/******************************************************************************
* 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. 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)
*
* For an in-place radix 2 Cooley-Tukey FFT, see
* https://introcs.cs.princeton.edu/java/97data/InplaceFFT.java.html
*
******************************************************************************/
import edu.princeton.cs.algs4.StdDraw;
import edu.princeton.cs.algs4.StdOut;
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 IllegalArgumentException("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].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);
}
// 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]);
int n = 4;
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)");
}
}
