe a.navbar-item.is-active,html.theme--catppuccin-frappe .navbar-link.is-active{color:#8caaee}html.theme--catppuccin-frappe a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-frappe .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-frappe .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-frappe .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-frappe .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-frappe .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-frappe .pagination.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-frappe .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-previous,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-frappe .pagination.is-rounded .pagination-next,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-frappe .pagination.is-rounded .pagination-link,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-frappe .pagination,html.theme--catppuccin-frappe .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link{border-color:#626880;color:#8caaee;min-width:2.5em}html.theme--catppuccin-frappe .pagination-previous:hover,html.theme--catppuccin-frappe .pagination-next:hover,html.theme--catppuccin-frappe .pagination-link:hover{border-color:#737994;color:#99d1db}html.theme--catppuccin-frappe .pagination-previous:focus,html.theme--catppuccin-frappe .pagination-next:focus,html.theme--catppuccin-frappe .pagination-link:focus{border-color:#737994}html.theme--catppuccin-frappe .pagination-previous:active,html.theme--catppuccin-frappe .pagination-next:active,html.theme--catppuccin-frappe .pagination-link:active{box-shadow:inset 0 1px 2px 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768px){html.theme--catppuccin-frappe .pagination{flex-wrap:wrap}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--catppuccin-frappe .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis{margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination-previous{order:2}html.theme--catppuccin-frappe .pagination-next{order:3}html.theme--catppuccin-frappe .pagination{justify-content:space-between;margin-bottom:0;margin-top:0}html.theme--catppuccin-frappe .pagination.is-centered .pagination-previous{order:1}html.theme--catppuccin-frappe .pagination.is-centered 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a.is-active{border-bottom-color:#0a0a0a}html.theme--catppuccin-frappe .panel.is-black .panel-block.is-active .panel-icon{color:#0a0a0a}html.theme--catppuccin-frappe .panel.is-light .panel-heading{background-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .panel.is-light .panel-tabs a.is-active{border-bottom-color:#f5f5f5}html.theme--catppuccin-frappe .panel.is-light .panel-block.is-active .panel-icon{color:#f5f5f5}html.theme--catppuccin-frappe .panel.is-dark .panel-heading,html.theme--catppuccin-frappe .content kbd.panel .panel-heading{background-color:#414559;color:#fff}html.theme--catppuccin-frappe .panel.is-dark .panel-tabs a.is-active,html.theme--catppuccin-frappe .content kbd.panel .panel-tabs a.is-active{border-bottom-color:#414559}html.theme--catppuccin-frappe .panel.is-dark .panel-block.is-active .panel-icon,html.theme--catppuccin-frappe .content kbd.panel .panel-block.is-active .panel-icon{color:#414559}html.theme--catppuccin-frappe .panel.is-primary 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.button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus,html.theme--catppuccin-latte .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus:not(:active),html.theme--catppuccin-latte .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-latte .button.is-black:active,html.theme--catppuccin-latte .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:hover,html.theme--catppuccin-latte .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus,html.theme--catppuccin-latte .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus:not(:active),html.theme--catppuccin-latte .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-latte .button.is-light:active,html.theme--catppuccin-latte .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-latte 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.button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark,html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:hover,html.theme--catppuccin-latte .content kbd.button:hover,html.theme--catppuccin-latte .button.is-dark.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-hovered{background-color:#c5c9d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus,html.theme--catppuccin-latte .content kbd.button:focus,html.theme--catppuccin-latte .button.is-dark.is-focused,html.theme--catppuccin-latte .content kbd.button.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte 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kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-focused{background-color:#ccd0da;border-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-primary,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:hover,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte 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.docstring>section>a.button.is-active.docs-sourcelink{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary[disabled],html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-primary.is-inverted,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-inverted:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-hovered,html.theme--catppuccin-latte 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.button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte 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.file.is-boxed .file-name{border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-boxed .file-icon{height:1.5em;width:1.5em}html.theme--catppuccin-latte .file.is-boxed .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-boxed.is-small .file-icon .fa,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-boxed .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .file.is-boxed.is-medium .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.is-boxed.is-large .file-icon .fa{font-size:35px}html.theme--catppuccin-latte .file.is-boxed.has-name .file-cta{border-radius:.4em .4em 0 0}html.theme--catppuccin-latte .file.is-boxed.has-name .file-name{border-radius:0 0 .4em .4em;border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-centered{justify-content:center}html.theme--catppuccin-latte .file.is-fullwidth .file-label{width:100%}html.theme--catppuccin-latte .file.is-fullwidth .file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-radius:.4em;font-size:1em;padding-left:1em;padding-right:1em;white-space:nowrap}html.theme--catppuccin-latte .file-cta{background-color:#ccd0da;color:#4c4f69}html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-style:solid;border-width:1px 1px 1px 0;display:block;max-width:16em;overflow:hidden;text-align:inherit;text-overflow:ellipsis}html.theme--catppuccin-latte .file-icon{align-items:center;display:flex;height:1em;justify-content:center;margin-right:.5em;width:1em}html.theme--catppuccin-latte .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .label{color:#41445a;display:block;font-size:1rem;font-weight:700}html.theme--catppuccin-latte .label:not(:last-child){margin-bottom:0.5em}html.theme--catppuccin-latte .label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.label{font-size:.75rem}html.theme--catppuccin-latte .label.is-medium{font-size:1.25rem}html.theme--catppuccin-latte 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.field.has-addons{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.has-addons .control:not(:last-child){margin-right:-1px}html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .button,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .select select{border-radius:0}html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select select{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-hovered:not([disabled]){z-index:2}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]){z-index:3}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]):hover{z-index:4}html.theme--catppuccin-latte .field.has-addons .control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.has-addons.has-addons-centered{justify-content:center}html.theme--catppuccin-latte .field.has-addons.has-addons-right{justify-content:flex-end}html.theme--catppuccin-latte .field.has-addons.has-addons-fullwidth 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.field.is-grouped.is-grouped-multiline:not(:last-child){margin-bottom:0}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field.is-horizontal{display:flex}}html.theme--catppuccin-latte .field-label .label{font-size:inherit}@media screen and (max-width: 768px){html.theme--catppuccin-latte .field-label{margin-bottom:0.5rem}}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .field-label{flex-basis:0;flex-grow:1;flex-shrink:0;margin-right:1.5rem;text-align:right}html.theme--catppuccin-latte .field-label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.field-label{font-size:.75rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-normal{padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-medium{font-size:1.25rem;padding-top:0.375em}html.theme--catppuccin-latte .field-label.is-large{font-size:1.5rem;padding-top:0.375em}}html.theme--catppuccin-latte .field-body .field 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form.docs-search>input~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-small~.icon{font-size:.75rem}html.theme--catppuccin-latte .control.has-icons-left .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input.is-medium~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-medium~.icon{font-size:1.25rem}html.theme--catppuccin-latte .control.has-icons-left .input.is-large~.icon,html.theme--catppuccin-latte .control.has-icons-left #documenter .docs-sidebar form.docs-search>input.is-large~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-left form.docs-search>input.is-large~.icon,html.theme--catppuccin-latte .control.has-icons-left .select.is-large~.icon,html.theme--catppuccin-latte .control.has-icons-right .input.is-large~.icon,html.theme--catppuccin-latte .control.has-icons-right #documenter .docs-sidebar form.docs-search>input.is-large~.icon,html.theme--catppuccin-latte #documenter .docs-sidebar .control.has-icons-right form.docs-search>input.is-large~.icon,html.theme--catppuccin-latte .control.has-icons-right .select.is-large~.icon{font-size:1.5rem}html.theme--catppuccin-latte .control.has-icons-left .icon,html.theme--catppuccin-latte .control.has-icons-right .icon{color:#acb0be;height:2.5em;pointer-events:none;position:absolute;top:0;width:2.5em;z-index:4}html.theme--catppuccin-latte 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.pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link,html.theme--catppuccin-latte .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link{border-color:#acb0be;color:#1e66f5;min-width:2.5em}html.theme--catppuccin-latte .pagination-previous:hover,html.theme--catppuccin-latte .pagination-next:hover,html.theme--catppuccin-latte .pagination-link:hover{border-color:#9ca0b0;color:#04a5e5}html.theme--catppuccin-latte .pagination-previous:focus,html.theme--catppuccin-latte .pagination-next:focus,html.theme--catppuccin-latte .pagination-link:focus{border-color:#9ca0b0}html.theme--catppuccin-latte 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kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small 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.container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-macchiato .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-macchiato .content li+li{margin-top:0.25em}html.theme--catppuccin-macchiato .content p:not(:last-child),html.theme--catppuccin-macchiato .content dl:not(:last-child),html.theme--catppuccin-macchiato .content ol:not(:last-child),html.theme--catppuccin-macchiato .content ul:not(:last-child),html.theme--catppuccin-macchiato .content blockquote:not(:last-child),html.theme--catppuccin-macchiato .content pre:not(:last-child),html.theme--catppuccin-macchiato .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-macchiato .content h1,html.theme--catppuccin-macchiato .content h2,html.theme--catppuccin-macchiato .content h3,html.theme--catppuccin-macchiato .content h4,html.theme--catppuccin-macchiato .content h5,html.theme--catppuccin-macchiato .content 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ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-macchiato .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-macchiato .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-macchiato .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-macchiato .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-macchiato .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-macchiato .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-macchiato .content ul ul ul{list-style-type:square}html.theme--catppuccin-macchiato .content dd{margin-left:2em}html.theme--catppuccin-macchiato .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-macchiato 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form.docs-search>input.is-active{box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-macchiato .is-black.textarea,html.theme--catppuccin-macchiato .is-black.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black{border-color:#0a0a0a}html.theme--catppuccin-macchiato .is-black.textarea:focus,html.theme--catppuccin-macchiato .is-black.input:focus,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:focus,html.theme--catppuccin-macchiato .is-black.is-focused.textarea,html.theme--catppuccin-macchiato .is-black.is-focused.input,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--catppuccin-macchiato .is-black.textarea:active,html.theme--catppuccin-macchiato .is-black.input:active,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-black:active,html.theme--catppuccin-macchiato 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus:not(:active),html.theme--catppuccin-mocha .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(148,226,213,0.25)}html.theme--catppuccin-mocha .button.is-info:active,html.theme--catppuccin-mocha .button.is-info.is-active{background-color:#80ddcd;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:#94e2d5;box-shadow:none}html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-mocha .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-mocha .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-mocha .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-mocha .content blockquote{background-color:#181825;border-left:5px solid #585b70;padding:1.25em 1.5em}html.theme--catppuccin-mocha .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-mocha .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-mocha .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-mocha .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-mocha .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-mocha .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-mocha .content ul ul ul{list-style-type:square}html.theme--catppuccin-mocha .content dd{margin-left:2em}html.theme--catppuccin-mocha .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-mocha .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-mocha .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-mocha .content figure img{display:inline-block}html.theme--catppuccin-mocha .content figure figcaption{font-style:italic}html.theme--catppuccin-mocha .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-mocha .content sup,html.theme--catppuccin-mocha .content sub{font-size:75%}html.theme--catppuccin-mocha .content table{width:100%}html.theme--catppuccin-mocha .content table td,html.theme--catppuccin-mocha .content table th{border:1px solid #585b70;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-mocha .content table th{color:#b8c5ef}html.theme--catppuccin-mocha .content table th:not([align]){text-align:inherit}html.theme--catppuccin-mocha .content table thead td,html.theme--catppuccin-mocha .content table thead th{border-width:0 0 2px;color:#b8c5ef}html.theme--catppuccin-mocha .content table tfoot td,html.theme--catppuccin-mocha .content table tfoot th{border-width:2px 0 0;color:#b8c5ef}html.theme--catppuccin-mocha .content table tbody tr:last-child td,html.theme--catppuccin-mocha .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-mocha .content .tabs li+li{margin-top:0}html.theme--catppuccin-mocha .content.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-mocha .content.is-normal{font-size:1rem}html.theme--catppuccin-mocha .content.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .content.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-mocha .icon.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-mocha .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-mocha .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-mocha .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-mocha .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-mocha .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-mocha .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-mocha div.icon-text{display:flex}html.theme--catppuccin-mocha .image,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-mocha .image img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-mocha .image img.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-mocha .image.is-fullwidth,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-mocha .image.is-square img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-mocha .image.is-square .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-mocha .image.is-1by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar 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.has-ratio,html.theme--catppuccin-mocha .image.is-5by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-mocha .image.is-5by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-mocha .image.is-16by9 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-mocha .image.is-16by9 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-mocha .image.is-2by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-mocha .image.is-2by1 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-mocha .image.is-3by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-mocha 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img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-mocha .image.is-3by5 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-mocha .image.is-9by16 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-mocha .image.is-9by16 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-mocha .image.is-1by2 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-mocha .image.is-1by2 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-mocha .image.is-1by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-mocha .image.is-1by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-mocha .image.is-square,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-mocha .image.is-1by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-mocha .image.is-5by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-mocha .image.is-4by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-mocha .image.is-3by2,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--catppuccin-mocha .image.is-5by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--catppuccin-mocha .image.is-16by9,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--catppuccin-mocha .image.is-2by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-mocha .image.is-3by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-mocha .image.is-4by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-mocha .image.is-3by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--catppuccin-mocha .image.is-2by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--catppuccin-mocha .image.is-3by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--catppuccin-mocha .image.is-9by16,html.theme--catppuccin-mocha #documenter 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark 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.button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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GitHub

Convolutions - similarity methods

DSP.conv — Function
conv(u, v; algorithm)

Convolution of two arrays. A convolution algorithm is automatically chosen among direct convolution, FFT, or FFT overlap-save, depending on the size of the input, unless explicitly specified with the algorithm keyword argument; see conv! for details.

source
conv(u,v,A)

2-D convolution of the matrix A with the 2-D separable kernel generated by the vector u and row-vector v. Uses 2-D FFT algorithm.

source
DSP.conv! — Function
conv!(out, u, v; algorithm=:auto)

Convolution of two arrays u and v with the result stored in out. out must be large enough to store the entire result; if it is even larger, the excess entries will be zeroed.

out, u, and v can be N-dimensional arrays, with arbitrary indexing offsets. If none of them has offset axes, size(out,d) ≥ size(u,d) + size(v,d) - 1 must hold. If both input and output have offset axes, firstindex(out,d) ≤ firstindex(u,d) + firstindex(v,d) and lastindex(out,d) ≥ lastindex(u,d) + lastindex(v,d) must hold (for d = 1,...,N). A mix of offset and non-offset axes is not permitted.

The algorithm keyword allows choosing the algorithm to use:

  • :direct: Evaluates the convolution sum in time domain.
  • :fft_simple: Evaluates the convolution as a product in the frequency domain.
  • :fft_overlapsave: Evaluates the convolution block-wise as a product in the frequency domain, overlapping the resulting blocks.
  • :fft: Selects the faster of :fft_simple and :fft_overlapsave (as estimated from the input size).
  • :fast: Selects the fastest of :direct, :fft_simple and :fft_overlapsave (as estimated from the input size).
  • :auto (default): Equivalent to :fast if the data type is known to be suitable for FFT-based computation, equivalent to :direct otherwise.
Warning

The choices made by :fft, :fast, and :auto are based on performance heuristics which may not result in the fastest algorithm in all cases. If best performance for a certain size/type combination is required, it is advised to do individual benchmarking and explicitly specify the desired algorithm.

source
DSP.deconv — Function
deconv(b,a) -> c

Construct vector c such that b = conv(a,c) + r. Equivalent to polynomial division.

source
DSP.xcorr — Function
xcorr(u; padmode::Symbol=:none, scaling::Symbol=:none)
-xcorr(u, v; padmode::Symbol=:none, scaling::Symbol=:none)

With two arguments, compute the cross-correlation of two vectors, by calculating the similarity between u and v with various offsets of v. Delaying u relative to v will shift the result to the right. If one argument is provided, calculate xcorr(u, u; kwargs...).

The size of the output depends on the padmode keyword argument: with padmode = :none the length of the result will be length(u) + length(v) - 1, as with conv. With padmode = :longest, the shorter of the arguments will be padded so they are of equal length. This gives a result with length 2*max(length(u), length(v))-1, with the zero-lag condition at the center.

The keyword argument scaling can be provided. Possible arguments are the default :none and :biased. :biased is valid only if the vectors have the same length, or only one vector is provided, dividing the result by length(u).

Examples

julia> xcorr([1,2,3],[1,2,3])
-5-element Vector{Int64}:
-  3
-  8
- 14
-  8
-  3
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diff --git a/previews/PR604/estimation/index.html b/previews/PR604/estimation/index.html deleted file mode 100644 index 76a8c003b..000000000 --- a/previews/PR604/estimation/index.html +++ /dev/null @@ -1,15 +0,0 @@ - -Estimation - parametric estimation functions · DSP.jl
GitHub

Estimation - parametric estimation functions

DSP.Estimation.esprit — Function
esprit(x::AbstractArray, M::Integer, p::Integer, Fs::Real=1.0)

ESPRIT [Roy1986] algorithm for frequency estimation. Estimation of Signal Parameters via Rotational Invariance Techniques

Given a signal x of length N that is the sum of p complex exponentials (cisoids) with unknown constant amplitudes and phases, estimate and return an array of the p frequencies.

\[x(t)=\sum_{i=1}^p A_ie^{j2\pi f_i t}\]

where $A_i$ and $f_i$ are the amplitude and frequency of the $i$th cisoid.

Arguments

  • x::AbstractArray: complex length N signal array
  • M::Integer: size of correlation matrix, must be <= N. The signal subspace is computed from the SVD of an M x (N-M+1) signal matrix formed from N-M+1 length-M shifts of the signal x in its columns. For best performance for 1 cisoid, use M = (N+1)/3 (according to van der Veen and Leus). For faster execution (due to smaller SVD), use small M or small N-M
  • p::Integer: number of complex exponentials/frequencies to estimate.
  • Fs::Float64: sampling frequency, in Hz.

Returns

length p real array of frequencies in units of Hz.

Examples

Suppose a signal $x(t)=2e^{j2\pi(2500)t} + 5e^{j2\pi(400)t}$ is corrupted by noise. The goal is to retrieve the frequencies in the signal. Let the sampling frequency be $8kHz$.

julia> Fs = 8000;                 # sampling frequency (Hz)
-julia> true_freq = [2500 400];    # true frequencies (Hz)
-julia> t = collect(1:Fs)/Fs;      # time vector
-julia> x = 2exp.(1im*2π*true_freq[1]*t) + 5exp.(1im*2π*true_freq[2]*t); # original signal x

Add gaussian noise in complex form to the signal $x$ to mimic noise corruption.

julia> noise = randn(Fs, 2)*[1; 1im];    # complex random noise
-julia> x += noise;                       # add noise to signal x

Run the ESPRIT algorithm to retrieve approximate frequencies.

julia> M = 5;    # window length of 5
-julia> p = 2;    # 2 frequencies to estimate
-julia> esprit(x, M, p, Fs)
-2-element Vector{Float64}:
-2502.2704154274957
-399.8420984461712
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DSP.Estimation.jacobsen — Function
jacobsen(x::AbstractVector, Fs::Real = 1.0)

Estimate the largest frequency in the complex signal x using Jacobsen's algorithm [Jacobsen2007]. Argument Fs is the sampling frequency. All frequencies are expressed in Hz.

If the signal x is real, the estimated frequency is guaranteed to be positive, but it may be highly inaccurate (especially for frequencies close to zero or to Fs/2).

If the sampling frequency Fs is not provided, then it is assumed that Fs = 1.0.

source
DSP.Estimation.quinn — Function
quinn(x::Vector, f0::Real, Fs::Real = 1.0 ; tol = 1e-6, maxiters = 20)
-
-quinn(x::Vector, Fs::Real = 1.0 ; kwargs...)
-
-quinn(x::Vector ; kwargs...)

Algorithms by Quinn and Quinn & Fernandes for frequency estimation. Given a signal x and an initial guess f0, estimate and return the frequency of the largest sinusoid in x. Fs is the sampling frequency. All frequencies are expressed in Hz.

If the initial guess f0 is not provided, then a guess is calculated using Jacobsen's estimator. If the sampling frequency Fs is not provided, then it is assumed that Fs = 1.0.

The following keyword arguments control the algorithm's behavior:

  • tol: the algorithm stops when the absolute value of the difference between two consecutive estimates is less than tol. Defaults to 1e-6.
  • maxiters: the maximum number of iterations to run. Defaults to 20.

Returns a tuple (estimate, reachedmaxiters), where estimate is the estimated frequency, and reachedmaxiters is true if the algorithm finished after running for maxiters iterations (this may indicate that the algorithm did not converge).

If the signal x is real, then the algorithm used is [Quinn1991]. If the signal is complex, the algorithm is [Quinn2009].

source
  • Roy1986R Roy, A Paulraj and T Kailath, ESPRIT - A subspace approach to estimation of parameters of cisoids in noise, IEEE Trans. Acoustics, Speech, Signal Process., 34, 1340-1342 (1986). http://ieeexplore.ieee.org/abstract/document/1164935/.
  • Jacobsen2007E Jacobsen and P Kootsookos, "Fast, Accurate Frequency Estimators", Chapter 10 in "Streamlining Digital Signal Processing", edited by R. Lyons, 2007, IEEE Press.
  • Quinn1991B Quinn and J Fernandes, "A fast efficient technique for the estimation of frequency", Biometrika, Vol. 78 (1991).
  • Quinn2009B Quinn, "Recent advances in rapid frequency estimation", Digital Signal Processing, Vol. 19 (2009), Elsevier.
diff --git a/previews/PR604/filters/index.html b/previews/PR604/filters/index.html deleted file mode 100644 index 3ce479bd9..000000000 --- a/previews/PR604/filters/index.html +++ /dev/null @@ -1,68 +0,0 @@ - -Filters - filter design and filtering · DSP.jl
GitHub

Filters - filter design and filtering

DSP.jl differentiates between filter coefficients and stateful filters. Filter coefficient objects specify the response of the filter in one of several standard forms. Stateful filter objects carry the state of the filter together with filter coefficients in an implementable form (PolynomialRatio, Biquad, or SecondOrderSections). When invoked on a filter coefficient object, filt does not preserve state.

Filter coefficient objects

DSP.jl supports common filter representations. Filter coefficients can be converted from one type to another using convert.

DSP.Filters.ZeroPoleGain — Type
ZeroPoleGain(z::Vector, p::Vector, k::Number)

Filter representation in terms of zeros z, poles p, and gain k:

\[H(x) = k\frac{(x - \verb!z[1]!) \ldots (x - \verb!z[m]!)}{(x - \verb!p[1]!) \ldots (x - \verb!p[n]!)}\]

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DSP.Filters.PolynomialRatio — Type
PolynomialRatio(b::Union{Number, Vector{<:Number}}, a::Union{Number, Vector{<:Number}})

Filter representation in terms of the coefficients of the numerator b and denominator a in the z or s domain where b and a are vectors ordered from highest power to lowest.

Inputs that are Numbers are treated as one-element Vectors.

Filter with:

  • Transfer function in z domain (zero & negative z powers):

\[H(z) = \frac{\verb!b[1]! + \ldots + \verb!b[m]! z^{-m+1}}{\verb!a[1]! + \ldots + \verb!a[n]! z^{-n+1}}\]

returns PolynomialRatio object with a[1] = 1 and other specified coefficients divided by a[1].

julia> PolynomialRatio([1,1],[1,2])
-PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0*z⁻¹ + 1.0), LaurentPolynomial(2.0*z⁻¹ + 1.0))
-julia> PolynomialRatio{:z}([1,2,3],[2,3,4])
-PolynomialRatio{:z, Float64}(LaurentPolynomial(1.5*z⁻² + 1.0*z⁻¹ + 0.5), LaurentPolynomial(2.0*z⁻² + 1.5*z⁻¹ + 1.0))
  • Transfer function in s domain (zero & positive s powers):

\[H(s) = \frac{\verb!b[1]! s^{m-1} + \ldots + \verb!b[m]!}{\verb!a[1]! s^{n-1} + \ldots + \verb!a[n]!}\]

returns PolynomialRatio object with specified b and a coefficients.

julia> PolynomialRatio{:s}([1,2,3],[2,3,4])
-PolynomialRatio{:s, Int64}(LaurentPolynomial(3 + 2*s + s²), LaurentPolynomial(4 + 3*s + 2*s²))
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DSP.Filters.Biquad — Type
Biquad(b0::T, b1::T, b2::T, a1::T, a2::T) where T <: Number

Filter representation in terms of the transfer function of a single second-order section given by:

\[H(s) = \frac{\verb!b0! s^2+\verb!b1! s+\verb!b2!}{s^2+\verb!a1! s + \verb!a2!}\]

or equivalently:

\[H(z) = \frac{\verb!b0!+\verb!b1! z^{-1}+\verb!b2! z^{-2}}{1+\verb!a1! z^{-1} + \verb!a2! z^{-2}}\]

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DSP.Filters.SecondOrderSections — Type
SecondOrderSections(biquads::Vector{<:Biquad}, gain::Number)

Filter representation in terms of a cascade of second-order sections and gain. biquads must be specified as a vector of Biquads.

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These filter coefficient objects support the following arithmetic operations: inversion (inv), multiplication (*) for series connection, and integral power (^) for repeated multiplication with itself. For example:

julia> H = PolynomialRatio([1.0], [1.0, 0.3])
-PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.3*z⁻¹ + 1.0))
-
-julia> inv(H)
-PolynomialRatio{:z, Float64}(LaurentPolynomial(0.3*z⁻¹ + 1.0), LaurentPolynomial(1.0))
-
-julia> H * H
-PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))
-
-julia> H^2
-PolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))
-
-julia> H^-2
-PolynomialRatio{:z, Float64}(LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0), LaurentPolynomial(1.0))
-

Stateful filter objects

DSP.Filters.DF2TFilter — Type
DF2TFilter(coef::FilterCoefficients{:z})
-DF2TFilter(coef::FilterCoefficients{:z}, coldims::Tuple)
-DF2TFilter(coef::FilterCoefficients{:z}, sitype::Type, coldims::Tuple = ())
-DF2TFilter(coef::FilterCoefficients{:z}, si)

Construct a stateful direct form II transposed filter with coefficients coef.

The initial filter state defaults to zeros (of a type derived from coef) suitable for vector input. Another element type of the state can be specified with sitype.

To allow column-wise filtering of higher-dimensional input, the size of the extra dimensions have to be given in coldims. To e.g. column-wise filter an input with size (L, N1, N2), set coldims to (N1, N2).

Alternatively, an array representing the initial filter state can be passed as si.

If coef is a PolynomialRatio, Biquad, or SecondOrderSections, filtering is implemented directly. If coef is a ZeroPoleGain object, it is first converted to a SecondOrderSections object.

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DSP.jl's FIRFilter type maintains state between calls to filt, allowing you to filter a signal of indefinite length in RAM-friendly chunks. FIRFilter contains nothing more that the state of the filter, and a FIRKernel. There are five different kinds of FIRKernel for single rate, up-sampling, down-sampling, rational resampling, and arbitrary sample-rate conversion. You need not specify the type of kernel. The FIRFilter constructor selects the correct kernel based on input parameters.

DSP.Filters.FIRFilter — Type
FIRFilter(h::Vector[, ratio::Union{Integer,Rational}])

Construct a stateful FIRFilter object from the vector of filter taps h. ratio is an optional rational integer which specifies the input to output sample rate relationship (e.g. 147//160 for converting recorded audio from 48 KHz to 44.1 KHz).

