Chris@0: /* 
Chris@0:  * Free FFT and convolution (JavaScript)
Chris@0:  * 
Chris@0:  * Copyright (c) 2014 Project Nayuki
Chris@0:  * http://www.nayuki.io/page/free-small-fft-in-multiple-languages
Chris@0:  * 
Chris@0:  * (MIT License)
Chris@0:  * Permission is hereby granted, free of charge, to any person obtaining a copy of
Chris@0:  * this software and associated documentation files (the "Software"), to deal in
Chris@0:  * the Software without restriction, including without limitation the rights to
Chris@0:  * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
Chris@0:  * the Software, and to permit persons to whom the Software is furnished to do so,
Chris@0:  * subject to the following conditions:
Chris@0:  * - The above copyright notice and this permission notice shall be included in
Chris@0:  *   all copies or substantial portions of the Software.
Chris@0:  * - The Software is provided "as is", without warranty of any kind, express or
Chris@0:  *   implied, including but not limited to the warranties of merchantability,
Chris@0:  *   fitness for a particular purpose and noninfringement. In no event shall the
Chris@0:  *   authors or copyright holders be liable for any claim, damages or other
Chris@0:  *   liability, whether in an action of contract, tort or otherwise, arising from,
Chris@0:  *   out of or in connection with the Software or the use or other dealings in the
Chris@0:  *   Software.
Chris@0:  */
Chris@0: 
Chris@0: "use strict";
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
Chris@0:  * The vector can have any length. This is a wrapper function.
Chris@0:  */
Chris@0: function transform(real, imag) {
Chris@0:     if (real.length != imag.length)
Chris@0:         throw "Mismatched lengths";
Chris@0:     
Chris@0:     var n = real.length;
Chris@0:     if (n == 0)
Chris@0:         return;
Chris@0:     else if ((n & (n - 1)) == 0)  // Is power of 2
Chris@0:         transformRadix2(real, imag);
Chris@0:     else  // More complicated algorithm for arbitrary sizes
Chris@0:         transformBluestein(real, imag);
Chris@0: }
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
Chris@0:  * 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.
Chris@0:  */
Chris@0: function inverseTransform(real, imag) {
Chris@0:     transform(imag, real);
Chris@0: }
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
Chris@0:  * The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
Chris@0:  */
Chris@0: function transformRadix2(real, imag) {
Chris@0:     // Initialization
Chris@0:     if (real.length != imag.length)
Chris@0:         throw "Mismatched lengths";
Chris@0:     var n = real.length;
Chris@0:     if (n == 1)  // Trivial transform
Chris@0:         return;
Chris@0:     var levels = -1;
Chris@0:     for (var i = 0; i < 32; i++) {
Chris@0:         if (1 << i == n)
Chris@0:             levels = i;  // Equal to log2(n)
Chris@0:     }
Chris@0:     if (levels == -1)
Chris@0:         throw "Length is not a power of 2";
Chris@0:     var cosTable = new Array(n / 2);
Chris@0:     var sinTable = new Array(n / 2);
Chris@0:     for (var i = 0; i < n / 2; i++) {
Chris@0:         cosTable[i] = Math.cos(2 * Math.PI * i / n);
Chris@0:         sinTable[i] = Math.sin(2 * Math.PI * i / n);
Chris@0:     }
Chris@0:     
Chris@0:     // Bit-reversed addressing permutation
Chris@0:     for (var i = 0; i < n; i++) {
Chris@0:         var j = reverseBits(i, levels);
Chris@0:         if (j > i) {
Chris@0:             var temp = real[i];
Chris@0:             real[i] = real[j];
Chris@0:             real[j] = temp;
Chris@0:             temp = imag[i];
Chris@0:             imag[i] = imag[j];
Chris@0:             imag[j] = temp;
Chris@0:         }
Chris@0:     }
Chris@0:     
Chris@0:     // Cooley-Tukey decimation-in-time radix-2 FFT
Chris@0:     for (var size = 2; size <= n; size *= 2) {
Chris@0:         var halfsize = size / 2;
Chris@0:         var tablestep = n / size;
Chris@0:         for (var i = 0; i < n; i += size) {
Chris@0:             for (var j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
Chris@0:                 var tpre =  real[j+halfsize] * cosTable[k] + imag[j+halfsize] * sinTable[k];
Chris@0:                 var tpim = -real[j+halfsize] * sinTable[k] + imag[j+halfsize] * cosTable[k];
Chris@0:                 real[j + halfsize] = real[j] - tpre;
Chris@0:                 imag[j + halfsize] = imag[j] - tpim;
Chris@0:                 real[j] += tpre;
Chris@0:                 imag[j] += tpim;
Chris@0:             }
Chris@0:         }
Chris@0:     }
Chris@0:     
Chris@0:     // Returns the integer whose value is the reverse of the lowest 'bits' bits of the integer 'x'.
