THE z-TRANSFORM - Signals, Systems, and Transforms - page 546

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THE z -TRANSFORM

In this chapter, we study the z-transform , which is one of several important trans-
forms used in linear-system analysis and design. The z -transform offers significant
advantages relative to time-domain procedures. When possible, we model discrete-
time physical systems with linear difference equations with constant coefficients;
one example is a linear time-invariant digital filter. The z -transform of a difference
equation gives us a good description of the characteristics of the equation (the
model) and, hence, of the physical system. In addition, transformed difference
equations are algebraic, and therefore easier to manipulate; in particular, the trans-
formed equations are easier to solve.
Using the z -transform to solve a difference equation yields the solution as a
function of the transform variable z . As a consequence, we must have a method for
converting functions of the transform variable back to functions of the discrete-time
variable; the inverse z-transform is used for this purpose.
Several important properties of the z -transform are derived in this chapter. These
derivations are not mathematically rigorous; such derivations are generally beyond the
scope of this topic. Thus, for some properties, certain constraints apply that are not ev-
ident from the derivations. However, these constraints will be stated; see Refs. 1 and 2
for rigorous mathematical derivations related to all aspects of the z -transform.

z -Transform A transform of a sampled signal or sequence was defined in 1947 by W.
Hurewicz as

Z[f(kT)] = q
k= 0

f(kT)z -k .

This equation was denoted in 1952 as a “ z transform” by a sampled-data control group
at Columbia University led by professor John R. Raggazini and including L.A. Zadeh,
E.I. Jury, R.E. Kalman, J.E. Bertram, B. Friedland, and G.F. Franklin.

Reference: Robert D. Strum and Donald E. Kirk, Contemporary Linear Systems, PWS
Publishing Company, Boston, 1994.

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