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FIRFilter(h::Vector, rate::AbstractFloat[, Nϕ::Integer=32])

Returns a polyphase FIRFilter object from the vector of filter taps h. rate is a floating point number that specifies the input to output sample-rate relationship $\frac{fs_{out}}{fs_{in}}$. Nϕ is an optional parameter which specifies the number of phases created from h. Nϕ defaults to 32.

source

Filter application

DSP.filt — Function
filt(b::Union{AbstractVector,Number},
-     a::Union{AbstractVector,Number},
-     x::AbstractArray,
-     [si::AbstractArray])

Apply filter described by vectors a and b to vector x, with an optional initial filter state vector si (defaults to zeros).

Inputs that are Numbers are treated as one-element Vectors.

source
filt(f::FilterCoefficients{:z}, x::AbstractArray[, si])

Apply filter or filter coefficients f along the first dimension of array x. If f is a filter coefficient object, si is an optional array representing the initial filter state (defaults to zeros). If f is a PolynomialRatio, Biquad, or SecondOrderSections, filtering is implemented directly. If f is a ZeroPoleGain object, it is first converted to a SecondOrderSections object. If f is a Vector, it is interpreted as an FIR filter, and a naïve or FFT-based algorithm is selected based on the data and filter length.

source
filt(f::DF2TFilter{<:FilterCoefficients{:z},<:Array{T}}, x::AbstractArray{V}) where {T,V}

Apply the stateful filter f on x.

Warning

The output array has eltype promote_type(T, V), where T is the eltype of the filter state.

For more control over the output type, provide a preallocated output array out to filt!(out, f, x).

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DSP.filt! — Function
filt!(out, b, a, x, [si])

Same as filt but writes the result into the out argument, which may alias the input x to modify it in-place.

source
filt!(out, f, x[, si])

Same as filt() but writes the result into the out argument. Output array out may not be an alias of x, i.e. filtering may not be done in place.

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DSP.Filters.filtfilt — Function
filtfilt(coef::FilterCoefficients, x::AbstractArray)
-filtfilt(b::AbstractVector, x::AbstractArray)
-filtfilt(b::AbstractVector, a::AbstractVector, x::AbstractArray)

Filter x in the forward and reverse directions using either a FilterCoefficients object coef, or the coefficients b and optionally a as in filt. The initial state of the filter is computed so that its response to a step function is steady state. Before filtering, the data is extrapolated at both ends with an odd-symmetric extension of length min(3*(max(length(b), length(a))-1), size(x, 1) - 1).

Because filtfilt applies the given filter twice, the effective filter order is twice the order of coef. The resulting signal has zero phase distortion.

source
DSP.Filters.fftfilt — Function
fftfilt(h::AbstractVector{<:Real}, x::AbstractArray{<:Real})

Apply FIR filter taps h along the first dimension of array x using an FFT-based overlap-save algorithm.

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DSP.Filters.fftfilt! — Function
fftfilt!(out::AbstractArray, h::AbstractVector, x::AbstractArray)

Like fftfilt but writes result into out array.

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DSP.Filters.tdfilt — Function
tdfilt(h::AbstractVector, x::AbstractArray)

Apply filter or filter coefficients h along the first dimension of array x using a naïve time-domain algorithm

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DSP.Filters.tdfilt! — Function
tdfilt!(out::AbstractArray, h::AbstractVector, x::AbstractArray)

Like tdfilt, but writes the result into array out. Output array out may not be an alias of x, i.e. filtering may not be done in place.

source
DSP.Filters.resample — Function
resample(x::AbstractVector, rate::Real[, coef::Vector])

Resample x at rational or arbitrary rate. coef is an optional vector of FIR filter taps. If coef is not provided, the taps will be computed using a Kaiser window.

Internally, resample uses a polyphase FIRFilter object, but performs additional operations to make resampling a signal easier. It compensates for the FIRFilter's delay (ramp-up), and appends zeros to x. The result is that when the input and output signals are plotted on top of each other, they correlate very well, but one signal will have more samples than the other.

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resample(x::AbstractArray, rate::Real, h::Vector = resample_filter(rate); dims)

Resample an array x along dimension dims.

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Filter design

Most analog and digital filters are constructed by composing response types, which determine the frequency response of the filter, with design methods, which determine how the filter is constructed.

The response type is Lowpass, Highpass, Bandpass or Bandstop and includes the edges of the bands.

The design method is Butterworth, Chebyshev1, Chebyshev2, Elliptic, or FIRWindow, and includes any necessary parameters for the method that affect the shape of the response, such as filter order, ripple, and attenuation.

Filter order estimation methods are available in buttord, cheb1ord, cheb2ord, and ellipord if the corner frequencies for different IIR filter types are known. remezord can be used for an initial FIR filter order estimate.

DSP.Filters.analogfilter — Function
analogfilter(responsetype::FilterType, designmethod::FilterCoefficients)

Construct an analog filter. See below for possible response and filter types.

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DSP.Filters.digitalfilter — Function
digitalfilter(responsetype::FilterType, designmethod::FilterCoefficients[; fs::Real])

Construct a digital filter. See below for possible response and filter types.

source

For some filters, the design method is more general or inherently implies a response type; these direct design methods include remez which designs equiripple FIR filters of all types, and iirnotch which designs a 2nd order "biquad" IIR notch filter.

For a more general application of creating a digital filter from s-domain representation of an analog filter, one can use bilinear:

DSP.Filters.bilinear — Function
bilinear(f::FilterCoefficients{:s}, fs::Real)

Calculate the digital filter (z-domain) ZPK representation of an analog filter defined in s-domain using bilinear transform with sampling frequency fs. The s-domain representation is first converted to a ZPK representation in s-domain and then transformed to z-domain using bilinear transform.

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bilinear(f::ZeroPoleGain{:s,Z,P,K}, fs::Real) where {Z,P,K}

Calculate the digital filter (z-domain) ZPK representation of an analog filter defined as a ZPK representation in s-domain using bilinear transform with sampling frequency fs.

Input s-domain representation must be a ZeroPoleGain{:s, Z, P, K} object:

\[H(s) = f.k\frac{(s - \verb!f.z[1]!) \ldots (s - \verb!f.z[m]!)}{(s - \verb!f.p[1]!) \ldots (s - \verb!f.p[n]!)}\]

Output z-domain representation is a ZeroPoleGain{:z, Z, P, K} object:

\[H(z) = K\frac{(z - \verb!Z[1]!) \ldots (z - \verb!Z[m]!)}{(z - \verb!P[1]!) \ldots (z - \verb!P[n]!)}\]

where Z, P, K are calculated as:

\[Z[i] = \frac{(2 + \verb!f.z[i]!/\verb!fs!)}{(2 - \verb!f.z[i]!/\verb!fs!)} \quad \text{for } i = 1, \ldots, m\]

\[P[i] = \frac{(2 + \verb!f.p[i]!/\verb!fs!)}{(2 - \verb!f.p[i]!/\verb!fs!)} \quad \text{for } i = 1, \ldots, n\]

\[K = f.k \ \mathcal{Re} \left[ \frac{\prod_{i=1}^m (2*fs - f.z[i])}{\prod_{i=1}^n (2*fs - f.p[i])} \right]\]

Here, m and n are respectively the numbers of zeros and poles in the s-domain representation. If m < n, then additional n-m zeros are added at z = -1.

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Filter response types

The interpretation of the frequencies Wn, Wn1 and Wn2 depends on wether an analog or a digital filter is designed.

  1. If an analog filter is designed using analogfilter, the frequencies are interpreted as analog frequencies in radians/second.
  2. If a digital filter is designed using digitalfilter and the sampling frequency fs is specified, the frequencies of the filter response type are normalized to fs. This requires that the sampling frequency and the filter response type use the same frequency unit (Hz, radians/second, ...). If fs is not specified, the frequencies of the filter response type are interpreted as normalized frequencies in half-cycles/sample.

Filter design methods

IIR filter design methods

DSP.Filters.Chebyshev1 — Function
Chebyshev1(n::Integer, ripple::Real)

n pole Chebyshev type I filter with ripple dB ripple in the passband.

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DSP.Filters.Chebyshev2 — Function
Chebyshev2(n::Integer, ripple::Real)

n pole Chebyshev type II filter with ripple dB ripple in the stopband.

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DSP.Filters.Elliptic — Function
Elliptic(n::Integer, rp::Real, rs::Real)

n pole elliptic (Cauer) filter with rp dB ripple in the passband and rs dB attentuation in the stopband.

source

Filter order estimation methods

IIR filter order estimation methods

DSP.Filters.buttord — Function
(N, ωn) = buttord(Wp::Tuple{Real,Real}, Ws::Tuple{Real,Real}, Rp::Real, Rs::Real; domain=:z)

Butterworth order estimate for bandpass and bandstop filter types. Wp and Ws are 2-element pass and stopband frequency edges, with no more than Rp dB passband ripple and at least Rs dB stopband attenuation. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or "-3 dB" frequencies.

If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving an order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

source
(N, ωn) = buttord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain=:z)

LPF/HPF Butterworth filter order and -3 dB frequency approximation. Wp and Ws are the passband and stopband frequencies, whereas Rp and Rs are the passband and stopband ripple attenuations in dB. If the passband is greater than stopband, the filter type is inferred to be for estimating the order of a highpass filter. N specifies the lowest possible integer filter order, whereas ωn is the cutoff or "-3 dB" frequency.

If a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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DSP.Filters.cheb1ord — Function
(N, ωn) = cheb1ord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Chebyshev Type I Filters. Wp and Ws indicate the passband and stopband frequency edges, and Rp and Rs indicate the maximum loss in the passband and the minimum attenuation in the stopband, (in dB.)

Based on the ordering of passband and stopband edges, the Lowpass or Highpass filter type is inferred. N indicates the smallest integer filter order that achieves the desired specifications, and ωn contains the natural frequency of the filter, (in this case, simply the passband edge.)

If a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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(N, ωn) = cheb1ord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Chebyshev Type I Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or "-3 dB" frequencies. If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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DSP.Filters.cheb2ord — Function
(N, ωn) = cheb2ord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Chebyshev Type II (inverse) Filters. Wp and Ws are the frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB.

Based on the ordering of the passband and stopband edges, the Lowpass or Highpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the "-3 dB" cutoff frequency.

If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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(N, ωn) = cheb2ord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Chebyshev Type II (inverse) Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB.

Based on the ordering of the passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the "-3 dB" cutoff frequency.

If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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DSP.Filters.ellipord — Function
(N, ωn) = ellipord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Elliptic (Cauer) Filters. Wp and Ws indicate the passband and stopband frequency edges, and Rp and Rs indicate the maximum loss in the passband and the minimum attenuation in the stopband, (in dB.)

Based on the ordering of passband and stopband edges, the Lowpass or Highpass filter type is inferred. N indicates the smallest integer filter order that achieves the desired specifications, and ωn contains the natural frequency of the filter, (in this case, simply the passband edge.)

If a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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(N, ωn) = ellipord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)

Integer and natural frequency order estimation for Elliptic (Cauer) Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or "-3 dB" frequencies. If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.

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FIR filter order estimation methods

DSP.Filters.remezord — Function
N = remezord(Wp::Real, Ws::Real, Rp::Real, Rs::Real)

Order estimation for lowpass digital filter cases based on the equations and coefficients in [Rabiner]. The original equation returned the minimum filter length, whereas this implementation returns the order (N=L-1). Wp and Ws are the normalized passband and stopband frequencies, with Rp indicating the passband ripple and Rs is the stopband attenuation (linear.)

NOTE: The value of N is an initial approximation. If under-estimated, the order should be increased until the design specifications are met.

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FIR filter design methods

DSP.Filters.FIRWindow — Type
FIRWindow(window::Vector; scale=true)

FIR filter design using window window, a vector whose length matches the number of taps in the resulting filter.

If scale is true (default), the designed FIR filter is scaled so that the following holds:

  • For Lowpass and Bandstop filters, the frequency response is unity at 0 (DC).
  • For Highpass filters, the frequency response is unity at the Nyquist frequency.
  • For Bandpass filters, the frequency response is unity in the center of the passband.
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FIRWindow(; transitionwidth::Real, attenuation::Real=60, scale::Bool=true)

Kaiser window FIR filter design. The required number of taps is calculated based on transitionwidth (in half-cycles/sample) and stopband attenuation (in dB). attenuation defaults to 60 dB.

source

Direct filter design methods

DSP.Filters.remez — Function
remez(numtaps::Integer, band_defs;
-      Hz::Real=1.0,
-      neg::Bool=false,
-      maxiter::Integer=25,
-      grid_density::Integer=16)

Calculate the minimax optimal filter using the Remez exchange algorithm [McClellan1973a] [McClellan1973b].

This is the simplified API that accepts just 2 required arguments (numtaps, band_defs). For a scipy compatible version see the 3 arguments version (numtaps, bands, desired).

Calculate the filter-coefficients for the finite impulse response (FIR) filter whose transfer function minimizes the maximum error between the desired gain and the realized gain in the specified frequency bands using the Remez exchange algorithm.

Arguments

  • numtaps::Integer: The desired number of taps in the filter. The number of taps is the number of terms in the filter, or the filter order plus one.
  • band_defs: A sequence of band definitions. This sequence defines the bands. Each entry is a pair. The pair's first item is a tuple of band edges (low, high). The pair's second item defines the desired response and weight in that band. The weight is optional and defaults to 1.0. Both the desired response and weight may be either scalars or functions. If a function, the function should accept a real frequency and return the real desired response or real weight. Examples:
    • LPF with unity weights. [(0, 0.475) => 1, (0.5, 1.0) => 0]
    • LPF with weight of 2 in the stop band. [(0, 0.475) => (1, 1), (0.5, 1.0) => (0, 2)]
    • BPF with unity weights. [(0, 0.375) => 0, (0.4, 0.5) => 1, (0.525, 1.0) => 0]
    • Hilbert transformer. [(0.1, 0.95) => 1]; neg=true
    • Differentiator. [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true
  • Hz::Real: The sampling frequency in Hz. Default is 1.
  • neg::Bool: Whether the filter has negative symmetry or not. Default is false. If false, the filter is even-symmetric. If true, the filter is odd-symmetric. neg=true means that h[n]=-h[end+1-n]; neg=false means that h[n]=h[end+1-n].
  • maxiter::Integer: (optional) Maximum number of iterations of the algorithm. Default is 25.
  • grid_density:Integer: (optional) Grid density. The dense grid used in remez is of size (numtaps + 1) * grid_density. Default is 16.

Returns

  • h::Array{Float64,1}: A rank-1 array containing the coefficients of the optimal (in a minimax sense) filter.

Examples

Construct a length 35 filter with a passband at 0.15-0.4 Hz (desired response of 1), and stop bands at 0-0.1 Hz and 0.45-0.5 Hz (desired response of 0). Note: the behavior in the frequency ranges between those bands - the transition bands - is unspecified.

julia> bpass = remez(35, [(0, 0.1)=>0, (0.15, 0.4)=>1, (0.45, 0.5)=>0]);

You can trade-off maximum error achieved for transition bandwidth. The wider the transition bands, the lower the maximum error in the bands specified. Here is a bandpass filter with the same passband, but wider transition bands.

julia> bpass2 = remez(35, [(0, 0.08)=>0, (0.15, 0.4)=>1, (0.47, 0.5)=>0]);

Here we compute the frequency responses and plot them in dB.

julia> using PyPlot
-julia> b = PolynomialRatio(bpass, [1.0])
-julia> b2 = PolynomialRatio(bpass2, [1.0])
-julia> f = range(0, stop=0.5, length=1000)
-julia> plot(f, 20*log10.(abs.(freqresp(b,f,1.0))))
-julia> plot(f, 20*log10.(abs.(freqresp(b2,f,1.0))))
-julia> grid()

Examples from the unittests - standard (even) symmetry.

Length 151 LPF (Low Pass Filter).

julia> h = remez(151, [(0, 0.475) => 1, (0.5, 1.0) => 0]; Hz=2.0);

Length 152 LPF. Non-default "weight" input.

julia> h = remez(152, [(0, 0.475) => (1, 1), (0.5, 1.0) => (0, 2)]; Hz=2.0);

Length 51 HPF (High Pass Filter).

julia> h = remez(51, [(0, 0.75) => 0, (0.8, 1.0) => 1]; Hz=2.0);

Length 180 BPF (Band Pass Filter).

julia> h = remez(180, [(0, 0.375) => 0, (0.4, 0.5) => 1, (0.525, 1.0) => 0]; Hz=2.0, maxiter=30);

Examples from the unittests - Odd-symmetric filters - hilbert and differentiators type.

Even length - has a much better approximation since the response is not constrained to 0 at the nyquist frequency. Length 20 Hilbert transformer.

julia> h = remez(20, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);

Length 21 Hilbert transformer.

julia> h = remez(21, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);

Length 200 differentiator.

julia> h = remez(200, [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);

Length 201 differentiator.

julia> h = remez(201, [(0.05, 0.95) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);

Inverse sinc filter - custom response function

julia> L = 64; Fs = 4800*L;
-julia> passband_response_function = f -> (f==0) ? 1.0 : abs.((π*f/4800) ./ sin.(π*f/4800));
-julia> h = remez(201, [(    0.0, 2880.0) => (passband_response_function, 1.0),
-                (10000.0,   Fs/2) => (0.0, 100.0)]; Hz=Fs);
source
remez(numtaps::Integer,
-      bands::Vector,
-      desired::Vector;
-      weight::Vector=[],
-      Hz::Real=1.0,
-      filter_type::RemezFilterType=filter_type_bandpass,
-      maxiter::Integer=25,
-      grid_density::Integer=16)

This is the scipy compatible version that requires 3 arguments (numtaps, bands, desired). For a simplified API, see the 2 argument version (numtaps, band_defs). The filters designed are equivalent, the inputs are just specified in a different way. Below the arguments and examples are described that differ from the simplified API version.

Arguments

  • bands::Vector: A monotonic sequence containing the band edges in Hz. All elements must be non-negative and less than half the sampling frequency as given by Hz.
  • desired::Vector:A sequence half the size of bands containing the desired gain in each of the specified bands.
  • weight::Vector: (optional) A relative weighting to give to each band region. The length of weight has to be half the length of bands.
  • filter_type::RemezFilterType: Default is filter_type_bandpass. The type of filter:
    • filter_type_bandpass : flat response in bands. This is the default.
    • filter_type_differentiator : frequency proportional response in bands. Odd symetric as in filter_type_hilbert case, but with a linear sloping desired response.
    • filter_type_hilbert : filter with odd symmetry, that is, type III (for even order) or type IV (for odd order) linear phase filters.

Examples

Compare the examples with the simplified API and the Scipy API. Each of the following blocks first designs a filter using the simplified (recommended) API, and then designs the same filter using the Scipy-compatible API.

julia> bpass = remez(35, [(0, 0.1)=>0, (0.15, 0.4)=>1, (0.45, 0.5)=>0]);
-
-julia> bpass = remez(35, [0, 0.1, 0.15, 0.4, 0.45, 0.5], [0, 1, 0]);
-
julia> bpass2 = remez(35, [(0, 0.08)=>0, (0.15, 0.4)=>1, (0.47, 0.5)=>0]);
-
-julia> bpass2 = remez(35, [0, 0.08, 0.15, 0.4, 0.47, 0.5], [0, 1, 0]);
-
julia> h = remez(20, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);
-
-julia> h = remez(20, [0.1, 0.95], [1]; filter_type=filter_type_hilbert, Hz=2.0);
-
julia> h = remez(200, [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);
-
-julia> h = remez(200, [0.01, 0.99], [1]; filter_type=filter_type_differentiator, Hz=2.0);
-
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DSP.Filters.iirnotch — Function
iirnotch(Wn::Real, bandwidth::Real[; fs=2])

Second-order digital IIR notch filter [Orfanidis] at frequency Wn with bandwidth bandwidth. If fs is not specified, Wn is interpreted as a normalized frequency in half-cycles/sample.

source

Filter response

DSP.Filters.freqresp — Function
H, w = freqresp(filter::FilterCoefficients)

Frequency response H of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.

source
freqresp(filter::FilterCoefficients{:z}, w)

Frequency response of digital filter at normalized frequency or frequencies w in radians/sample.

source
freqresp(filter::FilterCoefficients{:s}, w)

Frequency response of analog filter at frequency or frequencies w in radians/second.

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DSP.Filters.phaseresp — Function
phi, w = phaseresp(filter::FilterCoefficients)

Phase response phi of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.

source
phaseresp(filter::FilterCoefficients, w)

Phase response of a filter at (normalized) frequency or frequencies w in radians/sample for a digital filter or radians/second for an analog filter.

source
DSP.Filters.grpdelay — Function
tau, w = grpdelay(filter::FilterCoefficients)

Group delay tau of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.

source
grpdelay(filter::FilterCoefficients{:z}, w)

Group delay of a digital filter at normalized frequency or frequencies w in radians/sample.

source
DSP.Filters.impresp — Function
impresp(filter::FilterCoefficients{:z}, n=100)

Impulse response of a digital filter with n points.

source
DSP.Filters.stepresp — Function
stepresp(filter::FilterCoefficients{:z}, n=100)

Step response of a digital filter with n points.

source

Miscellaneous

DSP.Filters.coefb — Function
coefb(f::PolynomialRatio)

Coefficients of the numerator of a PolynomialRatio object, highest power first, i.e., the b passed to filt()

source
DSP.Filters.coefa — Function
coefa(f::PolynomialRatio)

Coefficients of the denominator of a PolynomialRatio object, highest power first, i.e., the a passed to filt()

source

Examples

Construct a 4th order elliptic lowpass filter with normalized cutoff frequency 0.2, 0.5 dB of passband ripple, and 30 dB attentuation in the stopband and extract the coefficients of the numerator and denominator of the transfer function:

responsetype = Lowpass(0.2)
-designmethod = Elliptic(4, 0.5, 30)
-tf = convert(PolynomialRatio, digitalfilter(responsetype, designmethod))
-numerator_coefs = coefb(tf)
-denominator_coefs = coefa(tf)

Filter the data in x, sampled at 1000 Hz, with a 4th order Butterworth bandpass filter between 10 and 40 Hz:

responsetype = Bandpass(10, 40)
-designmethod = Butterworth(4)
-filt(digitalfilter(responsetype, designmethod; fs=1000), x)

Filter the data in x, sampled at 50 Hz, with a 64 tap Hanning window FIR lowpass filter at 5 Hz:

responsetype = Lowpass(5)
-designmethod = FIRWindow(hanning(64))
-filt(digitalfilter(responsetype, designmethod; fs=50), x)

Estimate a Lowpass Elliptic filter order with a normalized passband cutoff frequency of 0.2, a stopband cutoff frequency of 0.4, 3 dB of passband ripple, and 40 dB attenuation in the stopband:

(N, ωn) = ellipord(0.2, 0.4, 3, 40)
  • RabinerHerrmann, O., Lawrence R. Rabiner, and D. S. K. Chan. "Practical design rules for optimum finite impulse response lowpass digital filters." Bell System Technical Journal 52.6 (1973): 769-799.
  • McClellan1973aJ. H. McClellan and T. W. Parks, A unified approach to the design of optimum FIR linear phase digital filters, IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973.
  • McClellan1973bJ. H. McClellan, T. W. Parks and L. R. Rabiner, A Computer Program for Designing Optimum FIR Linear Phase Digital Filters, IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 506-525, 1973.
  • OrfanidisOrfanidis, S. J. (1996). Introduction to signal processing. Englewood Cliffs, N.J: Prentice Hall, p. 370.
diff --git a/previews/PR604/index.html b/previews/PR604/index.html deleted file mode 100644 index 1b0dd382c..000000000 --- a/previews/PR604/index.html +++ /dev/null @@ -1,2 +0,0 @@ - -Home · DSP.jl diff --git a/previews/PR604/internals/index.html b/previews/PR604/internals/index.html deleted file mode 100644 index f83cd1e91..000000000 --- a/previews/PR604/internals/index.html +++ /dev/null @@ -1,10 +0,0 @@ - -Internals · DSP.jl
GitHub

Internals

Functions here are not exported.

DSP.os_fft_complexity — Function

Estimate the number of floating point multiplications per output sample for an overlap-save algorithm with fft size nfft, and filter size nb.

source
DSP.os_prepare_conv — Function

Prepare buffers and FFTW plans for convolution. The two buffers, tdbuff and fdbuff may be an alias of each other, because complex convolution only requires one buffer. The plans are mutating where possible, and the inverse plan is unnormalized.

source
DSP.os_conv_block! — Function

Take a block of data, and convolve it with the smaller convolution input. This may modify the contents of tdbuff and fdbuff, and the result will be in tdbuff.

source

Like the real version, but only operates on one buffer

source
DSP.os_filter_transform! — Function

Transform the smaller convolution input to frequency domain, and return it in a new array. However, the contents of buff may be modified.

source
DSP.Windows.makewindow — Function
makewindow(winfunc::Function, n::Integer, padding::Integer, zerophase::Bool)

Generate a discrete window vector of the given length with padding zeros. winfunc should be a function giving the window value in the range of [-0.5, 0.5]. The function is assumed to be 0 outside of this range.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

Example:

function hanning(n::Integer; padding::Integer=0, zerophase::Bool=false)
-    makewindow(n, padding, zerophase) do x
-        0.5*(1+cos(2pi*x))
-    end
-end
source
DSP.Windows.padplot — Function
padplot(plotstr)

Takes a multiline string and pre-pads it so that it shows up as preformatted code when included in a docstring.

source
DSP._zeropad — Function
_zeropad(u, padded_size, [data_dest, data_region])

Creates and returns a new base-1 index array of size padded_size, with the section of u specified by data_region copied into the region of the new array as specified by data_dest. All other values will be initialized to zero.

If either data_dest or data_region is not specified, then the defaults described in _zeropad! will be used.

source
DSP._zeropad! — Function
_zeropad!(padded::AbstractVector,
-          u::AbstractVector,
-          padded_axes = axes(padded),
-          data_dest::Tuple = (1,),
-          data_region = CartesianIndices(u))

Place the portion of u specified by data_region into padded, starting at location data_dest. Sets the rest of padded to zero. This will mutate padded. padded_axes must correspond to the axes of padded.

source
DSP.Filters._polyprep — Function
_polyprep(D::Symbol, x::Union{T,Vector{T}}, ::Type) where {T<:Number}

Converts x to polynomial form. If x is a Number, it has to be converted into a Vector, otherwise LaurentPolynomial dispatch goes into stack overflow trying to collect a 0-d array into a Vector.

Warning

The DSP convention for Laplace domain is highest power first.

