Next: Compression Up: Fourier Theory Previous: 2D Case

### The Discrete Fourier Transform (DFT)

Images and Digital Audio are digitised !!

Thus, we need a discrete formulation of the Fourier transform, which takes such regularly spaced data values, and returns the value of the Fourier transform for a set of values in frequency space which are equally spaced.

This is done quite naturally by replacing the integral by a summation, to give the discrete Fourier transform or DFT for short.

In 1D it is convenient now to assume that x goes up in steps of 1, and that there are N samples, at values of x from 0 to N-1.

So the DFT takes the form
 (6)
while the inverse DFT is
 (7)

NOTE: Minor changes from the continuous case are a factor of 1/N in the exponential terms, and also the factor 1/N in front of the forward transform which does not appear in the inverse transform.

The 2D DFT works is similar. So for an grid in x and y we have
 (8)
and
 (9)

Often N=M, and it is then it is more convenient to redefine F(u,v) by multiplying it by a factor of N, so that the forward and inverse transforms are more symmetrical:
 (10)
and
 (11)

Next: Compression Up: Fourier Theory Previous: 2D Case
Dave Marshall
10/4/2001