1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

338 CHAPTER, 9 • Z-TRANSFORMS AND APPLICATIONS

Remark 9.1
The z-transform is defined at points z EC where the Laurent series (9-1) con-
verges. The z-transform region of convergence (ROC) for the Laurent series is
chosen to be

lzl 2: R,

where

R = lim sup y/j;J.

n - oo

Remark 9.2

The sequence notation {xn}::'=o is used in mathematics to study difference equa-

tions, and the function notation {x[n)}::O=o is used by engineers for signal pro-

cessing. It's a good idea to know both notations. •

Remark 9.3

In the applications, the sequence {xn = x[n)}::'=o will be used for inputs and the

sequence {vn = y[n)}::'=o will be used for outputs. We will also use the notations

3[xn] = 3[x[n)] = X(z),

and

3 [y,,J = 3 {y[n)J = Y(z).