Chris@0:     function reverseBits(x, bits) {
Chris@0:         var y = 0;
Chris@0:         for (var i = 0; i < bits; i++) {
Chris@0:             y = (y << 1) | (x & 1);
Chris@0:             x >>>= 1;
Chris@0:         }
Chris@0:         return y;
Chris@0:     }
Chris@0: }
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
Chris@0:  * The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
Chris@0:  * Uses Bluestein's chirp z-transform algorithm.
Chris@0:  */
Chris@0: function transformBluestein(real, imag) {
Chris@0:     // Find a power-of-2 convolution length m such that m >= n * 2 + 1
Chris@0:     if (real.length != imag.length)
Chris@0:         throw "Mismatched lengths";
Chris@0:     var n = real.length;
Chris@0:     var m = 1;
Chris@0:     while (m < n * 2 + 1)
Chris@0:         m *= 2;
Chris@0:     
Chris@0:     // Trignometric tables
Chris@0:     var cosTable = new Array(n);
Chris@0:     var sinTable = new Array(n);
Chris@0:     for (var i = 0; i < n; i++) {
Chris@0:         var j = i * i % (n * 2);  // This is more accurate than j = i * i
Chris@0:         cosTable[i] = Math.cos(Math.PI * j / n);
Chris@0:         sinTable[i] = Math.sin(Math.PI * j / n);
Chris@0:     }
Chris@0:     
Chris@0:     // Temporary vectors and preprocessing
Chris@0:     var areal = new Array(m);
Chris@0:     var aimag = new Array(m);
Chris@0:     for (var i = 0; i < n; i++) {
Chris@0:         areal[i] =  real[i] * cosTable[i] + imag[i] * sinTable[i];
Chris@0:         aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
Chris@0:     }
Chris@0:     for (var i = n; i < m; i++)
Chris@0:         areal[i] = aimag[i] = 0;
Chris@0:     var breal = new Array(m);
Chris@0:     var bimag = new Array(m);
Chris@0:     breal[0] = cosTable[0];
Chris@0:     bimag[0] = sinTable[0];
Chris@0:     for (var i = 1; i < n; i++) {
Chris@0:         breal[i] = breal[m - i] = cosTable[i];
Chris@0:         bimag[i] = bimag[m - i] = sinTable[i];
Chris@0:     }
Chris@0:     for (var i = n; i <= m - n; i++)
Chris@0:         breal[i] = bimag[i] = 0;
Chris@0:     
Chris@0:     // Convolution
Chris@0:     var creal = new Array(m);
Chris@0:     var cimag = new Array(m);
Chris@0:     convolveComplex(areal, aimag, breal, bimag, creal, cimag);
Chris@0:     
Chris@0:     // Postprocessing
Chris@0:     for (var i = 0; i < n; i++) {
Chris@0:         real[i] =  creal[i] * cosTable[i] + cimag[i] * sinTable[i];
Chris@0:         imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
Chris@0:     }
Chris@0: }
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the circular convolution of the given real vectors. Each vector's length must be the same.
Chris@0:  */
Chris@0: function convolveReal(x, y, out) {
Chris@0:     if (x.length != y.length || x.length != out.length)
Chris@0:         throw "Mismatched lengths";
Chris@0:     var zeros = new Array(x.length);
Chris@0:     for (var i = 0; i < zeros.length; i++)
Chris@0:         zeros[i] = 0;
Chris@0:     convolveComplex(x, zeros, y, zeros.slice(0), out, zeros.slice(0));
Chris@0: }
Chris@0: 
Chris@0: 
Chris@0: /* 
Chris@0:  * Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
Chris@0:  */
Chris@0: function convolveComplex(xreal, ximag, yreal, yimag, outreal, outimag) {
Chris@0:     if (xreal.length != ximag.length || xreal.length != yreal.length || yreal.length != yimag.length || xreal.length != outreal.length || outreal.length != outimag.length)
Chris@0:         throw "Mismatched lengths";
Chris@0:     
Chris@0:     var n = xreal.length;
Chris@0:     xreal = xreal.slice(0);
Chris@0:     ximag = ximag.slice(0);
Chris@0:     yreal = yreal.slice(0);
Chris@0:     yimag = yimag.slice(0);
Chris@0:     
Chris@0:     transform(xreal, ximag);
Chris@0:     transform(yreal, yimag);
Chris@0:     for (var i = 0; i < n; i++) {
Chris@0:         var temp = xreal[i] * yreal[i] - ximag[i] * yimag[i];
Chris@0:         ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
Chris@0:         xreal[i] = temp;
Chris@0:     }
Chris@0:     inverseTransform(xreal, ximag);
Chris@0:     for (var i = 0; i < n; i++) {  // Scaling (because this FFT implementation omits it)
Chris@0:         outreal[i] = xreal[i] / n;
Chris@0:         outimag[i] = ximag[i] / n;
Chris@0:     }
Chris@0: }