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
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.
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.
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
][][][
LCCDE ingcorrespond theand
1)(
function system heRemember t
01
1
0
both or hardware software,
knxbknyany
za
zbzH
M
kk
N
kk
N
k
kk
M
k
kk
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
are theredistinct, are and all generalin where
][][][
LCCDEscalar generalour For
)(
are elementstion implementa )(primitive lfundamenta The
01
MNMNMN
ba
knxbknyany
memorydelay Unit
Multiplier
ractorAdder/Subt
kk
M
kk
N
kk
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
branches exitingonly have
: typesnode threeare There
nodes. theconnecting branches directed and nodes of consistsSFG A
:notation and sDefinition
mwyx
nodes Network
nodes Sink
nodes Source
mmm
1w
1y1x
2x
2w 3w
4w
EENG 751 04/22/23 9-7
SFG Fundamentals (Cont)SFG Fundamentals (Cont)
:branches entering all of ouputs thesums andadder an is nodeeach
delay timeArbitrary
delay e Unit tim2.
1by tion Multiplica
by tion Multiplica 1.
types twoof one toconfined be lbranch wil directedEach
a
][]1[][][ 321 nbxnxnaxny ][1 nx
][2 nx
1z
a
][3 nx
kz
a
b1z
EENG 751 04/22/23 9-8
SFG GenerationSFG Generation
says""equation what thedoingjust by SFG its draw lets and
]1[][][
toingcorrespond )(
system FIRorder first simpleour Consider
10
110
nxbnxbny
zbbzH
]1[][][ 10 nxbnxbny][nx0b
1z
1b]1[ nx ]1[1 nxb
.
only that branches hasfilter FIR simpleour for SFG that theNotice
forward feed
EENG 751 04/22/23 9-9
SFG Generation (Cont)SFG Generation (Cont)
says""equation what thedoingjust by SFG its draw letsagain and
][]1[][
toingcorrespond 1
1)(
system IIRorder first simplest our consider Now
1
11
nxnyany
zazH
][]1[][ 1 nxnyany ][nx
1z
1a]1[ ny]1[1 nya
.and
that branches hasfilter IIR simpleour for SFG that theNotice
backward forward feed
EENG 751 04/22/23 9-10
SFG Generation (Cont)SFG Generation (Cont)
:(why?) diagramblock with
]1[][][ and ][]1[][
form cascadein or ]1[][]1[][
toingcorrespond)()(1
11
)(
systemorder first simpleour Consider
101
101
2111
1101
1
110
nxbnxbnwnwnyany
nxbnxbnyany
zHzHza
zbbzazbb
zH
)(1 zH )(2 zH][nx][nw
][ny
][ny][nw1z
1a]1[ ny]1[1 nya
][nx 0b1z
1b
]1[ nx ]1[1 nxb
EENG 751 04/22/23 9-11
SFG Generation (Cont)SFG Generation (Cont)closely moreSFG combinedour Examine
][ny][nw1z
1a]1[ ny]1[1 nya
][nx 0b1z
1b
]1[ nx ]1[1 nxb
SFGour simplify can then we][]1[]1[][
as LCCDEour rewrite weIf
011 nxbnyanxbny
][ny1z
1a]1[ ny
][nx 0b1z
1b
]1[ nx]1[]1[ 11 nyanxbdelays! two theNotice
EENG 751 04/22/23 9-12
SFG Generation (Cont)SFG Generation (Cont)
givesorder reverse in the
SFG thengimplementi so)()(1
1)(
have also wesystems LTIfor
)()(1
11
)(
Since
121
1011
2111
1101
1
110
zHzHzbbza
zH
zHzHza
zbbzazbb
zH
][ny
1z
1a]1[ nw]1[1 nwa
][nw
][nx
0b1z
1b]1[ nw
]1[1 nwb
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
][ny
1a
]1[ nw
]1[1 nwa
][nx
0b1z
1b
]1[1 nwb
][nw
]1[][]1[][or )(1
)(
gives)( geliminatin and Transforms Using
].1[][][ and ][]1[][ now Where
10111
110
101
nxbnxbnyanyzXzazbb
zY
zWz
nwbnwbnynxnwanw
EENG 751 04/22/23 9-14
SFG Generation (Cont)SFG Generation (Cont)
N
k
kk
M
k
kk
N
k
kk
M
k
kkN
k
kk
M
k
kk
zazH
zbzH
zHzHza
zbza
zbzH
1
2
01
21
1
0
1
0
1
1)(
)(
where
)()(1
1
1)(
again function system heRemember t
EENG 751 04/22/23 9-15
SFG Generation (Cont)SFG Generation (Cont)
][][][ and ][][
where][][][
are LCCDEs ingcorrespond theand
10
01
nwknyanyknxbnw
knxbknyany
N
kk
M
kk
M
kk
N
kk
][ny][nw 1z1a]1[ ny
]2[ ny
][nx
0b
1z 1b]1[ nx
]2[ nx1z 2b
1z 1Mb
1z Mb]1[ Mnx
][ Mnx
2a 1z
1z1Na
Na 1z
]1[ Nny
][ Nny
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
NMIform direct
][nw
][nx
0b
1z 1b]1[ nx
]2[ nx1z 2b
1z 1Mb
1z Mb]1[ Mnx
][ Mnx
][ny1z1a
]1[ ny
]2[ ny2a 1z
1z1Na
Na 1z
]1[ Nny
][ Nny
EENG 751 04/22/23 9-17
SFG Generation (Cont)SFG Generation (Cont)gives (z) and (z) inginterchang delays, have weSince 21 HHM N
][ny1z1a
2a 1z
1z1Na
Na 1z
][nw][nx0b
1z 1b]1[ nw
]2[ nw1z 2b
1z 1Mb
1z Mb][ Mnw
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
Na 1z
][ny1z1a
2a 1z
][nw
][nx
0b
1b
2b
1Mb
Mb
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.
