/* This is the java implementation of performing Discrete Fourier Transform using Fast Fourier Transform algorithm. This class finds the DFT of N (power of 2) complex elements, generated randomly, using FFT. Further verification is done by taking the Inverse Discrete Fourier Transform, again using FFT. */ // This is a sample program to perform DFT using FFT, FFT is performed on random input sequence public class FFT { public static class Complex { private final double re; // the real part private final double im; // the imaginary part // create a new object with the given real and imaginary parts public Complex(double real, double imag) { re = real; im = imag; } // return a string representation of the invoking Complex object public String toString() { if (im == 0) return re + ""; if (re == 0) return im + "i"; if (im < 0) return re + " - " + (-im) + "i"; return re + " + " + im + "i"; } // return abs/modulus/magnitude and angle/phase/argument public double abs() { return Math.hypot(re, im); } // Math.sqrt(re*re + im*im) public double phase() { return Math.atan2(im, re); } // between -pi and pi // return a new Complex object whose value is (this + b) public Complex plus(Complex b) { Complex a = this; // invoking object double real = a.re + b.re; double imag = a.im + b.im; return new Complex(real, imag); } // return a new Complex object whose value is (this - b) public Complex minus(Complex b) { Complex a = this; double real = a.re - b.re; double imag = a.im - b.im; return new Complex(real, imag); } // return a new Complex object whose value is (this * b) public Complex times(Complex b) { Complex a = this; double real = a.re * b.re - a.im * b.im; double imag = a.re * b.im + a.im * b.re; return new Complex(real, imag); } // scalar multiplication // return a new object whose value is (this * alpha) public Complex times(double alpha) { return new Complex(alpha * re, alpha * im); } // return a new Complex object whose value is the conjugate of this public Complex conjugate() { return new Complex(re, -im); } // return a new Complex object whose value is the reciprocal of this public Complex reciprocal() { double scale = re * re + im * im; return new Complex(re / scale, -im / scale); } // return the real or imaginary part public double re() { return re; } public double im() { return im; } // return a / b public Complex divides(Complex b) { Complex a = this; return a.times(b.reciprocal()); } // return a new Complex object whose value is the complex exponential of // this public Complex exp() { return new Complex(Math.exp(re) * Math.cos(im), Math.exp(re) * Math.sin(im)); } // return a new Complex object whose value is the complex sine of this public Complex sin() { return new Complex(Math.sin(re) * Math.cosh(im), Math.cos(re) * Math.sinh(im)); } // return a new Complex object whose value is the complex cosine of this public Complex cos() { return new Complex(Math.cos(re) * Math.cosh(im), -Math.sin(re) * Math.sinh(im)); } // return a new Complex object whose value is the complex tangent of // this public Complex tan() { return sin().divides(cos()); } // a static version of plus public static Complex plus(Complex a, Complex b) { double real = a.re + b.re; double imag = a.im + b.im; Complex sum = new Complex(real, imag); return sum; } // 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].plus(wk.times(r[k])); y[k + N / 2] = q[k].minus(wk.times(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].times(1.0 / N); } return y; } // display an array of Complex numbers to standard output public static void show(Complex[] x, String title) { System.out.println(title); for (int i = 0; i < x.length; i++) { System.out.println(x[i]); } System.out.println(); } public static void main(String[] args) { int N = 8;//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)"); } } } /* x 0.5568836254037923 0.8735842104393365 0.6099699812709252 0.5631502515566189 -0.518857260970139 -0.5946393148293805 0.47144753318047794 -0.3501597962417593 y = fft(x) 1.6113792298098721 1.4681239692650163 - 1.8225209872296184i -1.0433911500177497 - 0.06595444029509645i 0.6833578034828462 - 1.545476091048724i 0.6275085279602408 0.6833578034828462 + 1.545476091048724i -1.0433911500177497 + 0.06595444029509645i 1.4681239692650163 + 1.8225209872296184i z = ifft(y) 0.5568836254037923 0.8735842104393365 - 5.652078740871965E-17i 0.6099699812709252 - 4.24102681660054E-18i 0.5631502515566189 - 5.4501515053796015E-17i -0.518857260970139 -0.5946393148293805 + 5.4501515053796015E-17i 0.47144753318047794 + 4.24102681660054E-18i -0.3501597962417593 + 5.652078740871965E-17i