Considering a continuous function *f*(*x*) of a single variable *x*
representing distance.

The Fourier
transform of that function is denoted *F*(*u*), where *u* represents spatial
frequency is defined by

(1) |

**Note**: In general *F*(*u*) will be a complex quantity *even though* the
original data is purely **real**.

The meaning of this is that not only is the magnitude of each frequency present important, but that its phase relationship is too.

The inverse Fourier transform for regenerating *f*(*x*) from *F*(*u*) is given by

(2) |

Let's see how we compute a Fourier Transform: consider a particular function *f*(*x*)
defined as

(3) |

** A top hat function **

So its Fourier transform is:

In this case *F*(*u*) is purely real, which is a consequence of the
original data being symmetric in *x* and -*x*. A graph of
*F*(*u*) is shown in Fig. 7.7. This function is often referred to as
the Sinc function.

** Fourier transform of a top hat function **