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 correspondance 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
=
eiw
=∑
k
=0M
b
k
e−(iwk)∑
k
=0N
a
k
e−(iwk)
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
eiww we arrive at the following equations:
Hw=
b
0
a
0
∏
k
=1Meiw−
c
k
∏
k
=1Neiw−
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
eiw−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
eiw
w
is a vector from the origin to its location on the unit circle. The vector connecting these two points,
eiw−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.