Digital filters - Summary

Like an analog filter, a digital filter can be made up of interconnected elements, in this case shift registers, multipliers, adders, etc. The discrete nature of the operations performed to calculate the output signal means that it can also be implemented in the form of a program or firmware on a digital computer in the broadest sense of the term (specialized integrated circuit, signal processor, general-purpose computer, etc.). The analysis of a digital filter consists in determining its response to a given excitation. Design is the process of synthesizing and realizing the filter in such a way that it meets given constraints (response in amplitude, phase, etc.). For reasons of physical feasibility and real-time use, the filters under consideration will necessarily have a causal impulse response, meaning that a sample of the output signal can be calculated from samples of the input and/or output depending only on instants in the present and/or past, but never in the future. It is, however, possible to simulate a non-causal usage with a processing delay, if the application allows it. In this section, we will focus on linear invariant filters.

Although the frequency response of a filter is defined by a modulus and a phase, synthesis methods generally satisfy only one of the two constraints. Either the phase has a particular property (e.g. linear for non-recursive filters) and we're only interested in the constraint on the modulus, or the modulus has a particular property (e.g. constant for all-pass filters) and we're only interested in the constraint on the phase, or in most cases, we're only interested in the modulus and the resulting phase will either be satisfactory or will have to be corrected using an additional filter.