When building a model to describe the behavior of some of the most commonly used systems, we often rely on the *superposition principle*. It amounts to assuming *linearity* (the use of Kirchoff’s laws are an example). Usually, *time invariance* is also assumed. It consists of saying that, on the time scales that are used, the characteristics of these systems remain unchanged.

Linear filters are defined in the following section by these two characteristics. Because of their importance in the field of signal processing, this chapter presents their main properties, as well as a few design methods.

**Definition 4.1 (Linear filter)** *A discrete-time linear filter*^{1} *is a system whose output sequence results from the input sequence* {*x*(*n*)} *according to the expression:*

(4.1)

*where the sequence* {*h*(*n*)} *that characterizes the filter is called the* impulse response. *The* (*x* * *h*) *operation is called* convolution *(Figure 4.1).*

For example, the processing defined by *y*(*n*) = *x*(*n*) + *x*(*n* – 1) is therefore a linear filtering. The sequence {*h*(*n*)} is defined by *h*(0) = , *h*(1) = and *h*(*n*) = 0 for any value of *n* ≠ {0, 1}.

For commonly used classes of signals, expression ...

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