If this discrete function

Assume then the Fourier coefficient of

The mapping , defined by

with

is called the

If we use the

then we can write (14) as

It follows, that

i.e.

The inverse mapping

(15) |

**Linearity**-

**Parseval**-

IF N is even (N=2M), then

We divide *c* in its odd and even indices

and

y is splitted in

and

It follows (

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;

}

}

}