The Polynomials.jl convention is lowest power first.

source
DSP.Filters.freq_eval — Function
function freq_eval(xf, x::AbstractVector, y::AbstractVector, ad::AbstractVector)

FUNCTION: freq_eval (gee) FUNCTION TO EVALUATE THE FREQUENCY RESPONSE USING THE LAGRANGE INTERPOLATION FORMULA IN THE BARYCENTRIC FORM

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DSP.Filters.lagrange_interp — Function
lagrange_interp(k::Integer, n::Integer, m::Integer, x::AbstractVector)

CALCULATE THE LAGRANGE INTERPOLATION COEFFICIENTS

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DSP.Periodograms.coherence_from_cs! — Function
coherence_from_cs!(output::AbstractArray, cs_matrix)

Compute the pairwise channel coherences from a cross spectral matrix, storing the result in output.

source
DSP.optimalfftfiltlength — Function

Determine the length of FFT that minimizes the number of multiplications per output sample for an overlap-save convolution of vectors of size nb and nx.

source
diff --git a/previews/PR604/lpc/index.html b/previews/PR604/lpc/index.html deleted file mode 100644 index 72149e563..000000000 --- a/previews/PR604/lpc/index.html +++ /dev/null @@ -1,8 +0,0 @@ - -LPC - Linear Predictive Coding · DSP.jl
GitHub

LPC - Linear Predictive Coding

DSP.LPC.lpc — Function
lpc(x::AbstractVector, p::Integer, [LPCBurg()])

Given input signal x and prediction order p, returns IIR coefficients a and average reconstruction error prediction_err. Note that this method does NOT return the leading $1$ present in the true autocorrelative estimate; it omits it as it is implicit in every LPC estimate, and must be manually reintroduced if the returned vector should be treated as a polynomial.

The algorithm used is determined by the last optional parameter, and can be either LPCBurg (arburg) or LPCLevinson (levinson).

source
DSP.LPC.arburg — Function
arburg(x::AbstractVector, p::Integer)

LPC (Linear Predictive Coding) estimation, using the Burg method. This function implements the mathematics published in [Lagrange], and the recursion relation as noted in [Vos], in turn referenced from [Andersen].

source
DSP.LPC.levinson — Function
levinson(x::AbstractVector, p::Integer)

Implements Levinson recursion, as described in [Levinson], to find the solution a of the linear equation

\[\mathbf{T} (-\vec{a}) -= -\begin{bmatrix} - x_2 \\ - \vdots \\ - x_{p+1} -\end{bmatrix}\]

in the case where $\mathbf{T}$ is Hermitian and Toeplitz, with first column x[1:p]. This function can be used for LPC (Linear Predictive Coding) estimation, by providing LPCLevinson() as an argument to lpc.

source
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GitHub

Periodograms - periodogram estimation

Basic functions

Common procedures like computing the short-time Fourier transform, periodograms, and spectrograms are documented below.

DSP.Periodograms.arraysplit — Function
arraysplit(s, n, noverlap, nfft=n, window=nothing; buffer=zeros(eltype(s), nfft))

Split an array s into arrays of length n with overlapping regions of length noverlap.

Arguments

  • s: Input array.
  • n: Specifies the required subarray length (n ≤ length(s)).
  • noverlap: Number of overlapping elements between subarrays.
  • nfft: Specifies the length of the split arrays. If length(s) < nfft, then the input is padded with zeros.
  • window: An optional scaling vector to be applied to the split arrays.
  • buffer: An optional buffer of length nfft. Iterating or indexing the returned AbstractVector always yields the same Vector with different contents. The last result after iteration or indexing calls is stored into buffer.

Returns

An ArraySplit object with split subarrays.

Examples

julia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 1)
-2-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Nothing}:
- [0.1, 0.2, 0.3]
- [0.3, 0.4, 0.5]
-
-julia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 2, 8)
-3-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Nothing}:
- [0.1, 0.2, 0.3, 0.0, 0.0, 0.0, 0.0, 0.0]
- [0.2, 0.3, 0.4, 0.0, 0.0, 0.0, 0.0, 0.0]
- [0.3, 0.4, 0.5, 0.0, 0.0, 0.0, 0.0, 0.0]
-
-julia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 1, 3, [1, 2, 1])
-2-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Vector{Int64}}:
- [0.1, 0.4, 0.3]
- [0.3, 0.8, 0.5]

arraysplit function with buffer

julia> x = [1, 2, 3, 4, 5];
-
-julia> sub_arr, n_overlap, nfft = 3, 1, 3;
-
-julia> x_split = arraysplit(x, sub_arr, n_overlap, nfft, nothing; buffer=zeros(nfft));
-
-julia> x_split[2]   # Returns AbstractVector result and stores it in ArraySplit.buf and buffer
-3-element Vector{Float64}:
- 3.0
- 4.0
- 5.0
-
-julia> x_split.buf  # Returns stored results from previous x_split[2] call
-3-element Vector{Float64}:
- 3.0
- 4.0
- 5.0
source
DSP.Periodograms.ArraySplit — Type
ArraySplit{T<:AbstractVector,S,W} <: AbstractVector{Vector{S}}

ArraySplit object with fields

  • s::T
  • buf::Vector{S}
  • n::Int
  • noverlap::Int
  • window::W
  • k::Int

where buf: buffer, k: number of split arrays, and s, n, noverlap, window are as per arguments in arraysplit.

source
DSP.Periodograms.periodogram — Function
periodogram(s::AbstractVector; onesided=eltype(s)<:Real, nfft=nextfastfft(length(s)), fs=1, window=nothing)

Computes periodogram of a 1-d signal s by FFT and returns a Periodogram object.

Arguments

  • onesided: For real signals, the two-sided periodogram is symmetric and this function returns a one-sided (real only) periodogram by default. A two-sided periodogram can be obtained by setting onesided=false.
  • nfft: Specifies the number of points to use for the Fourier transform. If length(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency.
  • fs: The sample rate of the original signal.
  • window: An optional window function or vector to be applied to the original signal before computing the Fourier transform. The computed periodogram is normalized so that the area under the periodogram is equal to the uncentered variance (or average power) of the original signal.

Returns

A Periodogram object with the 2 computed fields: power, freq.

Examples

Frequency estimate of cos(2π(25)t) with a 1-sided periodogram.

julia> Fs = 100;                     # Sampling frequency
-
-julia> t = (1:Fs)/Fs;                # 100 time samples
-
-julia> x = cos.(2π*25*t);            # Cosine signal with 25Hz frequency
-
-julia> pxx = periodogram(x; fs=Fs);                 # Periodogram
-
-julia> _, max_index = findmax(pxx.power);           # Find index of max power
-
-julia> pxx.power[max_index], pxx.freq[max_index]    # (power, frequency)
-(0.5, 25.0)

2-sided periodogram of a rectangle function with Hamming window.

julia> x = rect(50; padding=50);     # Rectangular pulse function
-
-julia> pxx = periodogram(x; onesided=false, nfft=512, fs=1000, window=hamming);
-
-julia> round(maximum(pxx.power), digits=4)  # Expect max power to be close to 0Hz for a DC signal
-0.0182
source
periodogram(s::AbstractMatrix; nfft::NTuple{2,Int}=nextfastfft(size(s)), fs=1, radialsum=false, radialavg=false)

Computes periodogram of a 2-d signal using the 2-d FFT and returns a Periodogram2 or Periodogram object.

Arguments

  • nfft: A 2-tuple specifying the number of points to use for the Fourier transform for each respective dimension. If size(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency.
  • fs: The sample rate of the original signal in both directions.
  • radialsum: For radialsum=true, the value of power[k] is proportional to $\frac{1}{N}\sum_{k\leq |k'|<k+1} |X[k']|^2$.
  • radialavg: For radialavg=true, the value of power[k] is proportional to $\frac{1}{N \#\{k\leq |k'|<k+1\}} \sum_{k\leq |k'|<k+1} |X[k']|^2$. The computation of |k'| takes into account non-square signals by scaling the coordinates of the wavevector accordingly.

Returns

  • A Periodogram2 object by default with the 3 computed fields: power, freq1, freq2.
  • A Periodogram object is returned for a radially summed or averaged periodogram (if radialsum or radialavg is true, respectively). Only one of radialsum or radialavg can be set to true in the function.

Examples

julia> x = [1 1; 0 1; 0 0];
-
-julia> pxx = periodogram(x; nfft=(3, 2));   # Returns Periodogram2
-
-julia> power(pxx)
-3×2 Matrix{Float64}:
- 1.5  0.166667
- 0.5  0.166667
- 0.5  0.166667
-
-julia> freq(pxx)
-([0.0, 0.3333333333333333, -0.3333333333333333], [0.0, -0.5])

Radial sum and radial average periodograms

julia> x = [1 3; 0 1];
-
-julia> periodogram(x; radialsum=true)  # Returns Periodogram
-DSP.Periodograms.Periodogram{Float64, AbstractFFTs.Frequencies{Float64}, Vector{Float64}}([6.25, 4.75], [0.0, 0.5])
-
-julia> periodogram(x; radialavg=true)  # Returns Periodogram
-DSP.Periodograms.Periodogram{Float64, AbstractFFTs.Frequencies{Float64}, Vector{Float64}}([6.25, 1.5833333333333333], [0.0, 0.5])
source
DSP.Periodograms.Periodogram2 — Type
Periodogram2{T, F1<:Union{Frequencies,AbstractRange}, F2<:Union{Frequencies,AbstractRange}, M<:AbstractMatrix{T}} <: TFR{T}

A Periodogram2 object with fields:

  • power::M
  • freq1::F1
  • freq2::F2

See power and freq for further details.

source
DSP.Periodograms.welch_pgram — Function
welch_pgram(s::AbstractVector, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real,
-            nfft=nextfastfft(n), fs=1, window)

Computes the Welch periodogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object.

welch_pgram by default divides s into 8 segments with 50% overlap between them. For a Bartlett periodogram, set noverlap=0.

See periodogram for description of optional keyword arguments.

Examples

julia> x = rect(10; padding=20);
-
-julia> power(welch_pgram(x; window=nothing))                    # 1-sided periodogram
-2-element Vector{Float64}:
- 0.9523809523809523
- 0.04761904761904761
-
-julia> power(welch_pgram(x; onesided=false, window=nothing))    # 2-sided periodogram
-3-element Vector{Float64}:
- 0.9523809523809523
- 0.023809523809523805
- 0.023809523809523805
-
-
-julia> power(welch_pgram(x, 5; onesided=false, window=hanning)) # 5 samples segment
-5-element Vector{Float64}:
- 0.8888888888888888
- 0.3807834425694269
- 0.008105446319461968
- 0.008105446319461968
- 0.3807834425694269
-

welch_pgram with $x = sin(2π(100)t) + 2sin(2π(150)t) + noise$

julia> Fs = 1000;                         # Sampling frequency
-
-julia> t = (1:Fs)/Fs;                     # 1000 time samples
-
-julia> f = [100 150];                     # 100Hz & 150Hz frequencies
-
-julia> A = [1; 2];                        # Amplitudes
-
-julia> x = sin.(2π*f.*t)*A + randn(1000); # Noise corrupted x signal
-
-julia> pxx = welch_pgram(x, 100; fs=Fs, nfft=512, window=hamming);
source
welch_pgram(s::AbstractVector, config::WelchConfig)

Computes the Welch periodogram of the given signal s using a predefined WelchConfig object.

Examples

julia> x = rect(10; padding=20);
-
-julia> wconfig = WelchConfig(x; fs=1000, window=hamming);
-
-julia> welch_pgram(x, wconfig);
source
DSP.Periodograms.welch_pgram! — Function
welch_pgram!(out::AbstractVector, s::AbstractVector, n=div(length(s), 8),
-             noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n),
-             fs=1, window=nothing)

Computes the Welch periodogram of a signal s, based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object. The Power Spectral Density (PSD) of Welch periodogram is stored in out.

welch_pgram! by default divides s into 8 segments with 50% overlap between them. For a Bartlett periodogram, set noverlap=0.

See periodogram for description of optional keyword arguments.

Examples

julia> Fs = 100;                     # Sampling frequency
-
-julia> t = (1:Fs)/Fs;                # 100 time samples
-
-julia> x = cos.(2π*25*t);            # Cosine signal with 25Hz frequency
-
-julia> nfft = 512;
-
-julia> y = vec(zeros(1, nfft));
-
-julia> pxx = welch_pgram!(y, x, 10; onesided=false, nfft=nfft, fs=Fs);
-
-julia> y == power(pxx)
-true
source
welch_pgram!(out::AbstractVector, s::AbstractVector, config::WelchConfig)

Computes the Welch periodogram of the given signal s, using a predefined WelchConfig object. The Power Spectral Density (PSD) of Welch periodogram is stored in out.

Examples

julia> Fs = 100;                     # Sampling frequency
-
-julia> t = (1:Fs)/Fs;                # 100 time samples
-
-julia> x = cos.(2π*25*t);            # Cosine signal with 25Hz frequency
-
-julia> nfft = 512;
-
-julia> y = vec(zeros(1, nfft));
-
-julia> wconfig = WelchConfig(x; n=10, onesided=false, nfft=nfft, fs=Fs, window=hamming);
-
-julia> pxx = welch_pgram!(y, x, wconfig);
-
-julia> y == power(pxx)
-true
source
DSP.Periodograms.spectrogram — Function
spectrogram(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes the spectrogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Spectrogram object.

spectrogram by default divides s into 8 segments with 50% overlap between them. See periodogram for description of optional keyword arguments.

The returned Spectrogram object stores the 3 computed fields: power, freq and time. See power, freq and time for further details.

Examples

julia> Fs = 1000;
-
-julia> t = 0:1/Fs:1-1/Fs;
-
-julia> x = sin.(2π*100*t.*t);
-
-julia> spec = spectrogram(x; fs=Fs);
-
-julia> size(power(spec))
-(63, 14)
-
-julia> size(freq(spec))
-(63,)
-
-julia> time(spec)
-0.0625:0.063:0.8815
source
DSP.Periodograms.Spectrogram — Type
Spectrogram{T, F<:Union{Frequencies,AbstractRange}, M<:AbstractMatrix{T}} <: TFR{T}

A Spectrogram object with fields:

  • power::M
  • freq::F
  • time::Float64Range

See power, freq and time for further details.

source
DSP.Periodograms.stft — Function
stft(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)

Computes the Short Time Fourier Transform (STFT) of a signal s based on segments with n samples with overlap of noverlap samples, and returns a matrix containing the STFT coefficients.

stft by default divides s into 8 segments with 50% overlap between them. See periodogram for description of optional keyword arguments.

The STFT computes the DFT over K sliding windows (segments) of the signal s. This returns a J x K matrix where J is the number of DFT coefficients and K the number of windowed segments. The kth column of the returned matrix contains the DFT coefficients for the kth segment.

Examples

julia> Fs = 1000;
-
-julia> t = 0:1/Fs:5-1/Fs;
-
-julia> x = sin.(2π*t.*t);
-
-julia> size(stft(x; window=hamming))
-(313, 14)
-
-julia> size(stft(x, 500, 250; onesided=false, window=hanning))
-(500, 19)
source
DSP.Periodograms.power — Function
power(p)

For a Periodogram, returns the computed power spectral density (PSD) as a Vector.

For a Spectrogram, returns the computed power spectral density (PSD) at each frequency and time bin as a Matrix. Dimensions are frequency × time.

For a CrossPowerSpectra, returns the pairwise cross power spectral density (CPSD) between each pair of channels at each frequency. Dimensions are channel x channel x frequency.

Examples

julia> power(periodogram([1, 2, 3]; fs=210))
-2-element Vector{Float64}:
- 0.05714285714285714
- 0.009523809523809525
source
Base.Libc.time — Function
time(p)

Returns the time bin centers for a given Spectrogram object.

Examples

julia> time(spectrogram(0:1/20:1; fs=8000))
-0.000125:0.000125:0.0025
source
DSP.Periodograms.coherence — Function
coherence(c::Coherence)

Given an Coherence object, returns an n_channels x n_channels x length(freq(c)) array consisting of the pairwise coherences between each channel for each frequency.

source

Multitaper periodogram estimation

DSP.Periodograms.mt_pgram — Function
mt_pgram(s; onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, nw=4, ntapers=iceil(2nw)-1, window=dpss(length(s), nw, ntapers))
-mt_pgram(signal::AbstractVector, config::MTConfig)

Computes the multitaper periodogram of a signal s.

If window is not specified, the signal is tapered with ntapers discrete prolate spheroidal sequences with time-bandwidth product nw. Each sequence is equally weighted; adaptive multitaper is not (yet) supported.

If window is specified, each column is applied as a taper. The sum of periodograms is normalized by the total sum of squares of window.

Returns a Periodogram.

See also mt_pgram! and MTConfig.

source
DSP.Periodograms.mt_pgram! — Function
mt_pgram!(output, s::AbstractVector{T}; onesided::Bool=eltype(s)<:Real,
-    nfft::Int=nextfastfft(length(s)), fs::Real=1,
-    nw::Real=4, ntapers::Int=ceil(Int, 2nw)-1,
-    window::Union{AbstractMatrix,Nothing}=nothing) where T<:Number
-mt_pgram!(output::AbstractVector, signal::AbstractVector, config::MTConfig) -> Periodogram

Computes a multitapered periodogram, storing the output in output. Arguments:

  • signal::AbstractVector: should be of length config.n_samples
  • output::AbstractVector: should be of length length(config.freq)

Optionally pass an MTConfig object to preallocate temporary variables and choose configuration settings; otherwise, keyword arguments may be passed to choose those settings.

Returns a Periodogram.

See also mt_pgram and MTConfig.

source
DSP.Periodograms.mt_spectrogram — Function
mt_spectrogram(signal::AbstractVector{T}, n::Int=length(s) >> 3,
-                              n_overlap::Int=n >> 1; fs=1,
-                              onesided::Bool=T <: Real, kwargs...) where {T}
-mt_spectrogram(signal::AbstractVector, config::MTSpectrogramConfig)

Compute a multitaper spectrogram, returning a Spectrogram object. Optionally pass a MTSpectrogramConfig object; otherwise, any additional keyword arguments accepted by MTConfig may be passed to configure the tapering.

Returns a Spectrogram.

See also mt_spectrogram!.

source
DSP.Periodograms.mt_spectrogram! — Function
mt_spectrogram!(output, signal::AbstractVector{T}, n::Int=length(signal) >> 3,
-    n_overlap::Int=n >> 1; fs=1, onesided::Bool=T <: Real, kwargs...) where {T}
-mt_spectrogram!(destination::AbstractMatrix, signal::AbstractVector, config::MTSpectrogramConfig)

Computes a multitaper spectrogram using the parameters specified in config. Arguments:

  • destination: length(config.mt_config.freq) x length(config.time) matrix. This can be created by DSP.allocate_output(config).
  • signal: vector of length config.n_samples
  • config: optionally, pass an MTSpectrogramConfig object to hold temporary variables and configuration settings. Otherwise, settings arguments may be passed directly.

Returns a Spectrogram.

See also mt_spectrogram.

source
DSP.Periodograms.mt_cross_power_spectra — Function
mt_cross_power_spectra(signal::AbstractMatrix{T}; fs=1, kwargs...) where {T}
-mt_cross_power_spectra(signal::AbstractMatrix, config::MTCrossSpectraConfig)

Computes multitapered cross power spectra between channels of a signal. Arguments:

  • signal: n_channels x n_samples
  • Optionally pass an MTCrossSpectraConfig object to preallocate temporary variables

and choose configuration settings. Otherwise, any keyword arguments accepted by MTCrossSpectraConfig may be passed here.

Produces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array (accessed by power) and the corresponding frequencies (accessed by freq).

See also mt_cross_power_spectra! and MTCrossSpectraConfig.

source
DSP.Periodograms.mt_cross_power_spectra! — Function
mt_cross_power_spectra!(output, signal::AbstractMatrix; fs=1, kwargs...)
-mt_cross_power_spectra!(output, signal::AbstractMatrix, config::MTCrossSpectraConfig)

Computes multitapered cross power spectra between channels of a signal. Arguments:

  • output: n_channels x n_channels x length(config.freq). Can be created by DSP.allocate_output(config).
  • signal: n_channels x n_samples
  • config: MTCrossSpectraConfig{T}: optionally pass a MTCrossSpectraConfig to preallocate temporary and choose configuration settings. Otherwise, one may pass any keyword arguments accepted by this object.

Produces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array and the corresponding frequencies (accessed by freq).

See also mt_cross_power_spectra and MTCrossSpectraConfig.

source
DSP.Periodograms.mt_coherence — Function
mt_coherence(signal::AbstractMatrix{T}; fs=1, freq_range = nothing, demean=false, kwargs...) where T
-mt_coherence(signal::AbstractMatrix, config::MTCoherenceConfig)

Arguments:

  • signal: n_channels x n_samples matrix
  • Optionally pass an MTCoherenceConfig to pre-allocate temporary variables and choose configuration settings, otherwise, see MTCrossSpectraConfig for the meaning of the keyword arguments.

Returns a Coherence object.

See also mt_coherence and MTCoherenceConfig.

source
DSP.Periodograms.mt_coherence! — Function
mt_coherence!(output, signal::AbstractMatrix; fs=1, freq_range=nothing, demean=false, kwargs...)
-mt_coherence!(output, signal::AbstractMatrix, config::MTCoherenceConfig)

Computes the pairwise coherences between channels.

  • output: n_channels x n_channels matrix
  • signal: n_samples x n_channels matrix
  • config: optional configuration object that pre-allocates temporary variables and choose settings.

Returns a Coherence object.

See also mt_coherence and MTCoherenceConfig.

source

Configuration objects

DSP.Periodograms.WelchConfig — Type
WelchConfig(s::AbstractArray; n=size(s, ndims(s))>>3, noverlap=n>>1,
-         onesided=eltype(s)<:Real, nfft=nextfastfft(n),
-         fs=1, window)
-
-WelchConfig(nsamples, eltype; n=nsamples>>3, noverlap=n>>1,
-         onesided=eltype<:Real, nfft=nextfastfft(n),
-         fs=1, window)

Captures all configuration options for welch_pgram in a single struct (akin to MTConfig). When passed on the second argument of welch_pgram, computes the periodogram based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object. For a Bartlett periodogram, set noverlap=0. See periodogram for description of optional keyword arguments.

Note

WelchConfig precomputes an fft plan, and preallocates the necessary intermediate buffers. Thus, repeated calls to welch_pgram that use the same WelchConfig object will be more efficient than otherwise possible.

Examples

julia> x = 1:8;
-
-julia> wconfig1 = WelchConfig(x; n=length(x), noverlap=0, window=hanning, nfft=16);
-
-julia> wconfig2 = WelchConfig(8, Float64; n=length(x), noverlap=0, window=hanning, nfft=16);
source
DSP.Periodograms.MTConfig — Type
MTConfig{T}(n_samples; fs=1,
-        nfft = nextpow(2, n_samples),
-        window = nothing,
-        nw = 4,
-        ntapers = 2 * nw - 1,
-        taper_weights = fill(1/ntapers, ntapers),
-        onesided::Bool=T<:Real,
-        fft_flags = FFTW.MEASURE)

Creates a config object which holds the configuration state and temporary variables used in multitaper computations, e.g. mt_pgram!, mt_spectrogram, MTSpectrogramConfig, MTCrossSpectraConfig, and MTCoherenceConfig.

An MTConfig can be re-used between computations as long as none of the input arguments change.

  • n_samples: the number of samples to be used as input when computing multitaper periodograms with this configuration. Used for pre-allocating temporary buffers.
  • fs: the number of samples per second of the input signal
  • nfft: length of input vector to the FFT; if nfft > n_samples, then the input signal will be zero-padded until it is of length nfft.
  • window: window function to use for tapering. If left at the default of nothing, window will be set to dpss(n_samples, nw, ntapers).
  • ntapers: the number of tapers to use.
  • taper_weights = fill(1/ntapers, ntapers): how to weight the contribution of each taper. The default setting is to simply average them.
  • onesided: whether or not to compute a "one-sided" FFT by using that real signal data yields conjugate-symmetry in Fourier space.
  • fft_flags: flags to control how the FFT plan is generated.
source
DSP.Periodograms.MTSpectrogramConfig — Type
MTSpectrogramConfig(n_samples, mt_config::MTConfig{T}, n_overlap_samples) where {T}
-MTSpectrogramConfig{T}(n_samples, samples_per_window, n_overlap_samples; fs=1, kwargs...) where {T}

Creates a MTSpectrogramConfig which holds configuration and temporary variables for mt_spectrogram and mt_spectrogram!. Any keyword arguments accepted by MTConfig may be passed here, or an MTConfig object itself.

source
DSP.Periodograms.MTCrossSpectraConfig — Type
MTCrossSpectraConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing,
-                        ensure_aligned = T == Float32 || T == Complex{Float32}, kwargs...) where {T}
-MTCrossSpectraConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,
-                     ensure_aligned = T == Float32 || T == Complex{Float32})

Creates a configuration object used for mt_cross_power_spectra and mt_cross_power_spectra!.

  • n_channels: the number of channels of the input.
  • n_samples: the number of samples for each channel of the input.
  • demean: if true, the channelwise mean will be subtracted from the input signals before the cross spectral powers are computed.
  • freq_range: if nothing, all frequencies are retained. Otherwise, only frequencies between first(freq_range) and last(freq_range) are retained.
  • ensure_aligned = T == Float32 || T == Complex{Float32}: perform an extra copy to ensure that the FFT output is memory-aligned
  • Additionally, either pass an MTConfig object, or keyword arguments such as fs accepted by MTConfig.

Returns a CrossPowerSpectra object.

source
DSP.Periodograms.MTCoherenceConfig — Type
MTCoherenceConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing, kwargs...) where T
-MTCoherenceConfig(cs_config::MTCrossSpectraConfig{T}) where {T}
-MTCoherenceConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,
-    ensure_aligned = T == Float32 || T == Complex{Float32}) where {T}

Creates a configuration object for coherences from a MTCrossSpectraConfig. Provides a helper method with the same arugments as MTCrossSpectraConfig to construct the MTCrossSpectraConfig object.

