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418 lines
14 KiB
JavaScript
418 lines
14 KiB
JavaScript
/*
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* Free FFT and convolution (JavaScript)
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*
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* Copyright (c) 2014 Project Nayuki
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* https://www.nayuki.io/page/free-small-fft-in-multiple-languages
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*
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* (MIT License)
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* Permission is hereby granted, free of charge, to any person obtaining a copy of
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* this software and associated documentation files (the "Software"), to deal in
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* the Software without restriction, including without limitation the rights to
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* use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
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* the Software, and to permit persons to whom the Software is furnished to do so,
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* subject to the following conditions:
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* - The above copyright notice and this permission notice shall be included in
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* all copies or substantial portions of the Software.
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* - The Software is provided "as is", without warranty of any kind, express or
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* implied, including but not limited to the warranties of merchantability,
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* fitness for a particular purpose and noninfringement. In no event shall the
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* authors or copyright holders be liable for any claim, damages or other
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* liability, whether in an action of contract, tort or otherwise, arising from,
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* out of or in connection with the Software or the use or other dealings in the
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* Software.
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*/
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"use strict";
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/* https://github.com/stanford-futuredata/ASAP/blob/master/ASAP-optimized.js */
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/*
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* Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
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* The vector can have any length. This is a wrapper function.
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*/
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function transform(real, imag) {
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if (real.length != imag.length)
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throw "Mismatched lengths";
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var n = real.length;
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if (n == 0)
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return;
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else if ((n & (n - 1)) == 0) // Is power of 2
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transformRadix2(real, imag);
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else // More complicated algorithm for arbitrary sizes
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transformBluestein(real, imag);
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}
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/*
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* Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
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* The vector can have any length. This is a wrapper function. This transform does not perform scaling, so the inverse is not a true inverse.
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*/
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function inverseTransform(real, imag) {
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transform(imag, real);
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}
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/*
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* Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
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* The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
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*/
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function transformRadix2(real, imag) {
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// Initialization
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if (real.length != imag.length)
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throw "Mismatched lengths";
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var n = real.length;
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if (n == 1) // Trivial transform
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return;
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var levels = -1;
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for (var i = 0; i < 32; i++) {
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if (1 << i == n)
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levels = i; // Equal to log2(n)
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}
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if (levels == -1)
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throw "Length is not a power of 2";
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var cosTable = new Array(n / 2);
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var sinTable = new Array(n / 2);
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for (var i = 0; i < n / 2; i++) {
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cosTable[i] = Math.cos(2 * Math.PI * i / n);
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sinTable[i] = Math.sin(2 * Math.PI * i / n);
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}
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// Bit-reversed addressing permutation
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for (var i = 0; i < n; i++) {
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var j = reverseBits(i, levels);
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if (j > i) {
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var temp = real[i];
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real[i] = real[j];
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real[j] = temp;
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temp = imag[i];
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imag[i] = imag[j];
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imag[j] = temp;
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}
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}
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// Cooley-Tukey decimation-in-time radix-2 FFT
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for (var size = 2; size <= n; size *= 2) {
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var halfsize = size / 2;
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var tablestep = n / size;
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for (var i = 0; i < n; i += size) {
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for (var j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
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var tpre = real[j+halfsize] * cosTable[k] + imag[j+halfsize] * sinTable[k];
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var tpim = -real[j+halfsize] * sinTable[k] + imag[j+halfsize] * cosTable[k];
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real[j + halfsize] = real[j] - tpre;
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imag[j + halfsize] = imag[j] - tpim;
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real[j] += tpre;
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imag[j] += tpim;
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}
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}
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}
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// Returns the integer whose value is the reverse of the lowest 'bits' bits of the integer 'x'.
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function reverseBits(x, bits) {
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var y = 0;
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for (var i = 0; i < bits; i++) {
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y = (y << 1) | (x & 1);
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x >>>= 1;
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}
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return y;
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}
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}
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/*
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* Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
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* The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
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* Uses Bluestein's chirp z-transform algorithm.