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.
EENG 751 04/22/23 9-21
SFG Application ExampleSFG Application Example
LCCDE.order first a iswhich
]1[]1[][
]1[][][
]1[][][
exercises, theof one similar to that Note . and
][, where][][
as drepresente becan ][
(26)equation
2
0
2
2
0
1
11
1
1
0
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1
0
1
1
nxnayny
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nxamxny
aW
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WkXjy
n
m
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n
m
mn
mnmjkN
pmQ
n
m
mn
j
m
mjkNpmQk
EENG 751 04/22/23 9-22
SFG Application ExampleSFG Application Example
isSFG II formdirect the,1 and 0 with 1
)(
LCCDEorder first general the to thiscomparing and1
)(
can write we]1[]1[][
withStarting
1011
`110
1
1
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azz
zH
nxnayny
][ny
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00 b1z
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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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or]1[]1[][
So
1
1
azz
zH
nxnayny
][nya
][nx
1z
EENG 751 04/22/23 9-24
SFG Application ExampleSFG Application Example
22
11
22
110
21
21
2
/2
221
11
1
1
1
1
1
1
1)(
biquad general the to thisComparing
/2cos21)(
??
so but
1
1)(
11
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back to Going
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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
][ny1z1a
2a 1z
][nx
0b
1b
2b
paper. in the 6 figure as same eexactly th is
SFG which following theyieldswhich 1,/2cos2
,,1,0 then problemour For
21
210
aNka
Wabbb kN
][ny1z Nk /2cos2
1 1z
][nx
a
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
][ny][nx
0b
2a 1z 110 bab
1z1Na
Na 1z
1Mb
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2a 1z 110 bab
EENG 751 04/22/23 9-27
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
NML
za
zbabbzH
zP
zBzPbb
zP
zBzPbzPbbzH
zazPzbzB
zP
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za
zbzH
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,max here w1
)(
)(1
)()(
)(1
)()()()(
)( and )(
where)(1
)(
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:follows asit rewrite andagain function system heRemember t
1
10
0
00
000
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0
1
0
EENG 751 04/22/23 9-28
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
N
k
kk
L
k
kkk
za
zbabzHzHbzH
1
10
110
1)()()(
:situation following thehave weSo
)(zY][nx0b
)(1 zH 1z1Na
Na 1z
][ny1z1a
2a 1z
][nx110 bab
220 bab
MM bab 0
EENG 751 04/22/23 9-29
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
get todiagramblock in the (z)for Substitute 1H
1z1Na
Na 1z
][ny
1z1a
2a 1z
][nx
110 bab
220 bab
MM bab 0
0b
EENG 751 04/22/23 9-30
Alternate Canonic Forms (Cont)Alternate Canonic Forms (Cont)
:flowgraph final thecreate Finally to
1z1Na
Na 1z
)(zY
1z1a
2a 1z
)(zX
110 bab
220 bab
MM bab 0
0b
EENG 751 04/22/23 9-31
Cascade FormCascade Form
difficult. very bemay ion factorizat thiswhere
1)(
1)(
:function system theof form cascasde thegives )( Factoring
12
21
1
22
110
1
1
0
P
k kk
kkkP
kkN
k
kk
M
k
kk
zzzz
zHza
zbzH
zH
][nx)(1 zH ][ny)(2 zH )(zH P
EENG 751 04/22/23 9-32
Optionan is Pipelining
Costs Hardware Reduced
Slow ation,Standardiz
1)( 2
21
1
22
110
:Form Cascade the of Properties
zzzz
zHkk
kkkk
Cascade FormCascade Form
1zk1
1zk2
k1
k0
k0
)(zH k
EENG 751 04/22/23 9-33
Parallel FormParallel Form
Nkzz
z
MkNzc
zH
zzz
zczH
zHza
zbzH
zH
kk
kk
kk
k
N
k kk
kkM
k
kk
L
kkN
k
kk
M
k
kk
1for 1
for
)(
where
1)(
)(1
)(
:function system
theof form parallel thegives )(on fractions partial Using
22
11
110
2/1
12
21
1
110
0
1
1
0
EENG 751 04/22/23 9-34
Hardware of Lots
Fast
ation,Standardiz
1
)(2
21
1
110
:Form Parallel the of Properties
zzz
zc
zH
kk
kk
kk
k
Parallel Form (Cont)Parallel Form (Cont)
][nx)(2 zH ][ny
1zk1
1zk2
k1
k0
)(zH k
)(zH L
)(1 zH
)(zH k
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
same.the
function systemthe leaves also output and inputthe reversing SFGs,
output-inputsingle For
The Transposition TheoremThe Transposition Theorem
1z
a][nx
1z
b c
1z
a][ny
][nx
1z
b c
1z ][ny][nx 1zbc
21
21
)(
)()()()(
czbzazH
zXczzXbzzaXzY
21
11
)(