See also mt_coherence and mt_coherence!.

source
diff --git a/previews/PR604/search_index.js b/previews/PR604/search_index.js deleted file mode 100644 index 4caf585a6..000000000 --- a/previews/PR604/search_index.js +++ /dev/null @@ -1,3 +0,0 @@ -var documenterSearchIndex = {"docs": -[{"location":"util/#Util-utility-functions","page":"Util - utility functions","title":"Util - utility functions","text":"","category":"section"},{"location":"util/","page":"Util - utility functions","title":"Util - utility functions","text":"note: Note\nAs of version 0.6.1 of DSP.jl, fftfreq and rfftfreq have been moved from DSP.jl to AbstractFFTs.jl version 0.5 and above. You can also access these functions through FFTW.jl version 1.1 and above.","category":"page"},{"location":"util/","page":"Util - utility functions","title":"Util - utility functions","text":"unwrap\nunwrap!\nhilbert\nnextfastfft\npow2db\namp2db\ndb2pow\ndb2amp\nrms\nrmsfft\nmeanfreq\nfinddelay\nshiftin!\nshiftsignal\nshiftsignal!\nalignsignals\nalignsignals!\ndiric","category":"page"},{"location":"util/#DSP.Unwrap.unwrap","page":"Util - utility functions","title":"DSP.Unwrap.unwrap","text":"unwrap(m; kwargs...)\n\nAssumes m to be a sequence of values that has been wrapped to be inside the given range (centered around zero), and undoes the wrapping by identifying discontinuities. If a single dimension is passed to dims, then m is assumed to have wrapping discontinuities only along that dimension. If a range of dimensions, as in 1:ndims(m), is passed to dims, then m is assumed to have wrapping discontinuities across all ndims(m) dimensions.\n\nA common usage for unwrapping across a singleton dimension is for a phase measurement over time, such as when comparing successive frames of a short-time-fourier-transform, as each frame is wrapped to stay within (-pi, pi].\n\nA common usage for unwrapping across multiple dimensions is for a phase measurement of a scene, such as when retrieving the phase information of of an image, as each pixel is wrapped to stay within (-pi, pi].\n\nArguments\n\nm::AbstractArray{T, N}: Array to unwrap.\ndims=nothing: Dimensions along which to unwrap. If dims is an integer, then unwrap is called on that dimension. If dims=1:ndims(m), then m is unwrapped across all dimensions.\nrange=2pi: Range of wrapped array.\ncircular_dims=(false, ...): When an element of this tuple is true, the unwrapping process will consider the edges along the corresponding axis of the array to be connected.\nrng=default_rng(): Unwrapping of arrays with dimension > 1 uses a random initialization. A user can pass their own RNG through this argument.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Unwrap.unwrap!","page":"Util - utility functions","title":"DSP.Unwrap.unwrap!","text":"unwrap!(m; kwargs...)\n\nIn-place version of unwrap.\n\n\n\n\n\nunwrap!(y, m; kwargs...)\n\nUnwrap m storing the result in y, see unwrap.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.hilbert","page":"Util - utility functions","title":"DSP.Util.hilbert","text":"hilbert(x)\n\nComputes the analytic representation of x, x_a = x + j hatx, where hatx is the Hilbert transform of x, along the first dimension of x.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.nextfastfft","page":"Util - utility functions","title":"DSP.Util.nextfastfft","text":"nextfastfft(n::Integer)\nnextfastfft(ns::Tuple)\n\nReturn an estimate for the padded size of an array that is optimal for the performance of an n-dimensional FFT.\n\nFor Tuple inputs, this returns a Tuple that is the size of the padded array.\n\nFor Integer inputs, this returns the closest product of 2, 3, 5, and 7 greater than or equal to n. FFTW contains optimized kernels for these sizes and computes Fourier transforms of inputs that are a product of these sizes faster than for inputs of other sizes.\n\nwarning: Warning\nCurrently, nextfastfft(ns::Tuple) is implemented as nextfastfft.(ns). This may change in future releases if a better estimation model is found.It is recommended to use nextfastfft.(ns) to obtain a tuple of padded lengths for 1D FFTs.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.pow2db","page":"Util - utility functions","title":"DSP.Util.pow2db","text":"pow2db(a)\n\nConvert a power ratio to dB (decibel), or 10log_10(a). The inverse of db2pow.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.amp2db","page":"Util - utility functions","title":"DSP.Util.amp2db","text":"amp2db(a)\n\nConvert an amplitude ratio to dB (decibel), or 20 log_10(a)=10log_10(a^2). The inverse of db2amp.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.db2pow","page":"Util - utility functions","title":"DSP.Util.db2pow","text":"db2pow(a)\n\nConvert dB to a power ratio. This function call also be called using a*dB, i.e. 3dB == db2pow(3). The inverse of pow2db.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.db2amp","page":"Util - utility functions","title":"DSP.Util.db2amp","text":"db2amp(a)\n\nConvert dB to an amplitude ratio. This function call also be called using a*dBa, i.e. 3dBa == db2amp(3). The inverse of amp2db.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.rms","page":"Util - utility functions","title":"DSP.Util.rms","text":"rms(s; dims)\n\nReturn the root mean square (rms) of signal s. Optional keyword parameter dims can be used to specify the dimensions along which to compue the rms.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.rmsfft","page":"Util - utility functions","title":"DSP.Util.rmsfft","text":"rmsfft(f)\n\nReturn the root mean square of signal s given the FFT transform f = fft(s). Equivalent to rms(ifft(f)).\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.meanfreq","page":"Util - utility functions","title":"DSP.Util.meanfreq","text":"meanfreq(x, fs)\n\nCalculate the mean power frequency of x with a sampling frequency of fs, defined as:\n\nMPF = fracsum_i=1^F f_i X_i^2 sum_i=0^F X_i^2 Hz\n\nwhere F is the Nyquist frequency, and X is the power spectral density.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.finddelay","page":"Util - utility functions","title":"DSP.Util.finddelay","text":"finddelay(x, y)\n\nEstimate the delay of x with respect to y by locating the peak of their cross-correlation.\n\nThe output delay will be positive when x is delayed with respect y, negative if advanced, 0 otherwise.\n\nExample\n\njulia> finddelay([0, 0, 1, 2, 3], [1, 2, 3])\n2\n\njulia> finddelay([1, 2, 3], [0, 0, 1, 2, 3])\n-2\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.shiftin!","page":"Util - utility functions","title":"DSP.Util.shiftin!","text":"shiftin!(a::AbstractVector{T}, b::AbstractVector{T}) where T\n\nShifts b into the end of a.\n\njulia> shiftin!([1,2,3,4], [5, 6])\n4-element Vector{Int64}:\n 3\n 4\n 5\n 6\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.shiftsignal","page":"Util - utility functions","title":"DSP.Util.shiftsignal","text":"shiftsignal(x, s)\n\nShift elements of signal x in time by a given amount s of samples and fill the spaces with zeros. For circular shifting, use circshift.\n\nExample\n\njulia> shiftsignal([1, 2, 3], 2)\n3-element Vector{Int64}:\n 0\n 0\n 1\n\njulia> shiftsignal([1, 2, 3], -2)\n3-element Vector{Int64}:\n 3\n 0\n 0\n\nSee also shiftsignal!.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.shiftsignal!","page":"Util - utility functions","title":"DSP.Util.shiftsignal!","text":"shiftsignal!(x, s)\n\nMutating version of shiftsignals(): shift x of s samples and fill the spaces with zeros in-place.\n\nSee also shiftsignal.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.alignsignals","page":"Util - utility functions","title":"DSP.Util.alignsignals","text":"alignsignals(x, y)\n\nUse finddelay() and shiftsignal() to time align x to y. Also return the delay of x with respect to y.\n\nExample\n\njulia> alignsignals([0, 0, 1, 2, 3], [1, 2, 3])\n([1, 2, 3, 0, 0], 2)\n\njulia> alignsignals([1, 2, 3], [0, 0, 1, 2, 3])\n([0, 0, 1], -2)\n\nSee also alignsignals!.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.Util.alignsignals!","page":"Util - utility functions","title":"DSP.Util.alignsignals!","text":"alignsignals!(x, y)\n\nMutating version of alignsignals(): time align x to y in-place.\n\nSee also alignsignals.\n\n\n\n\n\n","category":"function"},{"location":"util/#DSP.diric","page":"Util - utility functions","title":"DSP.diric","text":"kernel = diric(Ω::Real, n::Integer)\n\nDirichlet kernel, also known as periodic sinc function, where n should be a positive integer.\n\nTypically returns fracsin(Ω cdot n2)n * sin(Ω 2), but returns 1 when Ω is a multiple of 2π.\n\nIn the usual case where n is odd, the output is equivalent to frac1n sum_k=-(n-1)2^(n-1)2 e^i k Ω, which is the discrete-time Fourier transform (DTFT) of a n-point moving average filter.\n\nWhen n is odd or even, the function is 2π or 4π periodic, respectively. The formula for general n is\n\nmathrmdiric(Ωn) = frace^-i (n-1) Ω2n sum_k=0^n-1 e^i k Ω\n\nwhich is a real and symmetric function of Ω.\n\nAs of 2021-03-19, the Wikipedia definition has different factors. The definition here is consistent with scipy and other software frameworks.\n\nExamples\n\njulia> round.(diric.((-2:0.5:2)*π, 5), digits=9)'\n1×9 adjoint(::Vector{Float64}) with eltype Float64:\n 1.0 -0.2 0.2 -0.2 1.0 -0.2 0.2 -0.2 1.0\n\njulia> diric(0, 4)\n1.0\n\n\n\n\n\n","category":"function"},{"location":"appendix/#Index","page":"Index","title":"Index","text":"","category":"section"},{"location":"appendix/","page":"Index","title":"Index","text":"","category":"page"},{"location":"windows/#Windows-window-functions","page":"Windows - window functions","title":"Windows - window functions","text":"","category":"section"},{"location":"windows/","page":"Windows - window functions","title":"Windows - window functions","text":"rect\nhanning\nhamming\ntukey\ncosine\nlanczos\ntriang\nbartlett\ngaussian\nbartlett_hann\nblackman\nkaiser\ndpss\ndpsseig","category":"page"},{"location":"windows/#DSP.Windows.rect","page":"Windows - window functions","title":"DSP.Windows.rect","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀│ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n │ │ \n0 │ │ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nrect(n::Integer; padding::Integer=0, zerophase::Bool=false)\nrect(dims; padding=0, zerophase=false)\n\nRectangular window of length n, padded with padding zeros. This window is 1 within the window, and 0 outside of it.\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.hanning","page":"Windows - window functions","title":"DSP.Windows.hanning","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▄▀▀▀▀▀▀▀▀▄▄ │ \n │ ▗▄▀ ▀▄▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▀ ▀▖ │ \n │ ▞▘ ▝▚ │ \n │ ▗▀ ▀▖ │ \n │ ▗▄▘ ▝▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▞ ▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▀ ▀▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▀▘ ▝▀▄ │ \n0 │▄▄▄▄▀▀ ▀▀▄▄▄▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nhanning(n::Integer; padding::Integer=0, zerophase::Bool=false)\nhanning(dims; padding=0, zerophase=false)\n\nHanning window of length n with padding zeros. The Hanning (or Hann) window is a raised-cosine window that reaches zero at the endpoints.\n\nThe window is defined by sampling the continuous function:\n\n 1 + cos(2πx)\nw(x) = ──────────── = cos²(πx)\n 2\n\nin the range [-0.5, 0.5]\n\nThe hanning window satisfies the Constant Overlap-Add (COLA) property with a hop of 0.5, which means that adding together a sequence of delayed windows with 50% overlap will result in a constant signal. This is useful when synthesizing a signal from a number of overlapping frames (each with a roughly rectangular window), to eliminate windowing amplitude modulation.\n\nWhile this windowing function was originally named after its inventor Julius Von Hann, colloquial usage has adjusted to refer to it as the \"Hanning\" window, to better match the similarly-named \"Hamming\" window. Because of this, both hann() and hanning() are exported from this module.\n\nNote that the hanning window is the cosine window squared.\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.hamming","page":"Windows - window functions","title":"DSP.Windows.hamming","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▞▀▀▀▀▀▀▀▀▚▄ │ \n │ ▗▞▀ ▀▚▖ │ \n │ ▗▞▘ ▝▀▄ │ \n │ ▗▞▘ ▚▖ │ \n │ ▞▘ ▝▚▖ │ \n │ ▄▀ ▀▄ │ \n │ ▄▞ ▚▄ │ \n │ ▗▞ ▚▖ │ \n │ ▞▘ ▝▚ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▚▖ │ \n │ ▄▞▘ ▝▚▄ │ \n │ ▗▄▀▘ ▀▄▖ │ \n │▄▄▄▞▀▘ ▝▀▚▄▄▄│ \n0 │ │ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nhamming(n::Integer; padding::Integer=0, zerophase::Bool=false)\nhamming(dims; padding=0, zerophase=false)\n\nHamming window of length n with padding zeros. The Hamming window does not reach zero at the endpoints and so has a shallower frequency roll-off when compared to the Hanning window, but is designed to cancel the first side-lobe.\n\nThe window is defined by sampling the continuous function:\n\nw(x) = 0.54 + 0.46*cos(2pi*x)\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.tukey","page":"Windows - window functions","title":"DSP.Windows.tukey","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▗▞▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▜▖ │ \n │ ▗▘ ▝▖ │ \n │ ▗▘ ▝▖ │ \n │ ▞ ▚ │ \n │ ▐ ▌ │ \n │ ▌ ▝▖ │ \n │ ▐ ▚ │ \n │ ▗▘ ▝▖ │ \n │ ▌ ▚ │ \n │ ▗▘ ▌ │ \n │ ▌ ▐ │ \n │ ▞ ▚ │ \n │ ▞▘ ▝▖ │ \n │ ▗▘ ▝▖ │ \n0 │▄▄▘ ▝▚▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\ntukey(n::Integer, α::Real; padding::Integer=0, zerophase::Bool=false)\ntukey(dims, α; padding=0, zerophase=false)\n\nTukey window of length n with padding zeros. The Tukey window has a flat top and reaches zero at the endpoints, with a sinusoidal transition area parameterized by α. For α == 0, the window is equivalent to a rectangular window. For α == 1, the window is a Hann window.\n\nThe window is defined by sampling the continuous function:\n\n ⎛ ⎛2π ⎛ 1 - α⎞⎞\n ⎜ 1 + cos ⎜── ⎜x + ─────⎟⎟ 1 - α\n ⎜ ⎝ α ⎝ 2 ⎠⎠ x ≤ - ─────\n ⎜ ───────────────────────── 2\n ⎜ 2\n ⎜\n ⎜ 1 - α 1 - α\nw(x) = ⎜ 1 - ───── < x ≤ ─────\n ⎜ 2 2\n ⎜\n ⎜ ⎛2π ⎛ 1 - α⎞⎞\n ⎜ 1 + cos ⎜── ⎜x - ─────⎟⎟ 1 - α\n ⎜ ⎝ α ⎝ 2 ⎠⎠ x > ─────\n ⎜ ───────────────────────── 2\n ⎝ 2\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. α, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.cosine","page":"Windows - window functions","title":"DSP.Windows.cosine","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▄▀▀▀▀▀▀▀▀▀▀▀▀▄▄ │ \n │ ▗▄▀▀ ▀▀▄▖ │ \n │ ▄▀▘ ▝▀▄ │ \n │ ▗▞▀ ▀▚▖ │ \n │ ▄▀▘ ▝▀▄ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▚▖ │ \n │ ▞▘ ▝▚▖ │ \n │ ▄▀ ▝▄ │ \n │ ▗▞ ▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▞ ▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▞ ▚▖ │ \n0 │▄▘ ▝▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\ncosine(n::Integer; padding::Integer=0, zerophase::Bool=false)\ncosine(dims; padding=0, zerophase=false)\n\nCosine window of length n with padding zeros. The cosine window is the first lobe of a cosine function (with the zero crossings at +/- π as endpoints). Also called the sine window.\n\nThe window is defined by sampling the continuous function:\n\nw(x) = cos(πx)\n\nin the range [-0.5, 0.5]\n\nNote that the cosine window is the square root of the hanning window, so it is sometimes used when you are applying the window twice, such as the analysis and synthesis steps of an STFT.\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.lanczos","page":"Windows - window functions","title":"DSP.Windows.lanczos","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▄▀▀▀▀▀▀▀▀▀▀▄▄ │ \n │ ▄▞▀ ▀▀▄ │ \n │ ▗▞▀ ▀▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▀▘ ▝▀▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▘ ▝▚ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞▘ ▚▖ │ \n │ ▄▘ ▝▚ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▜▖ │ \n │ ▗▞▘ ▝▚▖ │ \n0 │▄▞▘ ▝▚▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nlanczos(n::Integer; padding::Integer=0, zerophase::Bool=false)\nlanczos(dims; padding=0, zerophase=false)\n\nLanczos window of length n with padding zeros. The Lanczos window is the main lobe of a sinc function.\n\nThe window is defined by sampling the continuous function:\n\n sin(2πx)\nw(x) = sinc(2x) = ────────\n 2πx\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.triang","page":"Windows - window functions","title":"DSP.Windows.triang","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▞▀▀▀▄▖ │ \n │ ▗▄▞▀ ▝▀▄▖ │ \n │ ▗▄▀▘ ▝▀▄▖ │ \n │ ▗▄▀▘ ▝▀▄▖ │ \n │ ▗▄▀▘ ▝▀▄▖ │ \n │ ▗▄▀▘ ▝▀▄▄ │ \n │ ▗▄▀▘ ▀▚▄ │ \n │ ▗▄▀▘ ▀▙▄ │ \n │ ▄▄▀▘ ▀▚▄ │ \n │ ▄▞▀ ▀▚▄ │ \n │ ▄▞▀ ▀▚▄ │ \n │▀ ▀│ \n │ │ \n │ │ \n0 │ │ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 7 \n\ntriang(n::Integer; padding::Integer=0, zerophase::Bool=false)\ntriang(dims; padding=0, zerophase=false)\n\nTriangular window of length n with padding zeros. The Triangular window does not reach zero at the endpoints. For odd n the triang window is the center n points of an n+2-point bartlett window (i.e. the samples just outside the window would be zero). For even n the window slope is the same as the n-1 window but delayed by a half sample so the zero points would be 1/2 sample past the ends of the window.\n\nThe window is defined by sampling the continuous function:\n\n ⎛ 2(n-1)\n ⎜1 - ────── abs(x) n is even\n ⎜ n\nw(x) = ⎜\n ⎜ 2(n-1)\n ⎜1 - ────── abs(x) n is odd\n ⎝ n+1\n\nin the range [-0.5, 0.5].\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\nWhen zerophase is true substitute n+1 for n in the above window expressions.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.bartlett","page":"Windows - window functions","title":"DSP.Windows.bartlett","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▗▞▀▀▚▖ │ \n │ ▄▞▘ ▝▚▄ │ \n │ ▄▀ ▀▄ │ \n │ ▗▄▀ ▀▄▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▞▘ ▝▚▄ │ \n │ ▄▀ ▀▄ │ \n │ ▄▄▀ ▀▚▄ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞▀ ▀▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▀▘ ▝▀▄ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞▀ ▀▚▖ │ \n0 │▄▞▘ ▝▚▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nbartlett(n::Integer; padding::Integer=0, zerophase::Bool=false)\nbartlett(dims; padding=0, zerophase=false)\n\nBartlett window of length n. The Bartlett window is a triangular window that reaches 0 at the endpoints. This is equivalent to convolving two rectangular windows of length (n-1)/2 and adding the zero endpoints. See triang for a window that does not reach zero at the endpoints.\n\nThe window is defined by sampling the continuous function:\n\n1 - abs(2x)\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.gaussian","page":"Windows - window functions","title":"DSP.Windows.gaussian","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▗▄▞▀▀▀▀▀▀▀▄▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▞ ▚▖ │ \n │ ▄▘ ▝▙ │ \n │ ▗▞ ▀▖ │ \n │ ▟▘ ▝▚ │ \n │ ▞▀ ▀▚ │ \n │ ▄▀ ▀▄ │ \n │ ▄▀ ▀▄ │ \n │ ▄▀ ▀▄ │ \n │ ▄▞▀ ▀▚▄ │ \n │ ▗▄▄▀▀ ▀▀▄▄▖ │ \n0 │▀▀▘ ▝▀▀│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\ngaussian(n::Integer, σ::Real; padding::Integer=0, zerophase::Bool=false)\ngaussian(dims, σ; padding=0, zerophase=false)\n\nGives an n-sample gaussian window defined by sampling the function:\n\n ⎛ 2⎞\n ⎜-1 ⎛x⎞ ⎟\n ⎜── ⋅ ⎜─⎟ ⎟\n ⎝ 2 ⎝σ⎠ ⎠\nw(x) = e\n\nin the range [-0.5,0.5]. This means that for σ=0.5 the endpoints of the window will correspond to 1 standard deviation away from the center.\n\nProviding a dims Tuple rather than a single n constructs a 2D window. σ, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.bartlett_hann","page":"Windows - window functions","title":"DSP.Windows.bartlett_hann","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▄▄▀▀▀▀▀▀▄▄ │ \n │ ▗▟▀ ▀▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▞▘ ▝▚ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▚▖ │ \n │ ▗▄▘ ▝▚▖ │ \n │ ▞▘ ▝▚ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▞▘ ▝▜▄ │ \n0 │▄▄▄▀▀ ▀▀▄▄▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nbartlett_hann(n::Integer; padding::Integer=0, zerophase::Bool=false)\nbartlett_hann(dims; padding=0, zerophase=false)\n\nBartlett-Hann window of length n with padding zeros. The Bartlett-Hann window is a weighted sum of the Bartlett and Hann windows.\n\nThe window is defined by sampling the continuous function:\n\nw(x) = 0.62 - 0.48*abs(x) + 0.38*cos(2π*x)\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.blackman","page":"Windows - window functions","title":"DSP.Windows.blackman","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▗▞▀▀▀▀▀▀▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▞▘ ▝▚ │ \n │ ▗▀ ▀▖ │ \n │ ▄▘ ▝▙ │ \n │ ▞ ▚ │ \n │ ▗▀ ▀▖ │ \n │ ▗▘ ▝▖ │ \n │ ▞▘ ▝▚ │ \n │ ▄▞ ▚▄ │ \n │ ▗▞ ▚▖ │ \n │ ▞▘ ▝▚ │ \n │ ▗▄▀ ▀▄▖ │ \n │ ▄▞▘ ▝▚▄▖ │ \n0 │▄▄▄▄▄▄▞▀▀ ▝▀▚▄▄▄▄▄▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nblackman(n::Integer; padding::Integer=0, zerophase::Bool=false)\nblackman(dims; padding=0, zerophase=false)\n\nApproximates the \"Exact\" Blackman window. This is the generalized Blackman window with α = 0.16.\n\nThe window is defined by sampling the continuous function:\n\nw(x) = 0.42 + 0.5*cos(2π*x) + 0.08*cos(4π*x)\n\nin the range [-0.5, 0.5]\n\nProviding a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.kaiser","page":"Windows - window functions","title":"DSP.Windows.kaiser","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n1 │ ▗▄▀▀▀▀▀▀▄▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▘ ▝▄ │ \n │ ▞ ▚ │ \n │ ▗▀ ▀▖ │ \n │ ▗▘ ▝▖ │ \n │ ▄▘ ▝▄ │ \n │ ▞ ▚ │ \n │ ▗▞ ▚▖ │ \n │ ▗▘ ▝▖ │ \n │ ▗▀▘ ▝▀▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▄▞▘ ▝▚▄▖ │ \n0 │▄▄▄▄▄▄▄▞▀▘ ▝▀▚▄▄▄▄▄▄▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\nkaiser(n::Integer, α::Real; padding::Integer=0, zerophase::Bool=false)\nkaiser(dims, α; padding=0, zerophase=false)\n\nKaiser window of length n parameterized by α. The Kaiser window approximates the DPSS window (given by dpss), using a simplified definition relying on a Bessel function. Larger values for α give a wider main lobe but have lower sidelobes. Typically α is set around 3.\n\nThe window is defined by sampling the continuous function:\n\n ⎛ ⎛ _________⎞⎞\n ⎜ ⎜ ╱ 2⎟⎟\nw(x) = I₀⎝πα⎝╲╱ 1 - (2x) ⎠⎠\n ────────────────────\n I₀(πα)\n\nin the range [-0.5, 0.5]\n\nWhere I₀(⋅) is the zeroth-order modified Bessel function of the first kind.\n\nProviding a dims Tuple rather than a single n constructs a 2D window. α, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.dpss","page":"Windows - window functions","title":"DSP.Windows.dpss","text":" ┌──────────────────────────────────────────────────────────────────────┐ \n0.2 │ ▗▄▞▀▀▀▀▀▀▚▄▖ │ \n │ ▄▀▘ ▝▀▄ │ \n │ ▄▀ ▀▄ │ \n │ ▗▞ ▀▖ │ \n │ ▞▘ ▝▚ │ \n │ ▗▀ ▀▄ │ \n │ ▗▞▘ ▚▖ │ \n │ ▗▄▀ ▝▄▖ │ \n │ ▄▘ ▝▄ │ \n │ ▗▞ ▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▗▞▘ ▝▚▖ │ \n │ ▄▞▘ ▝▚▄ │ \n │ ▄▄▀▘ ▀▚▄ │ \n 0 │▄▞▀▀ ▀▀▚▄│ \n └──────────────────────────────────────────────────────────────────────┘ \n 1 69 \n\ndpss(n::Integer, nw::Real, ntapers::Integer=iceil(2*nw)-1;\n padding::Integer=0, zerophase::Bool=false)\n\nThe first ntapers discrete prolate spheroid sequences (Slepian tapers) as an n × ntapers matrix. The signs of the tapers follow the convention that the first element of the skew-symmetric (odd) tapers is positive. The time-bandwidth product is given by nw.\n\nThe DPSS window maximizes the energy concentration in the main lobe.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\n\n\n\n\n","category":"function"},{"location":"windows/#DSP.Windows.dpsseig","page":"Windows - window functions","title":"DSP.Windows.dpsseig","text":"dpsseig(A, nw)\n\nEigenvalues of the DPSS matrix, representing the ratios of the power within the main lobe to the total power (main and sidelobes). A is the output of dpss, and nw is the time-bandwidth product provided to dpss as input.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#Convolutions-similarity-methods","page":"Convolutions - similarity methods","title":"Convolutions - similarity methods","text":"","category":"section"},{"location":"convolutions/","page":"Convolutions - similarity methods","title":"Convolutions - similarity methods","text":"conv\nconv!\ndeconv\nxcorr","category":"page"},{"location":"convolutions/#DSP.conv","page":"Convolutions - similarity methods","title":"DSP.conv","text":"conv(u, v; algorithm)\n\nConvolution of two arrays. A convolution algorithm is automatically chosen among direct convolution, FFT, or FFT overlap-save, depending on the size of the input, unless explicitly specified with the algorithm keyword argument; see conv! for details.\n\n\n\n\n\nconv(u,v,A)\n\n2-D convolution of the matrix A with the 2-D separable kernel generated by the vector u and row-vector v. Uses 2-D FFT algorithm.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#DSP.conv!","page":"Convolutions - similarity methods","title":"DSP.conv!","text":"conv!(out, u, v; algorithm=:auto)\n\nConvolution of two arrays u and v with the result stored in out. out must be large enough to store the entire result; if it is even larger, the excess entries will be zeroed.\n\nout, u, and v can be N-dimensional arrays, with arbitrary indexing offsets. If none of them has offset axes, size(out,d) ≥ size(u,d) + size(v,d) - 1 must hold. If both input and output have offset axes, firstindex(out,d) ≤ firstindex(u,d) + firstindex(v,d) and lastindex(out,d) ≥ lastindex(u,d) + lastindex(v,d) must hold (for d = 1,...,N). A mix of offset and non-offset axes is not permitted.\n\nThe algorithm keyword allows choosing the algorithm to use:\n\n:direct: Evaluates the convolution sum in time domain.\n:fft_simple: Evaluates the convolution as a product in the frequency domain.\n:fft_overlapsave: Evaluates the convolution block-wise as a product in the frequency domain, overlapping the resulting blocks.\n:fft: Selects the faster of :fft_simple and :fft_overlapsave (as estimated from the input size).\n:fast: Selects the fastest of :direct, :fft_simple and :fft_overlapsave (as estimated from the input size).\n:auto (default): Equivalent to :fast if the data type is known to be suitable for FFT-based computation, equivalent to :direct otherwise.\n\nwarning: Warning\nThe choices made by :fft, :fast, and :auto are based on performance heuristics which may not result in the fastest algorithm in all cases. If best performance for a certain size/type combination is required, it is advised to do individual benchmarking and explicitly specify the desired algorithm.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#DSP.deconv","page":"Convolutions - similarity methods","title":"DSP.deconv","text":"deconv(b,a) -> c\n\nConstruct vector c such that b = conv(a,c) + r. Equivalent to polynomial division.\n\n\n\n\n\n","category":"function"},{"location":"convolutions/#DSP.xcorr","page":"Convolutions - similarity methods","title":"DSP.xcorr","text":"xcorr(u; padmode::Symbol=:none, scaling::Symbol=:none)\nxcorr(u, v; padmode::Symbol=:none, scaling::Symbol=:none)\n\nWith two arguments, compute the cross-correlation of two vectors, by calculating the similarity between u and v with various offsets of v. Delaying u relative to v will shift the result to the right. If one argument is provided, calculate xcorr(u, u; kwargs...).\n\nThe size of the output depends on the padmode keyword argument: with padmode = :none the length of the result will be length(u) + length(v) - 1, as with conv. With padmode = :longest, the shorter of the arguments will be padded so they are of equal length. This gives a result with length 2*max(length(u), length(v))-1, with the zero-lag condition at the center.\n\nThe keyword argument scaling can be provided. Possible arguments are the default :none and :biased. :biased is valid only if the vectors have the same length, or only one vector is provided, dividing the result by length(u).