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![Page 1: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/1.jpg)
23/4/21 1Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Biomedical Signal processingChapter 5 Transform Analysis
of Linear Time-Invariant Systems
Zhongguo Liu, Biomedical Engineering
School of Control Science and Engineering, Shandong University
山东省精品课程山东省精品课程《《生物医学信号处理生物医学信号处理 (( 双语双语 )) 》》http://course.sdu.edu.cn/bdsp.htmlhttp://course.sdu.edu.cn/bdsp.html
![Page 2: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/2.jpg)
2
Chapter 5 Transform Analysis of Linear Time-
Invariant Systems5.0 Introduction5.1 Frequency Response of LTI Systems5.2 System Functions For Systems
Characterized by Linear Constant-coefficient Difference equation
5.3 Frequency Response for Rational System Functions
5.4 Relationship Between Magnitude and Phase5.5 All-Pass System5.6 Minimum-Phase Systems5.7 Linear Systems with Generalized Linear
Phase
![Page 3: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/3.jpg)
3
5.0 IntroductionAn LTI system can be characterized in
time domain by impulse responseOutput of the LTI system:
h n
k
knhkxnhnxny
zXzHzY
in Z-domain by system function
in frequency-domain by Frequency response
jwjwjw eXeHeY
With Fourier Transform and Z-transform, an LTI system can be characterized H z
jwH e
![Page 4: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/4.jpg)
4
5.1 Frequency Response of LTI Systems
jwjwjw eXeHeY
jwjwjw eXeHeY
Phase response (phase shift) jweH
jwjwjw eXeHeY
jweHFrequency response
jweHMagnitude response (gain)
distortions
change on useful signal
syst
em
Usefulinput signal
+deleteriou
ssignal
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5
5.1.1 Ideal Frequency-Selective Filters
Ideal lowpass filter
1,
0,cjw
lpc
w wH e
w w
jweH
0 cwcw 22
1
sin,
c
lp
w n
nnh n
Noncausal, not computationally realizable
no phase distortion
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6
5.1.1 Ideal Frequency-Selective Filters
Ideal highpass filter
0,
1,cjw
hpc
w wH e
w w
sin,c
hp
w nh n n n
n
0 cwcw 22
jweH
1
1 jwlpH e
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5.1.1 Ideal Frequency-Selective Filters
Ideal bandpass filter
1 21,
0,
c cjwbp
w w wH e
others
01cw
1cw
jweH
1
2cw2cw
![Page 8: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/8.jpg)
8
5.1.1 Ideal Frequency-Selective Filters
Ideal bandstop filter
1 20,
1,
c cjwbs
w w wH e
others
0
jweH
1
1cw1cw
2cw2cw
![Page 9: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/9.jpg)
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5.1.2 Phase Distortion and Delay
did nnnh
dnnxny
djwnjwid eeH
1jwid eH wwneH d
jwid ,
The frequency response
The impulse response
To understand the effect of the phase and the group delay of a linear system, first consider the ideal delay system:
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Group Delay(群延迟, grd )
For ideal delay system
argjw jwdw H e Hd eg
wr
d
d d
dwn n
dw
0arg jwdIf H e wn dthen w n
The group delay represents a convenient measure of the linearity of the phase.
arg arg djwnjwd dw H e e
dw dw
![Page 11: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/11.jpg)
Given a narrowband input x[n]=s[n]cos(w0n) for a system with frequency response H(ejw), it is assumed that X(ejw) is nonzero only around w =w0
11
Group Delay(群延迟, grd )
00 00[ ] cos( )d
jd
wy n H e s n wn wn n
0arg ,jwdI wf H e n dnthen w
it can be shown (see Problem 5.57) that the response y[n] to x[n] is
the time delay of the envelope s[n] is .
dn
Group Delay
![Page 12: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/12.jpg)
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25.0w85.0w
5.0w
Example 5.1 Effect of Attenuation and Group Delay
Three consecutive narrowband pulsesis applied to a filter
连贯的
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Ex. 5.1 Effect of Attenuation and Group Delay
Filter frequency response
Group Delay
magnitude
25.0w
85.0w5.0w
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25.0w85.0w 5.0w
Group Delay
50
Group Delay
200
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5.1.2 Phase Distortion and Delay
Ideal lowpass filter with linear phase delay
,
0,
djwncjw
lpc
e w wH e
w w
nnnn
nnwnh
d
dclp ,
sin
delay distortion is a rather mild form of phase distortion, its effect is to shift the sequence in time. we accept linear phase response rather than zero phase response.
The impulse response (delayed by time nd )
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5.2 System Functions For LTI Systems Characterized by Linear Constant-coefficient Difference
equation
Linear Constant-coefficient Difference equation
M
kk
N
kk knxbknya
00
M
k
kk
N
k
kk zXzbzYza
00
zXzbzYzaM
k
kk
N
k
kk
00
![Page 17: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/17.jpg)
17
5.2 LTI System Characterized by Linear Constant-coefficient
Difference equation
M
kk
N
kk knxbknya
00
M
k
kk
N
k
kk zXzbzYza
00
If a system is not LTI, then the following Z-transform cannot be derived. (see P37, example 2.16, for x[n]=kδ[n], y[n]=an+1c + Kanu[n], for all n, x[n] have z-transform K, y[n] have no z-transform.)
![Page 18: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/18.jpg)
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11 0 :k kz c zec roz
1
0 0 1
10
0 1
1
1
MMk
kkk kN N
kk k
k k
c zb zY z b
H zX z aa z d z
11 0 :k kz d pod lez
For an LTI system:
its poles and zeros:
5.2 System Functions For Systems Characterized by Linear Constant-coefficient
Difference equation
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19
Ex. 5.2 find difference equation for second-order System function
21
1 1
1
1 31 1
2 4
zH z
z z
zXzzzYzz 2121 218
3
4
11
21228
31
4
1 nxnxnxnynyny
1 2
1 2
1 21 3
14 8
Y zz z
X zz z
Solution:
![Page 20: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/20.jpg)
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5.2.1 Stability and Causality
The difference equation does not uniquely specify the impulse response of a linear time-invariant system.
Each possible choice for the ROC of the system function will lead to a different impulse response, but they will all correspond to the same difference equation.
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Causality
For a causal system the impulse response must be right-sided sequence.
nh
The region of convergence (ROC) of must be outside the outermost pole.
zH
0, 0h n for n
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Stability
n
nh
1
zforznhn
n
ROC of includes the unit circle
zH
For a stable system The impulse response must be absolutely summable, i.e.,
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23
Ex. 5.3 Determine the ROC, Stability and causality for LTI
system:
nxnynyny 212
5
1121 21
21
1
1
25
1
1
zzzz
zH
poles: 1/2, 2; zeros(two) : 0
Solution:
![Page 24: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/24.jpg)
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Example 5.3 Determining the ROC
2 : ,z causal not stable
12 :
2,
z
stable not causal
stablenotcausalnotz ,:2
1
1)
2)
3)
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Causal and Stable system
Causal: ROC must be outside the outermost pole
Stable: ROC includes the unit circle
Causal and stable: all the poles of the system function are inside the unit circle. ROC is outside the outermost pole, and includes the unit circle.
