One of the primary motivating factors for utilizing the ztransform and analyzing the pole/zero
plots is due to their relationship to the frequency response
of a discretetime system. Based on the position of the poles and zeros,
one can quickly determine the frequency response. This is a
result of the correspondence between the frequency response
and the transfer function evaluated on the unit circle in the
pole/zero plots. The frequency response, or DTFT, of the
system is defined as:
Hw=Hz
z
,
z
=
ejw
=∑
k
=0M
b
k
e−(jwk)∑
k
=0N
a
k
e−(jwk)
Hw
z
w
Hz
k
0
M
b
k
w
k
k
0
N
a
k
w
k
(1)
Next, by factoring the transfer function into poles and zeros
and multiplying the numerator and denominator by
ejww
we arrive at the following equations:
Hw=
b
0
a
0
∏
k
=1Mejw−
c
k
∏
k
=1Nejw−
d
k

Hw
b
0
a
0
k
1
M
w
c
k
k
1
N
w
d
k
(2)
From
Equation 2 we have the
frequency response in a form that can be used to interpret
physical characteristics about the filter's frequency
response. The numerator and denominator contain a product of
terms of the form
ejw−h
w
h
,
where
hh is either a zero, denoted by
c
k
c
k
or a pole, denoted by
d
k
d
k
. Vectors are commonly used to represent
the term and its parts on the complex plane. The pole or zero,
hh, is a vector from the origin
to its location anywhere on the complex plane and
ejw
w
is a vector from the origin to its
location on the unit circle. The vector connecting these two
points,
ejw−h
w
h
, connects the pole or zero location to a
place on the unit circle dependent on the value of
ww. From this, we can begin to
understand how the magnitude of the frequency response is a
ratio of the distances to the poles and zero present in the
zplane as
ww goes from zero to
pi. These characteristics allow us to interpret
HwHw
as follows:
Hw=
b
0
a
0
∏"distances from zeros"∏"distances from poles"
Hw
b
0
a
0
∏
"distances from zeros"
∏
"distances from poles"
(3)
"My introduction to signal processing course at Rice University."