Page 470 - Mechanical Engineers' Handbook (Volume 2)

4 z-Transforms 461

4.1 Single-Sided z-Transform
If a signal has discrete values ƒ ,ƒ ,...,ƒ ,...,we define the z-transform of the signal
1
0
k
as the function
F(z) Z{ƒ(k)}
ƒ(k)z k r z R (30)

k 0 0 0
and is assumed that one can find a range of values of the magnitude of the complex variable
z for which the series of Eq. (30) converges. This z-transform is referred to as the one-sided
z-transform. The symbol Z denotes ‘‘the z-transform of.’’ In the one-sided z-transform it is
assumed ƒ(k) 0 for k 0, which in the continuous-time case corresponds to ƒ(t) 0 for
t 0.
Expansion of the right-hand side of Eq. (30) gives
F(z) ƒ(0) ƒ(1)z 1 ƒ(2)z 2 ƒ(k)z k (31)
The last equation implies that the z-transform of any continuous-time function ƒ(t) may
be written in the series form by inspection. The z k in this series indicates the instant in time
at which the amplitude ƒ(k) occurs. Conversely, if F(z) is given in the series form of Eq.
(31), the inverse z-transform can be obtained by inspection as a sequence of the function
ƒ(k) that corresponds to the values of ƒ(t) at the respective instants of time. If the signal is
sampled at a fixed sampling period T, then the sampled version of the signal ƒ(t) given by
ƒ(0), ƒ(1), ...,ƒ(k) correspond to the signal values at the instants 0, T,2T, ..., kT.

4.2 Poles and Zeros in the z-Plane
If the z-transform of a function takes the form
m
bz bz m 1 b
F(z) 0 1 m (32)
n
z az n 1 a
1 n
or
b (z z )(z z ) (z z )
F(z) 0 1 2 m
(z p )(z p ) (z p )
n
2
1
then p ’s are the poles of F(z) and z ’s are the zeros of F(z).
i
i
In control studies, F(z) is frequently expressed as a ratio of polynomials in z 1 as
follows:
bz (n m) bz (n m 1) bz n
F(z) 0 1 m (33)
1 az 1 az 2 az n
2
n
1
where z 1 is interpreted as the unit delay operator.
4.3 z-Transforms of Some Elementary Functions
Unit Step Function
Consider the unit step function
u (t) 1 t 0
s
0
t 0
whose discrete representation is