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C H A P T E R 3TransformRepresentationofSignalsandLTISystemsAs you have seen in your prior studies ofsignals and systems, and as emphasizedin the review in Chapter 2, transforms play a central role in characterizing andrepresenting signals and LTI systems in both continuous and discrete time. Inthischapterwediscusssomespecificaspectsoftransformrepresentationsthatwillplay an important role in later chapters. These aspects include the interpreta-tionof Fourier transform phase through theconcept of group delay, and methods referred to as spectral factorization for obtaining a Fourier representation(magnitudeandphase)whenonlytheFouriertransformmagnitude isknown.

3.1 FOURIERTRANSFORMMAGNITUDEANDPHASETheFouriertransformofasignalorthefrequencyresponseofanLTIsystemisingeneralacomplexvaluedfunction. AmagnitudephaserepresentationofaFouriertransformX(j)takestheform

X(j) =|X(j)|ejX(j) . (3.1)In eq. (3.1), X(j) denotes the (nonnegative) magnitude andX(j) denotes| |the(realvalued)phase. Forexample,ifX(j)isthesincfunction,sin()/,then|X(j)|istheabsolutevalueofthisfunction,whileX(j)is0infrequencyrangeswherethesincispositive,and infrequencyrangeswherethesincisnegative. Analternativerepresentation isanamplitudephaserepresentation

A()ejAX(j) (3.2)in which A() = |X(j)| is real but can be positive for some frequencies andnegativeforothers. Correspondingly,AX(j) =X(j)whenA() = +X(j) ,andAX(j) =X(j)whenA() =|X(j)|.

| |Thisrepresentationisoften

preferred when its use can eliminate discontinuities of radians in the phase asA()changessign. Inthecaseofthesincfunctionabove,forinstance,wecanpickA()=sin()/andA =0. Itisgenerallyconvenientinthefollowingdiscussionforustoassumethatthetransformunderdiscussionhasnozerosonthejaxis,sothatwecantakeA() =|X(j)|forall (or,ifwewish,A() =|X(j)|forall). Asimilardiscussionappliesalso,ofcourse,indiscretetime.In eitheramagnitudephaserepresentationoranamplitudephaserepresentation,thephaseisambiguous,asanyintegermultipleof2canbeaddedatanyfrequencyc 47AlanV.OppenheimandGeorgeC.Verghese,2010


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48 Chapter3 TransformRepresentationofSignalsandLTISystemswithout changing X(j) in (3.1) or (3.2). A typical phase computation resolvesthisambiguitybygeneratingthephasemodulo2,i.e.,asthephasepassesthrough+ it wraps around to (or from wraps around to +). In Section 3.2we will find it convenient to resolve this ambiguity by choosing the phase to bea continuous function of frequency. This is referred to as the unwrapped phase,sincethediscontinuitiesatareunwrappedtoobtainacontinuousphasecurve.TheunwrappedphaseisobtainedfromX(j)byaddingstepsofheightequalto or2 whereverneeded, inordertoproduceacontinuousfunctionof. Thesteps of height are added at points where X(j) passes through 0, to absorbsignchangesasneeded;thestepsofheight2areaddedwhereverelseisneeded,invoking the fact that such steps make no difference to X(j), as is evident from(3.1). WeshallproceedasthoughX(j)isindeedcontinuous(anddifferentiable)atthepointsof interest, understandingthat continuitycan indeed beobtained inallcasesof interesttousbyadding intheappropriatestepsofheight or2.Typically,ourintuitionforthetimedomaineffectsoffrequencyresponsemagnitudeor amplitude on a signal is rather welldeveloped. For example, if the Fouriertransformmagnitudeissignificantlyattenuatedathighfrequencies,thenweexpectthe signaltovaryslowly andwithoutsharpdiscontinuities. Onthe otherhand,asignal in which the low frequencies are attenuated will tend to vary rapidly andwithoutslowlyvaryingtrends.Incontrast,visualizingtheeffectonasignalofthephaseofthefrequencyresponseofasystemismoresubtle,butequallyimportant. Webeginthediscussionbyfirstconsideringseveralspecificexampleswhicharehelpfulinthenconsideringthemoregeneralcase. Throughoutthisdiscussionwewillconsiderthesystemtobeanallpasssystemwithunitygain,i.e. theamplitudeofthefrequencyresponseA(j) = 1(continuoustime)orA(ej)=1(discretetime)sothatwecanfocusentirelyontheeffect

of

the

phase.