\n\nExamples\n\njulia> xcorr([1,2,3],[1,2,3])\n5-element Vector{Int64}:\n 3\n 8\n 14\n 8\n 3\n\n\n\n\n\n","category":"function"},{"location":"estimation/#Estimation-parametric-estimation-functions","page":"Estimation - parametric estimation functions","title":"Estimation - parametric estimation functions","text":"","category":"section"},{"location":"estimation/","page":"Estimation - parametric estimation functions","title":"Estimation - parametric estimation functions","text":"esprit\njacobsen\nquinn","category":"page"},{"location":"estimation/#DSP.Estimation.esprit","page":"Estimation - parametric estimation functions","title":"DSP.Estimation.esprit","text":"esprit(x::AbstractArray, M::Integer, p::Integer, Fs::Real=1.0)\n\nESPRIT [Roy1986] algorithm for frequency estimation. Estimation of Signal Parameters via Rotational Invariance Techniques\n\nGiven a signal x of length N that is the sum of p complex exponentials (cisoids) with unknown constant amplitudes and phases, estimate and return an array of the p frequencies.\n\nx(t)=sum_i=1^p A_ie^j2pi f_i t\n\nwhere A_i and f_i are the amplitude and frequency of the ith cisoid.\n\nArguments\n\nx::AbstractArray: complex length N signal array\nM::Integer: size of correlation matrix, must be <= N. The signal subspace is computed from the SVD of an M x (N-M+1) signal matrix formed from N-M+1 length-M shifts of the signal x in its columns. For best performance for 1 cisoid, use M = (N+1)/3 (according to van der Veen and Leus). For faster execution (due to smaller SVD), use small M or small N-M\np::Integer: number of complex exponentials/frequencies to estimate.\nFs::Float64: sampling frequency, in Hz.\n\nReturns\n\nlength p real array of frequencies in units of Hz.\n\n[Roy1986]: R Roy, A Paulraj and T Kailath, ESPRIT - A subspace approach to estimation of parameters of cisoids in noise, IEEE Trans. Acoustics, Speech, Signal Process., 34, 1340-1342 (1986). http://ieeexplore.ieee.org/abstract/document/1164935/.\n\nExamples\n\nSuppose a signal x(t)=2e^j2pi(2500)t + 5e^j2pi(400)t is corrupted by noise. The goal is to retrieve the frequencies in the signal. Let the sampling frequency be 8kHz.\n\njulia> Fs = 8000; # sampling frequency (Hz)\njulia> true_freq = [2500 400]; # true frequencies (Hz)\njulia> t = collect(1:Fs)/Fs; # time vector\njulia> x = 2exp.(1im*2π*true_freq[1]*t) + 5exp.(1im*2π*true_freq[2]*t); # original signal x\n\nAdd gaussian noise in complex form to the signal x to mimic noise corruption.\n\njulia> noise = randn(Fs, 2)*[1; 1im]; # complex random noise\njulia> x += noise; # add noise to signal x\n\nRun the ESPRIT algorithm to retrieve approximate frequencies.\n\njulia> M = 5; # window length of 5\njulia> p = 2; # 2 frequencies to estimate\njulia> esprit(x, M, p, Fs)\n2-element Vector{Float64}:\n2502.2704154274957\n399.8420984461712\n\n\n\n\n\n","category":"function"},{"location":"estimation/#DSP.Estimation.jacobsen","page":"Estimation - parametric estimation functions","title":"DSP.Estimation.jacobsen","text":"jacobsen(x::AbstractVector, Fs::Real = 1.0)\n\nEstimate the largest frequency in the complex signal x using Jacobsen's algorithm [Jacobsen2007]. Argument Fs is the sampling frequency. All frequencies are expressed in Hz.\n\nIf the signal x is real, the estimated frequency is guaranteed to be positive, but it may be highly inaccurate (especially for frequencies close to zero or to Fs/2).\n\nIf the sampling frequency Fs is not provided, then it is assumed that Fs = 1.0.\n\n[Jacobsen2007]: E Jacobsen and P Kootsookos, \"Fast, Accurate Frequency Estimators\", Chapter 10 in \"Streamlining Digital Signal Processing\", edited by R. Lyons, 2007, IEEE Press.\n\n\n\n\n\n","category":"function"},{"location":"estimation/#DSP.Estimation.quinn","page":"Estimation - parametric estimation functions","title":"DSP.Estimation.quinn","text":"quinn(x::Vector, f0::Real, Fs::Real = 1.0 ; tol = 1e-6, maxiters = 20)\n\nquinn(x::Vector, Fs::Real = 1.0 ; kwargs...)\n\nquinn(x::Vector ; kwargs...)\n\nAlgorithms by Quinn and Quinn & Fernandes for frequency estimation. Given a signal x and an initial guess f0, estimate and return the frequency of the largest sinusoid in x. Fs is the sampling frequency. All frequencies are expressed in Hz.\n\nIf the initial guess f0 is not provided, then a guess is calculated using Jacobsen's estimator. If the sampling frequency Fs is not provided, then it is assumed that Fs = 1.0.\n\nThe following keyword arguments control the algorithm's behavior:\n\ntol: the algorithm stops when the absolute value of the difference between two consecutive estimates is less than tol. Defaults to 1e-6.\nmaxiters: the maximum number of iterations to run. Defaults to 20.\n\nReturns a tuple (estimate, reachedmaxiters), where estimate is the estimated frequency, and reachedmaxiters is true if the algorithm finished after running for maxiters iterations (this may indicate that the algorithm did not converge).\n\nIf the signal x is real, then the algorithm used is [Quinn1991]. If the signal is complex, the algorithm is [Quinn2009].\n\n[Quinn1991]: B Quinn and J Fernandes, \"A fast efficient technique for the estimation of frequency\", Biometrika, Vol. 78 (1991).\n\n[Quinn2009]: B Quinn, \"Recent advances in rapid frequency estimation\", Digital Signal Processing, Vol. 19 (2009), Elsevier.\n\n\n\n\n\n","category":"function"},{"location":"filters/#Filters-filter-design-and-filtering","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"DSP.jl differentiates between filter coefficients and stateful filters. Filter coefficient objects specify the response of the filter in one of several standard forms. Stateful filter objects carry the state of the filter together with filter coefficients in an implementable form (PolynomialRatio, Biquad, or SecondOrderSections). When invoked on a filter coefficient object, filt does not preserve state.","category":"page"},{"location":"filters/#coefficient-objects","page":"Filters - filter design and filtering","title":"Filter coefficient objects","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"DSP.jl supports common filter representations. Filter coefficients can be converted from one type to another using convert.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"ZeroPoleGain\nPolynomialRatio\nBiquad\nSecondOrderSections","category":"page"},{"location":"filters/#DSP.Filters.ZeroPoleGain","page":"Filters - filter design and filtering","title":"DSP.Filters.ZeroPoleGain","text":"ZeroPoleGain(z::Vector, p::Vector, k::Number)\n\nFilter representation in terms of zeros z, poles p, and gain k:\n\nH(x) = kfrac(x - verbz1) ldots (x - verbzm)(x - verbp1) ldots (x - verbpn)\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.PolynomialRatio","page":"Filters - filter design and filtering","title":"DSP.Filters.PolynomialRatio","text":"PolynomialRatio(b::Union{Number, Vector{<:Number}}, a::Union{Number, Vector{<:Number}})\n\nFilter representation in terms of the coefficients of the numerator b and denominator a in the z or s domain where b and a are vectors ordered from highest power to lowest.\n\nInputs that are Numbers are treated as one-element Vectors.\n\nFilter with:\n\nTransfer function in z domain (zero & negative z powers):\n\nH(z) = fracverbb1 + ldots + verbbm z^-m+1verba1 + ldots + verban z^-n+1\n\nreturns PolynomialRatio object with a[1] = 1 and other specified coefficients divided by a[1].\n\njulia> PolynomialRatio([1,1],[1,2])\nPolynomialRatio{:z, Float64}(LaurentPolynomial(1.0*z⁻¹ + 1.0), LaurentPolynomial(2.0*z⁻¹ + 1.0))\njulia> PolynomialRatio{:z}([1,2,3],[2,3,4])\nPolynomialRatio{:z, Float64}(LaurentPolynomial(1.5*z⁻² + 1.0*z⁻¹ + 0.5), LaurentPolynomial(2.0*z⁻² + 1.5*z⁻¹ + 1.0))\n\nTransfer function in s domain (zero & positive s powers):\n\nH(s) = fracverbb1 s^m-1 + ldots + verbbmverba1 s^n-1 + ldots + verban\n\nreturns PolynomialRatio object with specified b and a coefficients.\n\njulia> PolynomialRatio{:s}([1,2,3],[2,3,4])\nPolynomialRatio{:s, Int64}(LaurentPolynomial(3 + 2*s + s²), LaurentPolynomial(4 + 3*s + 2*s²))\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.Biquad","page":"Filters - filter design and filtering","title":"DSP.Filters.Biquad","text":"Biquad(b0::T, b1::T, b2::T, a1::T, a2::T) where T <: Number\n\nFilter representation in terms of the transfer function of a single second-order section given by:\n\nH(s) = fracverbb0 s^2+verbb1 s+verbb2s^2+verba1 s + verba2\n\nor equivalently:\n\nH(z) = fracverbb0+verbb1 z^-1+verbb2 z^-21+verba1 z^-1 + verba2 z^-2\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.SecondOrderSections","page":"Filters - filter design and filtering","title":"DSP.Filters.SecondOrderSections","text":"SecondOrderSections(biquads::Vector{<:Biquad}, gain::Number)\n\nFilter representation in terms of a cascade of second-order sections and gain. biquads must be specified as a vector of Biquads.\n\n\n\n\n\n","category":"type"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"These filter coefficient objects support the following arithmetic operations: inversion (inv), multiplication (*) for series connection, and integral power (^) for repeated multiplication with itself. For example:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"julia> H = PolynomialRatio([1.0], [1.0, 0.3])\nPolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.3*z⁻¹ + 1.0))\n\njulia> inv(H)\nPolynomialRatio{:z, Float64}(LaurentPolynomial(0.3*z⁻¹ + 1.0), LaurentPolynomial(1.0))\n\njulia> H * H\nPolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))\n\njulia> H^2\nPolynomialRatio{:z, Float64}(LaurentPolynomial(1.0), LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0))\n\njulia> H^-2\nPolynomialRatio{:z, Float64}(LaurentPolynomial(0.09*z⁻² + 0.6*z⁻¹ + 1.0), LaurentPolynomial(1.0))\n","category":"page"},{"location":"filters/#stateful-filter-objects","page":"Filters - filter design and filtering","title":"Stateful filter objects","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"DF2TFilter","category":"page"},{"location":"filters/#DSP.Filters.DF2TFilter","page":"Filters - filter design and filtering","title":"DSP.Filters.DF2TFilter","text":"DF2TFilter(coef::FilterCoefficients{:z})\nDF2TFilter(coef::FilterCoefficients{:z}, coldims::Tuple)\nDF2TFilter(coef::FilterCoefficients{:z}, sitype::Type, coldims::Tuple = ())\nDF2TFilter(coef::FilterCoefficients{:z}, si)\n\nConstruct a stateful direct form II transposed filter with coefficients coef.\n\nThe initial filter state defaults to zeros (of a type derived from coef) suitable for vector input. Another element type of the state can be specified with sitype.\n\nTo allow column-wise filtering of higher-dimensional input, the size of the extra dimensions have to be given in coldims. To e.g. column-wise filter an input with size (L, N1, N2), set coldims to (N1, N2).\n\nAlternatively, an array representing the initial filter state can be passed as si.\n\nIf coef is a PolynomialRatio, Biquad, or SecondOrderSections, filtering is implemented directly. If coef is a ZeroPoleGain object, it is first converted to a SecondOrderSections object.\n\n\n\n\n\n","category":"type"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"DSP.jl's FIRFilter type maintains state between calls to filt, allowing you to filter a signal of indefinite length in RAM-friendly chunks. FIRFilter contains nothing more that the state of the filter, and a FIRKernel. There are five different kinds of FIRKernel for single rate, up-sampling, down-sampling, rational resampling, and arbitrary sample-rate conversion. You need not specify the type of kernel. The FIRFilter constructor selects the correct kernel based on input parameters.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"FIRFilter","category":"page"},{"location":"filters/#DSP.Filters.FIRFilter","page":"Filters - filter design and filtering","title":"DSP.Filters.FIRFilter","text":"FIRFilter(h::Vector[, ratio::Union{Integer,Rational}])\n\nConstruct a stateful FIRFilter object from the vector of filter taps h. ratio is an optional rational integer which specifies the input to output sample rate relationship (e.g. 147//160 for converting recorded audio from 48 KHz to 44.1 KHz).\n\n\n\n\n\nFIRFilter(h::Vector, rate::AbstractFloat[, Nϕ::Integer=32])\n\nReturns a polyphase FIRFilter object from the vector of filter taps h. rate is a floating point number that specifies the input to output sample-rate relationship fracfs_outfs_in. Nϕ is an optional parameter which specifies the number of phases created from h. Nϕ defaults to 32.\n\n\n\n\n\n","category":"type"},{"location":"filters/#Filter-application","page":"Filters - filter design and filtering","title":"Filter application","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"filt\nfilt!\nfiltfilt\nfftfilt\nfftfilt!\ntdfilt\ntdfilt!\nresample","category":"page"},{"location":"filters/#DSP.filt","page":"Filters - filter design and filtering","title":"DSP.filt","text":"filt(b::Union{AbstractVector,Number},\n a::Union{AbstractVector,Number},\n x::AbstractArray,\n [si::AbstractArray])\n\nApply filter described by vectors a and b to vector x, with an optional initial filter state vector si (defaults to zeros).\n\nInputs that are Numbers are treated as one-element Vectors.\n\n\n\n\n\nfilt(f::FilterCoefficients{:z}, x::AbstractArray[, si])\n\nApply filter or filter coefficients f along the first dimension of array x. If f is a filter coefficient object, si is an optional array representing the initial filter state (defaults to zeros). If f is a PolynomialRatio, Biquad, or SecondOrderSections, filtering is implemented directly. If f is a ZeroPoleGain object, it is first converted to a SecondOrderSections object. If f is a Vector, it is interpreted as an FIR filter, and a naïve or FFT-based algorithm is selected based on the data and filter length.\n\n\n\n\n\nfilt(f::DF2TFilter{<:FilterCoefficients{:z},<:Array{T}}, x::AbstractArray{V}) where {T,V}\n\nApply the stateful filter f on x.\n\nwarning: Warning\nThe output array has eltype promote_type(T, V), where T is the eltype of the filter state.For more control over the output type, provide a preallocated output array out to filt!(out, f, x).\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.filt!","page":"Filters - filter design and filtering","title":"DSP.filt!","text":"filt!(out, b, a, x, [si])\n\nSame as filt but writes the result into the out argument, which may alias the input x to modify it in-place.\n\n\n\n\n\nfilt!(out, f, x[, si])\n\nSame as filt() but writes the result into the out argument. Output array out may not be an alias of x, i.e. filtering may not be done in place.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.filtfilt","page":"Filters - filter design and filtering","title":"DSP.Filters.filtfilt","text":"filtfilt(coef::FilterCoefficients, x::AbstractArray)\nfiltfilt(b::AbstractVector, x::AbstractArray)\nfiltfilt(b::AbstractVector, a::AbstractVector, x::AbstractArray)\n\nFilter x in the forward and reverse directions using either a FilterCoefficients object coef, or the coefficients b and optionally a as in filt. The initial state of the filter is computed so that its response to a step function is steady state. Before filtering, the data is extrapolated at both ends with an odd-symmetric extension of length min(3*(max(length(b), length(a))-1), size(x, 1) - 1).\n\nBecause filtfilt applies the given filter twice, the effective filter order is twice the order of coef. The resulting signal has zero phase distortion.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.fftfilt","page":"Filters - filter design and filtering","title":"DSP.Filters.fftfilt","text":"fftfilt(h::AbstractVector{<:Real}, x::AbstractArray{<:Real})\n\nApply FIR filter taps h along the first dimension of array x using an FFT-based overlap-save algorithm.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.fftfilt!","page":"Filters - filter design and filtering","title":"DSP.Filters.fftfilt!","text":"fftfilt!(out::AbstractArray, h::AbstractVector, x::AbstractArray)\n\nLike fftfilt but writes result into out array.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.tdfilt","page":"Filters - filter design and filtering","title":"DSP.Filters.tdfilt","text":"tdfilt(h::AbstractVector, x::AbstractArray)\n\nApply filter or filter coefficients h along the first dimension of array x using a naïve time-domain algorithm\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.tdfilt!","page":"Filters - filter design and filtering","title":"DSP.Filters.tdfilt!","text":"tdfilt!(out::AbstractArray, h::AbstractVector, x::AbstractArray)\n\nLike tdfilt, but writes the result into array out. Output array out may not be an alias of x, i.e. filtering may not be done in place.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.resample","page":"Filters - filter design and filtering","title":"DSP.Filters.resample","text":"resample(x::AbstractVector, rate::Real[, coef::Vector])\n\nResample x at rational or arbitrary rate. coef is an optional vector of FIR filter taps. If coef is not provided, the taps will be computed using a Kaiser window.\n\nInternally, resample uses a polyphase FIRFilter object, but performs additional operations to make resampling a signal easier. It compensates for the FIRFilter's delay (ramp-up), and appends zeros to x. The result is that when the input and output signals are plotted on top of each other, they correlate very well, but one signal will have more samples than the other.\n\n\n\n\n\nresample(x::AbstractArray, rate::Real, h::Vector = resample_filter(rate); dims)\n\nResample an array x along dimension dims.\n\n\n\n\n\n","category":"function"},{"location":"filters/#Filter-design","page":"Filters - filter design and filtering","title":"Filter design","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Most analog and digital filters are constructed by composing response types, which determine the frequency response of the filter, with design methods, which determine how the filter is constructed.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"The response type is Lowpass, Highpass, Bandpass or Bandstop and includes the edges of the bands.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"The design method is Butterworth, Chebyshev1, Chebyshev2, Elliptic, or FIRWindow, and includes any necessary parameters for the method that affect the shape of the response, such as filter order, ripple, and attenuation.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Filter order estimation methods are available in buttord, cheb1ord, cheb2ord, and ellipord if the corner frequencies for different IIR filter types are known. remezord can be used for an initial FIR filter order estimate.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"analogfilter\ndigitalfilter","category":"page"},{"location":"filters/#DSP.Filters.analogfilter","page":"Filters - filter design and filtering","title":"DSP.Filters.analogfilter","text":"analogfilter(responsetype::FilterType, designmethod::FilterCoefficients)\n\nConstruct an analog filter. See below for possible response and filter types.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.digitalfilter","page":"Filters - filter design and filtering","title":"DSP.Filters.digitalfilter","text":"digitalfilter(responsetype::FilterType, designmethod::FilterCoefficients[; fs::Real])\n\nConstruct a digital filter. See below for possible response and filter types.\n\n\n\n\n\n","category":"function"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"For some filters, the design method is more general or inherently implies a response type; these direct design methods include remez which designs equiripple FIR filters of all types, and iirnotch which designs a 2nd order \"biquad\" IIR notch filter.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"For a more general application of creating a digital filter from s-domain representation of an analog filter, one can use bilinear:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"bilinear","category":"page"},{"location":"filters/#DSP.Filters.bilinear","page":"Filters - filter design and filtering","title":"DSP.Filters.bilinear","text":"bilinear(f::FilterCoefficients{:s}, fs::Real)\n\nCalculate the digital filter (z-domain) ZPK representation of an analog filter defined in s-domain using bilinear transform with sampling frequency fs. The s-domain representation is first converted to a ZPK representation in s-domain and then transformed to z-domain using bilinear transform.\n\n\n\n\n\nbilinear(f::ZeroPoleGain{:s,Z,P,K}, fs::Real) where {Z,P,K}\n\nCalculate the digital filter (z-domain) ZPK representation of an analog filter defined as a ZPK representation in s-domain using bilinear transform with sampling frequency fs.\n\nInput s-domain representation must be a ZeroPoleGain{:s, Z, P, K} object:\n\nH(s) = fkfrac(s - verbfz1) ldots (s - verbfzm)(s - verbfp1) ldots (s - verbfpn)\n\nOutput z-domain representation is a ZeroPoleGain{:z, Z, P, K} object:\n\nH(z) = Kfrac(z - verbZ1) ldots (z - verbZm)(z - verbP1) ldots (z - verbPn)\n\nwhere Z, P, K are calculated as:\n\nZi = frac(2 + verbfziverbfs)(2 - verbfziverbfs) quad textfor i = 1 ldots m\n\nPi = frac(2 + verbfpiverbfs)(2 - verbfpiverbfs) quad textfor i = 1 ldots n\n\nK = fk mathcalRe left fracprod_i=1^m (2*fs - fzi)prod_i=1^n (2*fs - fpi) right\n\nHere, m and n are respectively the numbers of zeros and poles in the s-domain representation. If m < n, then additional n-m zeros are added at z = -1.\n\n\n\n\n\n","category":"function"},{"location":"filters/#response-types","page":"Filters - filter design and filtering","title":"Filter response types","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Lowpass\nHighpass\nBandpass\nComplexBandpass\nBandstop","category":"page"},{"location":"filters/#DSP.Filters.Lowpass","page":"Filters - filter design and filtering","title":"DSP.Filters.Lowpass","text":"Lowpass(Wn::Real)\n\nLow pass filter with cutoff frequency Wn.\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.Highpass","page":"Filters - filter design and filtering","title":"DSP.Filters.Highpass","text":"Highpass(Wn::Real)\n\nHigh pass filter with cutoff frequency Wn.\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.Bandpass","page":"Filters - filter design and filtering","title":"DSP.Filters.Bandpass","text":"Bandpass(Wn1::Real, Wn2::Real)\n\nBand pass filter with pass band frequencies (Wn1, Wn2).\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.ComplexBandpass","page":"Filters - filter design and filtering","title":"DSP.Filters.ComplexBandpass","text":"ComplexBandpass(Wn1, Wn2)\n\nComplex band pass filter with pass band frequencies (Wn1, Wn2).\n\n\n\n\n\n","category":"type"},{"location":"filters/#DSP.Filters.Bandstop","page":"Filters - filter design and filtering","title":"DSP.Filters.Bandstop","text":"Bandstop(Wn1::Real, Wn2::Real)\n\nBand stop filter with stop band frequencies (Wn1, Wn2).\n\n\n\n\n\n","category":"type"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"The interpretation of the frequencies Wn, Wn1 and Wn2 depends on wether an analog or a digital filter is designed.","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"If an analog filter is designed using analogfilter, the frequencies are interpreted as analog frequencies in radians/second.\nIf a digital filter is designed using digitalfilter and the sampling frequency fs is specified, the frequencies of the filter response type are normalized to fs. This requires that the sampling frequency and the filter response type use the same frequency unit (Hz, radians/second, ...). If fs is not specified, the frequencies of the filter response type are interpreted as normalized frequencies in half-cycles/sample.","category":"page"},{"location":"filters/#design-methods","page":"Filters - filter design and filtering","title":"Filter design methods","text":"","category":"section"},{"location":"filters/#IIR-filter-design-methods","page":"Filters - filter design and filtering","title":"IIR filter design methods","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Butterworth\nChebyshev1\nChebyshev2\nElliptic","category":"page"},{"location":"filters/#DSP.Filters.Butterworth","page":"Filters - filter design and filtering","title":"DSP.Filters.Butterworth","text":"Butterworth(n::Integer)\n\nn pole Butterworth filter.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.Chebyshev1","page":"Filters - filter design and filtering","title":"DSP.Filters.Chebyshev1","text":"Chebyshev1(n::Integer, ripple::Real)\n\nn pole Chebyshev type I filter with ripple dB ripple in the passband.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.Chebyshev2","page":"Filters - filter design and filtering","title":"DSP.Filters.Chebyshev2","text":"Chebyshev2(n::Integer, ripple::Real)\n\nn pole Chebyshev type II filter with ripple dB ripple in the stopband.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.Elliptic","page":"Filters - filter design and filtering","title":"DSP.Filters.Elliptic","text":"Elliptic(n::Integer, rp::Real, rs::Real)\n\nn pole elliptic (Cauer) filter with rp dB ripple in the passband and rs dB attentuation in the stopband.\n\n\n\n\n\n","category":"function"},{"location":"filters/#order-est-methods","page":"Filters - filter design and filtering","title":"Filter order estimation methods","text":"","category":"section"},{"location":"filters/#IIR-filter-order-estimation-methods","page":"Filters - filter design and filtering","title":"IIR filter order estimation methods","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"buttord\ncheb1ord\ncheb2ord\nellipord","category":"page"},{"location":"filters/#DSP.Filters.buttord","page":"Filters - filter design and filtering","title":"DSP.Filters.buttord","text":"(N, ωn) = buttord(Wp::Tuple{Real,Real}, Ws::Tuple{Real,Real}, Rp::Real, Rs::Real; domain=:z)\n\nButterworth order estimate for bandpass and bandstop filter types. Wp and Ws are 2-element pass and stopband frequency edges, with no more than Rp dB passband ripple and at least Rs dB stopband attenuation. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or \"-3 dB\" frequencies.\n\nIf a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving an order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n(N, ωn) = buttord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain=:z)\n\nLPF/HPF Butterworth filter order and -3 dB frequency approximation. Wp and Ws are the passband and stopband frequencies, whereas Rp and Rs are the passband and stopband ripple attenuations in dB. If the passband is greater than stopband, the filter type is inferred to be for estimating the order of a highpass filter. N specifies the lowest possible integer filter order, whereas ωn is the cutoff or \"-3 dB\" frequency.\n\nIf a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.cheb1ord","page":"Filters - filter design and filtering","title":"DSP.Filters.cheb1ord","text":"(N, ωn) = cheb1ord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Chebyshev Type I Filters. Wp and Ws indicate the passband and stopband frequency edges, and Rp and Rs indicate the maximum loss in the passband and the minimum attenuation in the stopband, (in dB.)\n\nBased on the ordering of passband and stopband edges, the Lowpass or Highpass filter type is inferred. N indicates the smallest integer filter order that achieves the desired specifications, and ωn contains the natural frequency of the filter, (in this case, simply the passband edge.)\n\nIf a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n(N, ωn) = cheb1ord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Chebyshev Type I Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or \"-3 dB\" frequencies. If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.cheb2ord","page":"Filters - filter design and filtering","title":"DSP.Filters.cheb2ord","text":"(N, ωn) = cheb2ord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Chebyshev Type II (inverse) Filters. Wp and Ws are the frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB.\n\nBased on the ordering of the passband and stopband edges, the Lowpass or Highpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the \"-3 dB\" cutoff frequency.\n\nIf a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n(N, ωn) = cheb2ord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Chebyshev Type II (inverse) Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB.\n\nBased on the ordering of the passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the \"-3 dB\" cutoff frequency.