Chapter 5 transform analysis of linear
time-invariant system
5.1 the frequency response of LTI system
5.2 system function
5.3 frequency response for rational system function
5.4 relationship between magnitude and phase
5.5 all-pass system
5.6 minimum-phase system
5.7 linear system with generalized linear phase
5.1 the frequency response of LTI system
H ( e j ) 


h[ n]e  jn
n  
H (e
j
) | H (e
j
)|e
jH ( e j )
magnitude-frequency characteristic：
1. | H (e j ) |
magnitude response or gain
2. | H (e j ) |2  H (e j ) H  (e j )
magnitude square function
3.20 log10 | H (e j ) |, unit : dB
log magnitude
 20 log 10 | H (e j ) |
magnitude attenuation
log magnitude
linear magnitude
transform curve
from linear to log
magnitude
phase-frequency characteristic：
4.H (e j )
phase response
5.    ARG[ H (e j )]  
principal phase
6. arg[ H (e j )]
continuous phase
d arg[ H (e j )] group delay
7.grd [ H (e )]  
d
j
Figure 5.7
EXAMPLE
Figure 5.1
understand group delay
  0.85 ,0.25 ,0.5
Figure 5.2
  0.5 ,0.25

5.2 system function
H ( z) 

h[n]z  n
n  
Characteristics of zeros and poles：
（1）take origin and zeros and poles at infinite into consideration, the numbers of
zeros and poles are the same.
（2）for real coefficient, complex zeros and poles are conjugated, respectively.
（3）if causal and stable, poles are all in the unit circle.
（4）FIR：have no nonzero poles, called all-zeros type, steady
IIR：have nonzero pole; if no nonzero zeros , called all-poles type
EXAMPLE
y[n]  x[n]  x[n  1]  y[n  1]
H ( z) 
1  z 2
 1  z 1
1  z 1
 y[n]  x[n]  x[n  1]
Difference about
zeros and poles in
FIR and IIR
5.3 frequency response for rational system function
H ( z) 
 H (e j )
1.formular method
H (e j )  H ( z ) |z e j
2. Geometrical method
M
H ( z )  Bz N  M
 (z  c
)
k
k 1
N
 (z  d
k
)
k 1
M
| H (e j ) || H ( z ) | z e j | B |
j
|
e
  ck |
k 1
N
| e   d
j
k
|
k 1
M
arg[ H (e )]  arg[ B]   arg[ e
j
k 1
j
N
 ck ]   arg[ e j  d k ] | ( N  M )
k 1
EXAMPLE
| H (e j ) || B |
L1
, arg[ H (e j )]    1   2   3
L2 L3
magnitude response in w near zeros is minimum, there are zeros in unit circle, then
the magnitude is 0；
magnitude response in w near poles is maximum；zeros and poles counteracted each
other and in origin does not influence the magnitude.
EXAMPLE
H ( z) 
1
1  az
1
, | z || a |
| H (e j ) |
1 /(1  a)
1 /(1  a )
a
0

2
ω
arg{ H (e j )]
ω
0

2
EXAMPLE
y[n]  x[n]  x[n  4]
3.matlab method
EXAMPLE
B=1
A=[1,-0.5]
figure(1)
zplane(B,A)
figure(2)
freqz(B,A)
figure(3)
grpdelay(B,A,10)
H ( z) 
1
1  0.5 z 1
5.4 relationship between magnitude and phase
| H (e j ) | H ( z )
| H (e j ) |2  H ( z ) H * (1 / z * ) | H (e j ) |2 e j  z
uniform

4 conjugate reciprocal zeros
4 conjugate reciprocal poles

nonuniform
H ( z)
EXAMPLE Pole-zero plot for H ( z ) H * (1/ z* ) ，H(z): causal and stable，
Confirm the poles and zeros
 z1 , z 2 , z3
z , z , z
 1 2 6
p1 , p2 , p3 
 z 4 , z5 , z3
 z 4 , z5 , z 6
Figure 5.20
5.5 all-pass system
| H ap (e j ) | cons tan t
Mr
1
 dk
Mc
 ( z 1  e * ) ( z 1  e ) 
k
k 

