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5.1 the frequency response of LTI system5.2 system function5.3 frequency response for rational system function5.4 relationship between magnitude and phase5.5 all-pass system5.6 minimum-phase system5.7 linear system with generalized linear phase
Chapter 5 transform analysis of linear time-invariant system
5.1 the frequency response of LTI system
)(|)(|)(
][)(
jeHjjj
n
njj
eeHeH
enheH
|)(|.1 jeH magnitude response or gain
)()(|)(|.2 2 jjj eHeHeH magnitude square function
dBuniteH j :|,)(|log20.3 10 log magnitude
|)(|log20 10jeH magnitude attenuation
magnitude-frequency characteristic：
log magnitude
linear magnitude
transform curve from linear to log
magnitude
phase-frequency characteristic：
)(.4 jeH phase response
)]([.5 jeHARG principal phase
)](arg[.6 jeH continuous phase
d
eHdeHgrd
jj )](arg[
)]([.7 group delay
Figure 5.7
Figure 5.1EXAMPLEunderstand group delay
Figure 5.2
5.0,25.0,85.0
25.0,5.0
5.2 system function
n
nznhzH ][)(
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
]1[][][
11
1)(
]1[]1[][][
11
2
nxnxny
zz
zzH
nynxnxnyEXAMPLE
Difference about zeros and poles in
FIR and IIR
)()( jeHzH
5.3 frequency response for rational system function
jezj zHeH |)()(
1.formular method
2. Geometrical method
N
kk
M
kk
MN
dz
czBzzH
1
1
)(
)()(
)(|]arg[]arg[]arg[)](arg[
||
|||||)(||)(|
11
1
1
MNdeceBeH
de
ceBzHeH
N
kk
jM
kk
jj
N
kk
j
M
kk
j
ez
jj
32132
1 )](arg[,|||)(| jj eHLL
LBeHEXAMPLE
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.
ω
|)(| jeH
)1/(1 a
)1/(1 a
20
||||,1
1)(
1az
azzH
EXAMPLE
ω
20
)](arg{ jeH
a
]4[][][ nxnxny
EXAMPLE
B=1A=[1,-0.5]figure(1)zplane(B,A)figure(2)freqz(B,A)figure(3)grpdelay(B,A,10)
15.01
1)(
zzHEXAMPLE3.matlab method
5.4 relationship between magnitude and phase
)(|)(| zHeH j
poles reciprocal conjugate 4
zeros reciprocal conjugate 4
)(|)(|)/1()(|)(|nonuniform
2**
uniform
2
zHeHzHzHeH zejj
j
Figure 5.20
654
354
621
321
321
,,
,,
,,
,,
,,
zzz
zzz
zzz
zzz
ppp
EXAMPLE )/1()( ** zHzHPole-zero plot for ， H(z): causal and stable，Confirm the poles and zeros
5.5 all-pass system
tconseH jap tan|)(|
cr M
k k
k
k
kM
k k
kap
ze
ez
ze
ez
zd
dzAzH
11*
1
1
*1
11
1
)1(
)(
)1(
)(
1)(
Zeros and poles are conjugate reciprocalFor real coefficient, zeros are conjugated , poles are conjugated.
4/3 3/4
EXAMPLE
Y
Y Y
N
0)]([ jap eHgrd
0,0)](arg[ foreH jap
Characteristics of causal and stable all-pass system:
|)('||)(|),(')()( jjap eHeHzHzHzH
application： 1. compensate the phase distortion
2. compensate the magnitude distortion together with minimum-phase system
)()().( min zHzHzH ap
5.6 minimum-phase system
][][*][*][,
],[][*][:
)()()()(,
)(/1)(,,1)()(
nxnhnhnxthen
nnhnhor
zXzHzHzXthen
zHzHisthatzHzH
i
i
i
ii
inverse system:
onintersecti havemust )( and )( of
:,][][*][ ofcondition the
zHzHROC
nnhnh
i
i
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 minimum-phase 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。
|)(||)(|
)()()(
min
min
jj
ap
eHeH
zHzHzH
zeros outside the unit circle
poles outside the unit circle
minimum-phase system: conjugate reciprocal
zeros and poles
all-pass system: counteracted zeros and poles, zeros and
poles outside the circle
minimum-phase and all-pass decomposition：If H(z) is rational, then :
)()()()(:
,)(
1)(),()()(
minmin
zHzHzHzHsystemtotal
zHzHzHzHzH
apcd
capd
Figure 5.25
Application of minimum-phase and all-pass decomposition：
Compensate for amplitude distortion
Properties of minimum-phase systems:
)()()(min zHzHzH ap |)(||)(| min jj eHeH
（ 1） minimum phase-delay
0,0)](arg[
)](arg[)](arg[ min
jap
jj
eH
eHeH
（ 2） minimum group-delay
0)]([
)]([)]([ min
jap
jj
eHgrd
eHgrdeHgrd
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.