\n\nIf a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.ellipord","page":"Filters - filter design and filtering","title":"DSP.Filters.ellipord","text":"(N, ωn) = ellipord(Wp::Real, Ws::Real, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Elliptic (Cauer) Filters. Wp and Ws indicate the passband and stopband frequency edges, and Rp and Rs indicate the maximum loss in the passband and the minimum attenuation in the stopband, (in dB.)\n\nBased on the ordering of passband and stopband edges, the Lowpass or Highpass filter type is inferred. N indicates the smallest integer filter order that achieves the desired specifications, and ωn contains the natural frequency of the filter, (in this case, simply the passband edge.)\n\nIf a domain of :s is specified, the passband and stopband edges are interpreted as radians/second, giving an order and natural frequency result for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n(N, ωn) = ellipord(Wp::Tuple{Real, Real}, Ws::Tuple{Real, Real}, Rp::Real, Rs::Real; domain::Symbol=:z)\n\nInteger and natural frequency order estimation for Elliptic (Cauer) Filters. Wp and Ws are 2-element frequency edges for Bandpass/Bandstop cases, with Rp and Rs representing the ripple maximum loss in the passband and minimum ripple attenuation in the stopband in dB. Based on the ordering of passband and bandstop edges, the Bandstop or Bandpass filter type is inferred. N is an integer indicating the lowest estimated filter order, with ωn specifying the cutoff or \"-3 dB\" frequencies. If a domain of :s is specified, the passband and stopband frequencies are interpreted as radians/second, giving the order and natural frequencies for an analog filter. The default domain is :z, interpreting the input frequencies as normalized from 0 to 1, where 1 corresponds to π radians/sample.\n\n\n\n\n\n","category":"function"},{"location":"filters/#FIR-filter-order-estimation-methods","page":"Filters - filter design and filtering","title":"FIR filter order estimation methods","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"remezord","category":"page"},{"location":"filters/#DSP.Filters.remezord","page":"Filters - filter design and filtering","title":"DSP.Filters.remezord","text":"N = remezord(Wp::Real, Ws::Real, Rp::Real, Rs::Real)\n\nOrder estimation for lowpass digital filter cases based on the equations and coefficients in [Rabiner]. The original equation returned the minimum filter length, whereas this implementation returns the order (N=L-1). Wp and Ws are the normalized passband and stopband frequencies, with Rp indicating the passband ripple and Rs is the stopband attenuation (linear.)\n\nNOTE: The value of N is an initial approximation. If under-estimated, the order should be increased until the design specifications are met.\n\n[Rabiner]: Herrmann, O., Lawrence R. Rabiner, and D. S. K. Chan. \"Practical design rules for optimum finite impulse response lowpass digital filters.\" Bell System Technical Journal 52.6 (1973): 769-799.\n\n\n\n\n\n","category":"function"},{"location":"filters/#FIR-filter-design-methods","page":"Filters - filter design and filtering","title":"FIR filter design methods","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"FIRWindow","category":"page"},{"location":"filters/#DSP.Filters.FIRWindow","page":"Filters - filter design and filtering","title":"DSP.Filters.FIRWindow","text":"FIRWindow(window::Vector; scale=true)\n\nFIR filter design using window window, a vector whose length matches the number of taps in the resulting filter.\n\nIf scale is true (default), the designed FIR filter is scaled so that the following holds:\n\nFor Lowpass and Bandstop filters, the frequency response is unity at 0 (DC).\nFor Highpass filters, the frequency response is unity at the Nyquist frequency.\nFor Bandpass filters, the frequency response is unity in the center of the passband.\n\n\n\n\n\nFIRWindow(; transitionwidth::Real, attenuation::Real=60, scale::Bool=true)\n\nKaiser window FIR filter design. The required number of taps is calculated based on transitionwidth (in half-cycles/sample) and stopband attenuation (in dB). attenuation defaults to 60 dB.\n\n\n\n\n\n","category":"type"},{"location":"filters/#direct-design-methods","page":"Filters - filter design and filtering","title":"Direct filter design methods","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"remez\niirnotch","category":"page"},{"location":"filters/#DSP.Filters.remez","page":"Filters - filter design and filtering","title":"DSP.Filters.remez","text":"remez(numtaps::Integer, band_defs;\n Hz::Real=1.0,\n neg::Bool=false,\n maxiter::Integer=25,\n grid_density::Integer=16)\n\nCalculate the minimax optimal filter using the Remez exchange algorithm [McClellan1973a] [McClellan1973b].\n\nThis is the simplified API that accepts just 2 required arguments (numtaps, band_defs). For a scipy compatible version see the 3 arguments version (numtaps, bands, desired).\n\nCalculate the filter-coefficients for the finite impulse response (FIR) filter whose transfer function minimizes the maximum error between the desired gain and the realized gain in the specified frequency bands using the Remez exchange algorithm.\n\nArguments\n\nnumtaps::Integer: The desired number of taps in the filter. The number of taps is the number of terms in the filter, or the filter order plus one.\nband_defs: A sequence of band definitions. This sequence defines the bands. Each entry is a pair. The pair's first item is a tuple of band edges (low, high). The pair's second item defines the desired response and weight in that band. The weight is optional and defaults to 1.0. Both the desired response and weight may be either scalars or functions. If a function, the function should accept a real frequency and return the real desired response or real weight. Examples:\nLPF with unity weights. [(0, 0.475) => 1, (0.5, 1.0) => 0]\nLPF with weight of 2 in the stop band. [(0, 0.475) => (1, 1), (0.5, 1.0) => (0, 2)]\nBPF with unity weights. [(0, 0.375) => 0, (0.4, 0.5) => 1, (0.525, 1.0) => 0]\nHilbert transformer. [(0.1, 0.95) => 1]; neg=true\nDifferentiator. [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true\nHz::Real: The sampling frequency in Hz. Default is 1.\nneg::Bool: Whether the filter has negative symmetry or not. Default is false. If false, the filter is even-symmetric. If true, the filter is odd-symmetric. neg=true means that h[n]=-h[end+1-n]; neg=false means that h[n]=h[end+1-n].\nmaxiter::Integer: (optional) Maximum number of iterations of the algorithm. Default is 25.\ngrid_density:Integer: (optional) Grid density. The dense grid used in remez is of size (numtaps + 1) * grid_density. Default is 16.\n\nReturns\n\nh::Array{Float64,1}: A rank-1 array containing the coefficients of the optimal (in a minimax sense) filter.\n\n[McClellan1973a]: J. H. McClellan and T. W. Parks, A unified approach to the design of optimum FIR linear phase digital filters, IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973.\n\n[McClellan1973b]: J. H. McClellan, T. W. Parks and L. R. Rabiner, A Computer Program for Designing Optimum FIR Linear Phase Digital Filters, IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 506-525, 1973.\n\nExamples\n\nConstruct a length 35 filter with a passband at 0.15-0.4 Hz (desired response of 1), and stop bands at 0-0.1 Hz and 0.45-0.5 Hz (desired response of 0). Note: the behavior in the frequency ranges between those bands - the transition bands - is unspecified.\n\njulia> bpass = remez(35, [(0, 0.1)=>0, (0.15, 0.4)=>1, (0.45, 0.5)=>0]);\n\nYou can trade-off maximum error achieved for transition bandwidth. The wider the transition bands, the lower the maximum error in the bands specified. Here is a bandpass filter with the same passband, but wider transition bands.\n\njulia> bpass2 = remez(35, [(0, 0.08)=>0, (0.15, 0.4)=>1, (0.47, 0.5)=>0]);\n\nHere we compute the frequency responses and plot them in dB.\n\njulia> using PyPlot\njulia> b = PolynomialRatio(bpass, [1.0])\njulia> b2 = PolynomialRatio(bpass2, [1.0])\njulia> f = range(0, stop=0.5, length=1000)\njulia> plot(f, 20*log10.(abs.(freqresp(b,f,1.0))))\njulia> plot(f, 20*log10.(abs.(freqresp(b2,f,1.0))))\njulia> grid()\n\nExamples from the unittests - standard (even) symmetry.\n\nLength 151 LPF (Low Pass Filter).\n\njulia> h = remez(151, [(0, 0.475) => 1, (0.5, 1.0) => 0]; Hz=2.0);\n\nLength 152 LPF. Non-default \"weight\" input.\n\njulia> h = remez(152, [(0, 0.475) => (1, 1), (0.5, 1.0) => (0, 2)]; Hz=2.0);\n\nLength 51 HPF (High Pass Filter).\n\njulia> h = remez(51, [(0, 0.75) => 0, (0.8, 1.0) => 1]; Hz=2.0);\n\nLength 180 BPF (Band Pass Filter).\n\njulia> h = remez(180, [(0, 0.375) => 0, (0.4, 0.5) => 1, (0.525, 1.0) => 0]; Hz=2.0, maxiter=30);\n\nExamples from the unittests - Odd-symmetric filters - hilbert and differentiators type.\n\nEven length - has a much better approximation since the response is not constrained to 0 at the nyquist frequency. Length 20 Hilbert transformer.\n\njulia> h = remez(20, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);\n\nLength 21 Hilbert transformer.\n\njulia> h = remez(21, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);\n\nLength 200 differentiator.\n\njulia> h = remez(200, [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);\n\nLength 201 differentiator.\n\njulia> h = remez(201, [(0.05, 0.95) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);\n\nInverse sinc filter - custom response function\n\njulia> L = 64; Fs = 4800*L;\njulia> passband_response_function = f -> (f==0) ? 1.0 : abs.((π*f/4800) ./ sin.(π*f/4800));\njulia> h = remez(201, [( 0.0, 2880.0) => (passband_response_function, 1.0),\n (10000.0, Fs/2) => (0.0, 100.0)]; Hz=Fs);\n\n\n\n\n\nremez(numtaps::Integer,\n bands::Vector,\n desired::Vector;\n weight::Vector=[],\n Hz::Real=1.0,\n filter_type::RemezFilterType=filter_type_bandpass,\n maxiter::Integer=25,\n grid_density::Integer=16)\n\nThis is the scipy compatible version that requires 3 arguments (numtaps, bands, desired). For a simplified API, see the 2 argument version (numtaps, band_defs). The filters designed are equivalent, the inputs are just specified in a different way. Below the arguments and examples are described that differ from the simplified API version.\n\nArguments\n\nbands::Vector: A monotonic sequence containing the band edges in Hz. All elements must be non-negative and less than half the sampling frequency as given by Hz.\ndesired::Vector:A sequence half the size of bands containing the desired gain in each of the specified bands.\nweight::Vector: (optional) A relative weighting to give to each band region. The length of weight has to be half the length of bands.\nfilter_type::RemezFilterType: Default is filter_type_bandpass. The type of filter:\nfilter_type_bandpass : flat response in bands. This is the default.\nfilter_type_differentiator : frequency proportional response in bands. Odd symetric as in filter_type_hilbert case, but with a linear sloping desired response.\nfilter_type_hilbert : filter with odd symmetry, that is, type III (for even order) or type IV (for odd order) linear phase filters.\n\nExamples\n\nCompare the examples with the simplified API and the Scipy API. Each of the following blocks first designs a filter using the simplified (recommended) API, and then designs the same filter using the Scipy-compatible API.\n\njulia> bpass = remez(35, [(0, 0.1)=>0, (0.15, 0.4)=>1, (0.45, 0.5)=>0]);\n\njulia> bpass = remez(35, [0, 0.1, 0.15, 0.4, 0.45, 0.5], [0, 1, 0]);\n\n\njulia> bpass2 = remez(35, [(0, 0.08)=>0, (0.15, 0.4)=>1, (0.47, 0.5)=>0]);\n\njulia> bpass2 = remez(35, [0, 0.08, 0.15, 0.4, 0.47, 0.5], [0, 1, 0]);\n\n\njulia> h = remez(20, [(0.1, 0.95) => 1]; neg=true, Hz=2.0);\n\njulia> h = remez(20, [0.1, 0.95], [1]; filter_type=filter_type_hilbert, Hz=2.0);\n\n\njulia> h = remez(200, [(0.01, 0.99) => (f -> f/2, f -> 1/f)]; neg=true, Hz=2.0);\n\njulia> h = remez(200, [0.01, 0.99], [1]; filter_type=filter_type_differentiator, Hz=2.0);\n\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.iirnotch","page":"Filters - filter design and filtering","title":"DSP.Filters.iirnotch","text":"iirnotch(Wn::Real, bandwidth::Real[; fs=2])\n\nSecond-order digital IIR notch filter [Orfanidis] at frequency Wn with bandwidth bandwidth. If fs is not specified, Wn is interpreted as a normalized frequency in half-cycles/sample.\n\n[Orfanidis]: Orfanidis, S. J. (1996). Introduction to signal processing. Englewood Cliffs, N.J: Prentice Hall, p. 370.\n\n\n\n\n\n","category":"function"},{"location":"filters/#Filter-response","page":"Filters - filter design and filtering","title":"Filter response","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"freqresp\nphaseresp\ngrpdelay\nimpresp\nstepresp","category":"page"},{"location":"filters/#DSP.Filters.freqresp","page":"Filters - filter design and filtering","title":"DSP.Filters.freqresp","text":"H, w = freqresp(filter::FilterCoefficients)\n\nFrequency response H of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.\n\n\n\n\n\nfreqresp(filter::FilterCoefficients{:z}, w)\n\nFrequency response of digital filter at normalized frequency or frequencies w in radians/sample.\n\n\n\n\n\nfreqresp(filter::FilterCoefficients{:s}, w)\n\nFrequency response of analog filter at frequency or frequencies w in radians/second.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.phaseresp","page":"Filters - filter design and filtering","title":"DSP.Filters.phaseresp","text":"phi, w = phaseresp(filter::FilterCoefficients)\n\nPhase response phi of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.\n\n\n\n\n\nphaseresp(filter::FilterCoefficients, w)\n\nPhase response of a filter at (normalized) frequency or frequencies w in radians/sample for a digital filter or radians/second for an analog filter.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.grpdelay","page":"Filters - filter design and filtering","title":"DSP.Filters.grpdelay","text":"tau, w = grpdelay(filter::FilterCoefficients)\n\nGroup delay tau of a filter at (normalized) frequencies w in radians/sample for a digital filter or radians/second for an analog filter chosen as a reasonable default.\n\n\n\n\n\ngrpdelay(filter::FilterCoefficients{:z}, w)\n\nGroup delay of a digital filter at normalized frequency or frequencies w in radians/sample.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.impresp","page":"Filters - filter design and filtering","title":"DSP.Filters.impresp","text":"impresp(filter::FilterCoefficients{:z}, n=100)\n\nImpulse response of a digital filter with n points.\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.stepresp","page":"Filters - filter design and filtering","title":"DSP.Filters.stepresp","text":"stepresp(filter::FilterCoefficients{:z}, n=100)\n\nStep response of a digital filter with n points.\n\n\n\n\n\n","category":"function"},{"location":"filters/#Miscellaneous","page":"Filters - filter design and filtering","title":"Miscellaneous","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"coefb\ncoefa","category":"page"},{"location":"filters/#DSP.Filters.coefb","page":"Filters - filter design and filtering","title":"DSP.Filters.coefb","text":"coefb(f::PolynomialRatio)\n\nCoefficients of the numerator of a PolynomialRatio object, highest power first, i.e., the b passed to filt()\n\n\n\n\n\n","category":"function"},{"location":"filters/#DSP.Filters.coefa","page":"Filters - filter design and filtering","title":"DSP.Filters.coefa","text":"coefa(f::PolynomialRatio)\n\nCoefficients of the denominator of a PolynomialRatio object, highest power first, i.e., the a passed to filt()\n\n\n\n\n\n","category":"function"},{"location":"filters/#Examples","page":"Filters - filter design and filtering","title":"Examples","text":"","category":"section"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Construct a 4th order elliptic lowpass filter with normalized cutoff frequency 0.2, 0.5 dB of passband ripple, and 30 dB attentuation in the stopband and extract the coefficients of the numerator and denominator of the transfer function:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"responsetype = Lowpass(0.2)\ndesignmethod = Elliptic(4, 0.5, 30)\ntf = convert(PolynomialRatio, digitalfilter(responsetype, designmethod))\nnumerator_coefs = coefb(tf)\ndenominator_coefs = coefa(tf)","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Filter the data in x, sampled at 1000 Hz, with a 4th order Butterworth bandpass filter between 10 and 40 Hz:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"responsetype = Bandpass(10, 40)\ndesignmethod = Butterworth(4)\nfilt(digitalfilter(responsetype, designmethod; fs=1000), x)","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Filter the data in x, sampled at 50 Hz, with a 64 tap Hanning window FIR lowpass filter at 5 Hz:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"responsetype = Lowpass(5)\ndesignmethod = FIRWindow(hanning(64))\nfilt(digitalfilter(responsetype, designmethod; fs=50), x)","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"Estimate a Lowpass Elliptic filter order with a normalized passband cutoff frequency of 0.2, a stopband cutoff frequency of 0.4, 3 dB of passband ripple, and 40 dB attenuation in the stopband:","category":"page"},{"location":"filters/","page":"Filters - filter design and filtering","title":"Filters - filter design and filtering","text":"(N, ωn) = ellipord(0.2, 0.4, 3, 40)","category":"page"},{"location":"lpc/#LPC-Linear-Predictive-Coding","page":"LPC - Linear Predictive Coding","title":"LPC - Linear Predictive Coding","text":"","category":"section"},{"location":"lpc/","page":"LPC - Linear Predictive Coding","title":"LPC - Linear Predictive Coding","text":"lpc\narburg\nlevinson","category":"page"},{"location":"lpc/#DSP.LPC.lpc","page":"LPC - Linear Predictive Coding","title":"DSP.LPC.lpc","text":"lpc(x::AbstractVector, p::Integer, [LPCBurg()])\n\nGiven input signal x and prediction order p, returns IIR coefficients a and average reconstruction error prediction_err. Note that this method does NOT return the leading 1 present in the true autocorrelative estimate; it omits it as it is implicit in every LPC estimate, and must be manually reintroduced if the returned vector should be treated as a polynomial.\n\nThe algorithm used is determined by the last optional parameter, and can be either LPCBurg (arburg) or LPCLevinson (levinson).\n\n\n\n\n\n","category":"function"},{"location":"lpc/#DSP.LPC.arburg","page":"LPC - Linear Predictive Coding","title":"DSP.LPC.arburg","text":"arburg(x::AbstractVector, p::Integer)\n\nLPC (Linear Predictive Coding) estimation, using the Burg method. This function implements the mathematics published in [Lagrange], and the recursion relation as noted in [Vos], in turn referenced from [Andersen].\n\n[Lagrange]: Enhanced Partial Tracking Using Linear Prediction. DAFX 2003 article, Lagrange et al\n\n[Vos]: A Fast Implementation of Burg’s Method. © 2013 Koen Vos, licensed under CC BY 3.0\n\n[Andersen]: N. Andersen. Comments on the performance of maximum entropy algorithms. Proceedings of the IEEE 66.11: 1581-1582, 1978.\n\n\n\n\n\n","category":"function"},{"location":"lpc/#DSP.LPC.levinson","page":"LPC - Linear Predictive Coding","title":"DSP.LPC.levinson","text":"levinson(x::AbstractVector, p::Integer)\n\nImplements Levinson recursion, as described in [Levinson], to find the solution a of the linear equation\n\nmathbfT (-veca)\n=\nbeginbmatrix\n x_2 \n vdots \n x_p+1\nendbmatrix\n\nin the case where mathbfT is Hermitian and Toeplitz, with first column x[1:p]. This function can be used for LPC (Linear Predictive Coding) estimation, by providing LPCLevinson() as an argument to lpc.\n\n[Levinson]: The Wiener (RMS) Error Criterion in Filter Design and Prediction. N. Levinson, Studies in Applied Mathematics 25(1946), 261-278.\nhttps://doi.org/10.1002/sapm1946251261\n\n\n\n\n\n","category":"function"},{"location":"#Welcome-to-DSP.jl's-documentation!","page":"Home","title":"Welcome to DSP.jl's documentation!","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"DSP.jl provides a number of common digital signal processing routines in Julia. Many methods provided in this package are validated against popular tools such as MATLAB's Signal Processing suite and Python's MNE-Python package. So far, the following submodules are implemented:","category":"page"},{"location":"#Table-of-contents","page":"Home","title":"Table of contents","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Pages = [\n \"index.md\",\n \"periodograms.md\",\n \"estimation.md\",\n \"windows.md\",\n \"filters.md\",\n \"util.md\",\n \"convolutions.md\",\n \"lpc.md\",\n \"internals.md\",\n \"appendix.md\"\n]","category":"page"},{"location":"internals/#Internals","page":"Internals","title":"Internals","text":"","category":"section"},{"location":"internals/","page":"Internals","title":"Internals","text":"Functions here are not exported.","category":"page"},{"location":"internals/","page":"Internals","title":"Internals","text":"DSP.os_fft_complexity\nDSP.os_prepare_conv\nDSP.os_conv_block!\nDSP.os_filter_transform!\nDSP.Windows.makewindow\nDSP.Windows.padplot\nDSP._zeropad\nDSP._zeropad!\nDSP._zeropad_keep_offset\nDSP.Filters._polyprep\nDSP.Filters.freq_eval\nDSP.Filters.build_grid\nDSP.Filters.lagrange_interp\nDSP.Periodograms.coherence_from_cs!\nDSP.optimalfftfiltlength","category":"page"},{"location":"internals/#DSP.os_fft_complexity","page":"Internals","title":"DSP.os_fft_complexity","text":"Estimate the number of floating point multiplications per output sample for an overlap-save algorithm with fft size nfft, and filter size nb.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.os_prepare_conv","page":"Internals","title":"DSP.os_prepare_conv","text":"Prepare buffers and FFTW plans for convolution. The two buffers, tdbuff and fdbuff may be an alias of each other, because complex convolution only requires one buffer. The plans are mutating where possible, and the inverse plan is unnormalized.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.os_conv_block!","page":"Internals","title":"DSP.os_conv_block!","text":"Take a block of data, and convolve it with the smaller convolution input. This may modify the contents of tdbuff and fdbuff, and the result will be in tdbuff.\n\n\n\n\n\nLike the real version, but only operates on one buffer\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.os_filter_transform!","page":"Internals","title":"DSP.os_filter_transform!","text":"Transform the smaller convolution input to frequency domain, and return it in a new array. However, the contents of buff may be modified.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Windows.makewindow","page":"Internals","title":"DSP.Windows.makewindow","text":"makewindow(winfunc::Function, n::Integer, padding::Integer, zerophase::Bool)\n\nGenerate a discrete window vector of the given length with padding zeros. winfunc should be a function giving the window value in the range of [-0.5, 0.5]. The function is assumed to be 0 outside of this range.\n\nIf zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.\n\nIf zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a \"periodic\" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.\n\nExample:\n\nfunction hanning(n::Integer; padding::Integer=0, zerophase::Bool=false)\n makewindow(n, padding, zerophase) do x\n 0.5*(1+cos(2pi*x))\n end\nend\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Windows.padplot","page":"Internals","title":"DSP.Windows.padplot","text":"padplot(plotstr)\n\nTakes a multiline string and pre-pads it so that it shows up as preformatted code when included in a docstring.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP._zeropad","page":"Internals","title":"DSP._zeropad","text":"_zeropad(u, padded_size, [data_dest, data_region])\n\nCreates and returns a new base-1 index array of size padded_size, with the section of u specified by data_region copied into the region of the new array as specified by data_dest. All other values will be initialized to zero.\n\nIf either data_dest or data_region is not specified, then the defaults described in _zeropad! will be used.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP._zeropad!","page":"Internals","title":"DSP._zeropad!","text":"_zeropad!(padded::AbstractVector,\n u::AbstractVector,\n padded_axes = axes(padded),\n data_dest::Tuple = (1,),\n data_region = CartesianIndices(u))\n\nPlace the portion of u specified by data_region into padded, starting at location data_dest. Sets the rest of padded to zero. This will mutate padded. padded_axes must correspond to the axes of padded.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP._zeropad_keep_offset","page":"Internals","title":"DSP._zeropad_keep_offset","text":"_zeropad_keep_offset(u, padded_size, [data_dest, dat_region])\n\nLike _zeropad, but retains the first index of u when creating a new array.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Filters._polyprep","page":"Internals","title":"DSP.Filters._polyprep","text":"_polyprep(D::Symbol, x::Union{T,Vector{T}}, ::Type) where {T<:Number}\n\nConverts x to polynomial form. If x is a Number, it has to be converted into a Vector, otherwise LaurentPolynomial dispatch goes into stack overflow trying to collect a 0-d array into a Vector.\n\nwarning: Warning\nThe DSP convention for Laplace domain is highest power first.The Polynomials.jl convention is lowest power first.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Filters.freq_eval","page":"Internals","title":"DSP.Filters.freq_eval","text":"function freq_eval(xf, x::AbstractVector, y::AbstractVector, ad::AbstractVector)\n\n\n\nFUNCTION: freq_eval (gee) FUNCTION TO EVALUATE THE FREQUENCY RESPONSE USING THE LAGRANGE INTERPOLATION FORMULA IN THE BARYCENTRIC FORM\n\n\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Filters.build_grid","page":"Internals","title":"DSP.Filters.build_grid","text":"build_grid(numtaps, band_defs, Hz, grid_density, neg)\n\nReturns grid, des, and wt arrays\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Filters.lagrange_interp","page":"Internals","title":"DSP.Filters.lagrange_interp","text":"lagrange_interp(k::Integer, n::Integer, m::Integer, x::AbstractVector)\n\nCALCULATE THE LAGRANGE INTERPOLATION COEFFICIENTS\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.Periodograms.coherence_from_cs!","page":"Internals","title":"DSP.Periodograms.coherence_from_cs!","text":"coherence_from_cs!(output::AbstractArray, cs_matrix)\n\nCompute the pairwise channel coherences from a cross spectral matrix, storing the result in output.\n\n\n\n\n\n","category":"function"},{"location":"internals/#DSP.optimalfftfiltlength","page":"Internals","title":"DSP.optimalfftfiltlength","text":"Determine the length of FFT that minimizes the number of multiplications per output sample for an overlap-save convolution of vectors of size nb and nx.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#Periodograms-periodogram-estimation","page":"Periodograms - periodogram estimation","title":"Periodograms - periodogram estimation","text":"","category":"section"},{"location":"periodograms/#Basic-functions","page":"Periodograms - periodogram estimation","title":"Basic functions","text":"","category":"section"},{"location":"periodograms/","page":"Periodograms - periodogram estimation","title":"Periodograms - periodogram estimation","text":"Common procedures like computing the short-time Fourier transform, periodograms, and spectrograms are documented below.","category":"page"},{"location":"periodograms/","page":"Periodograms - periodogram estimation","title":"Periodograms - periodogram estimation","text":"arraysplit\nDSP.Periodograms.ArraySplit\nperiodogram\nDSP.Periodograms.Periodogram\nDSP.Periodograms.Periodogram2\nwelch_pgram\nwelch_pgram!\nspectrogram\nDSP.Periodograms.Spectrogram\nstft\nfreq\npower\ntime\ncoherence\nDSP.Periodograms.Coherence","category":"page"},{"location":"periodograms/#DSP.Periodograms.arraysplit","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.arraysplit","text":"arraysplit(s, n, noverlap, nfft=n, window=nothing; buffer=zeros(eltype(s), nfft))\n\nSplit an array s into arrays of length n with overlapping regions of length noverlap.