The

unwrapped

phase

associated

with

the

frequency

response

willbedenotedasAH(j)(continuoustime)andAH(ej)(discretetime).

EXAMPLE3.1 LinearPhaseConsideranallpasssystemwithfrequencyresponse

H(j) =ej (3.3)i.e. in an amplitude/phase representation A(j)=1 andAH(j) =. Theunwrapped phase for this example is linear with respect to , with slope of .For inputx(t)withFouriertransformX(j),theFouriertransformoftheoutputisY(j) =X(j)ej andcorrespondinglytheoutputy(t)isx(t). Inwords,linear

phase

with

a

slope

of

corresponds

to

atime

delay

of

(or

atime

advance

if isnegative).Foradiscretetimesystemwith

H(ej) =ej ||< (3.4)the phase is again linear with slope . When is an integer, the time domaininterpretationoftheeffectonaninputsequencex[n]isagainstraightforwardandisAlanV.OppenheimandGeorgeC.Verghese,2010c


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Section3.1 FourierTransformMagnitudeandPhase 49asimpledelay(positive)oradvance(negative)of . Whenisnotaninteger,| |theeffect isstillcommonlyreferredtoasadelayof,butthe interpretation ismore subtle. If we think of x[n] as being the result of sampling a bandlimited,continuoustime signal x(t) with sampling period T, the output y[n] will be theresultofsamplingthesignaly(t) =x(tT)withsamplingperiodT. In factwesaw this result in Example 2.4 of chapter 2 for the specific case of a halfsampledelay,i.e. = 21.

EXAMPLE3.2 ConstantPhaseShiftAs a second example, we again consider an allpass system with A(j) = 1 andunwrappedphase

for >

0

0AH(j) = +0 for


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50 Chapter3 TransformRepresentationofSignalsandLTISystemsX(j )

0

- 0

0

0

S(j +j )e-j S(j -j0)e+j

- +

0 0

FIGURE3.2 Spectrumofx(t)withs(t)narrowband

With these assumptions on x(t), it is relatively straightforward to determine theoutputy(t). Specifically,thesystemfrequencyresponseH(j) is

ej0

>0H(j) = +j0 (3.7)e 0, it issimplymultiplied by ej , and similarly the term S(j+j0)ej is multiplied only bye+j . Consequently,theoutputfrequencyresponse,Y(j),isgivenby

Y(j) = X(j)H(j)=

1S(j

j0)e

+jej0

+

1S(j

+

j0)e

j

e+j0

(3.8)

2 2which we recognize as a simple phase shift by 0 of the carrier in eq. (3.5), i.e.replacing ineq. (3.6)by0. Consequently,

y(t) =s(t) cos(0t+0) (3.9)Thischange inphaseofthecarriercanalsobeexpressed intermsofatimedelayforthecarrierbyrewritingeq. (3.9)as

0

y(t) =s(t)cos 0 t0 + (3.10)

3.2 GROUPDELAYANDTHEEFFECTOFNONLINEARPHASEIn Example 3.1, we saw that a phase characteristic that is linear with frequencycorresponds in the time domain to a time shift. In this section we consider thecAlanV.OppenheimandGeorgeC.Verghese,2010


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Section3.2 GroupDelayandTheEffectofNonlinearPhase 51effect of a nonlinear phase characteristic. We again assume the system is an allpasssystemwithfrequencyresponse

H(j) =A(j)ejA[H(j)] (3.11)withA(j)=1. AgeneralnonlinearunwrappedphasecharacteristicisdepictedinFigure3.3

A

+ 1

- 1

-0

+0

FIGURE3.3 NonlinearUnwrappedPhaseCharacteristicAswedidinExample3.2,weagainassumethatx(t)isnarrowbandoftheformofequation(3.5)andasdepictedinFigure3.2. WenextassumethatinFigure3.2issufficientlysmallsothat inthevicinityof0,AH(j)canbeapproximatedsufficientlywellbythezeroth andfirst order termsof a Taylors seriesexpansion,i.e.

