FFT, Exponent Testing, Herz frequency, with 2pi.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Тестирование инварианта для преобразования Фурье, частоты в Герцах"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi}f(x)dx$,\n",
    "\n",
    "инвариант: $f(x) = e^{-x^2\\pi}$, константа: 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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F0CBJkrIYGiRJUhZDgyRJylLT2hOSjj2DI4N09nUyMDLAyUtOJiJ4dd+rtC5tpdBVoKWp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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6ca288ae80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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lHpO45Zmbg5NPhm99q+xKNEq2bYO3u0JoxZxySmOtycRE79tFvWThQupXRKOj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\n8wgLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6ca29285c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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lHpO45Zmbg5NPhm99q+xKNEq2bYO3u0JoxZxySmOtycRE79tFvWThQupXRKOj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\n8wgLAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6ca28aa5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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n1SW/FDG6O+/rky5cSL5u0SLphReSx6JAIswD2Ly5MipRbwQjbN/+/XNftllr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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f6ca28dad30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from numpy import fft\n",
    "from scipy import integrate\n",
    "from matplotlib import pyplot as plt\n",
    "\n",
    "M = 2**11\n",
    "x_space = np.linspace(-100, 100, num = M, endpoint = False)\n",
    "dx = abs(x_space[1] - x_space[0])\n",
    "f = np.exp(- np.pi * (x_space * x_space))\n",
    "\n",
    "plt.plot(x_space, f, 'g.')\n",
    "plt.show()\n",
    "\n",
    "f_hat = fft.fft(f)\n",
    "xi_space = fft.fftfreq(M, d = dx)\n",
    "\n",
    "offset_factor = np.exp(2*np.pi*1j* (M//2) * dx * xi_space )\n",
    "         \n",
    "f_hat = dx * offset_factor * f_hat\n",
    "\n",
    "# Рисуем только те же значения xi, для которых мы определяли функцию по x, \n",
    "# чтобы сравнение картинок получилось наглядным\n",
    "# Мы не использовали сдвиг ответа, но это то, что нужно сделать, если потом нужно применять не ifft\n",
    "\n",
    "plt.plot(xi_space[(xi_space>-10) & (xi_space<10)], f_hat.real[(xi_space>-10) & (xi_space<10)], 'b.')\n",
    "plt.show()\n",
    "\n",
    "plt.plot(xi_space, f_hat.real, 'b.')\n",
    "plt.show()\n",
    "\n",
    "# мнимая часть мала, но есть. Можно раскомментировать, чтобы на неё посмотреть.\n",
    "plt.plot(xi_space[(xi_space>-10) & (xi_space<10)], f_hat.imag[(xi_space>-10) & (xi_space<10)], 'r.')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Мы задаём промежуток от -10 до 10, на котором определяем функцию $f(x)$. Выбираем на промежутке $M+1$ точку, отбрасываем последнюю и смотрим на шаг по $x$, который обозначаем $dx$.\n",
    "\n",
    "Считаем от набора, который представляет из себя эту функцию, библиотечное преобразование Фурье.\n",
    "\n",
    "1. Известно, что оно не включает множитель $dx$, поэтому корректируем результат соответственно.\n",
    "\n",
    "2. Определяем спектр частот и достаём массив $\\xi_l$.\n",
    "\n",
    "3. Для того, чтобы скорректировать сдвиг функции $f$ влево на $x_0 = \\frac{M}{2} dx$, используя теорему о сдвиге, домножим результат на $e^{2\\pi i (\\frac{M}{2} dx) \\xi_l }$  <span style=\"color:green\">(формулировка: Для преобразования Фурье вида $\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi}f(x)dx$ справедливо $\\hat{f(x-x0)(\\xi)} = e^{2\\pi ix_0\\xi}\\hat{f(\\xi)}$)</span>\n",
    "\n",
    "4. Выберем нужное подмножество точек, чтобы увидеть сравнимые графики. Более точное сравнение включает интерполяцию, нежелательную для этого простого примера.\n",
    "\n",
    "Замечание: разберёмся, куда девается множитель $\\sqrt(\\pi)$, который должен возникать при вычислении интеграла Пуассона. Короткий ответ - погашается обратным при замене.\n",
    "\n",
    "Напомним, что\n",
    "\n",
    "$$\\hat{f(\\xi)} = \\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi}f(x)dx$$, действуя на $f(x) = e^{-x^2\\pi}$, принимает вид\n",
    "\n",
    "$\\hat{f(\\xi)} = \n",
    "\\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi} e^{-x^2\\pi}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{-2\\pi ix\\xi - x^2\\pi}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{-x^2\\pi - 2\\pi ix\\xi - (i\\xi)^2\\pi + (i\\xi)^2\\pi}dx = \n",
    "\\int_{-\\infty}^{\\infty}e^{-(x\\sqrt{\\pi} + i\\xi\\sqrt{\\pi})^2 - \\xi^2\\pi}dx = \n",
    "e^{-\\xi^2\\pi}\\int_{-\\infty}^{\\infty}e^{-(x\\sqrt{\\pi} + i\\xi\\sqrt{\\pi})^2}dx =\n",
    "\\color{blue}{// t = x\\sqrt{\\pi} + i\\xi\\sqrt{\\pi}, dt = \\sqrt{\\pi}dx//}=\n",
    "e^{-\\xi^2\\pi} \\cdot \\sqrt{\\pi}$"
   ]
  }
 ],
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