FFT

Often we are interested in properties of a function f, knowing only measured values of f at equally spaced time intervals

If this discrete function f has the period , then f is described by the vector

Definition 9 (Discrete Fourier coefficient)
Assume then the Fourier coefficient of y is defined

Definition 10 (Discrete Fourier transform (DFT))
The mapping , defined by

with

is called the discrete Fourier transform (DFT).
If we use the Fourier-Matrix

then we can write (14) as

Theorem 11
It follows, that

i.e.

Definition 11 (Inverse discrete Fourier transform (IDFT))
The inverse mapping y=F_N c is called the inverse discrete Fourier transform (IDFT)

Some properties of the DFT are:

Linearity

Parseval

Theorem 12 (Fast Fourier Transform (FFT))
IF N is even (N=2M), then y= F_N c (and analog ) can be put down to two discrete transforms.

We divide c in its odd and even indices

and

y is splitted in

and

It follows ( w_N^k+M=-w_N^k) that

w_N² is an Mth root of unity, so the above sums describe two IDFT

In order to perform a Fourier transform of length N, one need to do two Fourier transforms F_M e and F_M o of length M on the even and odd elements. We now have two transforms which take less time to work out. The two sub-transforms can then be combined with the appropriate factor w^k to give the IDFT. Applying this recursively leads to the algorithm of the Fast Fourier transform (FFT).

/*
  x and y are real and imaginary arrays of 2^m points.
  dir = 1 gives forward transform
  dir = -1 gives reverse transform
*/


FFT(int dir, int m, double *x, double *y)
{
  int n,i,i1,j,k,i2,l,l1,l2;
  double c1,c2,tx,ty,t1,t2,u1,u2,z;
 
  /* Number of points */
  n = 1;
  for (i=0;i<m;i++)
    n *= 2;

  /* Bit reversal */
  i2 = n >> 1;
  j = 0;
  for (i=0;i < n-1; i++) {
    if (i < j) {
      tx = x[i];
      ty = y[i];
      x[i] = x[j];
      y[i] = y[j];
      x[j] = tx;
      y[j] = ty;
    }
    k = i2;
    while (k <= j) {
      j -= k;
      k >>= 1;
    }
    j += k;
  }

  for (i=0;i < n; i++) {
    printf("x[%i] = %f   y[%i] = %f\n", i, x[i], i, y[i]);   
  }
  printf("-------------------------\n");

  /* compute the FFT */
  c1 = -1.0;
  c2 = 0.0;
  l2 = 1;
  for (l=0;l<m;l++) {
    l1 = l2;
    l2 <<= 1;
    u1 = 1.0;
    u2 = 0.0;
    for (j=0;j<l1;j++) {
      for (i=j;i<n;i+=l2) {
        i1 = i + l1;
        t1 = u1 * x[i1] - u2 * y[i1];
        t2 = u1 * y[i1] + u2 * x[i1];
        x[i1] = x[i] - t1;
        y[i1] = y[i] -t2;
        x[i]  += t1;
        y[i]  += t2;
      }
      z =  u1 * c1 -u2 * c2;
      u2 = u1 * c2 + u2 * c1; 
      u1 = z;
    } 
    c2 = sqrt((1.0 - c1) / 2.0);
    if (dir == 1)
        c2 = -c2;
    c1 = sqrt((1.0 + c1) / 2.0);
  }
  
  /* scaling for forward transform */
  if (dir == 1) {
    for (i=0;i<n;i++) {
      x[i] /= n;
      y[i] /= n;
    }
  }
}