d

AH(j)AH(j0) + (0) AH(j) (3.12)d =0

Definingg()asd

g() = AH(j) (3.13)d

ourapproximationtoAH(j)inasmallregionaround=0 isexpressedasAH(j)AH(j0)(0)g(0) (3.14)

Similarlyinasmallregionaround=0,wemaketheapproximationAH(j)

AH(j0)

(

+

0)g(0)

(3.15)

Aswewillseeshortly, thequantityg()playsakeyrole inour interpretationoftheeffectonasignalofanonlinearphasecharacteristic.With the Taylors series approximation of eqs. (3.14) and (3.15) and for inputsignalswithfrequencycontentforwhichtheapproximationisvalid,wecanreplaceFigure3.3withFigure3.4.AlanV.OppenheimandGeorgeC.Verghese,2010c


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52 Chapter3 TransformRepresentationofSignalsandLTISystems

0

slope= -g(

0)

+ 1

+ 0 +

-

0 - 0

- 1

slope= -g(

0)

FIGURE3.4 Taylorsseriesapproximationofnonlinearphaseinthevicinityof0

where

1 =AH(j0)and

0 =AH(j0) +0g(0)SinceforLTIsystemsincascade,thefrequencyresponsesmultiplyandcorrespond-ingly the phases add, we can represent the allpass frequency response H(j) asthe cascade of two allpass systems, HI(j) and HII(j), with unwrapped phaseasdepictedinFigure3.5.

AH

I(j )

HI(j ) H (j )II

xI(t)x(t) x

II(t)

+0

-0

slope = -g(

0)

AH

II(j )

FIGURE3.5 Anallpasssystemfrequencyresponse,H(j),representedasthecas-cadeoftwoallpasssystems,HI(j)andHII(j).AlanV.OppenheimandGeorgeC.Verghese,2010c


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Section3.2 GroupDelayandTheEffectofNonlinearPhase 53We recognize HI(j) as corresponding to Example 3.2. Consequently, with x(t)narrowband,wehave

x(t) = s(t)cos(0t+) 0

xI(t) = s(t)cos 0 t0 + (3.16)

Next we recognize HII(j) as corresponding to Example 3.1 with = g(0).Consequently,

xII(t) =xI (tg(0)) (3.17)orequivalently

0 +0g(0)

xII(t) =s(tg(0))cos 0 t0 + (3.18)

Since,fromFigure3.4,weseethat1 =0 +0g(0)

equation(3.18)canberewrittenas

1 xII(t) =s(tg(0)) cos 0 t

0 + (3.19a)or

xII(t) =s(tg(0))cos[0(tp(0))+] (3.19b)wherep,referredtoasthephasedelay, isdefinedasp =10.In summary, according to eqs. (3.18) and (3.19a), the timedomain effect of thenonlinearphaseforthenarrowbandgroupoffrequenciesaroundthefrequency0 istodelaythenarrowbandsignalbythegroupdelay,g(0),andapplyanadditionalphaseshiftof

10 tothecarrier. Anequivalent,alternate interpretation isthatthe

timedomainenvelopeofthefrequencygroupisdelayedbythegroupdelayandthecarrierisdelayedbythephasedelay.Thediscussionhasbeencarriedoutthusfarfornarrowbandsignals. Toextendthediscussiontobroadbandsignals,weneedonlyrecognizethatanybroadbandsignalcan be viewed as a superposition of narrowband signals. This representation canin fact be developed formally by recognizing that the system in Figure 3.6 is anidentitysystem,i.e. r(t) =x(t)aslongas

Hi(j)=1 (3.20)i=0

Bychoosingthe filtersHi(j)tosatisfyeq. (3.20) and tobenarrowband aroundcenter frequencies i, each of the output signals, yi(t), is a narrowband signal.Consequently the timedomain effect of the phase of G(j) is to apply the groupAlanV.OppenheimandGeorgeC.Verghese,2010c


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G(j )x(t) r(t)

x(t)r(t)

H0(j ) G(j )

Hi(j ) G(j ) ri(t)

r0(t)

gi(t)

g0(t)

FIGURE 3.6 Continuoustime allpass system with frequency response amplitude,phaseandgroupdelayasshown inFigure3.7

FIGURE3.7 Magnitude,(nonlinear)phase,andgroupdelayofanallpassfilter.