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*/
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function transformBluestein(real, imag) {
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// Find a power-of-2 convolution length m such that m >= n * 2 + 1
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if (real.length != imag.length)
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throw "Mismatched lengths";
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var n = real.length;
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var m = 1;
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while (m < n * 2 + 1)
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m *= 2;
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// Trignometric tables
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var cosTable = new Array(n);
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var sinTable = new Array(n);
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for (var i = 0; i < n; i++) {
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var j = i * i % (n * 2); // This is more accurate than j = i * i
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cosTable[i] = Math.cos(Math.PI * j / n);
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sinTable[i] = Math.sin(Math.PI * j / n);
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}
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// Temporary vectors and preprocessing
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var areal = new Array(m);
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var aimag = new Array(m);
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for (var i = 0; i < n; i++) {
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areal[i] = real[i] * cosTable[i] + imag[i] * sinTable[i];
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aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
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}
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for (var i = n; i < m; i++)
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areal[i] = aimag[i] = 0;
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var breal = new Array(m);
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var bimag = new Array(m);
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breal[0] = cosTable[0];
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bimag[0] = sinTable[0];
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for (var i = 1; i < n; i++) {
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breal[i] = breal[m - i] = cosTable[i];
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bimag[i] = bimag[m - i] = sinTable[i];
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}
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for (var i = n; i <= m - n; i++)
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breal[i] = bimag[i] = 0;
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// Convolution
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var creal = new Array(m);
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var cimag = new Array(m);
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convolveComplex(areal, aimag, breal, bimag, creal, cimag);
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// Postprocessing
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for (var i = 0; i < n; i++) {
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real[i] = creal[i] * cosTable[i] + cimag[i] * sinTable[i];
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imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
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}
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}
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/*
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* Computes the circular convolution of the given real vectors. Each vector's length must be the same.
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*/
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function convolveReal(x, y, out) {
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if (x.length != y.length || x.length != out.length)
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throw "Mismatched lengths";
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var zeros = new Array(x.length);
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for (var i = 0; i < zeros.length; i++)
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zeros[i] = 0;
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convolveComplex(x, zeros, y, zeros.slice(), out, zeros.slice());
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}
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/*
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* Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
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*/
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function convolveComplex(xreal, ximag, yreal, yimag, outreal, outimag) {
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if (xreal.length != ximag.length || xreal.length != yreal.length || yreal.length != yimag.length || xreal.length != outreal.length || outreal.length != outimag.length)
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throw "Mismatched lengths";
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var n = xreal.length;
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xreal = xreal.slice();
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ximag = ximag.slice();
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yreal = yreal.slice();
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yimag = yimag.slice();
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transform(xreal, ximag);
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transform(yreal, yimag);
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for (var i = 0; i < n; i++) {
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var temp = xreal[i] * yreal[i] - ximag[i] * yimag[i];
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ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
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xreal[i] = temp;
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}
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inverseTransform(xreal, ximag);
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for (var i = 0; i < n; i++) { // Scaling (because this FFT implementation omits it)
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outreal[i] = xreal[i] / n;
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outimag[i] = ximag[i] / n;
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}
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}
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/**********************************************************************************/
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function binarySearch(head, tail, data, minObj, originalKurt, windowSize) {
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while (head <= tail) {
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var w = Math.round((head + tail) / 2);
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var smoothed = SMA(data, w, 1);
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var metrics = new Metrics(smoothed);
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if (metrics.kurtosis() >= originalKurt) { /* Search second half if feasible */
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var roughness = metrics.roughness();
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if (roughness < minObj) {
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windowSize = w;
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minObj = roughness;
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}
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head = w + 1;
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} else { /* Search first half */
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tail = w - 1;
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}
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}
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return windowSize;
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}
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function smooth(data, resolution) {
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/* Ignore the last value if it's NaN */
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if (isNaN(data[data.length-1])) {
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data = data.slice(0, -1);
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}
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if (data.length >= 2 * resolution ) {
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data = SMA(data, Math.trunc(data.length / resolution),
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Math.trunc(data.length / resolution));
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}
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var acf = new ACF(data, Math.round(data.length / 10));
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var peaks = acf.findPeaks();
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var metrics = new Metrics(data);
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var originalKurt = metrics.kurtosis();
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var minObj = metrics.roughness();
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var windowSize = 1;
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var lb = 1;
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var largestFeasible = -1;
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var tail = data.length / 10;
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for (var i = peaks.length - 1; i >= 0; i -=1) {
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var w = peaks[i];
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if (w < lb || w == 1) {
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break;
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} else if (Math.sqrt(1 - acf.correlations[w]) * windowSize >
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Math.sqrt(1 - acf.correlations[windowSize]) * w) {
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continue;
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}
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var smoothed = SMA(data, w, 1);
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metrics = new Metrics(smoothed);
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var roughness = metrics.roughness();
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if (metrics.kurtosis() >= originalKurt) {
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if (roughness < minObj) {
