Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 80

Digital System

X s ( f ) = Y ( e j Ω ),
Input
Spectrum

Y s ( f ) = X ( e j Ω ),
Output
Spectrum

H s ( f ) = H ( e j Ω )

Y s ( f ) = H s ( f ) . X s ( f )
[or: Y ( e j Ω ) = H ( e j Ω ) . X ( e j Ω ); where

= T s ]

Ω ω

Fig. 2.11 Frequency-domain representation of a discrete-time system

Again, analogously to the continuous-time case, convolution in the discrete-
time domain is transformed into multiplication in both the frequency domain and
the z-domain:

x ð n Þ y ð n Þ Z

X ð z Þ Y ð z Þ ð 2 : 12 Þ

x ð n Þ y ð n Þ Z X ð z Þ Y ð z Þ ð 2 : 13 Þ

(See also Tables, z-Transform Pairs and Theorems).

2.3.4.1 Relationship Between the ZT Transfer Function and the Frequency
Response

The (periodic) frequency response H s ð x Þ¼ H ð e jxT s Þ of a digital system can be
obtained from its ZT-transfer function H(z) with the substitution z ¼ e jxT s as

follows:

H ð e jxT s Þ¼ H ð z Þj z ¼ e jxT s :

2.3.4.2 Stability of Digital Systems in the z-Domain

It has been shown previously that a digital system is BIBO-stable if its impulse
response is absolutely summable, i.e., if P k ¼1 j h ð k Þj \ 1: Practically it is often
difficult to analyze the system stability in this way. Equivalently, a causal digital
system is BIBO-stable if and only if all the poles of its z-transfer function H(z) are
inside the unit circle (i.e., |z p | \ 1, where z p is the location of the pth pole in the z-
plane). [Note that this condition is for causal digital systems only, otherwise the
condition would be that the ROC of H(z) should contain the unit circle].
Example (1) If the system impulse response is h(n) = a n u(n), then its z-transfer
function is H(z) = z/(z - a) [from Tables. This system has a zero at z = 0, and a
pole at z = a. It is BIBO-stable if |a| \ 1, i.e., if the pole is within the unit circle