Transcript Document

Relationship between Magnitude and Phase
Quote of the Day
Experience is the name everyone gives to their
mistakes.
Oscar Wilde
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Relation between Magnitude and Phase
• For general LTI system
– Knowledge about magnitude doesn’t provide any information
about phase
– Knowledge about phase doesn’t provide any information about
magnitude
• For linear constant-coefficient difference equations however
– There is some constraint between magnitude and phase
– If magnitude and number of pole-zeros are known
• Only a finite number of choices for phase
– If phase and number of pole-zeros are known
• Only a finite number of choices for magnitude (ignoring scale)
• A class of systems called minimum-phase
– Magnitude specifies phase uniquely
– Phase specifies magnitude uniquely
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
2
Square Magnitude System Function
• Explore possible choices of system function of the form
 
He
j
   


 H e j H e j  H* 1 / z* Hz
2

z  e j
• Restricting the system to be rational
1  c z 

M
 b0
Hz    
 a0 
1  c z

M
1
k
k 1
N
*
 1  d z 
k

H 1/z
1
*

 b0
  
 a0 
k 1

Cz   Hz H 1 / z
*

 1  d z
2
k
k 1
N
1
 
*
k
 1  d z 1  d z
k
1
k 1
j
*
k
 1  c z 1  c z
M
*
k 1
N
k 1
• The square system function
 b0 
  
 a0 
*
k
*
k
2
• Given H e
we can get C(z)
• What information on H(z) can we get from C(z)?
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
Poles and Zeros of Magnitude Square System Function
 1  c z 1  c z
M

Cz   Hz H 1 / z
*
*

 b0 
  
 a0 
2
k
k 1
N
1
*
k
 1  d z 1  d z
k
1
k 1
*
k
• For every pole dk in H(z) there is a pole of C(z) at dk and
(1/dk)*
• For every zero ck in H(z) there is a zero of C(z) at ck and
(1/ck)*
• Poles and zeros of C(z) occur in conjugate reciprocal pairs
• If one of the pole/zero is inside the unit circle the reciprocal
will be outside
– Unless there are both on the unit circle
• If H(z) is stable all poles have to be inside the unit circle
– We can infer which poles of C(z) belong to H(z)
• However, zeros cannot be uniquely determined
– Example to follow
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
4
Example
• Two systems with





1  z 1  2z 
H z 
1  0.8e z 1  0.8e
2 1  z1 1  0.5z1
H1 z 
1  0.8e j / 4z1 1  0.8e j / 4z1
1
2

1
j / 4 1
 j / 4 1
z

• Both share the same magnitude square system function
H1 z 
Copyright (C) 2005 Güner Arslan
H2 z
351M Digital Signal Processing
 
 H z H 1 / z 
Cz   H1 z H1* 1 / z*
2
*
2
*
5
All-Pass System
• A system with frequency response magnitude constant
• Important uses such as compensating for phase distortion
• Simple all-pass system
z 1  a*
Hap z 
1  az1
• Magnitude response constant
 
Hap e
j
* j
e j  a*
 j 1  a e

e
 j
1  ae
1  ae j
• Most general form with real impulse response




z1  dk Mc z1  ck* z1  ck
Hap z  A
1 
1
1  ck*z1
k 1 1  dk z
k 1 1  ck z
Mr


• A: positive constant, dk: real poles, ck: complex poles
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
6
Phase of All-Pass Systems
* j
e j  a*
 j 1  a e
Hap e 
e
 j
1  ae
1  ae j
• Let’s write the phase with a represented in polar form
 
j
 e j  re j 
 r sin   





2
arctan
1  r cos  
j  j 
1

re
e




• The group delay of this system can be written as
 e j  re j 
1  r2
1  r2
grd


j  j 
2
2
j  j


1

re
e
1

2
r
cos




r


1  re e
• For stable and causal system |r|<1
– Group delay of all-pass systems is always positive
• Phase between 0 and  is always negative
  
argHap ej  0 for 0    
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
7