delayandphasedelaytoeachofthenarrowbandcomponents(i.e. frequencygroups)yi(t). Ifthegroupdelayisdifferentatthedifferentcenter(i.e. carrier)frequenciesAlanV.OppenheimandGeorgeC.Verghese,2010c


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Section3.2 GroupDelayandTheEffectofNonlinearPhase 55

FIGURE3.8 ImpulseresponseforallpassfiltershowninFigure3.7

i, then the time domain effect is for different frequency groups to arrive at theoutputatdifferenttimes.Asan illustrationofthiseffect,considerG(j) inFigure3.6tobethecontinuoustimeallpasssystemwithfrequencyresponseamplitude,phaseandgroupdelayasshown inFigure3.7. Thecorresponding impulseresponseisshown inFigure3.8.IfthephaseofG(j)werelinearwithfrequency,theimpulseresponsewouldsimplybeadelayedimpulse,i.e. allthenarrowbandcomponentswouldbedelayedbythesameamountandcorrespondinglywouldadduptoadelayedimpulse. However,asweseeinFigure3.7,thegroupdelayisnotconstantsincethephaseisnonlinear. Inparticular,frequenciesaround1200Hzaredelayedsignificantlymorethanaroundother frequencies. Correspondingly, in Figure 3.8 we see that frequency groupappearing late inthe impulseresponse.A second example is shown in Figure 3.9, in which G(j) is again an allpasssystemwithnonlinearphaseandconsequentlynonconstantgroupdelay. Withthisexample, we would expect to see different delays in the frequency groups around = 2 50, = 2 100, and = 2 300with the group at = 2 50 having themaximumdelayandthereforeappearinglast intheimpulseresponse.Inbothoftheseexamples,theinputishighlyconcentratedintime(i.e. animpulse)andtheresponseisdispersedintimebecauseofthenonconstantgroupdelay,i.e.AlanV.OppenheimandGeorgeC.Verghese,2010c


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FIGURE3.9 Phase, group delay, and impulse response for an allpass system: (a)principalphase;(b)unwrappedphase;(c)groupdelay;(d)impulseresponse. (FromOppenheimandWillsky,SignalsandSystems,PrenticeHall,1997,Figure6.5.)AlanV.OppenheimandGeorgeC.Verghese,2010c

4

2

0

-2

-4 0 50 100 150 200 250 300 350 400

Frequency (Hz)

Phase(rad)

0 50 100 150 200 250 300 350 400

0

-5

-10

-15

-20

Frequency (Hz)

Phase(rad)

600

400

200

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-200

-400

-600

Time (sec)

0 50 100 150 200 250 300 350 400

0.10

0.08

0.04

0.06

0.02

0

Frequency (Hz)

Groupdelay(sec)

(a)

(b)

(c)

(d)

Image by MIT OpenCourseWare,adapted from Signals and Systems, AlanOppenheim and Alan Willsky. Prentice Hall, 1996.


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Section3.3 All-PassandMinimum-PhaseSystems 57thenonlinearphase. Ingeneral,theeffectofnonlinearphaseisreferredtoasdisper-sion. In communication systems and many other application contexts even whena channel has a relatively constant frequency response magnitude characteristic,nonlinearphasecanresultinsignificantdistortionandothernegativeconsequencesbecause of the resulting time dispersion. For this reason, it is often essential toincorporatephaseequalizationtocompensatefornonconstantgroupdelay.Asathirdexample,weconsideranallpasssystemwithphaseandgroupdelayasshown in Figure 3.101. The input for this example is the touchtone digit fivewhich consists of two very narrowband tones at center frequencies 770 and 1336Hz. ThetimedomainsignalanditstwonarrowbandcomponentsignalsareshowninFigure3.11.

FIGURE3.10 Phaseandgroupdelayforallpassfilterfortouchtonesignalexample.ThetouchtonesignalisprocessedwithmultiplepassesthroughtheallpasssystemofFigure3.10. Fromthegroupdelayplot,weexpectthat,inasinglepassthroughtheallpassfilter,thetoneat1336Hzwouldbedelayedbyabout2.5millisecondsrelativetothetoneat770Hz. After200passes,thiswouldaccumulatetoarelativedelayofabout0.5seconds.InFigure3.12,weshowtheresultofmultiplepassesthroughfiltersandtheaccu-mulationofthedelays.

