Signals and Systems - Electrical Engineering

5.7 Convolution and Filtering 337

5.7.3 Frequency Response from Poles and Zeros..................................

Given a rational transfer functionH(s)=B(s)/A(s), to calculate its frequency response we lets=j

and find the magnitude and phase for a discrete set of frequencies. This can be done using MATLAB.

A geometric way to obtain an approximate magnitude and phase frequency responses is using the

effects of zeros and poles on the frequency response of a system.

Consider a function

G(s)=

s−z

s−p

with a zerozand a polep, as shown in Figure 5.10. The frequency response corresponding toG(s)at

some frequency 0 is found by lettings=j 0 , or

G(s)|s=j 0 =

j 0 −z

j 0 −p

Representingj 0 ,z, andp, which are complex numbers, as vectors coming from the origin, then the

vectorZE( 0 )=j 0 −z(adding toZE( 0 )the vector corresponding tozgives a vector corresponding

toj 0 ) goes from the zeroztoj 0 , and likewise the vectorEP( 0 )=j 0 −pgoes from the polepto

j 0. The argument 0 in the vectors indicates that the magnitude and phase of these vectors depend

on the frequency at which we are finding the frequency response. As we change the frequency at

which we are finding the frequency response, the lengths and the phases of these vectors change.

Therefore,

G(j 0 )=

ZE( 0 )

PE( 0 )

=

|ZE( 0 )|

|PE( 0 )|

ej(∠ZE(^0 )−∠PE(^0 ))

and the magnitude response is

|G(j 0 )|=

|ZE( 0 )|

|PE( 0 )|

(5.23)

and the phase response is

∠G(j 0 )=∠ZE( 0 )−∠PE( 0 ) (5.24)

FIGURE 5.10
Geometric interpretation of poles and
zeros.

jΩ 0

P(Ω 0 ) Z(Ω 0 )

→→

pz

s-plane