output Yz XzHz,the pole±zero cancelation may occur in the product of system
transfer function H(z) with the z-transform of the input signal Xz. By proper selection
of the zeros of the system transfer function,it is possible to suppress one or more poles of
the input signal from the output of the system,or vice versa. When the zero is located
very close to the pole but not exactly at the same location to cancel the pole,the system
response has a very small amplitude.
The portion of the output yn that is due to the poles of Xz is called the forced
response of the system. The portion of the output that is due to the poles of H(z) is
called the natural response. If a system has all its poles within the unit circle,then its
natural response dies down as n 3 I,and this is referred to as the transient response. If
the input to such a system is a periodic signal,then the corresponding forced response is
called the steady-state response.
Consider the recursive power estimator given in (3.2.11) as an LTI system H(z) with
input wn x
2
n and output yn
P
x
n. As illustrated in Figure 4.8,Equation
(3.2.11) can be rewritten as
yn 1 À ayn À 1 awn:
Taking the z-transform of both sides,we obtain the transfer function that describes this
efficient power estimator as
Hz
Yz
Wz
a
1 À 1 À az
À1
:
4:3:15
This is a simple first-order IIR filter with a zero at the origin and a pole at z 1 À a. A
pole±zero plot of H(z) given in (4.3.15) is illustrated in Figure 4.9. Note that a 1=L
results in 1 À a L À 1=L,which is slightly less than 1. When L is large,i.e.,a longer
window,the pole is closer to the unit circle.
x(n)
(·)
2
w(n)
= x
2
(n)
H(z)
y(n)
= P
x
(n)
^
Figure 4.8 Block diagram of recursive power estimator
Re[z]
Im[z]
|z| = 1
Zero
Pole
Figure 4.9 Pole±zero diagram of the recursive power estimator
SYSTEMS CONCEPTS
147