\n\nArguments\n\ns: Input array.\nn: Specifies the required subarray length (n ≤ length(s)).\nnoverlap: Number of overlapping elements between subarrays.\nnfft: Specifies the length of the split arrays. If length(s) < nfft, then the input is padded with zeros.\nwindow: An optional scaling vector to be applied to the split arrays.\nbuffer: An optional buffer of length nfft. Iterating or indexing the returned AbstractVector always yields the same Vector with different contents. The last result after iteration or indexing calls is stored into buffer.\n\nReturns\n\nAn ArraySplit object with split subarrays.\n\nExamples\n\njulia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 1)\n2-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Nothing}:\n [0.1, 0.2, 0.3]\n [0.3, 0.4, 0.5]\n\njulia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 2, 8)\n3-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Nothing}:\n [0.1, 0.2, 0.3, 0.0, 0.0, 0.0, 0.0, 0.0]\n [0.2, 0.3, 0.4, 0.0, 0.0, 0.0, 0.0, 0.0]\n [0.3, 0.4, 0.5, 0.0, 0.0, 0.0, 0.0, 0.0]\n\njulia> arraysplit([0.1, 0.2, 0.3, 0.4, 0.5], 3, 1, 3, [1, 2, 1])\n2-element DSP.Periodograms.ArraySplit{Vector{Float64}, Float64, Vector{Int64}}:\n [0.1, 0.4, 0.3]\n [0.3, 0.8, 0.5]\n\narraysplit function with buffer\n\njulia> x = [1, 2, 3, 4, 5];\n\njulia> sub_arr, n_overlap, nfft = 3, 1, 3;\n\njulia> x_split = arraysplit(x, sub_arr, n_overlap, nfft, nothing; buffer=zeros(nfft));\n\njulia> x_split[2] # Returns AbstractVector result and stores it in ArraySplit.buf and buffer\n3-element Vector{Float64}:\n 3.0\n 4.0\n 5.0\n\njulia> x_split.buf # Returns stored results from previous x_split[2] call\n3-element Vector{Float64}:\n 3.0\n 4.0\n 5.0\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.ArraySplit","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.ArraySplit","text":"ArraySplit{T<:AbstractVector,S,W} <: AbstractVector{Vector{S}}\n\nArraySplit object with fields\n\ns::T\nbuf::Vector{S}\nn::Int\nnoverlap::Int\nwindow::W\nk::Int\n\nwhere buf: buffer, k: number of split arrays, and s, n, noverlap, window are as per arguments in arraysplit.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.periodogram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.periodogram","text":"periodogram(s::AbstractVector; onesided=eltype(s)<:Real, nfft=nextfastfft(length(s)), fs=1, window=nothing)\n\nComputes periodogram of a 1-d signal s by FFT and returns a Periodogram object.\n\nArguments\n\nonesided: For real signals, the two-sided periodogram is symmetric and this function returns a one-sided (real only) periodogram by default. A two-sided periodogram can be obtained by setting onesided=false.\nnfft: Specifies the number of points to use for the Fourier transform. If length(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency.\nfs: The sample rate of the original signal.\nwindow: An optional window function or vector to be applied to the original signal before computing the Fourier transform. The computed periodogram is normalized so that the area under the periodogram is equal to the uncentered variance (or average power) of the original signal.\n\nReturns\n\nA Periodogram object with the 2 computed fields: power, freq.\n\nExamples\n\nFrequency estimate of cos(2π(25)t) with a 1-sided periodogram.\n\njulia> Fs = 100; # Sampling frequency\n\njulia> t = (1:Fs)/Fs; # 100 time samples\n\njulia> x = cos.(2π*25*t); # Cosine signal with 25Hz frequency\n\njulia> pxx = periodogram(x; fs=Fs); # Periodogram\n\njulia> _, max_index = findmax(pxx.power); # Find index of max power\n\njulia> pxx.power[max_index], pxx.freq[max_index] # (power, frequency)\n(0.5, 25.0)\n\n2-sided periodogram of a rectangle function with Hamming window.\n\njulia> x = rect(50; padding=50); # Rectangular pulse function\n\njulia> pxx = periodogram(x; onesided=false, nfft=512, fs=1000, window=hamming);\n\njulia> round(maximum(pxx.power), digits=4) # Expect max power to be close to 0Hz for a DC signal\n0.0182\n\n\n\n\n\nperiodogram(s::AbstractMatrix; nfft::NTuple{2,Int}=nextfastfft(size(s)), fs=1, radialsum=false, radialavg=false)\n\nComputes periodogram of a 2-d signal using the 2-d FFT and returns a Periodogram2 or Periodogram object.\n\nArguments\n\nnfft: A 2-tuple specifying the number of points to use for the Fourier transform for each respective dimension. If size(s) < nfft, then the input is padded with zeros. By default, nfft is the closest size for which the Fourier transform can be computed with maximal efficiency.\nfs: The sample rate of the original signal in both directions.\nradialsum: For radialsum=true, the value of power[k] is proportional to frac1Nsum_kleq kk+1 Xk^2.\nradialavg: For radialavg=true, the value of power[k] is proportional to frac1N kleq kk+1 sum_kleq kk+1 Xk^2. The computation of |k'| takes into account non-square signals by scaling the coordinates of the wavevector accordingly.\n\nReturns\n\nA Periodogram2 object by default with the 3 computed fields: power, freq1, freq2.\nA Periodogram object is returned for a radially summed or averaged periodogram (if radialsum or radialavg is true, respectively). Only one of radialsum or radialavg can be set to true in the function.\n\nExamples\n\njulia> x = [1 1; 0 1; 0 0];\n\njulia> pxx = periodogram(x; nfft=(3, 2)); # Returns Periodogram2\n\njulia> power(pxx)\n3×2 Matrix{Float64}:\n 1.5 0.166667\n 0.5 0.166667\n 0.5 0.166667\n\njulia> freq(pxx)\n([0.0, 0.3333333333333333, -0.3333333333333333], [0.0, -0.5])\n\nRadial sum and radial average periodograms\n\njulia> x = [1 3; 0 1];\n\njulia> periodogram(x; radialsum=true) # Returns Periodogram\nDSP.Periodograms.Periodogram{Float64, AbstractFFTs.Frequencies{Float64}, Vector{Float64}}([6.25, 4.75], [0.0, 0.5])\n\njulia> periodogram(x; radialavg=true) # Returns Periodogram\nDSP.Periodograms.Periodogram{Float64, AbstractFFTs.Frequencies{Float64}, Vector{Float64}}([6.25, 1.5833333333333333], [0.0, 0.5])\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.Periodogram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.Periodogram","text":"Periodogram{T, F<:Union{Frequencies,AbstractRange}, V<:AbstractVector{T}} <: TFR{T}\n\nA Periodogram object with fields:\n\npower::V\nfreq::F.\n\nSee power and freq for further details.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.Periodogram2","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.Periodogram2","text":"Periodogram2{T, F1<:Union{Frequencies,AbstractRange}, F2<:Union{Frequencies,AbstractRange}, M<:AbstractMatrix{T}} <: TFR{T}\n\nA Periodogram2 object with fields:\n\npower::M\nfreq1::F1\nfreq2::F2\n\nSee power and freq for further details.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.welch_pgram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.welch_pgram","text":"welch_pgram(s::AbstractVector, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real,\n nfft=nextfastfft(n), fs=1, window)\n\nComputes the Welch periodogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object.\n\nwelch_pgram by default divides s into 8 segments with 50% overlap between them. For a Bartlett periodogram, set noverlap=0.\n\nSee periodogram for description of optional keyword arguments.\n\nExamples\n\njulia> x = rect(10; padding=20);\n\njulia> power(welch_pgram(x; window=nothing)) # 1-sided periodogram\n2-element Vector{Float64}:\n 0.9523809523809523\n 0.04761904761904761\n\njulia> power(welch_pgram(x; onesided=false, window=nothing)) # 2-sided periodogram\n3-element Vector{Float64}:\n 0.9523809523809523\n 0.023809523809523805\n 0.023809523809523805\n\n\njulia> power(welch_pgram(x, 5; onesided=false, window=hanning)) # 5 samples segment\n5-element Vector{Float64}:\n 0.8888888888888888\n 0.3807834425694269\n 0.008105446319461968\n 0.008105446319461968\n 0.3807834425694269\n\n\nwelch_pgram with x = sin(2π(100)t) + 2sin(2π(150)t) + noise\n\njulia> Fs = 1000; # Sampling frequency\n\njulia> t = (1:Fs)/Fs; # 1000 time samples\n\njulia> f = [100 150]; # 100Hz & 150Hz frequencies\n\njulia> A = [1; 2]; # Amplitudes\n\njulia> x = sin.(2π*f.*t)*A + randn(1000); # Noise corrupted x signal\n\njulia> pxx = welch_pgram(x, 100; fs=Fs, nfft=512, window=hamming);\n\n\n\n\n\nwelch_pgram(s::AbstractVector, config::WelchConfig)\n\nComputes the Welch periodogram of the given signal s using a predefined WelchConfig object.\n\nExamples\n\njulia> x = rect(10; padding=20);\n\njulia> wconfig = WelchConfig(x; fs=1000, window=hamming);\n\njulia> welch_pgram(x, wconfig);\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.welch_pgram!","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.welch_pgram!","text":"welch_pgram!(out::AbstractVector, s::AbstractVector, n=div(length(s), 8),\n noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n),\n fs=1, window=nothing)\n\nComputes the Welch periodogram of a signal s, based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object. The Power Spectral Density (PSD) of Welch periodogram is stored in out.\n\nwelch_pgram! by default divides s into 8 segments with 50% overlap between them. For a Bartlett periodogram, set noverlap=0.\n\nSee periodogram for description of optional keyword arguments.\n\nExamples\n\njulia> Fs = 100; # Sampling frequency\n\njulia> t = (1:Fs)/Fs; # 100 time samples\n\njulia> x = cos.(2π*25*t); # Cosine signal with 25Hz frequency\n\njulia> nfft = 512;\n\njulia> y = vec(zeros(1, nfft));\n\njulia> pxx = welch_pgram!(y, x, 10; onesided=false, nfft=nfft, fs=Fs);\n\njulia> y == power(pxx)\ntrue\n\n\n\n\n\nwelch_pgram!(out::AbstractVector, s::AbstractVector, config::WelchConfig)\n\nComputes the Welch periodogram of the given signal s, using a predefined WelchConfig object. The Power Spectral Density (PSD) of Welch periodogram is stored in out.\n\nExamples\n\njulia> Fs = 100; # Sampling frequency\n\njulia> t = (1:Fs)/Fs; # 100 time samples\n\njulia> x = cos.(2π*25*t); # Cosine signal with 25Hz frequency\n\njulia> nfft = 512;\n\njulia> y = vec(zeros(1, nfft));\n\njulia> wconfig = WelchConfig(x; n=10, onesided=false, nfft=nfft, fs=Fs, window=hamming);\n\njulia> pxx = welch_pgram!(y, x, wconfig);\n\njulia> y == power(pxx)\ntrue\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.spectrogram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.spectrogram","text":"spectrogram(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)\n\nComputes the spectrogram of a signal s based on segments with n samples with overlap of noverlap samples, and returns a Spectrogram object.\n\nspectrogram by default divides s into 8 segments with 50% overlap between them. See periodogram for description of optional keyword arguments.\n\nThe returned Spectrogram object stores the 3 computed fields: power, freq and time. See power, freq and time for further details.\n\nExamples\n\njulia> Fs = 1000;\n\njulia> t = 0:1/Fs:1-1/Fs;\n\njulia> x = sin.(2π*100*t.*t);\n\njulia> spec = spectrogram(x; fs=Fs);\n\njulia> size(power(spec))\n(63, 14)\n\njulia> size(freq(spec))\n(63,)\n\njulia> time(spec)\n0.0625:0.063:0.8815\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.Spectrogram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.Spectrogram","text":"Spectrogram{T, F<:Union{Frequencies,AbstractRange}, M<:AbstractMatrix{T}} <: TFR{T}\n\nA Spectrogram object with fields:\n\npower::M\nfreq::F\ntime::Float64Range\n\nSee power, freq and time for further details.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.stft","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.stft","text":"stft(s, n=div(length(s), 8), noverlap=div(n, 2); onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, window=nothing)\n\nComputes the Short Time Fourier Transform (STFT) of a signal s based on segments with n samples with overlap of noverlap samples, and returns a matrix containing the STFT coefficients.\n\nstft by default divides s into 8 segments with 50% overlap between them. See periodogram for description of optional keyword arguments.\n\nThe STFT computes the DFT over K sliding windows (segments) of the signal s. This returns a J x K matrix where J is the number of DFT coefficients and K the number of windowed segments. The kth column of the returned matrix contains the DFT coefficients for the kth segment.\n\nExamples\n\njulia> Fs = 1000;\n\njulia> t = 0:1/Fs:5-1/Fs;\n\njulia> x = sin.(2π*t.*t);\n\njulia> size(stft(x; window=hamming))\n(313, 14)\n\njulia> size(stft(x, 500, 250; onesided=false, window=hanning))\n(500, 19)\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.freq","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.freq","text":"freq(p)\n\nReturns the frequency bin centers for a given Periodogram, Spectrogram, CrossPowerSpectra, or Coherence object.\n\nReturns a tuple of frequency bin centers for a given Periodogram2 object.\n\nExamples\n\njulia> freq(periodogram([1, 2, 3]; fs=210))\n2-element AbstractFFTs.Frequencies{Float64}:\n 0.0\n 70.0\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.power","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.power","text":"power(p)\n\nFor a Periodogram, returns the computed power spectral density (PSD) as a Vector.\n\nFor a Spectrogram, returns the computed power spectral density (PSD) at each frequency and time bin as a Matrix. Dimensions are frequency × time.\n\nFor a CrossPowerSpectra, returns the pairwise cross power spectral density (CPSD) between each pair of channels at each frequency. Dimensions are channel x channel x frequency.\n\nExamples\n\njulia> power(periodogram([1, 2, 3]; fs=210))\n2-element Vector{Float64}:\n 0.05714285714285714\n 0.009523809523809525\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#Base.Libc.time","page":"Periodograms - periodogram estimation","title":"Base.Libc.time","text":"time(p)\n\nReturns the time bin centers for a given Spectrogram object.\n\nExamples\n\njulia> time(spectrogram(0:1/20:1; fs=8000))\n0.000125:0.000125:0.0025\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.coherence","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.coherence","text":"coherence(c::Coherence)\n\nGiven an Coherence object, returns an n_channels x n_channels x length(freq(c)) array consisting of the pairwise coherences between each channel for each frequency.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.Coherence","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.Coherence","text":"Coherence{T, F, A<:AbstractArray{T, 3}}\n\nHolds an n_channels x n_channels x length(freq) array consisting of the pairwise coherences between each channel for each frequency which is accessed by coherence, as well as the frequencies accessed by freq.\n\nSee also mt_coherence and mt_coherence!.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#Multitaper-periodogram-estimation","page":"Periodograms - periodogram estimation","title":"Multitaper periodogram estimation","text":"","category":"section"},{"location":"periodograms/","page":"Periodograms - periodogram estimation","title":"Periodograms - periodogram estimation","text":"mt_pgram\nmt_pgram!\nmt_spectrogram\nmt_spectrogram!\nmt_cross_power_spectra\nmt_cross_power_spectra!\nDSP.Periodograms.CrossPowerSpectra\nmt_coherence\nmt_coherence!","category":"page"},{"location":"periodograms/#DSP.Periodograms.mt_pgram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_pgram","text":"mt_pgram(s; onesided=eltype(s)<:Real, nfft=nextfastfft(n), fs=1, nw=4, ntapers=iceil(2nw)-1, window=dpss(length(s), nw, ntapers))\nmt_pgram(signal::AbstractVector, config::MTConfig)\n\nComputes the multitaper periodogram of a signal s.\n\nIf window is not specified, the signal is tapered with ntapers discrete prolate spheroidal sequences with time-bandwidth product nw. Each sequence is equally weighted; adaptive multitaper is not (yet) supported.\n\nIf window is specified, each column is applied as a taper. The sum of periodograms is normalized by the total sum of squares of window.\n\nReturns a Periodogram.\n\nSee also mt_pgram! and MTConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_pgram!","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_pgram!","text":"mt_pgram!(output, s::AbstractVector{T}; onesided::Bool=eltype(s)<:Real,\n nfft::Int=nextfastfft(length(s)), fs::Real=1,\n nw::Real=4, ntapers::Int=ceil(Int, 2nw)-1,\n window::Union{AbstractMatrix,Nothing}=nothing) where T<:Number\nmt_pgram!(output::AbstractVector, signal::AbstractVector, config::MTConfig) -> Periodogram\n\nComputes a multitapered periodogram, storing the output in output. Arguments:\n\nsignal::AbstractVector: should be of length config.n_samples\noutput::AbstractVector: should be of length length(config.freq)\n\nOptionally pass an MTConfig object to preallocate temporary variables and choose configuration settings; otherwise, keyword arguments may be passed to choose those settings.\n\nReturns a Periodogram.\n\nSee also mt_pgram and MTConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_spectrogram","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_spectrogram","text":"mt_spectrogram(signal::AbstractVector{T}, n::Int=length(s) >> 3,\n n_overlap::Int=n >> 1; fs=1,\n onesided::Bool=T <: Real, kwargs...) where {T}\nmt_spectrogram(signal::AbstractVector, config::MTSpectrogramConfig)\n\nCompute a multitaper spectrogram, returning a Spectrogram object. Optionally pass a MTSpectrogramConfig object; otherwise, any additional keyword arguments accepted by MTConfig may be passed to configure the tapering.\n\nReturns a Spectrogram.\n\nSee also mt_spectrogram!.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_spectrogram!","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_spectrogram!","text":"mt_spectrogram!(output, signal::AbstractVector{T}, n::Int=length(signal) >> 3,\n n_overlap::Int=n >> 1; fs=1, onesided::Bool=T <: Real, kwargs...) where {T}\nmt_spectrogram!(destination::AbstractMatrix, signal::AbstractVector, config::MTSpectrogramConfig)\n\nComputes a multitaper spectrogram using the parameters specified in config. Arguments:\n\ndestination: length(config.mt_config.freq) x length(config.time) matrix. This can be created by DSP.allocate_output(config).\nsignal: vector of length config.n_samples\nconfig: optionally, pass an MTSpectrogramConfig object to hold temporary variables and configuration settings. Otherwise, settings arguments may be passed directly.\n\nReturns a Spectrogram.\n\nSee also mt_spectrogram.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_cross_power_spectra","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_cross_power_spectra","text":"mt_cross_power_spectra(signal::AbstractMatrix{T}; fs=1, kwargs...) where {T}\nmt_cross_power_spectra(signal::AbstractMatrix, config::MTCrossSpectraConfig)\n\nComputes multitapered cross power spectra between channels of a signal. Arguments:\n\nsignal: n_channels x n_samples\nOptionally pass an MTCrossSpectraConfig object to preallocate temporary variables\n\nand choose configuration settings. Otherwise, any keyword arguments accepted by MTCrossSpectraConfig may be passed here.\n\nProduces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array (accessed by power) and the corresponding frequencies (accessed by freq).\n\nSee also mt_cross_power_spectra! and MTCrossSpectraConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_cross_power_spectra!","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_cross_power_spectra!","text":"mt_cross_power_spectra!(output, signal::AbstractMatrix; fs=1, kwargs...)\nmt_cross_power_spectra!(output, signal::AbstractMatrix, config::MTCrossSpectraConfig)\n\nComputes multitapered cross power spectra between channels of a signal. Arguments:\n\noutput: n_channels x n_channels x length(config.freq). Can be created by DSP.allocate_output(config).\nsignal: n_channels x n_samples\nconfig: MTCrossSpectraConfig{T}: optionally pass a MTCrossSpectraConfig to preallocate temporary and choose configuration settings. Otherwise, one may pass any keyword arguments accepted by this object.\n\nProduces a CrossPowerSpectra object holding the n_channels x n_channels x n_frequencies output array and the corresponding frequencies (accessed by freq).\n\nSee also mt_cross_power_spectra and MTCrossSpectraConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.CrossPowerSpectra","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.CrossPowerSpectra","text":"CrossPowerSpectra{T,F,A<:AbstractArray{T, 3}}\n\nAccess the cross power spectral density (an n_channels x n_channels x length(freq) array) via the function power, and the frequencies by the function freq.\n\nSee also mt_cross_power_spectra and mt_cross_power_spectra!.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.mt_coherence","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_coherence","text":"mt_coherence(signal::AbstractMatrix{T}; fs=1, freq_range = nothing, demean=false, kwargs...) where T\nmt_coherence(signal::AbstractMatrix, config::MTCoherenceConfig)\n\nArguments:\n\nsignal: n_channels x n_samples matrix\nOptionally pass an MTCoherenceConfig to pre-allocate temporary variables and choose configuration settings, otherwise, see MTCrossSpectraConfig for the meaning of the keyword arguments.\n\nReturns a Coherence object.\n\nSee also mt_coherence and MTCoherenceConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#DSP.Periodograms.mt_coherence!","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.mt_coherence!","text":"mt_coherence!(output, signal::AbstractMatrix; fs=1, freq_range=nothing, demean=false, kwargs...)\nmt_coherence!(output, signal::AbstractMatrix, config::MTCoherenceConfig)\n\nComputes the pairwise coherences between channels.\n\noutput: n_channels x n_channels matrix\nsignal: n_samples x n_channels matrix\nconfig: optional configuration object that pre-allocates temporary variables and choose settings.\n\nReturns a Coherence object.\n\nSee also mt_coherence and MTCoherenceConfig.\n\n\n\n\n\n","category":"function"},{"location":"periodograms/#Configuration-objects","page":"Periodograms - periodogram estimation","title":"Configuration objects","text":"","category":"section"},{"location":"periodograms/","page":"Periodograms - periodogram estimation","title":"Periodograms - periodogram estimation","text":"WelchConfig\nMTConfig\nMTSpectrogramConfig\nMTCrossSpectraConfig\nMTCoherenceConfig","category":"page"},{"location":"periodograms/#DSP.Periodograms.WelchConfig","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.WelchConfig","text":"WelchConfig(s::AbstractArray; n=size(s, ndims(s))>>3, noverlap=n>>1,\n onesided=eltype(s)<:Real, nfft=nextfastfft(n),\n fs=1, window)\n\nWelchConfig(nsamples, eltype; n=nsamples>>3, noverlap=n>>1,\n onesided=eltype<:Real, nfft=nextfastfft(n),\n fs=1, window)\n\nCaptures all configuration options for welch_pgram in a single struct (akin to MTConfig). When passed on the second argument of welch_pgram, computes the periodogram based on segments with n samples with overlap of noverlap samples, and returns a Periodogram object. For a Bartlett periodogram, set noverlap=0. See periodogram for description of optional keyword arguments.\n\nnote: Note\nWelchConfig precomputes an fft plan, and preallocates the necessary intermediate buffers. Thus, repeated calls to welch_pgram that use the same WelchConfig object will be more efficient than otherwise possible.\n\nExamples\n\njulia> x = 1:8;\n\njulia> wconfig1 = WelchConfig(x; n=length(x), noverlap=0, window=hanning, nfft=16);\n\njulia> wconfig2 = WelchConfig(8, Float64; n=length(x), noverlap=0, window=hanning, nfft=16);\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.MTConfig","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.MTConfig","text":"MTConfig{T}(n_samples; fs=1,\n nfft = nextpow(2, n_samples),\n window = nothing,\n nw = 4,\n ntapers = 2 * nw - 1,\n taper_weights = fill(1/ntapers, ntapers),\n onesided::Bool=T<:Real,\n fft_flags = FFTW.MEASURE)\n\nCreates a config object which holds the configuration state and temporary variables used in multitaper computations, e.g. mt_pgram!, mt_spectrogram, MTSpectrogramConfig, MTCrossSpectraConfig, and MTCoherenceConfig.\n\nAn MTConfig can be re-used between computations as long as none of the input arguments change.\n\nn_samples: the number of samples to be used as input when computing multitaper periodograms with this configuration. Used for pre-allocating temporary buffers.\nfs: the number of samples per second of the input signal\nnfft: length of input vector to the FFT; if nfft > n_samples, then the input signal will be zero-padded until it is of length nfft.\nwindow: window function to use for tapering. If left at the default of nothing, window will be set to dpss(n_samples, nw, ntapers).\nntapers: the number of tapers to use.\ntaper_weights = fill(1/ntapers, ntapers): how to weight the contribution of each taper. The default setting is to simply average them.\nonesided: whether or not to compute a \"one-sided\" FFT by using that real signal data yields conjugate-symmetry in Fourier space.\nfft_flags: flags to control how the FFT plan is generated.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.MTSpectrogramConfig","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.MTSpectrogramConfig","text":"MTSpectrogramConfig(n_samples, mt_config::MTConfig{T}, n_overlap_samples) where {T}\nMTSpectrogramConfig{T}(n_samples, samples_per_window, n_overlap_samples; fs=1, kwargs...) where {T}\n\nCreates a MTSpectrogramConfig which holds configuration and temporary variables for mt_spectrogram and mt_spectrogram!. Any keyword arguments accepted by MTConfig may be passed here, or an MTConfig object itself.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.MTCrossSpectraConfig","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.MTCrossSpectraConfig","text":"MTCrossSpectraConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing,\n ensure_aligned = T == Float32 || T == Complex{Float32}, kwargs...) where {T}\nMTCrossSpectraConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,\n ensure_aligned = T == Float32 || T == Complex{Float32})\n\nCreates a configuration object used for mt_cross_power_spectra and mt_cross_power_spectra!.\n\nn_channels: the number of channels of the input.\nn_samples: the number of samples for each channel of the input.\ndemean: if true, the channelwise mean will be subtracted from the input signals before the cross spectral powers are computed.\nfreq_range: if nothing, all frequencies are retained. Otherwise, only frequencies between first(freq_range) and last(freq_range) are retained.\nensure_aligned = T == Float32 || T == Complex{Float32}: perform an extra copy to ensure that the FFT output is memory-aligned\nAdditionally, either pass an MTConfig object, or keyword arguments such as fs accepted by MTConfig.\n\nReturns a CrossPowerSpectra object.\n\n\n\n\n\n","category":"type"},{"location":"periodograms/#DSP.Periodograms.MTCoherenceConfig","page":"Periodograms - periodogram estimation","title":"DSP.Periodograms.MTCoherenceConfig","text":"MTCoherenceConfig{T}(n_channels, n_samples; fs=1, demean=false, freq_range=nothing, kwargs...) where T\nMTCoherenceConfig(cs_config::MTCrossSpectraConfig{T}) where {T}\nMTCoherenceConfig(n_channels, mt_config::MTConfig{T}; demean=false, freq_range=nothing,\n ensure_aligned = T == Float32 || T == Complex{Float32}) where {T}\n\nCreates a configuration object for coherences from a MTCrossSpectraConfig. Provides a helper method with the same arugments as MTCrossSpectraConfig to construct the MTCrossSpectraConfig object.\n\nSee also mt_coherence and mt_coherence!.\n\n\n\n\n\n","category":"type"}] -} diff --git a/previews/PR604/siteinfo.js b/previews/PR604/siteinfo.js deleted file mode 100644 index 3684ef64a..000000000 --- a/previews/PR604/siteinfo.js +++ /dev/null @@ -1 +0,0 @@ -var DOCUMENTER_CURRENT_VERSION = "previews/PR604"; diff --git a/previews/PR604/util/index.html b/previews/PR604/util/index.html deleted file mode 100644 index 3ebacddcf..000000000 --- a/previews/PR604/util/index.html +++ /dev/null @@ -1,31 +0,0 @@ - -Util - utility functions · DSP.jl
GitHub