 (1  e z 1) (1  e * z 1) 
k
k

k 1 
1  d z 
H ap ( z )  A
k 1
z
k
1
Zeros and poles are conjugate reciprocal
For real coefficient, zeros are conjugated , poles are conjugated.
EXAMPLE
Y
3/4
N
4/3
Y
Y
Characteristics of causal and stable all-pass system:
grd[ H ap (e j )]  0
arg[ H ap (e j )]  0, for 0    
application：
1. compensate the phase distortion
H ( z ) H ap ( z )  H ' ( z ),| H (e j ) || H ' (e j ) |
2. compensate the magnitude distortion together with minimum-phase system
H ( z ).H min ( z )  H ap ( z )
5.6 minimum-phase system
inverse system:
H ( z ) H i ( z )  1, that
is , H i ( z )  1 / H ( z )
then , X ( z ) H ( z ) H i ( z )  X ( z )
or : h[n] * hi [n]   [n],
then , x[n] * h[n] * hi [n]  x[n]
the condition of h[n] * hi [n]   [n] :,
ROC of H ( z ) and H i ( z ) must have intersecti on
explanation：
（1）not all the systems have inverse system。
（2）inverse system may be nonuniform。
（3）the inverse system of causal and stable system may not be causal and stable。
the condition of both original and its inverse system causal and stable：
zeros and poles are all in the unit circle，such system is called minimumphase system,
corresponding h[n] is minimum-phase sequence。
poles are all in the unit circle, zeros are all outside the unit circle, such
system is called maximum-phase system。
minimum-phase and all-pass decomposition：
If H(z) is rational, then :
H ( z )  H min ( z ) H ap ( z )
| H (e j ) || H min (e j ) |
poles outside
the unit circle
all-pass system: counteracted
zeros and poles, zeros and
poles outside the circle
minimum-phase system:
conjugate reciprocal
zeros and poles
zeros outside the
unit circle
Application of minimum-phase and all-pass decomposition：
Compensate for amplitude distortion
Figure 5.25
H d ( z )  H min ( z ) H ap ( z ), H c ( z ) 
1
H min ( z )
,
total system : H ( z )  H d ( z ) H c ( z )  H ap ( z )
Minimum-phase system and some all-pass system in cascade can make up of another
system having the same magnitude response, so there are infinite systems having the
same magnitude response.
H min ( z ) H ap ( z )  H ( z )
| H min (e j ) || H (e j ) |
Properties of minimum-phase systems:
（1）minimum phase-delay
 arg[ H min (e j )]   arg[ H (e j )]
 arg[ H ap (e j )]  0,0    
（2）minimum group-delay
grd[ H min (e j )]  grd[ H (e j )]
 grd[ H ap (e j )]  0
（3）minimum energy-delay（i.e. the partial energy is most concentrated around n=0）
define : E[ n] 
n
2
|
h
[
m
]
|
partial energy , then

m 0
Emin [ n]  E[ n]
| H min (e j ) || H (e j ) |


n 0
n 0
  | h[ n] |2   | hmin [ n] |2
then, Emin []  E[]
but , Emin [ n]  E[ n], n  
EXAMPLE
最小相位
minimum phase
maximum phase
Systems having the same
magnitude response
Figure 5.30
minimum phase
Figure 5.31
Figure 5.32
5.7 linear system with generalized linear phase
5.7.1 definition
5.7.2 conditions of generalized linear phase system
5.7.3 causal generalized linear phase (FIR)system
5.7.1 definition
Strict:
H (e j ) | H (e j ) | e j
|  | 
arg[ H (e j )]   (line ), grd[ H (e j )]   (real )
Generalized:
H (e j )  A(e j )e  j  j
|  | 
A(e j ) is a real function
arg[ H (e j )]     ,
grd[ H (e j )]  
Systems having constant group delay
phase
EXAMPLE
ideal delay system
hid [n]   [n  m]
y[n]  x[n]  hid [n]  x[n  m]
H id (e j )  e  jm
EXAMPLE
|  | 
differentiator：magnitude and phase are all linear
H (e j )  j / T 
 j / 2

e
,  0,    / 2, A(e j ) 
T
T
physical meaning：
all components of input signal are delayed by the same amount in strict
linear phase system ，then there is only magnitude distortion, no phase distortion.
it is very important for image signal and high-fidelity audio signal to have no
phase distortion.
when B=0, for generalized linear phase, the phase in the whole band is not
linear, but is linear in the pass band, because the phase +PI only occurs when
magnitude is 0, and the magnitude in the pass band is not 0.
EXAMPLE
square wave with fundamental
frequency 100 Hz
linear phase filter：
lowpass filter with cut-off
frequency 400Hz
nonlinear phase filter：
lowpass filter with cut-off
frequency 400Hz
Generalized linear phase in the
pass band is strict linear phase
Generalized linear phase in the
pass band is strict linear phase
5.7.2 conditions of generalized linear phase system
   0 or 

