/*
https://idmil.gitlab.io/course-materials/mumt203/interactive-demos
The purpose of the digital audio workbench is to illustrate key concepts in digital audio theory with interactive visualizations of each stage of the analog-to-digial conversion (ADC) and digital-to-analog conversion (DAC) processes. These visualizations are inspired by demonstrations using oscilloscopes and spectrum analyzers to compare the analog signal input into the ADC process with the analog signal output by the DAC process, e.g. https://youtu.be/cIQ9IXSUzuM
By experimenting with the settings of the simulation, numerous key concepts in digital signal theory can be nicely illustrated, such as aliasing, quantization error, critical sampling, under and oversampling, and many others. The interactive interface allows the simulation to be explored freely; users can examine the signals both visually through numerous graphs, or by listening to the test signals directly.
Since our demonstration takes place purely in the digital domain, we unfortunately cannot use real continuous time analog inputs and outputs. Instead, we simulate the ADC-DAC processes in the discrete time domain. The analog input and output are represented as discrete time signals with a high sampling rate; at the time of writing, the maximum sampling rate supported by WebAudio is 96 kHz.
The ADC process consists of several steps, including antialiasing, sampling, and quantization. All of these are simulated in our model: antialiasing is achieved with a windowed sinc FIR lowpass filter of order specified by the user; sampling is approximated by downsampling the input signal by an integer factor; and quantization is achieved by multiplying the sampled signal (which ranges from -1.0 to 1.0) by the maximum integer value possible given the requested bit depth (e.g. 255 for a bit depth of 8 bits), and then rounding every sample to the nearest integer. The DAC process is simulated in turn by zero stuffing and lowpass filtering the sampled and quantized output of the ADC simultion.
In summary, the continuous time input is simulated by a 96 kHz discrete time signal, the sampled output of the ADC process is simulated by a downsampled and quantized signal, and the continuous time reconstruction output by the DAC is simulated by upsampling the "sampled" signal back to 96 kHz. In our tests we have found this model to be reasonable; many key concepts, such as critical sampling, aliasing, and quantization noise are well represented in our simulation.
For more details, the reader is encouraged to peruse the rest of the source code in this document. Many comments have been included to aid readers who are unfamiliar with javascript. Any questions you may have about the implementation of the simulation can only be definitively answered by understanding the source code, but please feel free to contact the project maintainers if you have any questions.
*/
// `renderWavesImpl` returns an anonymous function that is bound in the widget
// constructor. This is done in order to seperate the implementation of the
// simulation from the other implementation details so that this documentation
// can be more easily accessed.
const soundTimeSeconds = 1.5;
const fadeTimeSeconds = 0.125;
function renderWavesImpl(settings, fft, p) { return (playback = false) => {
// if we are not rendering for playback, we are rendering for simulation
let simulation = !playback;
// select the buffer to render to; playback buffer, or simulation buffer
var original = playback ? settings.original_pb : settings.original;
var reconstructed = playback ? settings.reconstructed_pb : settings.reconstructed;
var stuffed = settings.stuffed;
// calculate harmonics ------------------------------------------------------
// The signal is generated using simple additive synthesis. Because of this,
// the exact frequency content of the signal can be determined a priori based
// on the settings. We generate this information here so that it can be used
// not only by the synthesis process below, but also by several of the graphs
// used to illustrate the frequency domain content of the signal.
// We only calculate the harmonics for the simulation; it is assumed they will
// already have been calculated earlier when rendering for playback
if (simulation) {
let harmonic_number = 1;
let harmonic_amplitude = 1;
let invert = 1;
let harmInc = (settings.harmType =="Odd" || settings.harmType == "Even") ? 2 : 1;
for (let i = 0; simulation && i < settings.numHarm; i++) {
// the amplitude of each harmonic depends on the harmonic slope setting
if (settings.harmSlope == "lin") harmonic_amplitude = 1 - i/settings.numHarm;
else if (settings.harmSlope == "1/x") harmonic_amplitude = 1/harmonic_number;
else if (settings.harmSlope == "1/x2") harmonic_amplitude = 1/harmonic_number/harmonic_number;
else if (settings.harmSlope == "flat") harmonic_amplitude = 1;
// In case the harmonic slope is 1/x^2 and the harmonic type is "odd",
// by inverting every other harmonic we generate a nice triangle wave.
if (settings.harmSlope =="1/x2" && settings.harmType == "Odd") {
harmonic_amplitude = harmonic_amplitude * invert;
invert *= -1;
}
// the frequency of each partial is a multiple of the fundamental frequency
settings.harmonicFreqs[i] = harmonic_number*settings.fundFreq;
// The harmonic amplitude is calculated above according to the harmonic
// slope setting, taking into account the special case for generating a
// triangle.