nnEnEbut
EEthen
nhnh
eHeH
nEnE
thenmhnEdefine
nn
jj
n
m
],[][,
][][,
|][||][|
|)(||)(|
][][
,energy partial |][|][:
min
min
0
2min
0
2
min
min
0
2
（ 3） minimum energy-delay（ i.e. the partial energy is most concentrated around n=0）
Figure 5.30
最小相位maximum phase
EXAMPLE
minimum phase Systems having the same magnitude response
Figure 5.31
minimum phase
Figure 5.32
5.7 linear system with generalized linear phase
5.7.1 definition5.7.2 conditions of generalized linear phase system 5.7.3 causal generalized linear phase (FIR)system
5.7.1 definition
)()]([),()](arg[
|||)(|)(
realeHgrdlineeH
eeHeHjj
jjj
Strict:
)]([
,)](arg[
)(
||)()(
j
j
j
jjjj
eHgrd
eH
functionrealaiseA
eeAeH
Generalized:
Systems having constant group delay
phase
||)(
][][][][
][][
mjjid
id
id
eeH
mnxnhnxny
mnnh
EXAMPLE ideal delay system
TeAe
TTjeH jjj )(,2/,0,/)( 2/
differentiator： magnitude and phase are all linearEXAMPLE
physical meaning：all components of input signal are delayed by the same amount in strict line
ar 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 li
near, 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.
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
EXAMPLE
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
][]2[
)(int2
2/32/
)2(
][]2[
)(int2
0
)1(
nhnh
egerM
or
nhnh
egerM
or
2/
2/32/
:,int
,,...],[][:)2(
2/
0
:,int
,,...],[][:)1(
M
or
thenegeraisM
nnhnMhif
M
or
thenegeraisM
nnhnMhif
Or:
Figure 5.35
M:even
M:odd
M:not integer
EXAMPLE
M:not integer
determine whether these system is linear phase,generalized or strict?a and ß=?
2
2
3
1
2
1
3
EXAMPLE
(1) (2)
(3) (4)
5.7.3 causal generalized linear phase (FIR)system
Mnornfornh
MnnMhnh
0,0][
0],[][
oddMnMhnh
typeIV
evenMnMhnh
typeIII
oddMnMhnh
typeII
evenMnMhnh
typeI
:],[][
:)4(
:],[][
:)3(
:],[][
:)2(
:],[][
:)1(
Magnitude and phase characteristics of the 4 types：
2/2
)( M
M
n
j nM
nheA0
22cos)()(
2/...2,1],2/[2][],2/[]0[:
)cos(][2
cos][)(:2/
00
MkkMhkaMhawhere
kkanM
nheAtypeIM
k
M
n
j
2/)1...(2,1],2/)1[(2][:
))2/1(cos(][2
cos][)(:2/
00
MkkMhkbwhere
kkbnM
nheAtypeIIM
k
M
n
j
2/...2,1],2/[2][:
)sin(][2
cos][)(:2/
00
MkkMhkcwhere
kkcnM
nheAtypeIIIM
k
M
n
j
2/)1...(2,1],2/)1[(2][:
))2/1(sin(][2
cos][)(:2/
00
MkkMhkdwhere
kkdnM
nheAtypeIVM
k
M
n
j
|)(| jeH
I II
)}({ jeHARG
)}({ jeHgrd
III IV
|)(| jeH
)}({ jeHARG
)}({ jeHgrd
z5
z4
z3*
z3
1/z2
1/z1*
1/z1
z1
z1*
z2
Characteristic of zeros: commonness
Figure 5.41
Characteristic of every type：
0)( jeH
type I：
type II：
type III：
type IV：
0)(,0)( 0 jj eHeH
0)( 0 jeH
characteristic of magnitude get from characteristic of zeros：
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 Y Y Y N Y N (II)
h[n] is odd (III) N N Y N N Y Y N (IV)
Application of 4 types of linear phase system:
5.1 the frequency response of LTI system :5.2 system function5.3 frequency response for rational system function:5.4 relationship between magnitude and phase :5.5 all-pass system5.6 minimum-phase system5.7 linear system with generalized linear phase （ FIR) 5.7.1 definition: 5.7.2 conditions : h[n] is symmetrical 5.7.3 causal generalized linear phase system
1.condition2.classification3.characteristics of magnitude and phase , filters in point respectively4.analyse of characteristic of magnitude from the zeros of system function
)( jeH
)()( jeHzH
)(|)(| 2 zHeH j 确定
||)()( ，jjj eeAeH
summary
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 decomposition5.215.455.53
the first experiment
problem 1（ D）problem 11problem 13（ C）problem 22（ A）problem 24（ A）（ C）
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