Page 186 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS

expansion would have the form

If X(Z) has multiple-order poles, say pi is the multiple pole with multiplicity r, then
the expansion of X(z)/z will consist of terms of the form

where

4.6 THE SYSTEM FUNCTION OF DISCRETE-TIME LTI SYSTEMS
A. The System Function:
In Sec. 2.6 we showed that the output y[n] of a discrete-time LTI system equals the
convolution of the input x[n] with the impulse response h[n]; that is [Eq. (2.3511,

Applying the convolution property (4.26) of the z-transform, we obtain

where Y(z), X(z), and H(z) are the z-transforms of y[n], x[n], and h[n], respectively.
Equation (4.40) can be expressed as

The z-transform H(z) of h[n] is referred to as the system function (or the transfer
function) of the system. By Eq. (4.41) the system function H(z) can also be defined as the
ratio of the z-transforms of the output y[n] and the input x[n.l. The system function H(z)
completely characterizes the system. Figure 4-3 illustrates the relationship of Eqs. (4.39)
and (4.40).

t t t

X(Z) Y(z)=X(z)H(z)
+ H(z) t

Fig. 4-3 Impulse response and system function.