# 4Time-Invariant Discrete Linear Systems

Time-invariant discrete linear systems represent a very important area for digital signal processing – digital filters with fixed characteristics. These systems are characterized by the fact that their behavior is governed by a convolution equation. Their properties are analyzed using the *Z*-transform, which plays the same role in discrete systems as the Laplace or Fourier transforms do in continuous systems. In this chapter, the elements which are most useful for studying such systems will be briefly introduced. To supplement this discussion, reference should be made to References [1–5].

## 4.1 Definition and Properties

A discrete system is one which converts a set of input data *x*(*n*) into a set of output data *y*(*n*). It is linear if the set *x*_{1}(*n*) + *ax*_{2}(*n*) is converted to the set *y*_{1}(*n*) + *ay*_{2}(*n*), and it is time-invariant if the set *x*(*n* − *n*_{0}) is converted to the set *y*(*n* − *n*_{0}) for any integer *n*_{0}.

Assume *u*_{0}(*n*) is a unit set, as shown in Figure 4.1, and defined by:

The set *x*(*n*) can be decomposed into a sum of suitably shifted unit sets:

Further, if *h*(*n*) is the set forming the system’s response to the unit set *u*_{0}(*n*), then *h*(*n* − *m*) corresponds to *u*_{0}(*n* − *m*) because of the time-invariance. Linearity then implies the following relation: ...

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