9.4 One-Sided Z-Transform 539
9.4.5 Initial and Final Value Properties
In some control applications and to check a partial fraction expansion, it is useful to find the initial
or the final value of a discrete-time signalx[n] from its Z-transform. These values can be found as
shown in the following box.
IfX(z)is the Z-transform of a causal signalx[n], then
Initial value: x[0]=zlim→∞X(z)
Final value: nlim→∞x[n]=lim
z→ 1
(z− 1 )X(z) (9.30)
The initial value results from the definition of the one-sided Z-transform—that is,
lim
z→∞
X(z)=lim
z→∞
x[0]+
∑
n≥ 1
x[n]
zn
=x[0]
To show the final value, we have that
(z− 1 )X(z)=
∑∞
n= 0
x[n]z−n+^1 −
∑∞
n= 0
x[n]z−n
=x[0]z+
∑∞
n= 0
[x[n+1]−x[n]]z−n
and thus the limit
lim
z→ 1
(z− 1 )X(z)=x[0]+
∑∞
n= 0
(x[n+1]−x[n])
=x[0]+(x[1]−x[0])+(x[2]−x[1])+(x[3]−x[2])···
=lim
n→∞
x[n]
given that the entries in the sum cancel out asnincreases, leavingx[∞].
nExample 9.12
Consider a negative-feedback connection of a plant with a transfer functionG(z)= 1 /( 1 −0.5z−^1 )
and a constant feedback gainK(see Figure 9.8). If the reference signal is a unit step,x[n]=u[n],
determine the behavior of the error signale[n]. What is the effect of the feedback, from the error
point of view, on an unstable plantG(z)= 1 /( 1 −z−^1 )?