)()()()(
czbzazH
zaXzzbXzXczzY
described be
need structures network
the half Only :Note
][ny
a
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
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
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
M
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [
0
0 0 0
1 0 1 1 0
2 0 2 1 1 2 0
0 1 1 1 1 0
1 0 1 1 1 2 1
2 0 2 1 1 1 3 2
3 0 3 1 2 1 4 3
4 0 4
] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ]
h x M h M x h M x
y M h x M h x M h M x h M x
y M h x M h x M h M x h M x
y M h x M h x M h M x h M x[ ] [ ] [ ] [5] [ ] [ ]1 3 1 4
EENG 751 04/22/23 9-39
]3[]0[]2[]1[]1[]2[][]3[]3[
]2[]0[]1[]1[][]2[]2[
]1[]0[][]1[]1[][]1[
][]0[]1[]1[]0[][][
]2[]0[]1[]1[]0[]2[]2[
]1[]0[]0[]1[]1[
]0[]0[]0[
0][][][
MxhMxhMxhMxhMy
MxhMxhMxhMy
MxhMxhxMhMy
MxhxMhxMhMy
xhxhxhy
xhxhy
xhy
M
kknxkhny
Transpose FIR Filter Equations
EENG 751 04/22/23 9-40
The Transposition Theorem (Cont)The Transposition Theorem (Cont)
FiltersNotch Digitalon Paper Classic :examples Transpose
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
M
k
M
kk
1z
]0[h][nx ]1[h
1z
]2[h
1z
][Mh]1[ Mh
][ny
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
]1[h
1z
]2[ Mh
1z
][Mh ]1[ Mh
][ny
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
41
][nx21
1z
41 ][ny
:form sedin transpoOr
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
21b
1z
1z
sMb1
sMb0
sMb2
1z
1z
12b
02b
22b
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
21
21
kez
zzzH
zzH
nxnxny
jk
k
k
k
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
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 ][][
]2/[]2/[][][][][][
have wesymmetry,even andeven for
e.g. since, tionsmultiplica save toused becan symmetry thisand
)()( i.e. phase,linear have will
system then the][][or ][][ If
12/
0
0
12/
12/
0
12/
12/
00
M
MnxMhkMnxknxkhny
mMnxnMhMnxMhknxkhny
kMmknxnh
MnxMhknxnhknxnhny
M
eAeeH
nMhnhnMhnh
M
k
Mk
M
k
M
Mk
M
k
M
k
jbajj
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
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
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
2/
1
2/
nRnh
kkM
hheeH
MnMhnh
MnnhM
nnh
M
k
Mjj
0
symmetry ofCenter
22M 4M
1
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
1
2
1
00
givessumation oforder thereversing and ][][
conditionsymmetry theusing andk index back to Switching
then,or let sum, second In the
theninteger,oddan ],[][
:systems II Type
M
k
kMj
M
k
kjj
Mm
mMj
M
k
kjj
M
Mk
kj
M
k
kjM
k
kjj
ekhekheH
nMhnh
emMhekheH
mMkM-km
ekhekhekheH
MnMhnh
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/
2
1
0
2/
2/
2
1
0
kkM
heeH
kkM
kM
k
Mkkk
Mk
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kheeH
kM
eee
eekheH
M
k
Mjj
M
k
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MjkMjkj
M
k
kMjkjj
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
jj
0
symmetry ofCenter 25
2M
5M
1
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
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1
1
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
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
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
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k
M
k
kMnxknxkhny
nMhnh
kMnxknxkhny
nMhnh
kMnxkMhknxkhny
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
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
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z 1
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1]0[ h 5]1[ h 10]2[ h
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
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
0
0
00 0
00
0 0
00
000
000
000
00
zz
zHz
zkhzzH
nMhnh
zkMhzzH
zkMhzzkhzzkhzH
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MM
k
k
M
M
k
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k
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k
kMMM
k
k
M
k
k
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
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zHzHzH
zzzz
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nh
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
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z
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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
