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Overview
The z-transform is useful for the manipulation of discrete data sequences and
has acquired a new significance in the formulation and analysis of discrete-time
systems. It is used extensively today in the areas of applied mathematics, digital
signal processing, control theory, population science, and economics. These dis-
crete models are solved with difference equations in a manner that is analogous
to solving continuous models with differential equations. The role played by the
z-transform in the solution of difference equations corresponds to that played by
the Laplace transforms in the solution of differential equations.9.1 The z.-Transform
T he function notation for sequences is used in the study and application of z-
transforms. Consider a function x(t] defined for t 2: 0 that is sampled at t imes
t = T, T, 2T, 3T, ... , where T is the sampling period (or rate). We can write
the sample as a sequence using the notation {xn = x{nTJ}:=o· Without loss of
generality we will set T = 1 and consider real sequences such as {xn = x[n]}:=o·
The definition of the z-transform involves an infinite series of the reciprocals z-n.I Definition 9.1: z-transform Given the sequence {xn = x[n]}~ 0 , the z-transform
is defined as follows:
00 00X(z) = 3[{xn}:=oJ = L:x,,z-n = L:x[n]z-", (9-1)
n = O n = Owhich is a series involving powers of~·337