(1)2  M (int eger )
 h[2  n]  h[n]

    / 2 or 3 / 2

(2)  2  M (int eger )
 h[2  n]  h[n]

Or:
(1)if : h[ M  n]  h[ n], n  ,...,
M is a int eger , then :
   0 or 

  M / 2
( 2)if : h[ M  n]   h[ n], n  ,...,
M is a int eger , then :
    / 2 or 3 / 2

  M / 2
M:even
M:odd
Figure 5.35
M:not integer
EXAMPLE
M:not integer
EXAMPLE
determine whether these system is linear phase,generalized or strict?a and ß=?
(1)
(2)
3
2
2
(3)
(4)
3
2
1
1
5.7.3 causal generalized linear phase (FIR)system
h[n]   h[ M  n],0  n  M
h[n]  0, for
n  0 or n  M
(1)typeI :
h[n]  h[ M  n], M : even
(2)typeII :
h[n]  h[ M  n], M : odd
(3)typeIII :
h[n]   h[ M  n], M : even
(4)typeIV :
h[n]   h[ M  n], M : odd
Magnitude and phase characteristics of the 4 types：
M
( )      / 2
2
A(e j ) 
M
 M


h(n) cos 
 n   
2

 2
n 0

 M
  M /2
typeI : A(e )   h[n] cos   n     a[k ] cos( k )
  k 0
n 0
 2
where : a[0]  h[ M / 2], a[k ]  2h[ M / 2  k ], k  1,2...M / 2
j
M
 M
  M /2
typeII : A(e )   h[n] cos   n     b[k ] cos( (k  1 / 2))
  k 0
n 0
 2
where : b[k ]  2h[( M  1) / 2  k ], k  1,2...( M  1) / 2
j
M
 M
  M /2
typeIII : A(e )   h[n] cos   n     c[k ] sin( k )
  k 0
n 0
 2
where : c[k ]  2h[ M / 2  k ], k  1,2...M / 2
j
M
 M
  M /2
typeIV : A(e )   h[n] cos   n     d [k ] sin(  (k  1 / 2))
  k 0
n 0
 2
where : d [k ]  2h[( M  1) / 2  k ], k  1,2...( M  1) / 2
j
M
I
| H (e j ) |
ARG{H (e j )}
grd{H (e j )}
II
III
| H (e j ) |
ARG{H (e j )}
grd{H (e j )}
IV
Characteristic of zeros: commonness
1/z1*
z3
z5
z1
z2
1/z2
z4
z1*
1/z1
z3*
Characteristic of every type：
Figure 5.41
characteristic of magnitude get from characteristic of zeros：
type I：
H (e j )  0
type II：
type III：
H (e j 0 )  0, H (e j )  0
type IV：
H (e j 0 )  0
Application of 4 types of linear phase system:
M is even
M is odd
low high band pass band stop low high band pass band stop
h[n] is even (I)
Y Y
h[n] is odd (III) N N
Y
Y
Y
N
Y N
Y
N
(II)
N Y
Y
N (IV)
summary
5.1 the frequency response of LTI system :
H ( e j )
5.2 system function
5.3 frequency response for rational system function:
j
H
(
z
)


H
(
e
)
5.4 relationship between magnitude and phase :
5.5 all-pass system
| H (e j ) | 2 确定

 H ( z )
5.6 minimum-phase system
5.7 linear system with generalized linear phase （FIR)
5.7.1 definition:
5.7.2 conditions : h[n] is symmetrical
H (e j )  A(e j )e j ，
|  | 
5.7.3 causal generalized linear phase system
1.condition
2.classification
3.characteristics of magnitude and phase , filters in point respectively
4.analyse of characteristic of magnitude from the zeros of system function
requirement:
concept of magnitude and phase response, group delay;
transformation among system function, phase response and
difference equation;
concept of all-pass, minimum-phase and linear phase system and
characteristic of zeros and poles;
minimum-phase and all-pass decomposition;
conditions of linear phase system , restriction of using as filters
key and difficulty：
linear phase system
exercises
5.17 complementarity：minimum-phase and all-pass decomposition
5.21
5.45
5.53
the first experiment
Get subjects from the experiment instruction book or downloading from network：
problem 1（D）
problem 11
problem 13（C）
problem 22（A）
problem 24（A）（C）