package algs91; // section 9.9
import stdlib.*;
import algs12.Complex;
/* ***********************************************************************
 *  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 Error("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 Error("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 (Complex element : x) {
			StdOut.println(element);
		}
		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]);
		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)");
	}

}