1. Discrete Fourier Transform

Signal and filter representations in the form of Fourier and z-transforms are theoretical tools. They can only be used if the data under study have a formal representation. This is the case for a non-recursive linear filter or a sinusoidal signal. Signal processing studies involve representing and processing measured signals whose transform cannot be written as a formula dependent on a small number of parameters. Even when a formal formula exists, it is often necessary to calculate the Fourier transform of a signal or the frequency response of a filter. Computer-aided tools are used for this purpose.

The use of digital techniques to perform a Fourier transform calculation assumes that the number of data to be processed is finite, and that the number of frequencies for which the transform is calculated is also finite. To retain the same amount of information, we...