Phase Responce of Pole zero in MATLAB Digital Signal Processing

1. 1
Lab 07:
Phase Response of Pole/Zero
Submitted By :
Syed Abuzar Hussain Shah
SP15-BEE-096
Uzair Ahmed
SP15-BEE-106
Syed Hasnain Shah
SP15-BEE-100
Submitted To: Sir Usman
Class: BEE-5A
Dated: 11/05/2017

2. 2
Objective:
In today’s lab the objective will be to understand the of effect of pole and zeros on
frequency phase response using MATLAB.
Statement:
1. Take a zero at unit circle using equation 1- .
2. Now take its frequency response using freqz command and analyze the effect
phase of the zero.
3. Now change the magnitude and keep on observing the frequency response.
4. Keeping the phase of zero using the equation given on right hand side
𝟏 - 𝒂 : -𝝅 𝜽 .
5. Take the value of a=1 and from – 𝝅 and increase slowly towards 𝝅,
accordingly observe the phase response.
6. Now repeat the same steps for the analysis of behavior of a pole.
7. Repeat step 6 for All Zero, All Pole and ARMA Models.
8. Now for FIR model with equation 𝟏 - 𝒂 keeping 𝜽 = 𝟎, and changing the
value of a from 0.1 to 10, analyze the effect of phase response using ‘freqz’
command.
9. Repeat step 9 for All Pole and ARMA Models.
Procedure:
1. Phase Response for 𝜽 𝟏 𝟎 and a=1 for 𝟏 𝟏
Clc
close all
clear all
th=-pi;
a=1;
%All zero Model
n=[1 -a*exp(i*th)];
d=1;
subplot(321)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(322)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All zero model');

3. 3
%All pole model
d=[1 -a*exp(i*th)];
n=1;
subplot(323)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(324)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All pole model');
axis([-2 2 -1 100])
grid on;
%ARMA Model
n=[1 -a*exp(i*th)];
d=[1 -a*exp(i*th)];
subplot(325)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(326)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('ARMA model');
grid on;

4. 4
2. Phase Response for 𝜽 𝟎 and a=1 for 𝟏 𝟏
clc, clear all, close all;
th=0;
a=1;
%All zero Model
n=[1 -a*exp(i*th)];
d=1;
subplot(321)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(322)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All zero model');
grid on
%All pole model
d=[1 -a*exp(i*th)];
n=1;
subplot(323)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(324)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All pole model');
grid on
%ARMA Model
n=[1 -a*exp(i*th)];
d=[1 -a*exp(i*th)];
subplot(325)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(326)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('ARMA model');
grid on

5. 5
3. Phase Response for 𝜽 𝟏 𝟎 and a=1 for 𝟏 𝟏
clc, clear all, close all;
th=180;
a=1;
%All zero Model
n=[1 -a*exp(i*th)];
d=1;
subplot(321)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(322)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All zero model');
grid on
%All pole model
d=[1 -a*exp(i*th)];
n=1;
subplot(323)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(324)
plot((w-pi)./pi,fftshift(abs(H)))

6. 6
xlabel 'omega (x pi)'
title('All pole model');
grid on
%ARMA Model
n=[1 -a*exp(i*th)];
d=[1 -a*exp(i*th)];
subplot(325)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(326)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('ARMA model');
grid on
4. Phase Response for 𝜽 𝟎 and a=2 for 𝟏 𝟏
clc, clear all, close all;
th=0;
a=2;
%All zero Model
n=[1 -a*exp(i*th)];
d=1;
subplot(321)

7. 7
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(322)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All zero model');
grid on
%All pole model
d=[1 -a*exp(i*th)];
n=1;
subplot(323)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(324)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All pole model');
%ARMA Model
n=[1 -a*exp(i*th)];
d=[1 -a*exp(i*th)];
subplot(325)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(326)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('ARMA model');

8. 8
5. Phase Response for 𝜽 𝟎 and a=0.5 for 𝟏 𝟏
clc, clear all, close all;
th=0;
a=0.5;
%All zero Model
n=[1 -a*exp(i*th)];
d=1;
subplot(321)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(322)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All zero model');
grid on
%All pole model
d=[1 -a*exp(i*th)];
n=1;
subplot(323)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(324)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('All pole model');
axis([-2 2 -1 100])
grid on
%ARMA Model
n=[1 -a*exp(i*th)];
d=[1 -a*exp(i*th)];
subplot(325)
zplane(n,d)
[H,w]=freqz(n,d,'whole');
subplot(326)
plot((w-pi)./pi,fftshift(abs(H)))
xlabel 'omega (x pi)'
title('ARMA model');
grid on

9. 9
Questions:
Q1): Suggest when do you face a linear phase and in which case have you
observed a non-linear phase?
Ans: Using ‘fvtool(n,d)’ in command window and giving inputs numerator and
denominator co-efficients, we observe linear phase in case of all pole model and all
zero model only. In ARMA model, non-linear phase is observed.
Q2): What is the difference between the phase of All Pole and All Zero Model?
Ans: The phase response of pole and zero are 180 degree out of phase.

10. 10
Q3): What is the effect of change of 𝜽 on phase of a pole or zero?
Ans: When 𝜽=0 the phase response is straight line.
When we change 𝜽=-180 to 𝜽=180 the phase response will flipped. There will
be shift of .

11. 11
Q4): What is the effect of change of 𝒂 > 𝟏 on phase of a pole or zero?
Ans: When distance of pole or zero is increased from 1, it has no effect on the
phase response.
Q5): What is the effect of change of 𝒂 < 𝟏 on phase of a pole or zero?
Ans: When distance of pole or zero is decreased from unity, the phase response will
become 0 at 0 and 180 while it is maximum in middle.

12. 12
Q6): When a=1 what is the effect of phase in case of an ARMA model?
Ans: In case of ARMA model Phase response is non-linear because when pole and
zero both are combined, they will cancel the effect of each other.
Conclusion:
In this lab we learnt the phase response of :
 All zero model
 All pole model
 ARMA model
 We concluded that phase response is linear in case of All pole and All zero
model but it is non linear in ARMA model because when pole and zero both are
combined, they will cancel the effect of each other.
 Phase response of pole and zero are 180 degree out of phase.
 The point where zero lies has minimum intensity.
 The point where pole lies has maximum intensity.