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- package com.karma.freqsensor;
- /*************************************************************************
- * 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].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;
- }
- // 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].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);
- }
- // 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]);
- }
- 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 = 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)");
- }
- }
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