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.
Here is the source code of the Java Program to Compute Discrete Fourier Transform Using the Fast Fourier Transform Approach. The Java program is successfully compiled and run on a Windows system. The program output is also shown below.
// 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)"); } } }
Output:
$ javac FFT.java $ java FFT 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
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