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111
1Re2
11
so equal are of powers like of tscoefficien theNow
zz
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zzd
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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
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zz
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zzz
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
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
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
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
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
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
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
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
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
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
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
nxnwnw
x n[ ]z 1z 1
y n[ ]
a2
1
a1
w n1[ ]w n2[ ]
w n3[ ]
w n4[ ]
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
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zXzWzW
X z( )z 1z 1
Y z( )
a2
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W z1( )W z2 ( )
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EENG 751 04/22/23 9-78
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
)( and),(),(),( unknownsfour in equationsFour
0
0
1
1
gives grearrangin and )(by equation
each dividing So system. theof )()()(
)()(
)(
,)(
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432
32211
1
212
31
44
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11
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EENG 751 04/22/23 9-79
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
)!( is need weall case in this
But . offunction a be could where)()(
for solve and )(calculate could onein theory Then
0
0
1
1
1110
01
001
0101
system. thedescribing equationslinear ofset
theof tscoefficien ofmatrix theis )( where)()(
as formmatrix in equations of system this writecould One
4
11
1
4
3
2
1
21
1
2
zH
zzCzH
zC
H
H
H
H
aza
z
zCzHzC
EENG 751 04/22/23 9-80
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
22
11
21
1
2
2
21
1
2
21
1
2
21
1
2
4
11
1
01)(
1
01
101
1110
01
001
0101
)(
.)( oft determinan theis )( where
)(
0110
01
101
1101
)(
gives which Method) s(Cramer' methodsimpler a Use
zazaaza
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zz
zCz
z
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z
zH
EENG 751 04/22/23 9-81
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
(Why?)column second thecolumn to third theaddingby
100
11
121
110
1
111
)(
(Why?) row second the torowfirst theaddingby
0110
01
0111
1101
0110
01
101
1101
)(
let numerator, thecalculate To
21
1
2
21
1
2
21
1
2
21
1
2
aza
z
aza
z
zN
aza
z
aza
zzN
EENG 751 04/22/23 9-82
Signal Flow Graph Example (Cont)Signal Flow Graph Example (Cont)
filter. edunnormaliz
an is that thisminutes few ain see willWhich we
21121
)()(
121 )(
2111
21)(
giveslet numerator, thecalculate toContinuing
22
11
22
112
4
22
112
112
2
21
1
2
Notch
zazazazaa
zHzH
zazaazN
zaazaza
zzN
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)
.41
4
system heconsider t Now
.multiplier one uses that ofSFG a Draw (c)
. ofSFG II formdirect a Draw (b)
Why?system. heIdentify t
system. theof zeros and poles theDetermine (a)4
14
function system with system LTI causal heConsider t 9.1
2
2
2
12
1
1
2
2
1
zH
z
zzHzH
zH
zH
z
zzH
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
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.
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
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
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
EENG 751 04/22/23 9-89
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
EENG 751 04/22/23 9-90
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
EENG 751 04/22/23 9-91
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
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
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[ ]
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