3.3 ALL-PASSANDMINIMUM-PHASESYSTEMSTwoparticularlyinterestingclassesofstableLTIsystemsareallpasssystemsandminimumphasesystems. Wedefineanddiscussthem inthissection.

1This example was developed by Prof. Bernard Lesieutre of the University of Wisconsin,Madison,whenhetaughtthecoursewithusatMITcAlanV.OppenheimandGeorgeC.Verghese,2010


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FIGURE3.11 Touchtonesignalwithitstwonarrowbandcomponentsignals.3.3.1 All-PassSystems

An allpass system is a stable system for which the magnitude of the frequencyresponse is a constant, independent of frequency. The frequency response in thecaseofacontinuoustime allpasssystem isthusoftheform

Hap(j) =AejHap(j) , (3.21)whereAisaconstant,notvaryingwith. Assumingtheassociatedtransferfunc-tionH(s)isrational ins, itwillcorrespondinglyhavetheform

Ms+akHap(s) =A . (3.22)sak

k=1Notethat foreachpoleats= +ak thishasazeroatthemirror imageacrosstheimaginary axis, namely at s ; and if ak is complex and the system impulse=aresponseisrealvalued,everycomplexpoleandzerowilloccurinaconjugatepair,k

and a zero at s=ak. An example of apolezero diagram (in the splane) for a continuoustime allpass system is shownso there will also be a pole at s +a= kinFigure(3.13). Itisstraightforward toverifythateachoftheM factorsin(3.22)hasunitmagnitudefors=j .cAlanV.OppenheimandGeorgeC.Verghese,2010


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Section3.3 All-PassandMinimum-PhaseSystems 59

200passes

200passes

200passes

200passes

200passes

FIGURE3.12 Effectofpassingtouchtonesignal(Figure3.11)throughmultiplepassesofanallpassfilterandtheaccumulationofdelays

.Foradiscretetimeallpasssystem,thefrequencyresponseisoftheform

Hap(ej) =AejHap(ej) . (3.23)

IftheassociatedtransferfunctionH(z)isrationalinz,itwillhavetheformM

Hap(z) =A z1 bk . (3.24)1bkz1

k=1The poles and zeros in this case occur at conjugate reciprocal locations: for eachpoleatz=bk thereisazeroatz= 1/bk. Azeroatz=0(andassociatedpoleat)isobtainedbysettingbk =inthecorrespondingfactorabove,afterfirstdividingboththenumeratoranddenominatorbybk;thisresultsinthecorrespondingfactorin (3.24) beingjust z. Again, if the impulse response is realvalued then everycomplex pole and zeros will occur in a conjugate pair, so there will be a pole atz = bk and a zero at z = 1/bk. An example of a polezero diagram (in the zplane)foradiscretetimeallpasssystemisshowninFigure(3.14). ItisoncemoreAlanV.OppenheimandGeorgeC.Verghese,2010c


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Im

1

1 2 Re2 1

1

FIGURE3.13 Typicalpolezeroplotforacontinuoustime allpasssystem.

straightforward to verify that each of theM factors in(3.24) has unit magnitudeforz=ej.The phase of a continuoustime allpass system will be the sum of the phases as-sociated with each of the M factors in (3.22). Assuming the system is causal (inaddition to being stable), then for each of these factors Re{ak}


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Section3.3 All-PassandMinimum-PhaseSystems 61

0.8

Unit circle

3/44/3

Im

Re

FIGURE3.14 Typicalpolezeroplotforadiscretetimeallpasssystem.

finite zeros are in the lefthalfplane, i.e., have real parts that are negative. Thesystemisthereforenecessarilycausal. Ifthereareasmanyfinitezerosastherearepoles,thenaCTminimumphasesystemcanequivalentlybecharacterizedbythestatement that both the system and its inverse are stable and causal,just as wehad in the DT case. However, it is quite possible and indeed common fora CT minimumphase system to have fewer finite zeros than poles. (Note that astable CT system must have all its poles at finite locations in the splane, sincepolesat infinitywould implythattheoutputofthesystem involvesderivativesoftheinput,whichisincompatiblewithstability. Also,whereasintheDTcaseazeroat infinity is clearly outside the unit circle, in the CT case there is no way to tellifazeroat infinity is inthe lefthalfplaneornot,so itshouldbenosurprisethattheCTdefinitioninvolvesonlythefinitezeros.)Theuseofthetermminimumphaseishistorical,andthepropertyshouldperhapsmoreappropriatelybetermedminimumgroupdelay,forreasonsthatwewillbringoutnext. Todothis,weneedafactthatweshallshortlyestablish: thatanycausalandstableCTsystemwitharationaltransferfunctionHcs(s)andnozerosontheimaginaryaxiscanberepresentedasthecascadeofaminimumphasesystemandanallpasssystem,