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minObj = roughness;
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windowSize = w;
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}
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lb = Math.round(Math.max(w * Math.sqrt((acf.maxACF - 1) / (acf.correlations[w] - 1)), lb));
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if (largestFeasible < 0) { largestFeasible = i; }
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}
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}
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if (largestFeasible > 0) {
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if (largestFeasible < peaks.length - 2) { tail = peaks[largestFeasible + 1]; }
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lb = Math.max(lb, peaks[largestFeasible] + 1);
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}
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windowSize = binarySearch(lb, tail, data, minObj, originalKurt, windowSize);
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return SMA(data, windowSize, 1);
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}
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function SMA(data, range, slide) {
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var windowStart = 0;
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var sum = 0;
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var count = 0;
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var values = [];
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for (var i = 0; i < data.length; i ++) {
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if (isNaN(data[i])) { data[i] = 0; }
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if (i - windowStart >= range) {
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values.push(sum / count);
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var oldStart = windowStart;
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while (windowStart < data.length && windowStart - oldStart < slide) {
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sum -= data[windowStart];
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count -= 1;
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windowStart += 1;
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}
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}
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sum += data[i];
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count += 1;
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}
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if (count == range) {
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values.push(sum / count);
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}
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return values;
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}
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function ACF(values, maxLag) {
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this.mean = Metrics.mean(values);
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this.values = values;
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this.correlations = new Array(maxLag);
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this.CORR_THRESH = 0.2;
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this.maxACF = 0;
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this.calculate();
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}
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ACF.prototype.calculate = function() {
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/* Padding to the closest power of 2 */
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var len = Math.pow(2, Math.trunc(Math.log2(this.values.length)) + 1)
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var fftreal = new Array(len).fill(0);
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var fftimg = new Array(len).fill(0);
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for (var i = 0; i < this.values.length; i += 1) {
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fftreal[i] = this.values[i] - this.mean;
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}
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/* F_R(f) = FFT(X) */
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transform(fftreal, fftimg);
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/* S(f) = F_R(f)F_R*(f) */
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for (var i = 0; i < fftreal.length; i += 1) {
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fftreal[i] = Math.pow(fftreal[i], 2) + Math.pow(fftimg[i], 2);
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fftimg[i] = 0;
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}
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/* R(t) = IFFT(S(f)) */
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inverseTransform(fftreal, fftimg);
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for (var i = 1; i < this.correlations.length; i += 1) {
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this.correlations[i] = fftreal[i] / fftreal[0];
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}
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}
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ACF.prototype.findPeaks = function() {
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var peakIndices = [];
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if (this.correlations.length > 1) {
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var positive = this.correlations[1] > this.correlations[0];
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var max = 1;
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for (var i = 2; i < this.correlations.length; i += 1) {
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if (!positive && this.correlations[i] > this.correlations[i - 1]) {
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max = i;
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positive = !positive;
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} else if (positive && this.correlations[i] > this.correlations[max]) {
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max = i;
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} else if (positive && this.correlations[i] < this.correlations[i - 1]) {
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if (max > 1 && this.correlations[max] > this.CORR_THRESH) {
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peakIndices.push(max);
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if (this.correlations[max] > this.maxACF) {
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this.maxACF = this.correlations[max];
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}
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}
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positive = !positive;
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}
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}
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}
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/* If there is no autocorrelation peak within the MAX_WINDOW boundary,
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# try windows from the largest to the smallest */
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if (peakIndices.length <= 1) {
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for (var i = 2; i < this.correlations.length; i += 1) {
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peakIndices.push(i);
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}
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}
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return peakIndices;
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}
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function Metrics(values) {
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this.len = values.length;
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this.values = values;
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this.m = Metrics.mean(values);
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}
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Metrics.mean = function(values) {
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var m = 0;
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for (var i = 0; i < values.length; i += 1) {
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m += values[i];
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}
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return m / values.length;
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}
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Metrics.std = function(values) {
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var m = Metrics.mean(values);
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var std = 0;
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for (var i = 0; i < values.length; i += 1) {
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std += Math.pow((values[i] - m), 2);
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}
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return Math.sqrt(std / values.length);
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}
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Metrics.prototype.kurtosis = function() {
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var u4 = 0, variance = 0;
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for (var i = 0; i < this.len; i ++) {
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u4 += Math.pow((this.values[i] - this.m), 4);
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variance += Math.pow((this.values[i] - this.m), 2);
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}
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return this.len * u4 / Math.pow(variance, 2);
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}
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Metrics.prototype.roughness = function() {
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return Metrics.std(this.diffs());
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}
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Metrics.prototype.diffs = function() {
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var diff = new Array(this.len - 1);
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for (var i = 1; i < this.len; i += 1) {
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diff[i - 1] = this.values[i] - this.values[i - 1];
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}
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return diff;
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}
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