Signal theory and signal processing encompass all the operations performed on the signal, with the aim of modulating, filtering, performing spectral analysis, etc. In the case of digital processing, the original signal, which is generally analog and of continuous value, will be represented by a sequence of numbers and, for this, operations such as sampling, quantization and coding will be performed. The resulting signal will then be digitally processed, in order, for instance, to be filtered to retain its essential characteristics by eliminating parasitic frequencies when it comes to reproduce the initial signal, or to suppress negligible details for compressing data before storage or transmission.

In this chapter we introduce the mathematical bases involved in these operations, in the cases of deterministic as well as of random signals. First, the concepts of linear system and of filtering, and the representation of the signal in the frequency domain by means of Fourier transforms are discussed. Then we present several concepts specific to digital signal processing, such as sampling, quantization and discrete Fourier transform, in order to then introduce the basic principles of digital filtering, spectral analysis, prediction and adaptive filtering.