Page 187 - Schaum's Outline of Theory and Problems of Signals and Systems

176 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

B. Characterization of Discrete-Time LTI Systems:

Many properties of discrete-time LTI systems can be closely associated with the
characteristics of H(z) in the z-plane and in particular with the pole locations and the
ROC.
1. Causality:

For a causal discrete-time LTI system, we have [Eq. (2.4411

since h[n] is a right-sided signal, the corresponding requirement on H(z) is that the ROC
of H(z) must be of the form

That is, the ROC is the exterior of a circle containing all of the poles of H(z) in the
z-plane. Similarly, if the system is anticausal, that is,

then h[n] is left-sided and the ROC of H(z) must be of the form

That is, the ROC is the interior of a circle containing no poles of H(z) in the z-plane.
2. Stability:

In Sec. 2.7 we stated that a discrete-time LTI system is BIB0 stable if and only if [Eq.
(2.4911

The corresponding requirement on H(z) is that the ROC of H(z1 contains the unit circle
(that is, lzl= 1). (See Prob. 4.30.)
3. Ctzusal and Stable Systems:
If the system is both causal and stable, then all of the poles of H(z) must lie inside the
unit circle of the z-plane because the ROC is of the form lzl> r,,, and since the unit
circle is included in the ROC, we must have r,, < 1.

C. System Function for LTI Systems Described by Linear Constant-Coefficient Difference
Equations:
!n Sec. 2.9 we considered a discrete-time LTI system for which input x[n] and output
y[n] satisfy the general linear constant-coefficient difference equation of the form