Page 182 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 171

4.4 PROPERTIES OF THE 2-TRANSFORM
Basic properties of the z-transform are presented in the following discussion. Verifica-
tion of these properties is given in Probs. 4.8 to 4.14.

A. Linearity:

If
xlb] ++X1(z) ROC = R,

I
~ 2 b -Xz(z) ROC= R,
then
QIXI[~] + a,xz[n] ++alXl(z) + a2XAz) Rr~Rl nR2 (4.1 7)
where a, and a, are arbitrary constants.

B. Time Shifting:

If
+I ++X(z) ROC = R
then

x[n - n,] -z-"oX(z) R' = R n {O < (21 < m} (4.18)
Special Cases:

Because of these relationships [Eqs. (4.19) and (4.20)1, z-' is often called the unit-delay
operator and z is called the unit-advance operator. Note that in the Laplace transform the
operators s - ' = 1 /s and s correspond to time-domain integration and differentiation,
respectively [Eqs. (3.22) and (3.2011.

C. Multiplication by z,":

If

then

In particular, a pole (or zero) at z = z, in X(z) moves to z = zoz, after multiplication by
2," and .the ROC expands or contracts by the factor (z,(.

Special Case: