fft.js

/* 
 * Free FFT and convolution (JavaScript)
 * 
 * Copyright (c) 2014 Project Nayuki
 * http://www.nayuki.io/page/free-small-fft-in-multiple-languages
 * 
 * (MIT License)
 * Permission is hereby granted, free of charge, to any person obtaining a copy of
 * this software and associated documentation files (the "Software"), to deal in
 * the Software without restriction, including without limitation the rights to
 * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
 * the Software, and to permit persons to whom the Software is furnished to do so,
 * subject to the following conditions:
 * - The above copyright notice and this permission notice shall be included in
 *   all copies or substantial portions of the Software.
 * - The Software is provided "as is", without warranty of any kind, express or
 *   implied, including but not limited to the warranties of merchantability,
 *   fitness for a particular purpose and noninfringement. In no event shall the
 *   authors or copyright holders be liable for any claim, damages or other
 *   liability, whether in an action of contract, tort or otherwise, arising from,
 *   out of or in connection with the Software or the use or other dealings in the
 *   Software.
 */

"use strict";


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This is a wrapper function.
 */
function transform(real, imag) {
    if (imag != undefined && real.length != imag.length && imag.length != 0)
        throw "Mismatched lengths";
    
    var n = real.length;
    if (n == 0)
        return;
    else if ((n & (n - 1)) == 0)  // Is power of 2
        transformRadix2(real, imag);
    else  // More complicated algorithm for arbitrary sizes
        transformBluestein(real, imag);
}


/* 
 * Computes the inverse discrete Fourier transform (IDFT) of the given complex vector, storing the result back into the vector.
 * 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.
 */
function inverseTransform(real, imag) {
    transform(imag, real);
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector's length must be a power of 2. Uses the Cooley-Tukey decimation-in-time radix-2 algorithm.
 */
function transformRadix2(real, imag) {
    // Initialization
    if (imag != undefined && real.length != imag.length && imag.length != 0)
        throw "Mismatched lengths";
    var n = real.length;
    if (n == 1)  // Trivial transform
        return;
    var levels = -1;
    for (var i = 0; i < 32; i++) {
        if (1 << i == n)
            levels = i;  // Equal to log2(n)
    }
    if (levels == -1)
        throw "Length is not a power of 2";
    var cosTable = new Array(n / 2);
    var sinTable = new Array(n / 2);
    for (var i = 0; i < n / 2; i++) {
        cosTable[i] = Math.cos(2 * Math.PI * i / n);
        sinTable[i] = Math.sin(2 * Math.PI * i / n);
    }
    
    // Bit-reversed addressing permutation
    for (var i = 0; i < n; i++) {
        var j = reverseBits(i, levels);
        if (j > i) {
            var temp = real[i];
            real[i] = real[j];
            real[j] = temp;
            temp = imag[i];
            imag[i] = imag[j];
            imag[j] = temp;
        }
    }
    
    // Cooley-Tukey decimation-in-time radix-2 FFT
    for (var size = 2; size <= n; size *= 2) {
        var halfsize = size / 2;
        var tablestep = n / size;
        for (var i = 0; i < n; i += size) {
            for (var j = i, k = 0; j < i + halfsize; j++, k += tablestep) {
                imag[j+halfsize] = (imag[j+halfsize]==undefined)? 0 : imag[j+halfsize];
                imag[j] = (imag[j]==undefined)? 0 : imag[j];
                var tpre =  real[j+halfsize] * cosTable[k] + imag[j+halfsize] * sinTable[k];
                var tpim = -real[j+halfsize] * sinTable[k] + imag[j+halfsize] * cosTable[k];
                real[j + halfsize] = real[j] - tpre;
                imag[j + halfsize] = imag[j] - tpim;
                real[j] += tpre;
                imag[j] += tpim;
            }
        }
    }
    
    // Returns the integer whose value is the reverse of the lowest 'bits' bits of the integer 'x'.
    function reverseBits(x, bits) {
        var y = 0;
        for (var i = 0; i < bits; i++) {
            y = (y << 1) | (x & 1);
            x >>>= 1;
        }
        return y;
    }
}


/* 
 * Computes the discrete Fourier transform (DFT) of the given complex vector, storing the result back into the vector.
 * The vector can have any length. This requires the convolution function, which in turn requires the radix-2 FFT function.
 * Uses Bluestein's chirp z-transform algorithm.
 */
function transformBluestein(real, imag) {
    // Find a power-of-2 convolution length m such that m >= n * 2 + 1
    if (real.length != imag.length)
        throw "Mismatched lengths";
    var n = real.length;
    var m = 1;
    while (m < n * 2 + 1)
        m *= 2;
    
    // Trignometric tables
    var cosTable = new Array(n);
    var sinTable = new Array(n);
    for (var i = 0; i < n; i++) {
        var j = i * i % (n * 2);  // This is more accurate than j = i * i
        cosTable[i] = Math.cos(Math.PI * j / n);
        sinTable[i] = Math.sin(Math.PI * j / n);
    }
    
    // Temporary vectors and preprocessing
    var areal = new Array(m);
    var aimag = new Array(m);
    for (var i = 0; i < n; i++) {
        areal[i] =  real[i] * cosTable[i] + imag[i] * sinTable[i];
        aimag[i] = -real[i] * sinTable[i] + imag[i] * cosTable[i];
    }
    for (var i = n; i < m; i++)
        areal[i] = aimag[i] = 0;
    var breal = new Array(m);
    var bimag = new Array(m);
    breal[0] = cosTable[0];
    bimag[0] = sinTable[0];
    for (var i = 1; i < n; i++) {
        breal[i] = breal[m - i] = cosTable[i];
        bimag[i] = bimag[m - i] = sinTable[i];
    }
    for (var i = n; i <= m - n; i++)
        breal[i] = bimag[i] = 0;
    
    // Convolution
    var creal = new Array(m);
    var cimag = new Array(m);
    convolveComplex(areal, aimag, breal, bimag, creal, cimag);
    
    // Postprocessing
    for (var i = 0; i < n; i++) {
        real[i] =  creal[i] * cosTable[i] + cimag[i] * sinTable[i];
        imag[i] = -creal[i] * sinTable[i] + cimag[i] * cosTable[i];
    }
}


/* 
 * Computes the circular convolution of the given real vectors. Each vector's length must be the same.
 */
function convolveReal(x, y, out) {
    if (x.length != y.length || x.length != out.length)
        throw "Mismatched lengths";
    var zeros = new Array(x.length);
    for (var i = 0; i < zeros.length; i++)
        zeros[i] = 0;
    convolveComplex(x, zeros, y, zeros.slice(0), out, zeros.slice(0));
}


/* 
 * Computes the circular convolution of the given complex vectors. Each vector's length must be the same.
 */
function convolveComplex(xreal, ximag, yreal, yimag, outreal, outimag) {
    if (xreal.length != ximag.length || xreal.length != yreal.length || yreal.length != yimag.length || xreal.length != outreal.length || outreal.length != outimag.length)
        throw "Mismatched lengths";
    
    var n = xreal.length;
    xreal = xreal.slice(0);
    ximag = ximag.slice(0);
    yreal = yreal.slice(0);
    yimag = yimag.slice(0);
    
    transform(xreal, ximag);
    transform(yreal, yimag);
    for (var i = 0; i < n; i++) {
        var temp = xreal[i] * yreal[i] - ximag[i] * yimag[i];
        ximag[i] = ximag[i] * yreal[i] + xreal[i] * yimag[i];
        xreal[i] = temp;
    }
    inverseTransform(xreal, ximag);
    for (var i = 0; i < n; i++) {  // Scaling (because this FFT implementation omits it)
        outreal[i] = xreal[i] / n;
        outimag[i] = ximag[i] / n;
    }
}