EENG 751: Signal Processing I
description
Transcript of EENG 751: Signal Processing I
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EENG 751 04/22/23 9-1
EENG 751: Signal Processing IEENG 751: Signal Processing IClass # 9 Outline
Signal Flow Graph Implementation Fundamentals System Function Graph Construction Graph Analysis Applications Complex Coefficient Systems
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EENG 751 04/22/23 9-2
SFG ReferenceSFG Reference
IEEE Transactions on Signal; Processing, vol 41 No. 3 March 1993“Efficient Computation of the DFT with Only a Subset of Inputor Output Points” page 1184.
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EENG 751 04/22/23 9-3
SFG ReferenceSFG Reference
IEEE Transactions on Signal; Processing, vol 41 No. 3 March 1993“Efficient Computation of the DFT with Only a Subset of Inputor Output Points” page 1188.
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EENG 751 04/22/23 9-4
SFG FundamentalsSFG Fundamentals
tionImplementa sMathematic e. I.
. viaexecuted be
can that algorithman as viewedbecan LCCDE The LCCDE.
theandfunction system ebetween thduality a is e that therNote
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EENG 751 04/22/23 9-5
SFG Fundamentals (Cont)SFG Fundamentals (Cont)
Structure Lattice Subtract-Add
Subtract-Add-Multiply AccumulateMultiply
include primitives possibleOther
locationsmemory and additions ,multiplies 1
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EENG 751 04/22/23 9-6
SFG Fundamentals (Cont)SFG Fundamentals (Cont)
ly.respective nodenetwork and
node,sink node, sourceth - theof values theare and , ,
nodes.sink nor sourceneither are
branches. enteringonly have
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EENG 751 04/22/23 9-7
SFG Fundamentals (Cont)SFG Fundamentals (Cont)
:branches entering all of ouputs thesums andadder an is nodeeach
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EENG 751 04/22/23 9-8
SFG GenerationSFG Generation
says""equation what thedoingjust by SFG its draw lets and
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forward feed
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EENG 751 04/22/23 9-9
SFG Generation (Cont)SFG Generation (Cont)
says""equation what thedoingjust by SFG its draw letsagain and
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backward forward feed
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EENG 751 04/22/23 9-10
SFG Generation (Cont)SFG Generation (Cont)
:(why?) diagramblock with
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EENG 751 04/22/23 9-11
SFG Generation (Cont)SFG Generation (Cont)closely moreSFG combinedour Examine
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EENG 751 04/22/23 9-12
SFG Generation (Cont)SFG Generation (Cont)
givesorder reverse in the
SFG thengimplementi so)()(1
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have also wesystems LTIfor
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EENG 751 04/22/23 9-13
SFG Generation (Cont)SFG Generation (Cont)
).(canonicalbranch delay oneonly SFG with a yields This
equal. also are functionsbranch theand equal are nodes end twothe
since combined becan branches twomiddle that theNotice
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EENG 751 04/22/23 9-14
SFG Generation (Cont)SFG Generation (Cont)
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EENG 751 04/22/23 9-15
SFG Generation (Cont)SFG Generation (Cont)
][][][ and ][][
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EENG 751 04/22/23 9-16
SFG Generation (Cont)SFG Generation (Cont)
delays. SFG with standard
get the weorder,different ain SFG theof middle theaddingBy
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EENG 751 04/22/23 9-17
SFG Generation (Cont)SFG Generation (Cont)gives (z) and (z) inginterchang delays, have weSince 21 HHM N
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1z 1Mb
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EENG 751 04/22/23 9-18
SFG Generation (Cont)SFG Generation (Cont)
.or II thecall is This ).,max( thecount to
delay thereduce and ladders two thecombinecan welevel, same
at the equal aresection middle in the valuesnode theall Since
form canonicalform directNM
1z1Na
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EENG 751 04/22/23 9-19
SFG Application ReferenceSFG Application Reference
IEEE Transactions on Signal; Processing, vol 41 No. 3 March 1993“Efficient Computation of the DFT with Only a Subset of Inputor Output Points” page 1188.
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EENG 751 04/22/23 9-20
SFG Application ReferenceSFG Application Reference
IEEE Transactions on Signal; Processing, vol 41 No. 3 March 1993“Efficient Computation of the DFT with Only a Subset of Inputor Output Points” page 1189.