settings.harmonicAmps[i] = harmonic_amplitude;
// With harmonic type set to "even" we want the fundamental and even
// harmonics. To achieve this, we increment the harmonic number by 1 after
// the fundamental and by 2 after every other partial.
if (i == 0 && settings.harmType == "Even") harmonic_number += 1;
else harmonic_number += harmInc;
}
}
// render original wave -----------------------------------------------------
// initialize the signal buffer with all zeros (silence)
original.fill(0);
// For the sample at time `n` in the signal buffer `original`,
// generate the sum of all the partials based on the previously calculated
// frequency and amplitude values.
original.forEach( (_, n, arr) => {
for (let harmonic = 0; harmonic < settings.numHarm; harmonic++) {
let fundamental_frequency = settings.harmonicFreqs[0];
let frequency = settings.harmonicFreqs[harmonic];
let amplitude = settings.harmonicAmps[harmonic];
// convert phase offset specified in degrees to radians
let phase_offset = Math.PI / 180 * settings.phase;
// adjust phase offset so that harmonics are shifted appropriately
let phase_offset_adjusted = phase_offset * frequency / fundamental_frequency;
let radian_frequency = 2 * Math.PI * frequency;
let phase_increment = radian_frequency / WEBAUDIO_MAX_SAMPLERATE;
let phase = phase_increment * n + phase_offset_adjusted;
// accumulate the amplitude contribution from the current harmonic
arr[n] += amplitude * Math.sin( phase );
}
});
// linearly search for the maximum amplitude value (easy but not efficient)
let max = 0;
original.forEach( (x, n, y) => {if (x > max) max = x} );
// normlize and apply amplitude scaling
original.forEach( (x, n, y) => y[n] = settings.amplitude * x / max );
// apply antialiasing filter if applicable ----------------------------------
// The antialiasing and reconstruction filters are generated using Fili.js.
// (https://github.com/markert/fili.js/)
let firCalculator = new Fili.FirCoeffs();
// Fili uses the windowed sinc method to generate FIR lowpass filters.
// Like real antialiasing and reconstruction filters, the filters used in the
// simulation are not ideal brick wall filters, but approximations.
// apply antialiasing only if the filter order is set
if (settings.antialiasing > 1) {
// specify the filter parameters; Fs = sampling rate, Fc = cutoff frequency
// The cutoff for the antialiasing filter is set to the Nyquist frequency
// of the simulated sampling process. The sampling rate of the "sampled"
// signal is WEBAUDIO_MAX_SAMPLERATE / the downsampling factor. This is
// divided by 2 to get the Nyquist frequency.
var filterCoeffs = firCalculator.lowpass(
{ order: settings.antialiasing
, Fs: WEBAUDIO_MAX_SAMPLERATE
, Fc: (WEBAUDIO_MAX_SAMPLERATE / settings.downsamplingFactor) / 2
});
// generate the filter
var filter = new Fili.FirFilter(filterCoeffs);
// apply the filter
original.forEach( (x, n, y) => y[n] = filter.singleStep(x) );
// time shift the signal by half the filter order to compensate for the
// delay introduced by the FIR filter
original.forEach( (x, i, arr) => arr[i - settings.antialiasing/2] = x );
}
// downsample original wave -------------------------------------------------
// zero initialize the reconstruction, and zero stuffed buffers
reconstructed.fill(0);
stuffed.fill(0);
// generate new signal buffers for the downsampled signal and quantization
// noise whose sizes are initialized according to the currently set
// downsampling factor
if (playback) {
settings.downsampled_pb = new Float32Array(p.round(original.length / settings.downsamplingFactor));
settings.quantNoise_pb = new Float32Array(p.round(original.length / settings.downsamplingFactor));
} else {
settings.downsampled = new Float32Array(p.round(original.length / settings.downsamplingFactor));
settings.quantNoise = new Float32Array(p.round(original.length / settings.downsamplingFactor));
}
var downsampled = playback ? settings.downsampled_pb : settings.downsampled;
var quantNoise = playback ? settings.quantNoise_pb : settings.quantNoise;
var quantNoiseStuffed = settings.quantNoiseStuffed;
quantNoiseStuffed.fill(0);
// calculate the maximum integer value representable with the given bit depth
let maxInt = p.pow(2, settings.bitDepth) - 1;
let stepSize = (settings.quantType == "midTread") ? 2/(maxInt-1) : 2/(maxInt);
// generate the output of the simulated ADC process by "sampling" (actually
// just downsampling), and quantizing with dither. During this process, we