![Page 26: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/26.jpg)
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5.2.2 Inverse Systems
1,iG z H z H z 1
iH zH z
Time domain: nnhnhng i
Not all systems have an inverse. Ideal LPF hasn’t
For a LTI system , the inverse system
which cascaded with satisfies:
H z iH z zH
jwjw
i eHeH
1Frequency response
x[n] y [n] h n ih n( )H z ( )iH z
![Page 27: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/27.jpg)
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5.2.2 Inverse Systems
N
kk
M
kk
zd
zc
a
bzH
1
1
1
1
0
0
1
1
N
kk
M
kk
i
zc
zd
b
azH
1
1
1
1
0
0
1
1
ROC of and ROC of must overlap, for convolution theorem to hold:
zH zH i
zHzH i
1systems with
rational system functions:
ih n h n n 1,iH z H z
![Page 28: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/28.jpg)
28
Ex. 5.4 analyse Inverse System for First-Order
System
1
1
1 0.5:
1 0.0
9.9
zH z R zOC
z
1
1
1 0.9:
1 0.0.5
5i
zH z ROC
zz
15.09.05.0 1 nununh nni
:i stable and cH z ausal
Solution:
![Page 29: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/29.jpg)
29
Ex. 5.5 find Inverse for System with a Zero in the
ROC
1
1
0.5:
1 0.0.9
9
zH z R zOC
z
1
1
1
1
21
8.12
5.0
9.01
z
z
z
zzH i
1
1: 2 2 1 1.8 22n n
iROC h n uz n u n
1
2: 2 2 1.82 2 1n n
iROC h n u n u nz
20.9Solution:
: ,i nos tta caH blez usal
1)
2)
: ,i noc tau stH salz able
![Page 30: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/30.jpg)
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Minimum-phase Systems
A LTI system is stable and causal and also has a stable and causal inverse if and only if both the poles and the zeros of are inside the unit circle
zH
Such systems are referred as minimum-phase systems
1
9.0:9.01
5.011
1
zROCz
zzH
5.0:5.01
9.011
1
zROCz
zzH i
![Page 31: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/31.jpg)
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5.2.3 Impulse Response for Rational System Functions
For a LTI system
N
k k
kNM
r
rr zd
AzBzH
11
0 1
N
k
nkk
NM
rr nudArnBnh
10
can be infinite impulse response (IIR)
or finite impulse response (FIR)
If causal,
0
0
, M N
Mk
kkN
kk
k
b zH z
a z
![Page 32: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/32.jpg)
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FIR System
0
, 0
0,
M
kn
k
b n Mh n n k
otherb
wise
0
Mk
kk
H z b z
0
M
kk
y n b x n k
( )
( )
Y Z
X Z
0
[ ]M
k
nb k k
![Page 33: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/33.jpg)
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Ex.5.6 A First-Order IIR System
nxnayny 1
1
1, :
1H z ROC a
azz
1,a
nuanh n
Determine System function, condition of stability, h[n] for stable, causal System.
Solution:∵ it is causal
condition of stability:
![Page 34: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/34.jpg)
34
Ex.5.7 A Simple FIR System
otherwise
Mnanh
n
,0
0,
1
11
0 1
1
az
zazazH
MMM
n
nn
2 1: , 0,1, ,j k Mkz ae kzeros M
7M
Determine System function, zero-pole plot, stability, difference equation For h[n]:
Solution:
system is stable: kpole p a
, 0z
![Page 35: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/35.jpg)
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Example 5.7 A Simple FIR System
Difference equation
0
Mk
k
y n a x n k
11 1My n ay n x n a x n M
0
( )
( )
Mn n
n
aY
HZ
z zX Z
1 1
1
1
1
M Ma z
az
![Page 36: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/36.jpg)
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5.3 Frequency Response for Rational System
FunctionsIf a stable LTI system has a
rational system function
N
k
jwkk
M
k
jwkk
jw
ea
ebeH
0
0
Its frequency response is
N
k
kk
M
k
kk
za
zbzH
0
0
![Page 37: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/37.jpg)
37
5.3 Frequency Response for Rational System
Functions
N
k
jwk
M
k
jwk
jw
ed
ec
a
beH
1
1
0
0
1
1
N
k
jwk
M
k
jwk
jw
ed
ec
a
beH
1
1
0
0
1
1
2jw jw jwH e H e H e
22
0 1
0 *
1
1 1
1 1
Mjw
k kjw
j
jw kN
jwk k
w
k
c e c eb
H ea
d e d e
magnitude-squared function:
![Page 38: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/38.jpg)
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Log magnitude
010 10 10
1 10
20log 20log 1 20log 1M N
jw jwk k
k k
bc e d e
a
1020log jwGain in d H eB
1020log jwAtt H eenuatio Gain in dn in dB B
1020log jwH e log magnitude in dB
N
k
jwk
M
k
jwk
jw
ed
ec
a
beH
1
1
0
0
1
1
![Page 39: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/39.jpg)
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Output: Log magnitude, phase
jwjwjw eXeHeY
jwjwjw eXeHeY 101010 log20log20log20
jwjwjw eXeHeY
jwjwjw eXeHeY
![Page 40: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/40.jpg)
40
N
k
jwk
M
k
jwk
jw edeca
beH
110
0 11
1 1
arg 1 arg 1N M
jw jw jwk k
k k
gd d
H e d e c edw
rddw
0 1
0
1
1
1
Mjw
kjw k
Njw
kk
c eb
H ea
d e
Phase Response for a rational system function
group delay:
![Page 41: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/41.jpg)
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Phase Response
1 1
a arg 11rg jwN M
jwk
j
kk
k
wd dgrd H e
dwe c
dd
we
22
22
{ }cos
1 2 cos 1 2 { }
jwk k
jwk k
d Re d er r w
r r w d Re d e
( )11 1j jw jk
jw we red ree
sinarg 1 arctan
1 cosjw
k
r wd e
r w
1 cos sinr w jr w
sinarctan
1 cos
r wd
dw r w
2
1arctan '
1x
x
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Phase Response
1 1
arg 1 arg 1N M
jw jwjk
k k
wk
dd e
dgrd H e
ddwe c
w
2
2
{ }
1 2 { }
jwk k
jwk k
d Re d e
d Re d e
sinarctan
1 cos
r wd
dw r w
2 2
2 21 1
{ } { }
1 2 { } 1 2 { }
jw jwN Mk k k k
jw jwk kk k k k
d Re d e c Re c e
d Re d e c Re c e
arg 1 jwk
dd e
dw
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Principal Value(主值)Principal Value of the phase of jweH
jwHARG e
2
where r w is a positive or negative integer
jw jwH e H eA r wRG
wredARG
ecARGa
bARGeHARG
N
k
jwk
M
k
jwk
jw
21
1
1
10
0
0 1
0
1
1
1
Mjw
kjw k
Njw
kk
c eb
H ea
d e
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441 2 3 4 5 6 7
2
jw jw
jw
arg
AR
H e H e
H e r wG
we refer to ARG [H(ejw)] as the "wrapped" phase,
continuous (unwrapped) phase curve is denotedas arg [H (ejw)]
卷绕的
解卷绕的
