Page 184 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS 173

Table 4-2. Some Properties of the z-Transform

Property Sequence Transform ROC

Linearity
Time shifting

Multiplication by z,"

Multiplication by einon
Time reversal

&(z)
Multiplication by n -2-
d.?
Accumulation

Convolution

H. Summary of Some z-transform Properties
For convenient reference, the properties of the z-transform presented above are
summarized in Table 4-2.

4.5 THE INVERSE Z-TRANSFORM

Inversion of the z-transform to find the sequence x[n] from its z-transform X(z) is
called the inverse z-transform, symbolically denoted as

~[n] =s-'{X(z>} (4.27)

A. Inversion Formula:

As in the case of the Laplace transform, there is a formal expression for the inverse
z-transform in terms of an integration in the z-plane; that is,

where C is a counterclockwise contour of integration enclosing the origin. Formal
evaluation of Eq. (4.28) requires an understanding of complex variable theory.

B. Use of Tables of z-Transform Pairs:
In the second method for the inversion of X(z), we attempt to express X(z) as a sum

X(z) =X,(z) + . . . +X,(z) (4.29)