Java tutorial
/** * Copyright (C) 2016 Hurence (support@hurence.com) * * Licensed under the Apache License, Version 2.0 (the "License"); * you may not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ /* * To change this template, choose Tools | Templates * and open the template in the editor. */ package com.hurence.tmp; import org.apache.commons.math3.complex.Complex; /** * *********************************************************************** * Compilation: javac FFT.java Execution: java FFT N Dependencies: Complex.java * * Compute the FFT and inverse FFT of a length N complex sequence. Bare bones * implementation that runs in O(N log N) time. Our goal is to optimize the * clarity of the code, rather than performance. * * Limitations ----------- - assumes N is a power of 2 * * - not the most memory efficient algorithm (because it uses an object type for * representing complex numbers and because it re-allocates memory for the * subarray, instead of doing in-place or reusing a single temporary array) * ************************************************************************ */ public class FFT { // compute the FFT of x[], assuming its length is a power of 2 public static Complex[] fft(Complex[] x) { int N = x.length; // base case if (N == 1) { return new Complex[] { x[0] }; } // radix 2 Cooley-Tukey FFT if (N % 2 != 0) { throw new RuntimeException("N is not a power of 2"); } // fft of even terms Complex[] even = new Complex[N / 2]; for (int k = 0; k < N / 2; k++) { even[k] = x[2 * k]; } Complex[] q = fft(even); // fft of odd terms Complex[] odd = even; // reuse the array for (int k = 0; k < N / 2; k++) { odd[k] = x[2 * k + 1]; } Complex[] r = fft(odd); // combine Complex[] y = new Complex[N]; for (int k = 0; k < N / 2; k++) { double kth = -2 * k * Math.PI / N; Complex wk = new Complex(Math.cos(kth), Math.sin(kth)); y[k] = q[k].add(wk.multiply(r[k])); y[k + N / 2] = q[k].subtract(wk.multiply(r[k])); } return y; } // compute the inverse FFT of x[], assuming its length is a power of 2 public static Complex[] ifft(Complex[] x) { int N = x.length; Complex[] y = new Complex[N]; // take conjugate for (int i = 0; i < N; i++) { y[i] = x[i].conjugate(); } // compute forward FFT y = fft(y); // take conjugate again for (int i = 0; i < N; i++) { y[i] = y[i].conjugate(); } // divide by N for (int i = 0; i < N; i++) { y[i] = y[i].multiply(1.0 / N); } return y; } // compute the circular convolution of x and y public static Complex[] cconvolve(Complex[] x, Complex[] y) { // should probably pad x and y with 0s so that they have same length // and are powers of 2 if (x.length != y.length) { throw new RuntimeException("Dimensions don't agree"); } int N = x.length; // compute FFT of each sequence Complex[] a = fft(x); Complex[] b = fft(y); // point-wise multiply Complex[] c = new Complex[N]; for (int i = 0; i < N; i++) { c[i] = a[i].multiply(b[i]); } // compute inverse FFT return ifft(c); } // compute the linear convolution of x and y public static Complex[] convolve(Complex[] x, Complex[] y) { Complex ZERO = new Complex(0, 0); Complex[] a = new Complex[2 * x.length]; for (int i = 0; i < x.length; i++) { a[i] = x[i]; } for (int i = x.length; i < 2 * x.length; i++) { a[i] = ZERO; } Complex[] b = new Complex[2 * y.length]; for (int i = 0; i < y.length; i++) { b[i] = y[i]; } for (int i = y.length; i < 2 * y.length; i++) { b[i] = ZERO; } return cconvolve(a, b); } // display an array of Complex numbers to standard output public static void show(Complex[] x, String title) { System.out.println(title); System.out.println("-------------------"); for (int i = 0; i < x.length; i++) { System.out.println(x[i].getReal() + " " + x[i].getImaginary() + " i"); } System.out.println(); } /** * ******************************************************************* * Test client and sample execution * * % java FFT 4 x ------------------- -0.03480425839330703 * 0.07910192950176387 0.7233322451735928 0.1659819820667019 * * y = fft(x) ------------------- 0.9336118983487516 -0.7581365035668999 + * 0.08688005256493803i 0.44344407521182005 -0.7581365035668999 - * 0.08688005256493803i * * z = ifft(y) ------------------- -0.03480425839330703 0.07910192950176387 * + 2.6599344570851287E-18i 0.7233322451735928 0.1659819820667019 - * 2.6599344570851287E-18i * * c = cconvolve(x, x) ------------------- 0.5506798633981853 * 0.23461407150576394 - 4.033186818023279E-18i -0.016542951108772352 * 0.10288019294318276 + 4.033186818023279E-18i * * d = convolve(x, x) ------------------- 0.001211336402308083 - * 3.122502256758253E-17i -0.005506167987577068 - 5.058885073636224E-17i * -0.044092969479563274 + 2.1934338938072244E-18i 0.10288019294318276 - * 3.6147323062478115E-17i 0.5494685269958772 + 3.122502256758253E-17i * 0.240120239493341 + 4.655566391833896E-17i 0.02755001837079092 - * 2.1934338938072244E-18i 4.01805098805014E-17i * ******************************************************************** */ public static void main(String[] args) { int N = 4; //Integer.parseInt(args[0]); Complex[] x = new Complex[N]; // original data for (int i = 0; i < N; i++) { x[i] = new Complex(i, 0); x[i] = new Complex(-2 * Math.random() + 1, 0); } show(x, "x"); // FFT of original data Complex[] y = fft(x); show(y, "y = fft(x)"); // take inverse FFT Complex[] z = ifft(y); show(z, "z = ifft(y)"); // circular convolution of x with itself Complex[] c = cconvolve(x, x); show(c, "c = cconvolve(x, x)"); // linear convolution of x with itself Complex[] d = convolve(x, x); show(d, "d = convolve(x, x)"); } }