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5.3.1 Frequency Response of a Single Zero or Pole
1 1 1jw jwk kc e or d e
1 : , :j jw r radiusre e wher anglee
2
2
1 1 1
1 2 cos
j jw j jw j jwre e re e re e
r r w
wrrere jwj cos21log101log20 21010
1. formular method 2. Geometrical method
magnitude-squared function:
log magnitude in dB
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5.3.1 Frequency Response of a Single
Zero or Pole
wr
wrereARG jwj
cos1
sintan1 1
2 2
22
cos cos1
1 2 cos 1
j jw
j jw
r r w r r wgrd re e
r r w re e
1 1 cos sinj jwre e r w jr w
group delay:
Phase Response :
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Log Magnitude response for a single zero with r=0.9
0
2
wrrere jwj cos21log101log20 21010
1 j jwre e
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for a single zero with r=0.9
0
2
Group Delay
Phase response
1 sintan
1 cos
r w
r w
2
2
cos1
1 2 cosj jw r r w
grd re er r w
0.470.45 0.470.461 j jwre e
0
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frequency response is associated with vector diagrams in the complex plane and
pole-zero plots
2. Geometrical method
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frequency response is associated with vector diagrams in the complex plane and
pole-zero plots
2. Geometrical method
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Magnitude response for a single zero with
5.0r
7.0r
9.0r
1r
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Phase response for a single zero with
5.0r
7.0r9.0r
1r
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Group Delay for a single zero with
5.0r
7.0r
9.0r
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for a single zero outside the unit circle, with
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Magnitude response for a single zero outside the unit circle, with
9.01r
25.1r
0.2r
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Phase response for a single zero outside the unit circle,
with
9.01r
25.1r
0.2r
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Group Delay for a single zero outside the unit circle, with
9.01r
25.1r
0.2r
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5.3.2 Examples with Multiple Poles and Zeros
( self study)
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5.4 Relationship Between Magnitude and Phase
In general, knowledge about the magnitude provides no information about the phase, and vice versa.
If the magnitude of the frequency response and the number of poles and zeros are known, then there are only a finite number of choices for the associated phase.
For frequency response of LTI system
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5.4 Relationship Between Magnitude and Phase
Under a constraint referred to as minimum phase, the frequency-response magnitude specifies the phase uniquely, and the frequency-response phase specifies the magnitude to within a scale factor.
If the number of poles and zeros and the phase are known, then, to within a scale factor, there are only a finite number of choices for the magnitude.
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5.4 Relationship Between Magnitude and Phase
2* * *1
jw
jw jw jw
z eH e H e H e H z H z
N
kk
M
kk
zd
zc
a
bzH
1
1
1
1
0
0
1
1
*
* 0 1*
*0
1
11
1
M
kkN
kk
zb
ad z
cH
z
1 *2
* * 0 1
1 *0
1
1 11
1 1
M
k kkM
k kk
c z cb
zC z H z H z
azd z d
kc *1 kc
kd*1 kd
共轭倒数对conjugate reciprocal pairs
* *1 1 ( ) 1jw
jw jw
z e
jwz e e e z
magnitude-squared system
function
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5.4 Relationship Between Magnitude and Phase
If is causal and stable, then all its poles are inside the unit circle ,the poles of H (z) can be identified from the poles of C(z).
zH
The poles and zeros of occur in conjugate reciprocal pairs, with one element of each pair associated with and one element of each pair associated with
C z
* *1H z H z
* *1C z H z H z
but its zeros are not uniquely identified by C(z)
pz
*pz
1 pz
*1 pz
2z21 z
*11 z
1z
*1z
11 z
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Example 5.11
1
1 1 4 1
1
4
1
1 0.8 1 0.
2 1 5
8
0.j j
zH z
e z e z
z
1
2 1 1
1
4 4
1 21
1 0.8 1 0.8j j
zH z
e z e
z
z
* *2 2 2 1C z H z H z
Two systems with
have same magnitude-squared system function
* *1 1 1
1
4 1
1
4 1 4 4
1
1 1
1 0.8 1 0.
2 1 0
8 1 0.8 1 0
2 1
.8
0 5. .5j j j j
C z H z H z
z z
e z e
z
z e e z
z
z
11 12 1 0.5 1 22 1 0.5 1 2zzz zz z
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Example
1H z
1 1
2 4 1 4 1
1 1 2
1 0.8 1 0.8j j
z zH z
e z e z
2H z
* * * *1 1 2 21/ 1/C z H z H z H z H z
1 1
1 4 1 4 1
2 1 1 0.5
1 0.8 1 0.8j j
z zH z
e z e z
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Ex. 5.12 given
1 *
1 11
z aH z H z
az
1 *
11
z a
az
1
* 1 *
1
1
z a az
a z z a
* * * *1 11 1C H z H z H z H zz
apH z * *1apH z 1
Determine zeros and poles of stable,causal system H(z), if coefficients are real.
number of poles, zeros is known(3)
H(z): conjugate pole,zero pairs
* *1apH z apH z1
* *1C z H z H z
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Ex. 5.12 given
Determine zeros and poles of stable,causal system H(z), if coefficients are real.
number of poles, zeros is known(3)
H(z): conjugate pole,zero pairs
Solution:
H(z) poles: p1, p2, p3;H(z) Zeros: (z1, z2, z3),
or (z1, z2, z6), or (z4, z5, z3), or (z4, z5, z6),
* *1C z H z H z
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5.5 All-Pass(全通 ) SystemA stable system function of the form
1 *
1,
1ap
z aH z
az
z a*1z a , Pole:
Zero:
*1 az
az
apH z * *1apH z 1
jw
jwjw
jw
jwjw
ap ae
eae
ae
aeeH
1
1
1
**
11
1 *
jw
jwjwjw
ap ae
eaeeH
frequency-response
magnitude is unity
,1 c la ausa
,1 non la causa*1 az
az
unit circle
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General Form of All-Pass System
An all-pass system is always stable, since when frequency response characteristics (such as allpass) are discussed, it is naturally assumed that the Fourier transform exists, thus stability is implied.
all-pass system: A system for which the frequency-response magnitude is a constant.
