## 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

 (14)

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=FN c is called the inverse discrete Fourier transform (IDFT)

 (15)

Some properties of the DFT are:
Linearity

Parseval

Theorem 12 (Fast Fourier Transform (FFT))
IF N is even (N=2M), then y= FN 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 ( wNk+M=-wNk) that

wN2 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 FM e and FM 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 wk 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;    }  }}

Developed by Marian Prutscher: Email