content: /*************************************************************************
   *  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 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) {
          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]);
          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)");
      }
  
  }
  