It passes all frequency components of its input with a constant gain A (is not restricted to be unity in this text).
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General Form of All-Pass System
1 * 11
1 1 * 11 11 1 1
tan''
cr MMk kk
apk kk
k
k
ap
ap
k
k
z e z ez dH z A
d z e z e zwhere A is a pd real pol
ositive cons ts are the of H zs are thee complex pol o zs
se f H
e
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5.5 All-Pass System
( )
( )
(1 )
(1 )
jw j w
j w
e re
re
sin2arctan
1 cosjw
ap
r wH e w
r w
j
jwj
jjw
jw
jwjw
ap reaifere
ree
ae
aeeH
11
*
(1 cos sin )
1 cos sin
jwe r w jr w
r w jr w
phase response
![Page 72: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/72.jpg)
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5.5 All-Pass System group delay of a causal all-pass system is positive
1
jw j
j jw
e re
re e
sin2arctan
1 cos
r ww
r w
2
2
cos1 2
1 1 2 cos
jw j
j jw
r r we regrd
re e r r w
2 2
2
1 2 cos 2 2 cos
1 2 cos
r r w r r w
r r w
2
2
1
1 2 cos
r
r r w
2
2
1
1 j jw
r
re e
0
1then r
,if stable causal
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nonpositivity of the unwrapped Phase of All-Pass Systems
0
0arg argwjw j
ap ap apjH e grd H e d H e
0arg arg 0jap AeH 0jw
apgrd H e
0arg 0jwap fH o we r
1 * 11
1 1 * 11 11 1 1
cr MMk kk
apk kk k k
z e z ez dH z A
d z e z e z
0for w
*
*
0
1 1
1 11
1 1 1
crk kk
k k k
MMj
apk k
e ed
d e eH e A A
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Example 5.13 analyse First-Order All-Pass System
: 0.9, 0.9, 0rpole z
: 0.9, 0.9,rpole z
11
1 *
jw
jwjwjw
ap ae
eaeeH
Log magnitude 1
jw jjw
ap j jw
e reH e
re e
![Page 75: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/75.jpg)
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Example 5.13 First-Order All-Pass System: "wrapped"
phase
: 0.9, 0.9,pole z r
: 0.9, 0.9, 0pole z r
sin2arctan
1 cos
r ww
r w
1
jw jjw
ap j jw
e reH e
re e
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Example 5.13 First-Order All-Pass System: group
delay
,9.0
,9.0:
r
zpole
0,9.0
,9.0:
r
zpole
2
2
1
1 j jw
r
re e
2
2
1
1 2 cos
r
r r w
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Second-Order All-Pass System
with poles at and
.
( )
(
( )
(1 )1 )
jw jjw
ap
j
j jw
w j
j jw
e re
re e
e re
re eH e
1
1
sin( )( )2 2 tan
(1 )(1 ) 1 cos
sin 2 tan
1 cos
jw j jw j
j jw j jw
r we re e rew
re e re e r w
r w
r w
jz re jz re
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Ex. 5.13 Second-Order All-Pass System :
4: 0.9 jpole z e Magnitude
Phase"wrapped"
Group delay 4
2
4
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Fig. 5.21 fourth order all-pass system
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Frequency response of Fig. 5.21
Magnitude
Phase
Group delay
"wrapped"
24
4
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Application of All-Pass Systems
Used as compensators for phase or group delay (Chapter 7)
Be useful in the theory of minimum-phase systems (Section 5.6)
Be useful in transforming frequency-selective lowpass filters into other frequency-selective forms and in obtaining variable-cutoff frequency-selective filters (chapter 7)
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11 0 :k kz c zec roz
1
0 0 1
10
0 1
1
1
MMk
kkk kN N
kk k
k k
c zb zY z b
H zX z aa z d z
11 0 :k kz d pod lez
5.6 Minimum-Phase Systems
For an LTI system:
its poles and zeros:
Its inverse:
1invH z
H z
:kz d zero:kz c pole
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5.6 Minimum-Phase Systems
For a stable and causal LTI system, all the poles must be inside the unit circle.
If its inverse system is also stable and causal, all the zeros must be inside the unit circle.
1p
3z
1z
2z 2p
3pUnit
Circle
1p
3z1z
2z
2p
3p
Unit Circle
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5.6 Minimum-Phase Systems
Minimum-phase system: all the poles and zeros of an LTI system are inside unit circle, so the system and its inverse is stable and causal.
1p
3z
1z
2z 2p
3pUnit
Circle
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5.6.1 Minimum-Phase and All-Pass Decomposition
Any rational system function can be expressed as
zHzHzH apminSuppose has one zero outside the
unit circle at , , and the remaining poles and zeros are inside the unit circle.
zH*1z c 1c
1
1min
*
1
z
cz
cH z
11 *zH z H z c
stable, causal in the text, but it applies more generally.
reflect the zero to conjugate reciprocal locations inside the unit circle:
1
*
1
1
11
1cz
z
z
cH z
c
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Example 5.14 Minimum-Phase/All-Pass Decomposition
11
11 31
12
z
zH z
circleunittheinsidezpole2
1:
: 3 :z outside the unit czero ircle1
3z
1
1
1
11
3 ,
3
apH zz
z
1
min1
11
31
13
2
H zz
z
reflect this zero to conjugate reciprocal locations inside the unit circle:
(1)
1 min apH z H z H z
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Example 5.14 Minimum-Phase/ All-Pass Decomposition
4
21
1 4 13 31
2
3
21
1
1 j je z
z
zz
eH
circleunittheinsidezpole3
1:
43: :
2j outside the uzer nito z c re i cle
42
3jz e reflect two zeros to conjugate
reciprocal locations inside the unit circle:
(2)
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Example 5.14 Minimum-Phase/All-Pass Decomposition
4 1
4 1
1 1
1 4
41
1
44 2 29 3
21
32
1
21
321
31
3
3
34 1
j jj j
jj
e z
e z
ee z
e z
z
z
ez
4 1
21
413 31
2
3
21
11
j jz ze
z
eH z
min apH z H z
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5.6.2 Frequency-Response Compensation
When a signal has been distorted by an LTI system with an undesirable frequency response, perfect compensation:
nsnsc zHzH
dc
1If poles and zeros of Hd(z)
are inside the unit circle :
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assume that the distorting system is stable and causal and require the compensating system to be stable and causal,
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5.6.2 Frequency-Response Compensation
zH d
zH c
then perfect compensation is possible only if is a minimum-phase system.
dH z
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5.6.2 Frequency-Response Compensation
min
1
dc H
Hz
z
minmin
1d pc pd a
daH z H z HH z
H zG zzz H
min ,apdd H zH z H z
If isn’t minimum-phase, its inverse then isn’t stable, so we decompose
zH d
zH d
G z
cs n s n
stable
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Example 5.15 Compensation of an FIR System
0.6 1 0.6 1
0.8 1 0.8 11 1.25 1 1.