// also load the buffer for the reconstructed signal with the sampled values;
// this allows us to skip an explicit zero-stuffing step later
downsampled.forEach( (_, n, arr) => {
// keep only every kth sample where k is the integer downsampling factor
let y = original[n * settings.downsamplingFactor];
y = y > 1.0 ? 1.0 : y < -1.0 ? -1.0 : y; // apply clipping
// if the bit depth is set to the maximum, we skip quantization and dither
if (settings.bitDepth == BIT_DEPTH_MAX) {
// record the sampled output of the ADC process
arr[n] = y;
// sparsely fill the reconstruction and zero stuffed buffers to avoid
// having to explicitly zero-stuff
reconstructed[n * settings.downsamplingFactor] = y;
stuffed[n * settings.downsamplingFactor] = y * settings.downsamplingFactor;
return;
}
// generate dither noise
let dither = (2 * Math.random() - 1) * settings.dither;
let quantized;
// Add dither signal and quantize. Constrain so we dont clip after dither
switch(settings.quantType) {
case "midTread" :
quantized = stepSize*p.floor(p.constrain((y+dither),-1,0.99)/stepSize + 0.5);
break;
case "midRise" :
quantized = stepSize*(p.floor(p.constrain((y+dither),-1,0.99)/stepSize) + 0.5);
break;
}
// record the sampled and quantized output of the ADC process with clipping
arr[n] = quantized;
// sparsely fill the reconstruction buffer to avoid having to zero-stuff
reconstructed[n * settings.downsamplingFactor] = quantized;
stuffed[n * settings.downsamplingFactor] = quantized * settings.downsamplingFactor;
// record the quantization error
quantNoise[n] = quantized - y;
quantNoiseStuffed[n * settings.downsamplingFactor] = quantNoise[n];
});
// render reconstructed wave by low pass filtering the zero stuffed array----
// specify filter parameters; as before, the cutoff is set to the Nyquist
var filterCoeffs = firCalculator.lowpass(
{ order: 200
, Fs: WEBAUDIO_MAX_SAMPLERATE
, Fc: (WEBAUDIO_MAX_SAMPLERATE / settings.downsamplingFactor) / 2
});
// generate the filter
var filter = new Fili.FirFilter(filterCoeffs);
// apply the filter
reconstructed.forEach( (x, n, arr) => {
let y = filter.singleStep(x);
// To retain the correct amplitude, we must multiply the output of the
// filter by the downsampling factor.
arr[n] = y * settings.downsamplingFactor;
});
// time shift the signal by half the filter order to compensate for the delay
// introduced by the FIR filter
reconstructed.forEach( (x, n, arr) => arr[n - 100] = x );
// render FFTs --------------------------------------------------------------
// TODO: apply windows?
// The FFTs of the signals at the various stages of the process are generated
// using fft.js (https://github.com/indutny/fft.js). The call to
// `realTransform()` performs the FFT, and the call to `completeSpectrum`
// fills the upper half of the spectrum, which is otherwise not calculated
// since it is a redundant reflection of the lower half of the spectrum.
if (simulation) {
fft.realTransform(settings.originalFreq, original);
fft.completeSpectrum(settings.originalFreq);
fft.realTransform(settings.stuffedFreq, stuffed)
fft.completeSpectrum(settings.reconstructedFreq);
fft.realTransform(settings.reconstructedFreq, reconstructed)
fft.completeSpectrum(settings.reconstructedFreq);
fft.realTransform(settings.quantNoiseFreq, quantNoiseStuffed)
fft.completeSpectrum(settings.quantNoiseFreq);
}
// fade in and out and suppress clipping distortions ------------------------
// Audio output is windowed to prevent pops. The envelope is a simple linear
// ramp up at the beginning and linear ramp down at the end.
if (playback) {
// This normalization makes sure the original signal isn't clipped.
// The output is clipped during the simulation, so this may reduce its peak
// amplitude a bit, but since the clipping adds distortion the perceived
// loudness is relatively the same as the original signal in my testing.
let normalize = settings.amplitude > 1.0 ? settings.amplitude : 1.0;
// Define the fade function
let fade = (_, n, arr) => {
let fadeTimeSamps = Math.min(fadeTimeSeconds * WEBAUDIO_MAX_SAMPLERATE, arr.length / 2);
// The conditional ensures there is a fade even if the fade time is longer than the signal
if (n < fadeTimeSamps)
arr[n] = (n / fadeTimeSamps) * arr[n] / normalize;
else if (n > arr.length - fadeTimeSamps)
arr[n] = ((arr.length - n) / fadeTimeSamps) * arr[n] / normalize;
else arr[n] = arr[n] / normalize;
};
// Apply the fade function
original.forEach(fade);
reconstructed.forEach(fade);
quantNoise.forEach(fade);
}
}}
/**/