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The z Transform
+∞
The z transform of the sequence {x k } −∞ is defined as the generating function
∞
Xz() = x{} = ∑ x k z k – (23.48)
k=−∞
where the variable z has the essential interpretation of a forward shift operator so that

x k+1 } = z x k zX z() (23.49)
{} =
{
−1
The z transform is an infinite power series in the complex variable z where {x k } constitutes a sequence
of coefficients. As the z transform is an infinite power series, it exists only for those values of z for which
this series converges and the region of convergence of X(z) is the set of z for which X(z) takes on a finite
value. A sufficient condition for existence of the z transform is convergence of the power series

∞
∑ x k ⋅ z k – < ∞ (23.50)
k=−∞
The region of convergence for a finite-duration signal is the entire z plane except z = 0 and z = ∞. For
a one-sided infinite-duration sequence {x k } ∞ k=0 , a number r can usually be found so that the power series
converges for |z| > r. Then, the inverse z transform can be derived as

1
∫
x k = 2pi° k−1 dz (23.51)
-------- Xz()z
where the contour of integration encloses all singularities of X(z). In practice, it is standard procedure
to use tabulated results; some standard z transform pairs are to be found in Table 23.10.

Digital Systems and Discretized Data

Periodic sampling of signals and subsequent computation or storing of the results requires the computer
to schedule sampling and to handle the resulting sequences of numbers. A measured variable x(t) may
be available only as periodic observations of x(t) as sampled with a time interval T (the sampling period).
The sample sequence can be represented as

∞
(
{} , x k = xkT) for k = …, −1, 0, 1, 2,… (23.52)
∞
x k –
and it is important to ascertain that the sample sequence adequately represents the original variable x(t);
see Fig. 23.16. For ideal sampling it is required that the duration of each sampling be very short and the
sampled function may be represented by a sequence of infinitely short impulses δ(t) (the Dirac impulse).
Let the sampled function of time be expressed thus:

∞
x ∆ t() = xt() T ∑ d tkT) = xt() ⋅ T t() (23.53)
(
⋅
–
k=−∞
where

∆ ∞
T t() = T ∑ d tkT) (23.54)
(
–
k=−∞
©2002 CRC Press LLC