1 0.9 1
5
0.9
2j
j jd
j
H z e z e
e z e
z
z
for causal
unstable, is d
inverz
seH
zeros: outside the unit circle,
Solution:
min ,apdd H z HH z zdecomposition is needed:
min
1c
d
H zH z
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frequency response of
Magnitude
Phase
Group delay
zH d
"wrapped"
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Example 5.15 Compensation of an FIR System
18.018.0
16.016.0
25.1125.11
9.019.01
zeze
zezezHjj
jjd
1 0.8 1 0.8
0.8 1 0.8 11 0
0.8 0.
. 0
8
8 1 .8
j j
ap j j
zH
e
e z e
z ez
z
min ,apdd H z HH z z
2 1 0.8 1 0.81.25 0.8 0.8j jz e z e min ofCompe nsation , causal, stable: dinverse H z
min
1c
d
H zH z
0.6 1 0.
0.8 1 0.8 1
6min
2 11 0.9 1 0
1 0.8 1 0.
1.25
8
.9j j
j
d
j
H z e z e z
e z e z
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Frequency response of
Magnitude
Phase
Group delay
mindH z
"wrapped"
Minimum Phase-Lag
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Frequency response of apH z
Magnitude
Phase
Group delay
"wrapped"Maxmum Phase-Lag
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5.6.3 Properties of Minimum-Phase Systems
1. Minimum Phase-Lag Property
jwap
jwjw eHeHeH min
minarg arg argjw jw jwapH e H e H e
arg 0jwapH e for all w
minarg argjw jwH e H e For all systems that have a given magnitude response , minimum-phase system has the Minimum Phase-Lag.
minjwH e
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0 0j
n
H e h n
1. Minimum Phase-Lag Property
to make the interpretation of Minimum Phase-Lag systems more precise, it is necessary to impose the additional constraint that be positive at
jwH e
0.w
h n jwH eIts system function with same poles and
zeros, is also a minimum-phase system, according to its defination, but the phase is altered by π.
since
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5.6.3 Properties of Minimum-Phase Systems
2. Minimum Group-Delay Property
jwap
jwjw eHeHeH min
jwap
jwjw eHgrdeHgrdeHgrd min
0jwapgrd H e for all w
minjw jwgrd H e grd H e
For all systems that have a given magnitude response , minimum-phase system has the Minimum Group Delay.
minjwH e
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5.6.3 Properties of Minimum-Phase Systems
3. Minimum Energy-Delay PropertyFor any causal, stable, LTI systems
min ,jw jwH e H e min0 0h h
2
02
min
2
2
min0
1
21
2
n
w
jw
n
jH e dw
H e h nw
n
d
h
HW 5.65
If then
1 *
min 1lim ,
1z
z cH z H
zz
c
min0 0 ch h
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Minimum Energy-Delay Property
For any causal LTI systems, define the partial energy of the impulse response
n
m
mhnE0
2
2
mi
2
n00
n
m
n
m
h h mm
HW 5.66
For all systems that have a given magnitude response , minimum-phase system has the Minimum Energy-Delay.
minjwH e
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Four systems, all having the same frequency-response magnitude.
Zeros are at all combinations of the complex conjugate zero pairs and and their reciprocals.
0.60.9 je 0.80.9 je
Fig. 5.30
minimum-phase maximum-
phase
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Minimum-Phase System and Maximum-Phase System
A maximum-phase system is the opposite of a minimum-phase system. A causal and stable LTI system is a maximum-phase system if its inverse is causal and unstable. (From Wikipedia)
Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.
(noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).
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Maximum energy-delay systems are also often called maximum-phase systems.
105
Sequences corresponding to the pole-zero plots of Fig. 5.30
minimum-phase sequence ha[n]
maximum-phase sequence hb[n].
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Fig.5.32 Partial energies for the four sequences of Fig. 5.30. (Note that Ea[n] is for the minimum-phase sequence ha[n] and Eb[n] is for the maximum-phase sequence hb[n].
the maximum energy delay occurs for the system that has all its zeros outside the unit circle. Maximum energy-delay systems are also often called maximum-phase systems.
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5.7 Linear Systems with Generalized Linear Phase
In designing filters, it’s desired to have nearly constant magnitude response and zero phase in passband.
For causal systems, zero phase is not attainable, and some phase distortion must be allowed.
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5.7 Linear Systems with Generalized Linear Phase
The effect of linear phase (constant group delay) with integer slope is a simple time shift.
A nonlinear phase, on the other hand, can have a major effect on the shape of a signal, even when the frequency-response magnitude is constant
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5.7.1 System with Linear Phase
, :jw jwid reaH e w le
1jwid eH weH jw
id
jwid eHgrd
sin,
if is in t eger
id
nh n n
n
n
k kn
knkx
n
nnxny
sinsin
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5.7.1 System with Linear Phase
Specially, if : integerdn
did nnnh
dd nnxnnnxny
nn
nnhid ,
sin
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Interpretation of
ch t t T
Tjc ejH
( ) ,jw jwH e e w
, :jw jwid reaH e w le
sinC
n
nt T Tx n
nt x
t T T
c cy t x t T
c ny n x T T
sin
Ck
kt T Tx k
kt x
t T T
comes from sampling a continuous-time signal, if is not an integer *cx t t T
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Interpretation of
is not an integer
, :jw jwid reaH e w le
comes from sampling a continuous-time signal, if
sinC
k
kt T Tx k
kt x
t T T
c ny n x T T
sin
k
kxk
n
n
k
sin nx n
n
2T
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General frequency response with linear
phase
,jw jw jwH e H e e w
For nonconstant magnitude response
multiplication in frequency domain
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Linear-phase ideal lowpass filter
ww
wweeH
c
cjw
jwlp ,0
,
ww
wweH
c
cjwlp ,0
,1 jwe
n
nwnh c
sin
n
nwnh c
lp
sin
nx nw ny
sinid
nh n
n
The corresponding impulse response is
*
time domain
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Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear Phase in three cases:
α is integer; 2α is integer; 2α is not integerdn
sin
c dlp
d
nnw n
hn n
sin 22
2c d d
lp dd d
w n n nh n n
n n n
sin
clp
w nh n
n
0.4 , 5c dw n
sin c d
lpd
w n nh n
n n
Solution: (1)
:
2lp lp
even symmetric abouth n h n
lph n 2 lph n lph n
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:
2
lp
lp lp
h n is even
symmetric about
h n h n
, .50.4 4 cw
n
nwnh c
lp
sin(2)
Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear
Phase
is an integer2
4.5
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0. 3, 44 .cw
n
nwnh c
lp
sin(3) is not an integer2
4.3
Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear
Phase
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2
n
nwnh c
lp
sin
ww
wweeH
c
cjw
jwlp ,0
,
Ex. 5.16 symmetry of impulse response of Ideal Lowpass with Linear
Phase
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, constant
( )j
jw j j
w
w jw and are
A e is a real b
H
i
e A e
pol
e
possibly funct war ion of
119
5.7.2 Generalized Linear phase
jwjw eHdw
deHgrd arg
1 2 22 111 2
sin 1 21
sin 2jw M M
M M
w M M
we
For moving average system (Ex.2.20, Page 45)
1 2
21
sin ( 1) 21,
si,
n0
2
jwMM
w M
wM M Me
1 2 0,if M M M if negtive, it’s not, strictly speaking,a linear-phase system, since π is added to the phase. It’s the form:
arg , 0jwH e w w
it is referred to Generalized Linear phase system
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120
cos sin
wjjw jw
jw jw
H e A e e
A e w jA e w
cos sinjwn
n nn
h n wn j hh n wne n
tan( )
sinsin
cc ososn
n
hw
n wn
h n wn
w
w
cos sin sin cos 0
n n
h n wn w h n wn w
If a system with h[n] has Linear phase
sin 0 n
for allh n w n w
, ,condition o h nn
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j wjw jwH e A e e
sin 0n
for alln w n wh
This equation is a necessary condition
on h[n], for the system to have constant group delay.
