package com.example.androidtest;

/**

* **********************************************************************

* Compilation: javac FFT.java

* Execution: java FFT N

* Dependencies: Complex.java

* <p/>

* 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.

* <p/>

* Limitations

* -----------

* - assumes N is a power of 2

* <p/>

* - 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)

* <p/>

* ***********************************************************************

*/

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();

}

}