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package algs9; // 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)");
  }

}