and
It is not a sufficient condition, however, and, owing to its implicit nature,it does not tell us how to find a linear-phase system.
If a system with h[n] has Linear phase
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One set of condition: even symmetry
jw jw jw jH e A e e sin 0
n
h n w n for all w
2 : integerM 2h n h n
0 or sin 0
n
h n w n
:jwA e even function of w
0 M/2=α
sati
sf
y
Shown inType I,II FIR0 M/2=α M=5
M
M even
M odd sin i2 s nnw w n
sin sin 2 02n n n nwhwh
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Another set of condition : odd symmetry
2 : integerM 2h n h n
jw jw jw jH e A e e sin 0
n
h n w n for all w
2 3 2or cos 0
n
h n w n
woffunctionoddeA jw :
0
M
M/2
sati
sf
y
Shown inType III, IV FIR0 M/2
M=3
M even
M odd
cos cos 2 02n n n nwhwh
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5.7.3 Causal Generalized Linear-Phase Systems
0
sin 0n
h n w n for all w
0 0causal h n n
22 or n nh n hh hn
0 2, nh n M
0
sin 0M
n
h n w n for all w
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5.7.3 Causal Generalized Linear-Phase Systems
Causal FIR systems have generalized linear phase if they have impulse response length
and satisfy 1M
0 or
integer:2 M h M n h n
2 3 2or integer:2 M
h M n h n
jwjw jw jA e eH e
![Page 125: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/125.jpg)
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5.7.3 Causal Generalized Linear-Phase Systems
If
, 0
0,
n Mh n
otherw
h M
ise
n
, ,jwe real even periodwhere A e is a function of w
2jwj jw wMeA e eH e
then
It’s sufficient condition, not necessary condition
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5.7.3 Causal Generalized Linear-Phase Systems
If
, 0
0,
n Mh n
otherw
M
s
h
i e
n
, ,jwo real odd periodwhere A e is a function of w
then
2 2 2jw jwM jw j jwMo o
jwH A e e A ej ee
It’s sufficient condition, not necessary condition
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5.7.3 Causal Generalized Linear-Phase Systems
The above two FIR conditions are sufficient to guarantee a causal system with generalized linear phase.
Clements (1989) showed that causal IIR can also have Fourier transforms with generalized linear phase.
The corresponding system function, however, are not rational, and thus, the systems cannot be implemented with difference equations.
![Page 128: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/128.jpg)
M is even M is odd
h[Mn] = h[n]
h[Mn] = h[n]
5.7.3 Causal FIR Linear-Phase Systems
satisfies: symmetric or Antisymmetric impulse response
h[Mn] = ±h[n] for n = 0,1,…,M
Type I
Type III
Type II
Type VI
symmetric
Antisymmetric
0 M/2
M=3
0
M=6M/2
0 M/2 M=6 0 M/2 M=5
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Type I FIR Linear-Phase Systems
Symmetric impulse response
2M k
2
k 1
2 2 2( ) ( )
2 2 2
M M M Mjw k w jkj wh h hM M Mk ke e e
2M k
MnnMhnh 0,
0
Mjw jwn
n
H e h n e
2
k 1
2 2
2 2
M MM jwk jjw wk jwM Mh k he e ee
0 M/2 M=10K=0 1 2 M/212M/2
h[n] 2
Mh k 2Mh k
M: even integer,
M/2 : integer.
![Page 130: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/130.jpg)
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Type I FIR Linear-Phase Systems
2
k 1
2 2
2 2
MM Mjwk jjw wk jwM Mh k he eee
0 2
2 1, , , 22 2
where a h M
a k h M k k M
2
2 2
0
cosM
jwM jw jwM
kea k wk e A e e
0
Mjw jwn
n
H e h n e
k 1
22
2 22 cos cos( 0)
M MjwM Mh k hwk w e
, ,real even period
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Type I FIR Linear-Phase Systems
Symmetric impulse response
MnnMhnh 0,
0
Mjw jwn
n
H e h n e
0 2
2 1, , , 22 2
where a h M
a k h M k k M
2
2 2
0
cosM
jwM jw jwMe
k
wka k e A e e
0 M/2 M=10K=01 2 M/212M/2
, ,real even period
M: even integer,
M/2 : integer.
![Page 132: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/132.jpg)
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Ex. 5.17 determine H(ejw) of Type I FIR Linear-Phase Systems
2sin
25sin
1
1 254
0 w
we
e
eeeH jw
jw
jw
n
jwnjw
4 2
2 2
0
cos (1 2cos 2cos 2 )jwM j w
k
e a k wk e w w
Solution:
20 2 , 2 1,2, , 2 where a h M a k h M k k M
2
2
0
cosM
jw jwM
k
H e a ewk k
![Page 133: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/133.jpg)
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22
0
cos jw
k
a ekk w
Frequency response
Magnitude
Phase
Group delay
252sin
sin 2j we
w
w
"wrapped"
2
5
4
5
6
5
8
5
Type I
![Page 134: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/134.jpg)
M: odd integer. integer plus one-half.
