/******************************************************************************
 *  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)
 *  
 *
 *  % 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
 *
 ******************************************************************************/

package edu.princeton.cs.algs4;

/**
 *  The {@code FFT} class provides methods for computing the 
 *  FFT (Fast-Fourier Transform), inverse FFT, linear convolution,
 *  and circular convolution of a complex array.
 *  <p>
 *  It is a bare-bones implementation that runs in <em>n</em> log <em>n</em> time,
 *  where <em>n</em> is the length of the complex array. For simplicity,
 *  <em>n</em> must be a power of 2.
 *  Our goal is to optimize the clarity of the code, rather than performance.
 *  It is not the most memory efficient implementation because it uses
 *  objects to represents complex numbers and it it re-allocates memory
 *  for the subarray, instead of doing in-place or reusing a single temporary array.
 *  
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/99scientific">Section 9.9</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class FFT {

    private static final Complex ZERO = new Complex(0, 0);

    // Do not instantiate.
    private FFT() { }

    /**
     * Returns the FFT of the specified complex array.
     *
     * @param  x the complex array
     * @return the FFT of the complex array {@code x}
     * @throws IllegalArgumentException if the length of {@code x} is not 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;
    }


    /**
     * Returns the inverse FFT of the specified complex array.
     *
     * @param  x the complex array
     * @return the inverse FFT of the complex array {@code x}
     * @throws IllegalArgumentException if the length of {@code x} is not 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;

    }

    /**
     * Returns the circular convolution of the two specified complex arrays.
     *
     * @param  x one complex array
     * @param  y the other complex array
     * @return the circular convolution of {@code x} and {@code y}
     * @throws IllegalArgumentException if the length of {@code x} does not equal
     *         the length of {@code y} or if the length is not a power of 2
     */
    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);
    }

    /**
     * Returns the linear convolution of the two specified complex arrays.
     *
     * @param  x one complex array
     * @param  y the other complex array
     * @return the linear convolution of {@code x} and {@code y}
     * @throws IllegalArgumentException if the length of {@code x} does not equal
     *         the length of {@code y} or if the length is not a power of 2
     */
    public static Complex[] convolve(Complex[] x, Complex[] y) {
        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
    private 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.
    ***************************************************************************/

    /**
     * Unit tests the {@code FFT} class.
     *
     * @param args the command-line arguments
     */
    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(StdRandom.uniform(-1.0, 1.0), 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)");
    }

}

/******************************************************************************
 *  Copyright 2002-2016, Robert Sedgewick and Kevin Wayne.
 *
 *  This file is part of algs4.jar, which accompanies the textbook
 *
 *      Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
 *      Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
 *      http://algs4.cs.princeton.edu
 *
 *
 *  algs4.jar is free software: you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation, either version 3 of the License, or
 *  (at your option) any later version.
 *
 *  algs4.jar is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with algs4.jar.  If not, see http://www.gnu.org/licenses.
 ******************************************************************************/