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EENG 751 04/22/23 9-21
SFG Application ExampleSFG Application Example
LCCDE.order first a iswhich
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as drepresente becan ][
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EENG 751 04/22/23 9-22
SFG Application ExampleSFG Application Example
isSFG II formdirect the,1 and 0 with 1
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EENG 751 04/22/23 9-23
SFG Application ExampleSFG Application Example
paper. in the 5 figure as same theisSFG which II formdirect thehas1
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EENG 751 04/22/23 9-24
SFG Application ExampleSFG Application Example
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EENG 751 04/22/23 9-25
SFG Application Example (Cont)SFG Application Example (Cont):is biquad general for theSFG canonical theWhere
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EENG 751 04/22/23 9-26
Alternate Canonic FormsAlternate Canonic Forms
tion.implementa theand equations theof
entsrearrangembetween encecorrespond thesillustrate example This text.
theofedition 1985 theof 151 pageon appearsSFG following The
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EENG 751 04/22/23 9-27
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
NML
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EENG 751 04/22/23 9-28
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
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EENG 751 04/22/23 9-29
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
get todiagramblock in the (z)for Substitute 1H
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EENG 751 04/22/23 9-30
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
:flowgraph final thecreate Finally to
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EENG 751 04/22/23 9-31
Cascade FormCascade Form
difficult. very bemay ion factorizat thiswhere
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EENG 751 04/22/23 9-32
Optionan is Pipelining
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EENG 751 04/22/23 9-33
Parallel FormParallel Form
Nkzz
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EENG 751 04/22/23 9-34
Hardware of Lots
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EENG 751 04/22/23 9-35
nodes.sink and source of roles thereversing
and same) theances transmitt the(leaving branchesnetwork all of
direction thereversingby generated isSFG a of transposeThe :Definition
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The Transposition TheoremThe Transposition Theorem
1z
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EENG 751 04/22/23 9-36
The Transposition Theorem (Cont)The Transposition Theorem (Cont)
1992October 10, No 39 vol
II Systems and Circuitson nsTransactio IEEE :examples Transpose
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EENG 751 04/22/23 9-37
The Transposition Theorem (Cont)The Transposition Theorem (Cont)
1992October 10, No 39 vol
II Systems and Circuitson nsTransactio IEEE :examples Transpose
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EENG 751 04/22/23 9-38
FIR Filter Equations
y n h k x n k
y h x
y h x h x
y h x h x h x
y M h x M h x M h M x h M x
y M h x M
k
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0
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y M h x M h x M h M x h M x[ ] [ ] [ ] [5] [ ] [ ]1 3 1 4
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EENG 751 04/22/23 9-39
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MxhxMhxMhMy
xhxhxhy
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kknxkhny
Transpose FIR Filter Equations
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EENG 751 04/22/23 9-40
The Transposition Theorem (Cont)The Transposition Theorem (Cont)
FiltersNotch Digitalon Paper Classic :examples Transpose
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EENG 751 04/22/23 9-41
FIR SFGsFIR SFGs
toreduce II and I formdirect Then the
otherwise 0
0for ][
Define
][][][][
system Average) (Moving FIR general heConsider t
00
Mnbnh
knxkhknxbny
n
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k
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][Mh]1[ Mh
][ny
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EENG 751 04/22/23 9-42
FIR SFGs (Cont)FIR SFGs (Cont)
:istion implementa
filter FIR general theof transpose that theNote filters. ltransversa
or linedelay tappeda as toreferred sometimes are systems These
1z
]0[h
][nx
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1z
