4.2 zTransforms
• For a discretetime signal
, the twosided ztransform is defined by
. The onesided ztransform is defined by
. In both cases, the ztransform is a polynomial in the complex variable
.
• The inverse ztransform is obtained by contour integration in the complex plane
. This is usually avoided by partial fraction inversion techniques, similar to the Laplace transform.
• Along with a ztransform we associate its region of convergence (or ROC). These are the values of
for which
is bounded (i.e., of finite magnitude).
• Some common ztransforms are shown below.
Table 1: Common ztransforms
Signal

zTransform

ROC


1

All





































• The ztransform also has various properties that are useful. The table below lists properties for the twosided ztransform. The onesided ztransform properties can be derived from the ones below by considering the signal
instead of simply
.
Table 2: Twosided zTransform Properties
Property

Time Domain

zDomain

ROC

Notation




Linearity



At least the intersection of
and

Time Shifting



That of
, except
if
and
if

zDomain Scaling




Time Reversal




zDomain
Differentiation




Convolution



At least the intersection of
and

Multiplication



At least

Initial value theorem

causal


