Digital filters - Frequency conversion and filter banks

The sampling period or frequency is an essential element in many digital signal processing operations. It often determines the efficiency and precision with which processing can be carried out; in some cases, it needs to be changed to obtain algorithms that depend on the frequency band to be analyzed, in other cases it's simply a question of switching from one standard to another (from 44.1 kHz to 32 or 48 kHz in audio). Applications can be found in communications, speech processing, antenna processing, etc.

Transforming a signal gives it a new representation. The aim is often to obtain a better localization or representation of the information contained in the signal, the qualifier "better" being measured according to a criterion yet to be defined.

An elementary example of a transformation is the discrete Fourier transform, which associates a sequence of length N with a sequence of the same length. Such a transform is called lossless or orthogonal and can be represented by a unitary matrix, i.e. one whose inverse is the conjugate transpose matrix.

In many applications, it can be useful to separate the input signal into several sub-band components. This makes it possible to locate the frequency band(s) where the information can be found. Compression can be used to reduce the amount of information to be transmitted, by taking into account only those frequency bands in which the energy exceeds a certain threshold. The discrete Fourier transform is an example, but it is possible to envisage equivalent processing using a set of bandpass filters.