Hcs(s) =Hmin(s)Hap(s). (3.25)Similarly, in the DT case, provided the transfer function Hcs(z) has no zeros onAlanV.OppenheimandGeorgeC.Verghese,2010c


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62 Chapter3 TransformRepresentationofSignalsandLTISystemstheunitcircle,itcanbewrittenas

Hcs(z) =Hmin(z)Hap(z). (3.26)The frequency response magnitude of the allpass factor is constant, independentoffrequency,andforconvenienceletussetthisconstanttounity. Thenfrom(3.25)

|Hcs(j)|=|Hmin(j)|, and (3.27a)grpdelay[Hcs(j)]=grpdelay[Hmin(j)]+grpdelay[Hap(j)] (3.27b)

andsimilarequationshold intheDTcase.We will see in the next section that the minimumphase term in (3.25) or (3.26)canbeuniquelydeterminedfromthemagnitudeofHcs(j),respectivelyHcs(ej).Consequently all causal, stable systems with the same frequency response magni-tude

differ

only

in

the

choice

of

the

allpass

factor

in

(3.25)

or

(3.26).

However,

we

have shown previously that allpass factors must contribute positive group delay.Thereforeweconcludefrom(3.27b)thatamongallcausal,stablesystemswiththesameCT frequencyresponsemagnitude,theonewithnoallpassfactors in(3.25)willhavetheminimumgroupdelay. ThesameresultholdsintheDTcase.Weshallnowdemonstratethevalidityof(3.25);thecorrespondingresultin(3.26)fordiscretetimefollowsinaverysimilarmanner. Consideracausal,stabletransferfunctionHcs(s)expressedintheform

M1 (slk)M2 (sri)Hcs(s) =A k=1 i=1 (3.28)N

)n=1(sdnwhere

the

dns

are

the

poles

of

the

system,

the

lks

are

the

zeros

in

the

lefthalf

planeandtherisarethezeros intherighthalfplane. SinceHcs(s) isstableandcausal, allofthepolesare in the lefthalf planeandwouldbeassociatedwith thefactor Hmin(s) in (3.25), as would be all of the zeros lk. We next represent therighthalfplanezerosas

M2 M2 M2 (sri)(sri) = (s+ri)

(s+ri) (3.29)i=1 i=1 i=1

Since Re{ri} is positive, the first factor in (3.29) represents lefthalfplane zeros.Thesecondfactorcorrespondstoallpasstermswithlefthalfplanepoles,andwithzeros at mirror image locations to the poles. Thus, combining (3.28) and (3.29),Hcs(s)hasbeendecomposedaccordingto(3.25)where

M1 (slk)M2 (s+ri)Hmin(s) =A k=1 i=1 (3.30a)N

(sdn)n=1M2

Hap(s) =(sri) (3.30b)(s+ri)

i=1AlanV.OppenheimandGeorgeC.Verghese,2010c


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Section3.4 SpectralFactorization 63

EXAMPLE3.3

Causal,

stable

system

as

cascade

of

minimum-phase

and

all-pass

Consideracausal,stablesystemwithtransferfunction

Hcs = (s1) (3.31)(s+2)(s+3)

Thecorrespondingminimumphaseandallpassfactorsare(s+1)

Hmin(s)= (3.32)(s+2)(s+3)

