Digital Signal Processing Reference
In-Depth Information
The transfer function is the ratio of two polynomials in
z
−
1
where the degree
of the numerator polynomial is
M
−
1 and that of the denominator polyno-
mial is
N
1. As a consequence, the transfer function can be rewritten in
factored form as
−
l
g
r
,
y
i
d
.
,
©
,
L
s
M
−
1
z
n
z
−
1
1
(
1
−
)
n
=
H
(
z
)=
b
0
(6.15)
N
−
1
p
n
z
−
1
1
(
1
−
)
n
=
where the
z
n
are the
M
1 complex roots of the numerator polynomial and
are called the
zeros
of the system; the
p
n
are the
N
−
1 complex roots of the
denominator polynomial and, as we have seen, they are called the poles of
the system. Both poles and zeros can have arbitrary multiplicity. Clearly, if
z
i
−
p
k
for some
i
and
k
(i.e. if a pole and a zero coincide) the corresponding
first-order factors cancel each other out and the degrees of numerator and
denominator are both decreased by one. In general, it is assumed that such
factors have already been removed and that the numerator and denomina-
tor polynomials of a given rational transfer function are coprime.
The poles and the zeros of a filter are usually represented graphically
on the complex plane as crosses and dots, respectively. This allows for a
quick visual assessment of stability which, for a causal system, consists of
checking whether all the crosses are inside the unit circle (or, for anticausal
systems, outside).
=
6.2.1
Pole-Zero Patterns
The pole-zero plot can exhibit distinctive patterns according to the proper-
ties of the filter.
Real-Valued Filters.
If the filter coefficients are real-valued (and this is
the only case that we consider in this text book) both the numerator and de-
nominator polynomials are going to have real-valued coefficients. We can
now recall a fundamental result from complex algebra: the roots of a poly-
nomial with real-valued coefficients are either real or they occur in complex-
conjugate pairs. So, if
z
0
is a complex zero of the system,
z
∗
0
is a zero as well.
Similarly, if
p
0
is a complex pole, so is
p
∗
0
. The pole-zero plot will therefore
shows a symmetry around the real axis (Fig. 6.2a).
