**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) |

(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) |

(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) |

(11) |