eeng 751: signal processing i

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


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


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


: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


:(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


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


).(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


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


][][][ 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


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


.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


EENG 751 04/22/23 9-20

SFG Application ReferenceSFG Application Reference


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

1

1

0

1

1

nxnayny

nxamxany

nxamxny

aW

mxkXnjamxny

WkXjy

n

m

mn

n

m

mn

mnmjkN

pmQ

n

m

mn

j

m

mjkNpmQk

EENG 751 04/22/23 9-22


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

1

bbzazbb

zH

azz

zH

nxnayny

][ny

1a

][nx

00 b1z

11 b

EENG 751 04/22/23 9-23


paper. in the 5 figure as same theisSFG which II formdirect thehas1

)(

or]1[]1[][

So

1

1

azz

zH

nxnayny

][nya

][nx

1z

EENG 751 04/22/23 9-24


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

11)(

back to Going

zazazbzbb

zH

zzNkzaz

zH

aaa

eWa

zazaa

zazzH

zaza

azz

azz

zH

NjkkN

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

Mb

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

zBb

za

zbzH

N

k

kk

L

k

kkk

N

k

kk

M

k

kk

N

k

kk

M

k

kk

,max here w1

)(

)(1

)()(

)(1

)()()()(

)( and )(

where)(1

)(

1)(

:follows asit rewrite andagain function system heRemember t

1

10

0

00

000

11

0

1

0

EENG 751 04/22/23 9-28


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


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


: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


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[ ] [ ] [ ] [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


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


: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


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


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


)()()()(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


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


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

kM

kheeH

kM

eee

eekheH

M

k

Mjj

M

k

Mjj

MjkMjkj

M

k

kMjkjj

EENG 751 04/22/23 9-53


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


]2[][][ :example

sin2

2

integereven an ],[][ : systems III Type2/

1

2/

nnnh

kkM

hjeeH

MnMhnhM

k

jMj

0

symmetry ofCenter

12M

2M

1

1

EENG 751 04/22/23 9-55


]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


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


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

EENG 751 04/22/23 9-58


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


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

EENG 751 04/22/23 9-60


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


!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

HzkhH

MM

k

k

M

M

k

k

M

M

k

kMM

k

kMMM

k

k

M

k

k

EENG 751 04/22/23 9-62


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

EENG 751 04/22/23 9-63


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

EENG 751 04/22/23 9-64


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

EENG 751 04/22/23 9-65


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

EENG 751 04/22/23 9-66


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


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


: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


: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


: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


)()()()(

)()()()(

)()()(

)()()(

324

2211

13

12

2

31

zWzWzYzW

zWzazWzazW

zXzWzzW

zXzWzW

X z( )z 1z 1

Y z( )

a2

1

a1

W z1( )W z2 ( )

W z3( )

W z4 ( )

EENG 751 04/22/23 9-78


)( and),(),(),( unknownsfour in equationsFour

0

0

1

1

gives grearrangin and )(by equation

each dividing So system. theof )()()(

)()(

)(

,)(

)()(,

)(

)()(,

)(

)()(Let

4321

432

32211

1

212

31

44

33

22

11

zHzHzHzH

HHH

HHaHza

HHz

HH

zX

zHzXzY

zXzW

zH

zX

zWzH

zX

zWzH

zX

zWzH

EENG 751 04/22/23 9-79


)!( 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


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

z

az

aza

zaza

zz

zCz

z

aza

z

zH

EENG 751 04/22/23 9-81


(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


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[ ]

eeng 751: signal processing i

Documents