Below is the syntax highlighted version of FFT.java
from §9.7 Optimization.
/****************************************************************************** * Compilation: javac FFT.java * Execution: java FFT n * Dependencies: Complex.java * * Compute the FFT and inverse FFT of a length n complex sequence * using the radix 2 Cooley-Tukey algorithm. * Bare bones implementation that runs in O(n log n) time and O(n) * space. Our goal is to optimize the clarity of the code, rather * than performance. * * This implementation uses the primitive root of unity w = e^(-2 pi i / n). * Some resources use w = e^(2 pi i / n). * * Reference: https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/05DivideAndConquerII.pdf * * 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) * * For an in-place radix 2 Cooley-Tukey FFT, see * https://introcs.cs.princeton.edu/java/97data/InplaceFFT.java.html * ******************************************************************************/ public class FFT { // compute the FFT of x[], assuming its length n 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 IllegalArgumentException("n is not a power of 2"); } // compute FFT of even terms Complex[] even = new Complex[n/2]; for (int k = 0; k < n/2; k++) { even[k] = x[2*k]; } Complex[] evenFFT = fft(even); // compute FFT of odd terms Complex[] odd = even; // reuse the array (to avoid n log n space) for (int k = 0; k < n/2; k++) { odd[k] = x[2*k + 1]; } Complex[] oddFFT = 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] = evenFFT[k].plus (wk.times(oddFFT[k])); y[k + n/2] = evenFFT[k].minus(wk.times(oddFFT[k])); } return y; } // compute the inverse FFT of x[], assuming its length n 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].scale(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 IllegalArgumentException("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].times(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); } // compute the DFT of x[] via brute force (n^2 time) public static Complex[] dft(Complex[] x) { int n = x.length; Complex ZERO = new Complex(0, 0); Complex[] y = new Complex[n]; for (int k = 0; k < n; k++) { y[k] = ZERO; for (int j = 0; j < n; j++) { int power = (k * j) % n; double kth = -2 * power * Math.PI / n; Complex wkj = new Complex(Math.cos(kth), Math.sin(kth)); y[k] = y[k].plus(x[j].times(wkj)); } } return y; } // display an array of Complex numbers to standard output public static void show(Complex[] x, String title) { StdOut.println(title); StdOut.println("-------------------"); for (int i = 0; i < x.length; i++) { StdOut.println(x[i]); } StdOut.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 = Integer.parseInt(args[0]); Complex[] x = new Complex[n]; // original data for (int i = 0; i < n; i++) { x[i] = new Complex(i, 0); } show(x, "x"); // FFT of original data Complex[] y = fft(x); show(y, "y = fft(x)"); // FFT of original data Complex[] y2 = dft(x); show(y2, "y2 = dft(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)"); } }