# 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 [15].

## 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 x1(n) + ax2(n) is converted to the set y1(n) + ay2(n), and it is time-invariant if the set x(nn0) is converted to the set y(nn0) for any integer n0.

Assume u0(n) is a unit set, as shown in Figure 4.1, and defined by:

(4.1)

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

(4.2)

Further, if h(n) is the set forming the system’s response to the unit set u0(n), then h(nm) corresponds to u0(nm) because of the time-invariance. Linearity then implies the following relation: ...

Get Digital Signal Processing, 10th Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.