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![Page 1: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/1.jpg)
Generalized Linear Phase
Quote of the DayThe 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.
![Page 2: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/2.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 2
Linear Phase System• Ideal Delay System
• Magnitude, phase, and group delay
• Impulse response
• If =nd is integer
• For integer linear phase system delays the input
eeH jjid
jid
jid
jid
eHgrd
eH
1eH
nnsin
nhid
did nnnh
ddid nnxnnnxnhnxny
![Page 3: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/3.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 3
Linear Phase Systems• For non-integer the output is an interpolation of samples• Easiest way of representing is to think of it in continuous
• This representation can be used even if x[n] was not originally derived from a continuous-time signal
• The output of the system is
• Samples of a time-shifted, band-limited interpolation of the input sequence x[n]
• A linear phase system can be thought as
• A zero-phase system output is delayed by
Tjcc ejH and Ttth
TnTxny
jjj eeHeH
![Page 4: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/4.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 4
Symmetry of Linear Phase Impulse Responses• Linear-phase systems
• If 2 is integer – Impulse response symmetric
jjj eeHeH
nhn2h
=5
=4.5
=4.3
![Page 5: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/5.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 5
Generalized Linear Phase System• Generalized Linear Phase
• Additive constant in addition to linear term• Has constant group delay
• And linear phase of general form
jjjj eeAeH
constants and
of function Real :eA j
jj eHargdd
eHgrd
0 eHarg j
![Page 6: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/6.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 6
Condition for Generalized Linear Phase• We can write a generalized linear phase system response as
• The phase angle of this system is
• Cross multiply to get necessary condition for generalized linear phase
nsinnhjncosnhenheHnn
nj
n
j
sinejAcoseAeeAeH jjjjjj
ncosnh
nsinnh
cossin
n
n
0nsinnhnsinnh
0cosnsinsinncosnh
0cosnsinnhsinncosnh
nn
n
nn
![Page 7: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/7.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 7
Symmetry of Generalized Linear Phase• Necessary condition for generalized linear phase
• For =0 or
• For = /2 or 3/2
0nsinnhn
nhn2h0nsinnhn
nhn2h0ncosnhn
![Page 8: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/8.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 8
Causal Generalized Linear-Phase System• If the system is causal and generalized linear-phase
• Since h[n]=0 for n<0 we get
• An FIR impulse response of length M+1 is generalized linear phase if they are symmetric
• Here M is an even integer
Mn and 0n 0nh
nhnMh
![Page 9: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/9.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 9
Type I FIR Linear-Phase System• Type I system is defined with
symmetric impulse response
– M is an even integer
• The frequency response can be written as
• Where
Mn0for nMhnh
ncosnae
enheH
2/M
0n
2/Mj
njM
0n
j
M/21,2,...,kfor k2/Mh2ka
2/Mh0a
![Page 10: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/10.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 10
Type II FIR Linear-Phase System• Type I system is defined with
symmetric impulse response
– M is an odd integer
• The frequency response can be written as
• Where
Mn0for nMhnh
21
ncosnbe
enheH
2/1M
1n
2/Mj
njM
0n
j
/21M1,2,...,kfor
k2/1Mh2kb
![Page 11: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/11.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 11
Type III FIR Linear-Phase System• Type I system is defined with
symmetric impulse response
– M is an even integer
• The frequency response can be written as
• Where
Mn0for nMhnh
nsinncje
enheH
2/M
1n
2/Mj
njM
0n
j
M/21,2,...,kfor
k2/Mh2kc
![Page 12: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/12.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 12
Type IV FIR Linear-Phase System• Type I system is defined with
symmetric impulse response
– M is an odd integer
• The frequency response can be written as
• Where
Mn0for nMhnh
21
nsinndje
enheH
2/1M
1n
2/Mj
njM
0n
j
/21M1,2,...,kfor
k2/1Mh2kd
![Page 13: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/13.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 13
Location of Zeros for Symmetric Cases• For type I and II we have
• 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
– If M is odd implies that
– Cannot design high-pass filter with symmetric FIR filter and M odd
1Mz zHzzHnMhnh
1H11H M
01H
![Page 14: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/14.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 14
Location of Zeros for Antisymmetric Cases• For type III and IV we have
• All properties of symmetric systems holds• Particular importance of both z=+1 and z=-1
– If z=1
• Independent from M: odd or even
– If z=-1
• If M+1 is odd implies that
1Mz zHzzHnMhnh
1H11H 1M
01H
01H1H1H
![Page 15: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/15.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 15
Typical Zero Locations
![Page 16: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/16.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 16
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
• Where
• 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
zHzHzHzH maxucmin
iM1minmax zzHzH
![Page 17: Generalized Linear Phase Quote of the Day The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms.](https://reader035.fdocuments.us/reader035/viewer/2022072015/56649ec05503460f94bcbf5d/html5/thumbnails/17.jpg)
Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing 17
Example• Problem 5.45