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1z
][Mh ]1[ Mh
][ny
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EENG 751 04/22/23 9-43
FIR SFGs (Cont)FIR SFGs (Cont)
:likelook would41
21
41
function systemwith canceller pulse-3 normalizedA
21 zzzH
1z
41][nx
21
1z
41 ][ny
1z
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][nx21
1z
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:form sedin transpoOr
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EENG 751 04/22/23 9-44
FIR SFGs (Cont)FIR SFGs (Cont)
zero. be will
tscoefficien theof one then odd is If .2/1 where
][)(
from derived becan and
form general theof case special a is filters FIRfor form cascade The
2
1
22
11
0
k
s
M
kkkok
M
n
n
b
MMM
zbzbbznhzHs
1z
1z
][nx11b
01b
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1z
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sMb1
sMb0
sMb2
1z
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12b
02b
22b
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EENG 751 04/22/23 9-45
FIR SFGs (Cont)FIR SFGs (Cont)
system. theof zeros theare 7,...,1,0for where
1)(
as factored becan which
1)(
asy immediatelfunction system
the write toable be should everyone where
]7[][][
filter comb heConsider t
7/2
7
1
121
721
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21
kez
zzzH
zzH
nxnxny
jk
k
k
k
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EENG 751 04/22/23 9-46
FIR SFGs (Cont)FIR SFGs (Cont)
)()()()(7/6cos21
7/4cos217/2cos211)(
gives
1 and 7/2cos2
where
111
termsconjugatecomplex combining
4321
21
2121121
2
22111
zHzHzHzHzz
zzzzzzH
zkzz
zzzzzzzzz
kkk
kkkkk
1z
][nx 1 1z
7/2cos2
1z21
)(3 zH
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EENG 751 04/22/23 9-47
Linear Phase FIR SFGsLinear Phase FIR SFGs
equation.last in this required are multiplies 12/only that Note
]2/[]2/[][][][][
)]([][]2/[]2/[][][][
thensum, second in the Let ][][
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have wesymmetry,even andeven for
e.g. since, tionsmultiplica save toused becan symmetry thisand
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MnxMhkMnxknxkhny
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MnxMhknxnhknxnhny
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jbajj
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EENG 751 04/22/23 9-48
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
x n[ ]
y n[ ]
z 1z 1 z 1
z 1z 1z 1
h[ ]0 h[ ]1 h[ ]2
1
2M
h
2M
h
. degree,even offilter
FIRsymmetry even an of structure formDirect 6.34 Figure
M
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EENG 751 04/22/23 9-49
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
IVcOdd][][
IIIcEven][][
IIc Odd][][
IcEven][][
IVOdd][][
IIIEven][][
II Odd][][
IEven][][
Type Symmetry
nMhnh
nMhnh
nMhnh
nMhnh
nMhnh
nMhnh
nMhnh
nMhnh
M
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EENG 751 04/22/23 9-50
Causal Linear Phase SystemsCausal Linear Phase Systems
][][ Example.
cos2
2]0[
theninteger,evenan ][][
:systems I Type
.for 0][ then ,1
islength filter theIf .0for 0][ implies Causal
5
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kkM
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k
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0
symmetry ofCenter
22M 4M
1
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EENG 751 04/22/23 9-51
Causal Linear Phase Systems (Cont)Causal Linear Phase Systems (Cont)
2
1
0
2
1
0
0
2
1
2
1
0
2
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2
1
00
givessumation oforder thereversing and ][][
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then,or let sum, second In the
theninteger,oddan ],[][
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M
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k
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k
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ekhekheH
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emMhekheH
mMkM-km
ekhekhekheH
MnMhnh
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EENG 751 04/22/23 9-52
Causal Linear Phase Systems (Cont)Causal Linear Phase Systems (Cont)
5.151a eq 21
cos2
12
so ,21
2,1
21
21
0 then ,2
1let solution, text get the To
2cos2
2cos2 ,identitiesour fromBut
sums two theCombining :(Cont) systems II Type
2
1
1
2/
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kkM
heeH
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kM
k
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M
k
kMjkjj