Type II FIR Linear-Phase Systems
Symmetric impulse response MnnMhnh 0,
2 :M
0
M/2
M=9K 1 212(M+1)/2 (M+1)/2
12
M k 12
M k 1
2Mh k 1
2Mh k
1
2
k
1 12
12( ) ( )1
21
2jw
M MMjw kjwk MMH e h kk he e
( )
1 2
k 1
21
221
12
M Mjw kj jw k wMh k e e e
( ) (( ) )
1
2M
2M 1
2M
![Page 135: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/135.jpg)
Type II FIR Linear-Phase Systems
1
21 2 1
cos2
M
k
jwMb k w k e
1 2 1,2, , 22 1where b k h M k k M
2jw jwMeB e e
1 2
k 1
21 1
1 2 22
Mjw
Mjw k jw k jwMH e h k e e e
( ) ( () )
1 2
k 1
212 co
1s2
2
M MjwM kh k w e
( )
, ,real even period
![Page 136: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/136.jpg)
M: odd integer. integer plus one-half.
Type II FIR Linear-Phase Systems
Symmetric impulse response
MnnMhnh 0,
2
21
1
1s
2co
jw jwM
M
k
H e b k w ek
2 1 2 11 2, , 2,whe M Mre b k h k k
2M
2jw jwMeB e e
, ,real even period
0 M/2 M=5
![Page 137: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/137.jpg)
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Ex. 5.18 determine H(ejw) of Type II FIR Linear-Phase Systems
2sin
3sin
1
1 2565
0 w
we
e
eeeH jw
jw
jw
n
jwnjw
Solution:
![Page 138: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/138.jpg)
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Frequency response Type II
Magnitude
Phase
3
5 2
1
cos1
2jw
k
wb k ek
"wrapped"
5 2sin 3
sin 2 jww
ew
Group delay
![Page 139: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/139.jpg)
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Frequency response
Magnitude
Phase
Group delay
"wrapped"
Type I
22
0
cos jw
k
a ekk w
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Type III FIR Linear-Phase Systems
Antisymmetric impulse response
, 0h n h M n n M
2
1
2
sinM
jw jwM
k
H e ewkc kj
2,,2,122 MkkMhkcwhere
M: even integer.
integer.2M
2 22jw jwM jw jwMo
joA e e Aj e e
0 M/2
M=4
, ,real odd period
![Page 141: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/141.jpg)
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Ex. 5.19 determine H(ejw) of Type III FIR Linear-Phase
Systems
jwwjjw ewjeeH sin21 2
Solution:
![Page 142: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/142.jpg)
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Ex. 5.19 Frequency response Type III
Magnitude
Group delay
Phase"wrapped"
2sinjw jwH e j w e
![Page 143: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/143.jpg)
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Type IV FIR Linear-Phase Systems
Antisymmetric impulse response
MnnMhnh 0,
2 22jw jwM jw jwMo
joA e e Aj e e
21,,2,1212 MkkMhkdwhere
1 2
2
1
1sin
2jw jwM
k
M
H e d w ekk j
M: odd integer. integer plus one-
half.2M 0 M/2
M=3
, ,real odd period
![Page 144: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/144.jpg)
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Ex. 5.20 determine H(ejw) of Type IV FIR Linear-Phase Systems
22sin21 wjjwjw ewjeeH
Solution:
![Page 145: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/145.jpg)
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Ex. 5.20 Frequency response Type IV
Magnitude
Phase"wrapped"
22sin 2 jwj ew
Group delay
![Page 146: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/146.jpg)
M is even M is oddh[Mn] = h[n]
h[Mn] = h[n]
5.7.3 Causal FIR Linear-Phase Systems
satisfies:
h[Mn] = ±h[n] for n = 0,1,…,M
Type I
Type III
Type II
Type VI
![Page 147: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/147.jpg)
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Type I , I I FIR Linear-Phase Systems
Symmetric impulse response
MnnMhnh 0,
jwH e
2
2
0
cosM
jwM
k
a k wk e
0 M/2 M=10
1 2
2
1
1cos
2
Mjw jwM
k
H e b k w k e
M: odd integer. : integer plus one-half.
2M
0 M/2 M=7
M: even integer. integer.2M
, ,real even period
![Page 148: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/148.jpg)
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Type III , IV FIR Linear-Phase Systems
Antisymmetric impulse response
MnnMhnh 0,
2
2
1
sinM
jw jwM
k
H e c k wk ej
M: even integer.
integer.2M
0 M/2
M=4
1 2
2
1
1sin
2
Mjw jwM
k
jH e d k w k e
M: odd integer. :integer plus one-
half.2M 0 M/2
M=3
, ,real odd period
![Page 149: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/149.jpg)
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Locations of Zeros for FIR Linear-Phase Systems
For Type I and II, nhnMh
0
Mn
n
H z h n z
1 1
0
( )M
M k
k
Mz z zh k Hz
0
Mn
n
H z h M n z
0
( )M k
k M
h k z
M n k n M k
For Type III and IV,
1MH z z H z
![Page 150: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/150.jpg)
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Type I and II
If is a zero of , 0z zH
This implies that if is a zero of
, then is also a zero of
0jz re zH
1 10
jz r e zH
1 MH z z H z nhnMh
01
0 0MozH H zz then
oz z
The same result for Type III and IV
factor
10z z 1
0 0H z has factor H z
![Page 151: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/151.jpg)
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Type I and II
When is real and is a zero of
,
will also be a zero of
, so will .
h n 0z zH
0jz re zH
1 10
jz r e
0
Mn
n
H z h n z
1Mz H z
0jz re
0jz re 1 1
0jz r e
1 10
jz r e
real coefficient equation has conjugate complex roots pair:
So there are four possible complex zeros:
same result for Type III and IV
![Page 152: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/152.jpg)
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Type I and II ,Type III and IV
111111 1111 zerzerzrezre jjjj
When is real, each complex zero not on the unit circle will be part of a set of four conjugate reciprocal zeros of the form
h n
complex zeros on the unit circle
1 11 1j je z e z
10 0z z
1
0 0z z
Type IIType I
![Page 153: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/153.jpg)
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Type I and II ,Type III and IV
if a zero of is real, and not on the unit circle, the reciprocal is also a zero, and
have the factors of the form
H z
1 1 11 1rz r z
H z
Type IIType I
![Page 154: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/154.jpg)
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Type I and IIThe case of a zero at is important in
designing filter of some types of frequency responses (such as high-pass,low-pass filter).
1z
1 1 1M
H H
so z=-1 must be zero of Type II generalized linear-phase systems.
1MH z z H z If M is even 1 1H H
If M is odd,
1 1H H 01 HType I can be HP filter
Type II cannot be HP filter
Both can be LP filter
z=1 is not zero for both,
![Page 155: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/155.jpg)
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Type I and II
Type IIType I
cannot be HP filtercan be HP,LP filtercan be LP filter
![Page 156: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/156.jpg)
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Type III and IV
nhnMh 1 zHzzH M
111111 1111 zerzerzrezre jjjj
Type IVType III
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Type III and IVThe case of 1z
11 HH 01 HFor both M is even and
odd,1z must be zero of Type III and IV generalized linear-phase systems.
