Generalized Linear Phase
Quote of the Day
The mathematical sciences particularly exhibit
order, symmetry, and limitation; and these are
the greatest forms of the beautiful.
Aristotle
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, ©1999-2000 Prentice Hall
Inc.
Linear Phase System
• Ideal Delay System
 
Hid ej  e j

• Magnitude, phase, and group delay
 
H e   
grdH e   
Hid e j  1
id
j
id
• Impulse response
hid n 
• If =nd is integer
j
sinn   
n   
hid n  n  nd 
• For integer  linear phase system delays the input
yn  xn  hid n  xn  n  nd   xn  nd 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
2
Linear Phase Systems
• For non-integer  the output is an interpolation of samples
• Easiest way of representing is to think of it in continuous
hc t   t  T 
and
Hc j   e jT
• This representation can be used even if x[n] was not originally
derived from a continuous-time signal
• The output of the system is
yn  xnT  T 
• Samples of a time-shifted, band-limited interpolation of the
input sequence x[n]
• A linear phase system can be thought as
 
 
H e j  H e j e  j
• A zero-phase system output is delayed by 
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
Symmetry of Linear Phase Impulse Responses
• Linear-phase systems
 
 
H e j  H e j e  j
=5
• If 2 is integer
– Impulse response symmetric
h2  n  hn
=4.5
=4.3
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
4
Generalized Linear Phase System
• Generalized Linear Phase
   Ae e
He
j
j
 j  j
 
A e j : Real function of 
 and  constants
• Additive constant in addition to linear term
• Has constant group delay
d
  grd H e j  
arg H e j  
d
  
   
• And linear phase of general form
  
arg H e j    
Copyright (C) 2005 Güner Arslan
0
351M Digital Signal Processing
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Condition for Generalized Linear Phase
• We can write a generalized linear phase system response as
H e j  A e j e j  j  A e j cos     jA e j sin   
 
 
He    hne
j

n  
 
 jn

 


 hn cosn  j  hn sinn
n  
n  
• The phase angle of this system is 
  hn sinn
sin   
 n 
cos   
 hn cosn
n  
• Cross multiply to get necessary condition for generalized linear
phase


n  
n  
 hn cosn sin      hnsinn cos     0

 hncosn sin     sinn cos     0
n  


n  
n  
 hn sin    n   hnsin  n     0
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
6
Symmetry of Generalized Linear Phase
• Necessary condition for generalized linear phase

 hn sin  n     0
• For =0 or 
n  

 h2  n  hn
 hn sinn     0 
n  
• For = /2 or 3/2

 h2  n  hn
 hn cosn     0 
n  
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
7
Causal Generalized Linear-Phase System
• If the system is causal and generalized linear-phase
hM  n  hn
• Since h[n]=0 for n<0 we get
hn  0
n  0 and n  M
• An FIR impulse response of length M+1 is generalized linear
phase if they are symmetric
• Here M is an even integer
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
8
Type I FIR Linear-Phase System
• Type I system is defined with
symmetric impulse response
hn  hM  n
for 0  n  M
– M is an even integer
• The frequency response can
be written as
    hne
He
M
j
 jn
n0
e
 jM / 2
M / 2





a
n
cos

n


n

0


• Where
a0  hM / 2
ak   2hM / 2  k 
Copyright (C) 2005 Güner Arslan
for k  1,2,...,M/2
351M Digital Signal Processing
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Type II FIR Linear-Phase System
• Type I system is defined with
symmetric impulse response
hn  hM  n
for 0  n  M
– M is an odd integer
• The frequency response can
be written as
    hne
He
j
M
 jn
n0
e
 jM / 2
M1 / 2
 
1  
  bn cos  n   
2  
 
 n 1
• Where
bk   2hM  1 / 2  k 
for k  1,2,..., M  1/2
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
10
Type III FIR Linear-Phase System
• Type I system is defined with
symmetric impulse response
hn  hM  n
for 0  n  M
– M is an even integer
• The frequency response can
be written as
    hne
He
M
j
 jn
n0
 je
• Where
 jM / 2
M / 2





c
n
sin

n


n

1


ck   2hM / 2  k 
for k  1,2,..., M/2
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
11
Type IV FIR Linear-Phase System
• Type I system is defined with
symmetric impulse response
hn  hM  n
for 0  n  M
– M is an odd integer
• The frequency response can
be written as
    hne
He
j
M
 jn
n0
 jM / 2
 je
M1 / 2
 
1  
  dn sin  n   
2  
 
 n1
• Where
dk   2hM  1 / 2  k 
for k  1,2,..., M  1/2
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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Location of Zeros for Symmetric Cases
• For type I and II we have
z
hn  hM  n 

 Hz  zMH z1
• So if z0 is a zero 1/z0 is also a zero of the system
• If h[n] is real and z0 is a zero z0* is also a zero
• So for real and symmetric h[n] zeros come in sets of four
• Special cases where zeros come in pairs
 
– If a zero is on the unit circle reciprocal is equal to conjugate
– If a zero is real conjugate is equal to itself
• Special cases where a zero come by itself
– If z=1 both the reciprocal and conjugate is itself
• Particular importance of z=-1
H 1   1 H 1
M
– If M is odd implies that
H 1  0
– Cannot design high-pass filter with symmetric FIR filter and M odd
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
13
Location of Zeros for Antisymmetric Cases
• For type III and IV we have
 
z
hn  hM  n 

 Hz  zMH z1
• All properties of symmetric systems holds
• Particular importance of both z=+1 and z=-1
– If z=1
H1  H1  H1  0
• Independent from M: odd or even
– If z=-1
H 1   1
H 1
M1
• If M+1 is odd implies that
H 1  0
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
14
Typical Zero Locations
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351M Digital Signal Processing
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Relation of FIR Linear Phase to Minimum-Phase
• In general a linear-phase FIR system is not minimum-phase
• We can always write a linear-phase FIR system as
Hz  Hmin zHuc zHmax z
• Where
•
•
•
•
 
Hmax z  Hmin z1 zMi
And Mi is the number of zeros
Hmin(z) covers all zeros inside the unit circle
Huc(z) covers all zeros on the unit circle
Hmax(z) covers all zeros outside the unit circle
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351M Digital Signal Processing
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Example
• Problem 5.45
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