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EENG 751 04/22/23 9-53
Causal Linear Phase Systems (Cont)Causal Linear Phase Systems (Cont)
21
cos32
or
25
cos2
5],[][ :(Cont) systems II Type
3
1
2/5
2
0
2/5
6
kkheeH
kkheeH
MnRnh
k
jj
k
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0
symmetry ofCenter 25
2M
5M
1
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EENG 751 04/22/23 9-54
Causal Linear Phase Systems (Cont)Causal Linear Phase Systems (Cont)
]2[][][ :example
sin2
2
integereven an ],[][ : systems III Type2/
1
2/
nnnh
kkM
hjeeH
MnMhnhM
k
jMj
0
symmetry ofCenter
12M
2M
1
1
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EENG 751 04/22/23 9-55
Causal Linear Phase Systems (Cont)Causal Linear Phase Systems (Cont)
]3[][][ :example
21
sin2
12
integer oddan ],[][ : IVsystems Type2/1
1
2/
nnnh
kkM
hjeeH
MnMhnhM
k
jMj
0
symmetry ofCenter 23
2M
3M
1
1
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EENG 751 04/22/23 9-56
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
0
2/1
2/1
0
2/1
2/1
0
0
)]([][][][][
thensum, second in the Let
][][][][][
][][][
Then integer. oddan is wherecase heConsider t
Mk
M
k
M
Mk
M
k
M
k
mMnxmMhknxkhny
kMm
knxnhknxnhny
knxnhny
M
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EENG 751 04/22/23 9-57
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
2/1
0
2/1
0
2/1
0
][][][][
][][ :IV Type
][][][][
][][ :II Type
then
][][][][][
With
M
k
M
k
M
k
kMnxknxkhny
nMhnh
kMnxknxkhny
nMhnh
kMnxkMhknxkhny
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EENG 751 04/22/23 9-58
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
z 1
x n[ ]
y n[ ]
z 1z 1 z 1
z 1z 1z 1
h[ ]0 h[ ]1 h[ ]2 hM
[ ] 3
2h
M[ ]
12
odd for ][][ :System II Type MnMhnh
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EENG 751 04/22/23 9-59
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
32415
5432151
1051)(
and 10]3[]2[,5]4[]1[ ,1]5[]0[
5Mfor ][][
51010511)( :System II Type
zzzzzzH
hhhhhh
nMhnh
zzzzzzzH
z 1
x n[ ]
y n[ ]
z 1z 1
z 1z 1
1]0[ h 5]1[ h 10]2[ h
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EENG 751 04/22/23 9-60
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
z 1
x n[ ]
y n[ ]
z 1z 1 z 1
z 1z 1z 1
h[ ]0 h[ ]1 h[ ]2 hM
[ ] 3
2h
M[ ]
12
odd for ][][ :System IV Type MnMhnh
1 1 11 1
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EENG 751 04/22/23 9-61
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
!1
is so root, a is If
011
][)(
then,][][ If
1][)(
][][][)(
then,0)(z and ][(z) :Suppose
0
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00
0 0
00
000
000
000
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zz
zHz
zkhzzH
nMhnh
zkMhzzH
zkMhzzkhzzkhzH
HzkhH
MM
k
k
M
M
k
k
M
M
k
kMM
k
kMMM
k
k
M
k
k
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EENG 751 04/22/23 9-62
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
2
11
11
43211
1
1111
11
12,
1Re2
where1)(
is )(say ),( offactor a and )( of zeros all are
1,
1, then circle,unit on thenot zerocomplex a is If (1)
:situations following thehavecan weThus
0 then,0)(
if i.e. pairs, conjugate
complex in occur zeros then real, is ][ if Similarly,
zzd
zzc
zczdzczzH
zHzHzH
zzzz
zHzH
nh
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EENG 751 04/22/23 9-63
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
43112
111
2
1
2
11112
1
2
1
1
11111
1
1
1
1
11
111
111
111
111)(
givesout is th gMultiplyin
11
1111)(
Consider values?get these wedo How
zzzzzzz
z
zzz
zzz
z
zzz
zzzH
zz
zz
zzzzzH
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EENG 751 04/22/23 9-64
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
2
1
1
1
1
1
1
11
112
1
2
1
1
1
11
11
1
12
112
gives termscollecting and
numberscomplex of properties theusing out, thisgMultiplyin
111
1Re2
11
so equal are of powers like of tscoefficien theNow
zz
zz
zzd
zzzz
zzd
zz
zzzzc
z
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EENG 751 04/22/23 9-65
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
14
44
321
3
3
33
22
212
2
222
1)( is
)(say ),( offactor ingcorrespond thezero a is 1 If (4)
Re2 where1)(
is )(say ),( offactor ingcorrespond theand
zero a also is then circle,unit on the zerocomplex a is If (3)
1 where1)(
is )(say ),( offactor ingcorrespond
theand zero a also is 1
then 1 zero, real a is If (2)
listour with Continuing
zzH
zHzHz
zbzbzzH
zHzH
zz
zzazazzH
zHzH
zzz
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EENG 751 04/22/23 9-66
Linear Phase FIR SFGs (Cont)Linear Phase FIR SFGs (Cont)
H z h z az z bz z
cz dz cz z
a zz
b z c zz
d zz
( ) [ ]( )( )( )
( )
, Re{ }, Re , .
