Page 188 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 177

Applying the z-transform and using the time-shift property (4.18) and the linearity
property (4.17) of the z-transform, we obtain

or

Thus,

Hence, H(z) is always rational. Note that the ROC of H(z) is not specified by Eq. (4.44)
but must be inferred with additional requirements on the system such as the causality or
the stability.

D. Systems Interconnection:
For two LTI systems (with h,[n] and h2[n], respectively) in cascade, the overall
impulse response h[n] is given by
h[nl =h, [nl * h2bl (4.45)
Thus, the corresponding system functions are related by the product

Similarly, the impulse response of a parallel combination of two LTI systems is given
by
+I =h,[nl +h*lnl (4.47)

and

4.7 THE UNILATERAL Z-TRANSFORM

A. Definition:
The unilateral (or one-sided) z-transform X,(z) of a sequence x[n] is defined as [Eq.
(4.511
m
X,(z) = z x[n]z-" (4.49)
n-0
and differs from the bilateral transform in that the summation is carried over only n 2 0.
Thus, the unilateral z-transform of x[n] can be thought of as the bilateral transform of
x[n]u[n]. Since x[n]u[n] is a right-sided sequence, the ROC of X,(z) is always outside a
circle in the z-plane.