One of the motivating factors for analyzing the pole/zero
plots is due to their relationship to the frequency response
of the 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)
In conclusion, using the distances from the unit circle to the
poles and zeros, we can plot the frequency response of the
system. As
ww goes from
00 to
2π
2
, the following two properties, taken from the above
equations, specify how one should draw
Hw
Hw
.

if close to a zero, then the magnitude is small. If a
zero is on the unit circle, then the frequency response is
zero at that point.

if close to a pole, then the magnitude is large. If a
pole is on the unit circle, then the frequency response
goes to infinity at that point.
"My introduction to signal processing course at Rice University."