1MH z z H z
Type IVType IIIcannot be LP filter
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Type III and IVThe case of 1z
111 1 HH M
must be zero of Type III generalized linear-phase systems.
1z
1 zHzzH M
01 HIf M is even, 1 1H H
Type IVType III
If M is odd (Type IV) ,
1 1H H
cannot be HP filter
can be HP filter
![Page 159: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/159.jpg)
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Fig.5.41Typical plots of zeros for linear-phase systems
Type IV
(a) Type I
Type III
(b) Type II
z0=-1, 不能做 HP filter
z0=±1, 不能做 LP, HP filter
z0=1, 不能做 LP filter
能做 LP, HP filter
![Page 160: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/160.jpg)
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5.7.4 Relation of FIR Linear-Phase Systems to Minimum-
SystemsAll FIR linear-phase systems have zero of
111111 1111 zerzerzrezre jjjj
mi axn mucH z H z HH z z
has all zeros inside the unit circle.
zHmin iM
has all zeros on the unit circle.
zHuc oM
has all zeros outside the unit circle
zHmax iM
iMzzHzH 1minmax
2r 1rSame
magnitude
![Page 161: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/161.jpg)
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Example 5.21 Decomposition of a Linear-Phase System
18.018.0
16.016.02min
8.018.01
9.019.0125.1
zeze
zezezHjj
jj
For Minimum-Phase System of Page 287, Eq. 5.109
2 0.6 1 0.6 1max
0.8 1 0.8 1
0.9 1 1.1111 1 1.1111
1 1.25 1 1.25
j j
j j
H z e z e z
e z e z
iMzzHzH 1minmax
min maxH z H z H z
Determine the frequecny response of Maximum-Phase System and the system cascaded by two.Solution:
![Page 162: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/162.jpg)
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Example 5.21 Decomposition of a Linear-
Phase System zHzHzH maxmin
jw
jwjwjw
eH
eHeHeH
min10
max10min1010
log40
log20log20log20
iMzzHzH 1minmax
max min mini ijM w jM wjw jw jwH e H e e H e e
max minjw jwH e H e
![Page 163: 2015-10-201Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 5 Transform Analysis of Linear Time-Invariant Systems.](https://reader035.fdocuments.net/reader035/viewer/2022070410/56649edb5503460f94bea5a7/html5/thumbnails/163.jpg)
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Example 5.21 Decomposition of a Linear-Phase System
min minjw jw jw
i iH e wM H e H e wM
4jwigrd H e M
min maxjw jw jwH e H e H e
iMzzHzH 1minmax
jwi
jw eHwMeH minmax
zHzHzH maxmin
max min mini ijM w jM wjw jw jwH e H e e H e e
min maxjw jw jwH e H e H e
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Frequency response of
Magnitude
Phase
Group delay
minH z
"wrapped"
2 0.6 1 0.6 1min
0.8 1 0.8 1
1.25 1 0.9 1 0.9
1 0.8 1 0.8
j j
j j
H z e z e z
e z e z
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166
Frequency response of
Magnitude
maxH z
Phase
Group delay
"wrapped"
1max min
iMH z H z z
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167
Frequency response of H z
Magnitude
Phase
Group delay
"wrapped"
min maxH z H z H z
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Review
168
Minimum-Phase System and Maximum-Phase System
a LTI bsystem is said to be minimum-phase if the system and its inverse are causal and stable.
Minimum-Phase System: all the zeros and poles are in the unit circle.
Properties:The Minimum Phase-Lag PropertyThe Minimum Group-Delay PropertyThe Minimum Energy-Delay Property
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Review
169
Minimum-Phase System and Maximum-Phase System
A maximum-phase system is the opposite of a minimum-phase system. A causal and stable LTI system is a maximum-phase system if its inverse is causal and unstable. (From Wikipedia)
Maximum-Phase System: poles are all in the unit circle, zeros are all outside the unit circle. It’s causal and stable.
(noncausal)Maximum-Phase System: anti-causal, stable System whose System function has all its poles and zeros outside the unit circle. (problem5.63).
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170
Rational System Function has an equal number of poles and
zeros
zeroczzc kk :01 1
1
0 0 1
10
0 1
1
1
MMk
kkk kN N
kk k
k k
L L
c zb zY z b
H zX z aa z
z zd z
poledzzd kk :01 1
0H z
H z
Review
z=∞, z=0 may be poles or zeros
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171
1
0 0 1
10
0 1
1
1
MMk
kkk kN N
kk k
k k
L L
c zb zY z b
H zX z aa z
z zd z
If L=0 and M>N, then M−N extra poles at z = 0 are induced by the numerator.
If L=0 and M<N, then N−M zeros at z = 0 appear from the denominator.
if L<0 then H(z) has L zeros at z=∞, L poles at z=0 [+(M-N) ,↑for M>N, ↓for M<N].
If L>0, then H(z) has L poles at z=∞, L zeros at z = 0 [-(M-N) ,↓for M>N, ↑for M<N] .
Review
Rational System Function has an equal number of poles and zeros
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M is even M is odd
h[Mn] = h[n]
h[Mn] = h[n]
5.7.3 Causal FIR Linear-Phase Systems
satisfies: symmetric or Antisymmetric impulse response
h[Mn] = ±h[n] for n = 0,1,…,M
Type I
Type III
Type II
Type VI
symmetric
Antisymmetric
0 M/2
M=3
0
M=6M/2
0 M/2 M=6 0 M/2 M=5
Review
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Type I FIR Linear-Phase Systems
Symmetric impulse response
2M k
2
k 1
2 2 2( ) ( )
2 2 2
M M M Mjw k jw k jwM M Mh k h k he e e
2M k
MnnMhnh 0,
0
Mjw jwn
n
H e h n e
2
k 1
2 2
2 2
MM Mjwk jwkjw jwM Mh k he e e e
0 M/2 M=10K=0 1 2 M/212M/2
h[n] 2
Mh k 2Mh k
M: even integer,
M/2 : integer.
Review
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174
Type I FIR Linear-Phase Systems
2
k 1
2 2
2 2
MM Mjw jwjwk jwkM Mh k he e e e
2,,2,122
20
MkkMhka
Mhawhere
2
2 2
0
cosM
jwM jw jwMe
k
a k wk e A e e
0
Mjw jwn
n
H e h n e
2
k 1
22 cos cos( 0)2 2
M MjwM Mh k wk h w e
, ,real even period
Review
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23/4/21175Zhongguo Liu_Biomedical Engineering_Shandong U
niv.
Chapter 5 HW5.3, 5.4, 5.32, 5.12, 5.15, 5.19, 5.22, 5.43, 5.65, 5.66,
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