0 1 1 1
1
12 2
12
1
1 1 2 1 2
1 2 3 4
22
3 11
11
2
1z
1z
1
1z
1
1z
2z 2
1z
3z
3z
4z
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EENG 751 04/22/23 9-67
All Pass FiltersAll Pass Filters
(Why?) filter. pass allan is filters pass all of cascadeA :Note
111
11
1
1
1 :Zero, :Pole
]1[][]1[][1
)( 1
1
j
j
j
jj
j
jj
j
jj
aeae
aeae
eae
aeeH
aeae
eH
azaz
nxnxanaynyaz
azzH
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EENG 751 04/22/23 9-68
All Pass Filters (Cont)All Pass Filters (Cont)
:locationmemory one and multiplies 2
requires and belowshown istion implementa II formdirect The
being. timefor theparameter real a is where1
)(Let 1
1
aaz
azzH
1za
a][nx ][ny
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EENG 751 04/22/23 9-69
All Pass Filters (Cont)All Pass Filters (Cont)
locations.memory 2 andmultiply one
requires which ]1[][]1[][tion multiplica
single aget torearrange and ]1[][]1[][
i.e. ),( toingcorrespond LCCDE heConsider t
nxnxnyany
nxnaxnayny
zH
1z
a 1z
][nx ][ny
1
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EENG 751 04/22/23 9-70
All Pass Filters (Cont)All Pass Filters (Cont)
:is system cascaded thisoftion implementa II formdirect The
.parameters real are b and where1
1)(Let 1
1
1
1
abz
bzaz
azzH
1z
a 1z
][nx ][ny
1
1z
b 1z1
1z
b 1z1 b
1z1
1z
][ny
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EENG 751 04/22/23 9-71
All Pass Filters (Cont)All Pass Filters (Cont)
:is system cascaded thisoftion implementa II formdirect The
.parameters real are b and where1
1)(Let 1
1
1
1
abz
bzaz
azzH
1z
a 1z
][nx ][ny
1
1z
b 1z1
1z
b 1z1 b
1z1
1z
][ny
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EENG 751 04/22/23 9-72
All Pass Filters (Cont)All Pass Filters (Cont)Consider the second SFG
1z
b 1z1
Flip it over I.e.
1z
b1z
1
][ny
][ny
Pull down I.e.
1z b
1z1][ny
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EENG 751 04/22/23 9-73
All Pass Filters (Cont)All Pass Filters (Cont)filter pass all second for the form alternate theSubstitute
1z
a 1z
][nx
1 b
1z1
1z
][ny
1z
a 1z
][nx
1 b
1z1 ][ny
filter. pass allorder second for theSFG sharingdelay theis This
branches. middle two theCombine
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EENG 751 04/22/23 9-74
All Pass Filters (Cont)All Pass Filters (Cont)
:locationmemory one and multiplies 2 requires
and belowshown istion implementa )(canonical II formdirect The
parameter. real a is where1
)(Let 1
1
aaz
azzH
1za
a][nx ][ny
SFG.. canonicalnon locationsmemory 2ith multiply w single The
1z
a 1z
][nx ][ny
1
tion?implementamultiply single canonical a thereIs
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EENG 751 04/22/23 9-75
Signal Flow Graph ExampleSignal Flow Graph Example
z 1z 1
x n[ ]
y n[ ]
a2
1
a1
input. one than more having nodes the
only Label nodes.network 8 are thereNote ).( Calculate zH
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EENG 751 04/22/23 9-76
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
][][][][
][]1[][
][]2[][
][][][
324
22113
12
31
nwnwnynw
nwanwanw
nxnwnw
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x n[ ]z 1z 1
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1
a1
w n1[ ]w n2[ ]
w n3[ ]
w n4[ ]
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EENG 751 04/22/23 9-77
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
)()()()(
)()()()(
)()()(
)()()(
324
2211
13
12
2
31
zWzWzYzW
zWzazWzazW
zXzWzzW
zXzWzW
X z( )z 1z 1
Y z( )
a2
1
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W z1( )W z2 ( )
W z3( )
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Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
)( and),(),(),( unknownsfour in equationsFour
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EENG 751 04/22/23 9-79
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
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EENG 751 04/22/23 9-80
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
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EENG 751 04/22/23 9-81
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
(Why?)column second thecolumn to third theaddingby
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EENG 751 04/22/23 9-82
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
filter. edunnormaliz
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EENG 751 04/22/23 9-83
Exercise (To be Handed In)Exercise (To be Handed In)
11/09/07ok
delays. 6 and mulipliers with twofor SFG a Draw (d)
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system heconsider t Now
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. ofSFG II formdirect a Draw (b)
Why?system. heIdentify t
system. theof zeros and poles theDetermine (a)4
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function system with system LTI causal heConsider t 9.1
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EENG 751 04/22/23 9-84
Complex Filter ExampleComplex Filter Example
Consider the simple complex FIR filter given by the
system function:
) where , so
) = ), and
is determined from the which occurs when
, i.e.