Hap(s) = s1 (3.33)s+ 1

3.4 SPECTRALFACTORIZATIONTheminimumphase/allpassdecompositiondevelopedabove isuseful inavarietyofcontexts. Onethatisofparticularinteresttousinlaterchaptersariseswhenwewearegivenorhavemeasuredthemagnitudeofthefrequencyresponseofastablesystemwith a rationaltransfer function H(s) (andrealvalued impulseresponse),andourobjectiveistorecoverH(s)fromthis information. AsimilartaskmaybeposedintheDTcase,butwefocusontheCTversionhere. Wearethusgiven

|H(j)|2 =H(j)H(j) (3.34)or,sinceH(j) =H(j),

|H(j)|2 =H(j)H(j). (3.35)NowH(j) isH(s)fors=j ,andtherefore

H(j) 2 =H(s)H(s) (3.36)| | s=j

For any numerator or denominator factor (sa) in H(s), there will be a corre-sponding factor (sa) in H(s)H(s). Thus H(s)H(s) will consist of factorsinthenumeratorordenominatoroftheform(sa)(sa) =s2+a2,andwilltherefore be a rational function of s2. Consequently H(j) 2 will be a rational| |functionof2. Thus,ifwearegivenorcanexpress H(j) 2 asarationalfunction| |

2of2,wecanobtaintheproductH(s)H(s)bymakingthesubstitution2 =s .TheproductH(s)H(s)willalwayshaveitszerosinpairsthataremirroredacrosstheimaginaryaxisofthesplane,andsimilarlyforitspoles. ForanypoleorzeroofH(s)H(s)attherealvaluea,therewillbeanotheratthemirrorimagea,whileforanypoleorzeroatthecomplexvalueq,therewillbeothersatq,qandq,cAlanV.OppenheimandGeorgeC.Verghese,2010


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64 Chapter3 TransformRepresentationofSignalsandLTISystemsformingacomplexconjugatepair(q, q)and itsmirror image(q,q). Wethenneed to assign one of each mirrored real pole and zero and one of each mirroredconjugatepairofpolesandzerostoH(s),andthemirrorimagetoH(s).If we assume (or know) that H(s) is causal, in addition to being stable, thenwe would assign the lefthalf plane poles of each pair to H(s). With no furtherknowledgeorassumptionwehavenoguidanceontheassignmentofthezerosotherthan the requirement of assigning one of each mirror image pair to H(s) and theothertoH(s). Ifwefurtherknoworassumethatthesystemisminimumphase,thenthelefthalfplanezerosfromeachmirroredpairareassignedtoH(s),andtherighthalfplane zeros to H(s). This process of factoring H(s)H(s) to obtainH(s)isreferredtoasspectralfactorization.

EXAMPLE3.4 SpectralfactorizationConsiderafrequencyresponsemagnitudethathasbeenmeasuredorapproximatedas

2 + 1 2 + 1|H(j)|2 =

4 +132 + 36 = (2 +4)(2 +9) (3.37)Makingthesubstitution2 =s2,weobtain

s2 + 1H(s)H(s) =

(s2 +4)(s2 +9) (3.38)whichwefurtherfactoras

H(s)H(s) = (s+1)(s+1) (3.39)(s+2)(s+2)(s+3)(s+3)

Itnowremainstoassociateappropriate factors withH(s) andH(s). Assumingthe system is causal in addition to being stable, the two lefthalf plane poles ats=2ands=3 mustbeassociated with H(s). With no further assumptions,eitheroneofthenumeratorfactorscanbeassociatedwithH(s)andtheotherwithH(s). However, if we know or assume that H(s) is minimum phase, then wewouldassignthelefthalfplanezerotoH(s),resulting inthechoice

(s+1)H(s)= (3.40)

(s+2)(s+3)

Inthediscretetimecase,asimilardevelopmentleadstoanexpressionforH(z)H(1/z)fromknowledgeof|H(ej)|2. ThezerosofH(z)H(1/z)occurinconjugaterecipro-calpairs,andsimilarlyforthepoles. Weagainhavetosplitsuchconjugaterecip-rocalpairs,assigningoneofeachtoH(z),theothertoH(1/z),basedonwhateveradditional knowledge we have. For instance, if H(z) is known to be causal in ad-ditiontobeingstable,thenallthepolesofH(z)H(1/z)thatare intheunitcircleareassignedtoH(z);and ifH(z) isknowntobeminimumphaseaswell,thenallthezerosofH(z)H(1/z)thatareintheunitcircleareassignedtoH(z).AlanV.OppenheimandGeorgeC.Verghese,2010c


19/19

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6.011 Introduction to Communication, Control, and Signal Processing

Spring 2010

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