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- package com.feemers.android.fftdrawer.SignalProcessing;
-
-
- /******************************************************************************
- * 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;
- }
-
-
- /***************************************************************************
- * 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
- *
- ***************************************************************************/
-
- }
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