Util - utility functions

Note

As of version 0.6.1 of DSP.jl, fftfreq and rfftfreq have been moved from DSP.jl to AbstractFFTs.jl version 0.5 and above. You can also access these functions through FFTW.jl version 1.1 and above.

DSP.Unwrap.unwrap — Function
unwrap(m; kwargs...)

Assumes m to be a sequence of values that has been wrapped to be inside the given range (centered around zero), and undoes the wrapping by identifying discontinuities. If a single dimension is passed to dims, then m is assumed to have wrapping discontinuities only along that dimension. If a range of dimensions, as in 1:ndims(m), is passed to dims, then m is assumed to have wrapping discontinuities across all ndims(m) dimensions.

A common usage for unwrapping across a singleton dimension is for a phase measurement over time, such as when comparing successive frames of a short-time-fourier-transform, as each frame is wrapped to stay within (-pi, pi].

A common usage for unwrapping across multiple dimensions is for a phase measurement of a scene, such as when retrieving the phase information of of an image, as each pixel is wrapped to stay within (-pi, pi].

Arguments

  • m::AbstractArray{T, N}: Array to unwrap.
  • dims=nothing: Dimensions along which to unwrap. If dims is an integer, then unwrap is called on that dimension. If dims=1:ndims(m), then m is unwrapped across all dimensions.
  • range=2pi: Range of wrapped array.
  • circular_dims=(false, ...): When an element of this tuple is true, the unwrapping process will consider the edges along the corresponding axis of the array to be connected.
  • rng=default_rng(): Unwrapping of arrays with dimension > 1 uses a random initialization. A user can pass their own RNG through this argument.
source
DSP.Util.hilbert — Function
hilbert(x)

Computes the analytic representation of x, $x_a = x + j \hat{x}$, where $\hat{x}$ is the Hilbert transform of x, along the first dimension of x.

source
DSP.Util.nextfastfft — Function
nextfastfft(n::Integer)
-nextfastfft(ns::Tuple)

Return an estimate for the padded size of an array that is optimal for the performance of an n-dimensional FFT.

For Tuple inputs, this returns a Tuple that is the size of the padded array.

For Integer inputs, this returns the closest product of 2, 3, 5, and 7 greater than or equal to n. FFTW contains optimized kernels for these sizes and computes Fourier transforms of inputs that are a product of these sizes faster than for inputs of other sizes.

Warning

Currently, nextfastfft(ns::Tuple) is implemented as nextfastfft.(ns). This may change in future releases if a better estimation model is found.

It is recommended to use nextfastfft.(ns) to obtain a tuple of padded lengths for 1D FFTs.

source
DSP.Util.pow2db — Function
pow2db(a)

Convert a power ratio to dB (decibel), or $10\log_{10}(a)$. The inverse of db2pow.

source
DSP.Util.amp2db — Function
amp2db(a)

Convert an amplitude ratio to dB (decibel), or $20 \log_{10}(a)=10\log_{10}(a^2)$. The inverse of db2amp.

source
DSP.Util.db2pow — Function
db2pow(a)

Convert dB to a power ratio. This function call also be called using a*dB, i.e. 3dB == db2pow(3). The inverse of pow2db.

source
DSP.Util.db2amp — Function
db2amp(a)

Convert dB to an amplitude ratio. This function call also be called using a*dBa, i.e. 3dBa == db2amp(3). The inverse of amp2db.

source
DSP.Util.rms — Function
rms(s; dims)

Return the root mean square (rms) of signal s. Optional keyword parameter dims can be used to specify the dimensions along which to compue the rms.

source
DSP.Util.rmsfft — Function
rmsfft(f)

Return the root mean square of signal s given the FFT transform f = fft(s). Equivalent to rms(ifft(f)).

source
DSP.Util.meanfreq — Function
meanfreq(x, fs)

Calculate the mean power frequency of x with a sampling frequency of fs, defined as:

\[MPF = \frac{\sum_{i=1}^{F} f_i X_i^2 }{\sum_{i=0}^{F} X_i^2 } Hz\]

where $F$ is the Nyquist frequency, and $X$ is the power spectral density.

source
DSP.Util.finddelay — Function
finddelay(x, y)

Estimate the delay of x with respect to y by locating the peak of their cross-correlation.

The output delay will be positive when x is delayed with respect y, negative if advanced, 0 otherwise.

Example

julia> finddelay([0, 0, 1, 2, 3], [1, 2, 3])
-2
-
-julia> finddelay([1, 2, 3], [0, 0, 1, 2, 3])
--2
source
DSP.Util.shiftin! — Function
shiftin!(a::AbstractVector{T}, b::AbstractVector{T}) where T

Shifts b into the end of a.

julia> shiftin!([1,2,3,4], [5, 6])
-4-element Vector{Int64}:
- 3
- 4
- 5
- 6
source
DSP.Util.shiftsignal — Function
shiftsignal(x, s)

Shift elements of signal x in time by a given amount s of samples and fill the spaces with zeros. For circular shifting, use circshift.

Example

julia> shiftsignal([1, 2, 3], 2)
-3-element Vector{Int64}:
- 0
- 0
- 1
-
-julia> shiftsignal([1, 2, 3], -2)
-3-element Vector{Int64}:
- 3
- 0
- 0

See also shiftsignal!.

source
DSP.Util.alignsignals — Function
alignsignals(x, y)

Use finddelay() and shiftsignal() to time align x to y. Also return the delay of x with respect to y.

Example

julia> alignsignals([0, 0, 1, 2, 3], [1, 2, 3])
-([1, 2, 3, 0, 0], 2)
-
-julia> alignsignals([1, 2, 3], [0, 0, 1, 2, 3])
-([0, 0, 1], -2)

See also alignsignals!.

source
DSP.diric — Function
kernel = diric(Ω::Real, n::Integer)

Dirichlet kernel, also known as periodic sinc function, where n should be a positive integer.

Typically returns $\frac{\sin(Ω \cdot n/2)}{n * \sin(Ω / 2)}$, but returns $±1$ when Ω is a multiple of 2π.

In the usual case where n is odd, the output is equivalent to $\frac{1}{n} \sum_{k=-(n-1)/2}^{(n-1)/2} e^{i k Ω}$, which is the discrete-time Fourier transform (DTFT) of a n-point moving average filter.

When n is odd or even, the function is 2π or 4π periodic, respectively. The formula for general n is

\[\mathrm{diric}(Ω,n) = \frac{e^{-i (n-1) Ω/2}}{n} \sum_{k=0}^{n-1} e^{i k Ω}\]

which is a real and symmetric function of Ω.

As of 2021-03-19, the Wikipedia definition has different factors. The definition here is consistent with scipy and other software frameworks.

Examples

julia> round.(diric.((-2:0.5:2)*π, 5), digits=9)'
-1×9 adjoint(::Vector{Float64}) with eltype Float64:
- 1.0  -0.2  0.2  -0.2  1.0  -0.2  0.2  -0.2  1.0
-
-julia> diric(0, 4)
-1.0
source
diff --git a/previews/PR604/windows/index.html b/previews/PR604/windows/index.html deleted file mode 100644 index 579cd5865..000000000 --- a/previews/PR604/windows/index.html +++ /dev/null @@ -1,294 +0,0 @@ - -Windows - window functions · DSP.jl
GitHub

Windows - window functions

DSP.Windows.rect — Function
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-rect(n::Integer; padding::Integer=0, zerophase::Bool=false)
-rect(dims; padding=0, zerophase=false)

Rectangular window of length n, padded with padding zeros. This window is 1 within the window, and 0 outside of it.

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.hanning — Function
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-  │                ▗▄▘                                ▝▚▖                │ 
-  │               ▄▘                                    ▝▄               │ 
-  │             ▗▞                                        ▚▖             │ 
-  │            ▄▘                                          ▝▄            │ 
-  │          ▗▀                                              ▀▖          │ 
-  │        ▗▞▘                                                ▝▚▖        │ 
-  │      ▄▀▘                                                    ▝▀▄      │ 
-0 │▄▄▄▄▀▀                                                          ▀▀▄▄▄▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-hanning(n::Integer; padding::Integer=0, zerophase::Bool=false)
-hanning(dims; padding=0, zerophase=false)

Hanning window of length n with padding zeros. The Hanning (or Hann) window is a raised-cosine window that reaches zero at the endpoints.

The window is defined by sampling the continuous function:

       1 + cos(2πx)
-w(x) = ──────────── = cos²(πx)
-            2

in the range [-0.5, 0.5]

The hanning window satisfies the Constant Overlap-Add (COLA) property with a hop of 0.5, which means that adding together a sequence of delayed windows with 50% overlap will result in a constant signal. This is useful when synthesizing a signal from a number of overlapping frames (each with a roughly rectangular window), to eliminate windowing amplitude modulation.

While this windowing function was originally named after its inventor Julius Von Hann, colloquial usage has adjusted to refer to it as the "Hanning" window, to better match the similarly-named "Hamming" window. Because of this, both hann() and hanning() are exported from this module.

Note that the hanning window is the cosine window squared.

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.hamming — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                             ▄▞▀▀▀▀▀▀▀▀▚▄                             │ 
-  │                          ▗▞▀            ▀▚▖                          │ 
-  │                        ▗▞▘                ▝▀▄                        │ 
-  │                      ▗▞▘                     ▚▖                      │ 
-  │                     ▞▘                        ▝▚▖                    │ 
-  │                   ▄▀                            ▀▄                   │ 
-  │                 ▄▞                                ▚▄                 │ 
-  │               ▗▞                                    ▚▖               │ 
-  │              ▞▘                                      ▝▚              │ 
-  │            ▄▀                                          ▀▄            │ 
-  │          ▗▞                                              ▚▖          │ 
-  │        ▄▞▘                                                ▝▚▄        │ 
-  │     ▗▄▀▘                                                     ▀▄▖     │ 
-  │▄▄▄▞▀▘                                                          ▝▀▚▄▄▄│ 
-0 │                                                                      │ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-hamming(n::Integer; padding::Integer=0, zerophase::Bool=false)
-hamming(dims; padding=0, zerophase=false)

Hamming window of length n with padding zeros. The Hamming window does not reach zero at the endpoints and so has a shallower frequency roll-off when compared to the Hanning window, but is designed to cancel the first side-lobe.

The window is defined by sampling the continuous function:

w(x) = 0.54 + 0.46*cos(2pi*x)

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.tukey — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │           ▗▞▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▜▖           │ 
-  │          ▗▘                                              ▝▖          │ 
-  │         ▗▘                                                ▝▖         │ 
-  │         ▞                                                  ▚         │ 
-  │        ▐                                                    ▌        │ 
-  │        ▌                                                    ▝▖       │ 
-  │       ▐                                                      ▚       │ 
-  │      ▗▘                                                      ▝▖      │ 
-  │      ▌                                                        ▚      │ 
-  │     ▗▘                                                         ▌     │ 
-  │     ▌                                                          ▐     │ 
-  │    ▞                                                            ▚    │ 
-  │   ▞▘                                                            ▝▖   │ 
-  │  ▗▘                                                              ▝▖  │ 
-0 │▄▄▘                                                                ▝▚▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-tukey(n::Integer, α::Real; padding::Integer=0, zerophase::Bool=false)
-tukey(dims, α; padding=0, zerophase=false)

Tukey window of length n with padding zeros. The Tukey window has a flat top and reaches zero at the endpoints, with a sinusoidal transition area parameterized by α. For α == 0, the window is equivalent to a rectangular window. For α == 1, the window is a Hann window.

The window is defined by sampling the continuous function:

       ⎛              ⎛2π ⎛    1 - α⎞⎞
-       ⎜      1 + cos ⎜── ⎜x + ─────⎟⎟             1 - α
-       ⎜              ⎝ α ⎝      2  ⎠⎠       x ≤ - ─────
-       ⎜      ─────────────────────────              2
-       ⎜                  2
-       ⎜
-       ⎜                                       1 - α       1 - α
-w(x) = ⎜      1                              - ───── < x ≤ ─────
-       ⎜                                         2           2
-       ⎜
-       ⎜              ⎛2π ⎛    1 - α⎞⎞
-       ⎜      1 + cos ⎜── ⎜x - ─────⎟⎟           1 - α
-       ⎜              ⎝ α ⎝      2  ⎠⎠       x > ─────
-       ⎜      ─────────────────────────            2
-       ⎝                  2

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. α, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.cosine — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                           ▄▄▀▀▀▀▀▀▀▀▀▀▀▀▄▄                           │ 
-  │                       ▗▄▀▀                ▀▀▄▖                       │ 
-  │                     ▄▀▘                      ▝▀▄                     │ 
-  │                  ▗▞▀                            ▀▚▖                  │ 
-  │                ▄▀▘                                ▝▀▄                │ 
-  │              ▄▀                                      ▀▄              │ 
-  │            ▗▞                                          ▚▖            │ 
-  │           ▞▘                                            ▝▚▖          │ 
-  │         ▄▀                                                ▝▄         │ 
-  │       ▗▞                                                    ▚▖       │ 
-  │      ▄▘                                                      ▝▄      │ 
-  │    ▗▞                                                          ▚▖    │ 
-  │   ▄▘                                                            ▝▄   │ 
-  │ ▗▞                                                                ▚▖ │ 
-0 │▄▘                                                                  ▝▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-cosine(n::Integer; padding::Integer=0, zerophase::Bool=false)
-cosine(dims; padding=0, zerophase=false)

Cosine window of length n with padding zeros. The cosine window is the first lobe of a cosine function (with the zero crossings at +/- π as endpoints). Also called the sine window.

The window is defined by sampling the continuous function:

w(x) = cos(πx)

in the range [-0.5, 0.5]

Note that the cosine window is the square root of the hanning window, so it is sometimes used when you are applying the window twice, such as the analysis and synthesis steps of an STFT.

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.lanczos — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                            ▄▄▀▀▀▀▀▀▀▀▀▀▄▄                            │ 
-  │                         ▄▞▀              ▀▀▄                         │ 
-  │                      ▗▞▀                    ▀▚▖                      │ 
-  │                    ▗▞▘                        ▝▚▖                    │ 
-  │                  ▗▞▘                            ▝▚▖                  │ 
-  │                ▗▀▘                                ▝▀▖                │ 
-  │              ▗▞▘                                    ▝▚▖              │ 
-  │             ▄▘                                        ▝▚             │ 
-  │           ▄▀                                            ▀▄           │ 
-  │         ▗▞▘                                               ▚▖         │ 
-  │        ▄▘                                                  ▝▚        │ 
-  │      ▄▀                                                      ▀▄      │ 
-  │    ▗▞                                                          ▜▖    │ 
-  │  ▗▞▘                                                            ▝▚▖  │ 
-0 │▄▞▘                                                                ▝▚▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-lanczos(n::Integer; padding::Integer=0, zerophase::Bool=false)
-lanczos(dims; padding=0, zerophase=false)

Lanczos window of length n with padding zeros. The Lanczos window is the main lobe of a sinc function.

The window is defined by sampling the continuous function:

                  sin(2πx)
-w(x) = sinc(2x) = ────────
-                     2πx

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.triang — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                                ▄▞▀▀▀▄▖                               │ 
-  │                            ▗▄▞▀      ▝▀▄▖                            │ 
-  │                         ▗▄▀▘            ▝▀▄▖                         │ 
-  │                      ▗▄▀▘                  ▝▀▄▖                      │ 
-  │                   ▗▄▀▘                        ▝▀▄▖                   │ 
-  │                ▗▄▀▘                              ▝▀▄▄                │ 
-  │             ▗▄▀▘                                     ▀▚▄             │ 
-  │          ▗▄▀▘                                           ▀▙▄          │ 
-  │       ▄▄▀▘                                                 ▀▚▄       │ 
-  │    ▄▞▀                                                        ▀▚▄    │ 
-  │ ▄▞▀                                                              ▀▚▄ │ 
-  │▀                                                                    ▀│ 
-  │                                                                      │ 
-  │                                                                      │ 
-0 │                                                                      │ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                    7  
-
-triang(n::Integer; padding::Integer=0, zerophase::Bool=false)
-triang(dims; padding=0, zerophase=false)

Triangular window of length n with padding zeros. The Triangular window does not reach zero at the endpoints. For odd n the triang window is the center n points of an n+2-point bartlett window (i.e. the samples just outside the window would be zero). For even n the window slope is the same as the n-1 window but delayed by a half sample so the zero points would be 1/2 sample past the ends of the window.

The window is defined by sampling the continuous function:

        ⎛    2(n-1)
-        ⎜1 - ────── abs(x)     n is even
-        ⎜       n
-w(x) =  ⎜
-        ⎜    2(n-1)
-        ⎜1 - ────── abs(x)     n is odd
-        ⎝     n+1

in the range [-0.5, 0.5].

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

When zerophase is true substitute n+1 for n in the above window expressions.

source
DSP.Windows.bartlett — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                                ▗▞▀▀▚▖                                │ 
-  │                              ▄▞▘    ▝▚▄                              │ 
-  │                            ▄▀          ▀▄                            │ 
-  │                         ▗▄▀              ▀▄▖                         │ 
-  │                       ▗▞▘                  ▝▚▖                       │ 
-  │                     ▄▞▘                      ▝▚▄                     │ 
-  │                   ▄▀                            ▀▄                   │ 
-  │                ▄▄▀                                ▀▚▄                │ 
-  │              ▄▀                                      ▀▄              │ 
-  │           ▗▞▀                                          ▀▚▖           │ 
-  │         ▗▞▘                                              ▝▚▖         │ 
-  │       ▄▀▘                                                  ▝▀▄       │ 
-  │     ▄▀                                                        ▀▄     │ 
-  │  ▗▞▀                                                            ▀▚▖  │ 
-0 │▄▞▘                                                                ▝▚▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-bartlett(n::Integer; padding::Integer=0, zerophase::Bool=false)
-bartlett(dims; padding=0, zerophase=false)

Bartlett window of length n. The Bartlett window is a triangular window that reaches 0 at the endpoints. This is equivalent to convolving two rectangular windows of length (n-1)/2 and adding the zero endpoints. See triang for a window that does not reach zero at the endpoints.

The window is defined by sampling the continuous function:

1 - abs(2x)

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.gaussian — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                             ▗▄▞▀▀▀▀▀▀▀▄▖                             │ 
-  │                           ▗▞▘          ▝▚▖                           │ 
-  │                         ▗▞▘              ▝▚▖                         │ 
-  │                        ▄▘                  ▝▄                        │ 
-  │                      ▗▞                      ▚▖                      │ 
-  │                     ▄▘                        ▝▙                     │ 
-  │                   ▗▞                            ▀▖                   │ 
-  │                  ▟▘                              ▝▚                  │ 
-  │                ▞▀                                  ▀▚                │ 
-  │              ▄▀                                      ▀▄              │ 
-  │            ▄▀                                          ▀▄            │ 
-  │          ▄▀                                              ▀▄          │ 
-  │       ▄▞▀                                                  ▀▚▄       │ 
-  │  ▗▄▄▀▀                                                        ▀▀▄▄▖  │ 
-0 │▀▀▘                                                                ▝▀▀│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-gaussian(n::Integer, σ::Real; padding::Integer=0, zerophase::Bool=false)
-gaussian(dims, σ; padding=0, zerophase=false)

Gives an n-sample gaussian window defined by sampling the function:

        ⎛        2⎞
-        ⎜-1   ⎛x⎞ ⎟
-        ⎜── ⋅ ⎜─⎟ ⎟
-        ⎝ 2   ⎝σ⎠ ⎠
-w(x) = e

in the range [-0.5,0.5]. This means that for σ=0.5 the endpoints of the window will correspond to 1 standard deviation away from the center.

Providing a dims Tuple rather than a single n constructs a 2D window. σ, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.bartlett_hann — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                              ▄▄▀▀▀▀▀▀▄▄                              │ 
-  │                           ▗▟▀          ▀▚▖                           │ 
-  │                         ▗▞▘              ▝▚▖                         │ 
-  │                       ▗▞▘                  ▝▚▖                       │ 
-  │                      ▞▘                      ▝▚                      │ 
-  │                    ▄▀                          ▀▄                    │ 
-  │                  ▗▞                              ▚▖                  │ 
-  │                ▗▄▘                                ▝▚▖                │ 
-  │               ▞▘                                    ▝▚               │ 
-  │             ▄▀                                        ▀▄             │ 
-  │           ▗▞                                            ▚▖           │ 
-  │         ▗▞▘                                              ▝▚▖         │ 
-  │       ▗▞▘                                                  ▝▚▖       │ 
-  │     ▄▞▘                                                      ▝▜▄     │ 
-0 │▄▄▄▀▀                                                            ▀▀▄▄▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-bartlett_hann(n::Integer; padding::Integer=0, zerophase::Bool=false)
-bartlett_hann(dims; padding=0, zerophase=false)

Bartlett-Hann window of length n with padding zeros. The Bartlett-Hann window is a weighted sum of the Bartlett and Hann windows.

The window is defined by sampling the continuous function:

w(x) = 0.62 - 0.48*abs(x) + 0.38*cos(2π*x)

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.blackman — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                              ▗▞▀▀▀▀▀▀▚▖                              │ 
-  │                            ▗▞▘        ▝▚▖                            │ 
-  │                           ▞▘            ▝▚                           │ 
-  │                         ▗▀                ▀▖                         │ 
-  │                        ▄▘                  ▝▙                        │ 
-  │                       ▞                      ▚                       │ 
-  │                     ▗▀                        ▀▖                     │ 
-  │                    ▗▘                          ▝▖                    │ 
-  │                   ▞▘                            ▝▚                   │ 
-  │                 ▄▞                                ▚▄                 │ 
-  │               ▗▞                                    ▚▖               │ 
-  │              ▞▘                                      ▝▚              │ 
-  │           ▗▄▀                                          ▀▄▖           │ 
-  │         ▄▞▘                                              ▝▚▄▖        │ 
-0 │▄▄▄▄▄▄▞▀▀                                                    ▝▀▚▄▄▄▄▄▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-blackman(n::Integer; padding::Integer=0, zerophase::Bool=false)
-blackman(dims; padding=0, zerophase=false)

Approximates the "Exact" Blackman window. This is the generalized Blackman window with α = 0.16.

The window is defined by sampling the continuous function:

w(x) = 0.42 + 0.5*cos(2π*x) + 0.08*cos(4π*x)

in the range [-0.5, 0.5]

Providing a dims Tuple rather than a single n constructs a 2D window. padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.kaiser — Function
  ┌──────────────────────────────────────────────────────────────────────┐ 
-1 │                              ▗▄▀▀▀▀▀▀▄▖                              │ 
-  │                            ▗▞▘        ▝▚▖                            │ 
-  │                           ▄▘            ▝▄                           │ 
-  │                          ▞                ▚                          │ 
-  │                        ▗▀                  ▀▖                        │ 
-  │                       ▗▘                    ▝▖                       │ 
-  │                      ▄▘                      ▝▄                      │ 
-  │                     ▞                          ▚                     │ 
-  │                   ▗▞                            ▚▖                   │ 
-  │                  ▗▘                              ▝▖                  │ 
-  │                ▗▀▘                                ▝▀▖                │ 
-  │              ▗▞▘                                    ▝▚▖              │ 
-  │            ▗▞▘                                        ▝▚▖            │ 
-  │         ▗▄▞▘                                            ▝▚▄▖         │ 
-0 │▄▄▄▄▄▄▄▞▀▘                                                  ▝▀▚▄▄▄▄▄▄▄│ 
-  └──────────────────────────────────────────────────────────────────────┘ 
-   1                                                                   69  
-
-kaiser(n::Integer, α::Real; padding::Integer=0, zerophase::Bool=false)
-kaiser(dims, α; padding=0, zerophase=false)

Kaiser window of length n parameterized by α. The Kaiser window approximates the DPSS window (given by dpss), using a simplified definition relying on a Bessel function. Larger values for α give a wider main lobe but have lower sidelobes. Typically α is set around 3.

The window is defined by sampling the continuous function:

         ⎛  ⎛   _________⎞⎞
-         ⎜  ⎜  ╱        2⎟⎟
-w(x) = I₀⎝πα⎝╲╱ 1 - (2x) ⎠⎠
-       ────────────────────
-               I₀(πα)

in the range [-0.5, 0.5]

Where I₀(⋅) is the zeroth-order modified Bessel function of the first kind.

Providing a dims Tuple rather than a single n constructs a 2D window. α, padding and zerophase can then be given either as a single value for both horizontal and vertical or a 2-tuple to specify them separately.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.dpss — Function
    ┌──────────────────────────────────────────────────────────────────────┐ 
-0.2 │                             ▗▄▞▀▀▀▀▀▀▚▄▖                             │ 
-    │                           ▄▀▘          ▝▀▄                           │ 
-    │                         ▄▀                ▀▄                         │ 
-    │                       ▗▞                    ▀▖                       │ 
-    │                      ▞▘                      ▝▚                      │ 
-    │                    ▗▀                          ▀▄                    │ 
-    │                  ▗▞▘                             ▚▖                  │ 
-    │                ▗▄▀                                ▝▄▖                │ 
-    │               ▄▘                                    ▝▄               │ 
-    │             ▗▞                                        ▚▖             │ 
-    │           ▗▞▘                                          ▝▚▖           │ 
-    │         ▗▞▘                                              ▝▚▖         │ 
-    │       ▄▞▘                                                  ▝▚▄       │ 
-    │    ▄▄▀▘                                                       ▀▚▄    │ 
-  0 │▄▞▀▀                                                              ▀▀▚▄│ 
-    └──────────────────────────────────────────────────────────────────────┘ 
-     1                                                                   69  
-
-dpss(n::Integer, nw::Real, ntapers::Integer=iceil(2*nw)-1;
-     padding::Integer=0, zerophase::Bool=false)

The first ntapers discrete prolate spheroid sequences (Slepian tapers) as an n × ntapers matrix. The signs of the tapers follow the convention that the first element of the skew-symmetric (odd) tapers is positive. The time-bandwidth product is given by nw.

The DPSS window maximizes the energy concentration in the main lobe.

If zerophase is false (the default) the window is centered around index (n+1)/2, which is commonly used for FIR filter design. These are often used for FIR filter design, and usually odd-length. Note that for even-length windows this will cause the window center to be located between samples.

If zerophase is true the window is centered around index 1 (with the negative half wrapped to the end of the vector). Additionally this creates a "periodic" window, which means that if there is no padding then the left and right endpoints of the window wrap around to the same sample, so the window length is the same as an n+1-length non-zerophase window. Alternatively you can think of the continuous zerophase window being of width n and the non-zerophase window being of length n-1. zerophase windows are often used in FFT processing, and are usually even-length.

source
DSP.Windows.dpsseig — Function
dpsseig(A, nw)

Eigenvalues of the DPSS matrix, representing the ratios of the power within the main lobe to the total power (main and sidelobes). A is the output of dpss, and nw is the time-bandwidth product provided to dpss as input.

source