H z A az a re
H e A re e A re
H e A r r
A H e
K
H e A r r A r
Ar
j
j j j j
j
j
j
( ) (
( ) ( (
( ) cos( )
max ( )
( )
max ( ) ( )
( )
1
1 1
1 2
2 1
1 2 1 1
11
1
2
2
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EENG 751 04/22/23 9-85
Complex Filter Example(Cont)Complex Filter Example(Cont)
Note that
) where , and
) ,
and
If , i.e. , then
Since multiplying by shifts the frequency response by
and if is linear phase, i.e
H z G a re
G z A z g n A R n A A
h n a g n
a e r
h n e g n
H e G e
e
G e
G e e
za
j
n
n
j
j n
j j
j n
j
j j
( ) (
( ) ( [ ] ( ) [ ] ,
[ ] [ ].
[ ] [ ] and
( ) ( ).
( ) .
( )
( )
1 1
1
12
( )
( ( ) ) ( )
( ) ( )
( ) ( )
A e A e
H e e A e
j j
j j j
where is real, then
is also linear phase.
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EENG 751 04/22/23 9-86
Complex System Signal Flow GraphsComplex System Signal Flow Graphs
z 1
y n[ ]
ar1
x n[ ]
11 r
z 1
y n[ ]
b1
x n[ ]b0
The system function for this simple FIR filter is:
where
and
H z b b z
b hr
b ha
rre
r
j
( )
[ ] [ ]
0 1
1
0 101
11
1 1
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EENG 751 04/22/23 9-87
Complex System SFG(Cont)Complex System SFG(Cont)
Since the LCCDE corresponding to the system function
is
it can be broken up into the real and imaginary parts
yielding two coupled LCCDEs, i.e.
H zr
re z
y nr
x n re x n
y nr
x n r x n r x n
y nr
x n r x n r x n
j
j
R R R I
I I I R
( )
[ ] [ ] [ ]
[ ] [ ] cos [ ] sin [ ]
[ ] [ ] cos [ ] sin [ ]
11
1
11
1
11
1 1
11
1 1
1
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EENG 751 04/22/23 9-88
Complex System Signal Flow GraphsComplex System Signal Flow Graphs
z 1
Re( [ ])y n
r sin
r cosz 1
Re( [ ])x n
Im( [ ])y nIm( [ ])x n
r sin
r cos
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Application from IEEE Transactions on Application from IEEE Transactions on Signal Processing, Vol 46, No.2 Feb 98 Signal Processing, Vol 46, No.2 Feb 98
Page 364Page 364
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Application from IEEE Transactions on Application from IEEE Transactions on Signal Processing, Vol 46, No.2 Feb 98 Signal Processing, Vol 46, No.2 Feb 98
Page 368Page 368
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Application Example (Continued)Application Example (Continued)x n[ ] y n[ ]
z 1
z 1
H zz
z z( )
cos ( ) ( )cos ( )
1 0 0 1 0 11
01
0 1 0 12
2 1 11 2
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EENG 751 04/22/23 9-92
Application Example (Continued)Application Example (Continued)
y k[ ] y k k[ | ]1
z 1
z 1
H zz
z z( )
cos ( ) ( )cos ( )
1 0 0 1 0 11
01
0 1 0 12
2 1 11 2
2 1 0( ) cos
( )( )1 11 0
2 0 cos
( ) 0 1 0 1
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EENG 751 04/22/23 9-93
Application Example (Continued)Application Example (Continued)
z 1
z 1
z 11
0
2cos
1 0 1 1
1
W1
W2
W3
H zz
z z( )
cos ( ) ( )cos ( )
1 0 0 1 0 11
01
0 1 0 12
2 1 11 2
y k k[ | ]1
y k[ ]