Page 189 - Schaum's Outline of Theory and Problems of Signals and Systems
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178 THE 2-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4
B. Basic Properties:
Most of the properties of the unilateral z-transform are the same as for the bilateral
z-transform. The unilateral z-transform is useful for calculating the response of a causal
system to a causal input when the system is described by a linear constant-coefficient
difference equation with nonzero initial conditions. The basic property of the unilateral
z-transform that is useful in this application is the following time-shifting property which is
different from that of the bilateral transform.
Time-Shifting Property:
If x[n] t, X,( z ), then for m 2 0,
x[n -m] -Z-~X,(Z) +z-"+'x[-11 +z-"+~x[-~] + - +x[-m]
x[n + m] t,zmX,(z) -zmx[O] -zm-'x[l] - . . - -~[m - 11
The proofs of Eqs. (4.50) and (4.51) are given in Prob. 4.36.
D. System Function:
Similar to the case of the continuous-time LTI system, with the unilateral z-transform,
the system function H(z) = Y(z)/X(z) is defined under the condition that the system is
relaxed, that is, all initial conditions are zero.
Solved Problems
THE Z-TRANSFORM
4.1. Find the z-transform of
(a) From Eq. (4.3)
By Eq. (1.91)
1
(a-~z)~ if la-'zl< 1 or lz( < la1
=
I - a-'z
n =O
