《信息工程專業(yè)英語(yǔ)》課件第5章

Unit5UnderstandingDigitalSignalProcessing5.1Text5.2ReadingMaterials

5.1Text

Overviewofdigitalsignalprocessing

1．DigitalRepresentationofaWaveform

Adigitalsignalprocessingsystemtakesacontinuoussoundwaveasinput,feedsitthroughananaloglow-passfilter(ananti-aliasingfilter)toremoveallfrequenciesabovehalfthesamplingrate(seeNyquist’ssamplingtheorem).

ThisAnalog-to-DigitalConverter(ADC)filtersandsamplesthewaveamplitudeatequally-spacedtimeintervalsandgeneratesasimplelistoforderedsamplevaluesinsuccessivememorylocationsinthecomputer.Thesamplevalues,representingamplitudes,areencodedusingsomenumberofbitsthatdetermineshowaccuratelythesamplesaremeasured.TheProcessorisacomputerthatappliesnumericaloperationstothesampledwaves.InFig5.1,itseemstheprocessorhaslow-passfilteredthesignal,thusremovingthejumpyirregularitiesintheinputwave.

WhenthesignalisconvertedbackintoanaudiosignalbytheDigital-AnalogConverter(DAC),therewillbejaggedirregularitiesthatarequantizationerrors.Ofcourse,thesewilllieabovetheNyquistfrequency,sothenewanalogsignal(backinrealtime)needsanotheranalog,low-passfilteronoutputsinceeverythingabovetheNyquistfrequencyisanoisyartefact.

Fig5.1Adigitalsignalprocessingsystemandwaves

2.Samplingtheorem

‘Nyquistfreq’=(samplingrate)/2

Aliasingproblem(frequenciesaboveNyquistfrequencygetmappedtolowerfrequencies).ThreekindsofwaveformisshowninFig5.2.Theleftmostwavebelowhas8samplespercycle(thatis,thefrequencyis1/8thesamplerate(state)).Themiddlecurveisthehighestfrequencythatcanberepresentedbythissamplerate.Therightmostfigureismuchtoofast,sothesampledwavewillsoundlike(bealiasedas)theslowdottedcurve(whichisidenticaltotheleftmostcurve).

Fig5.2Threekindsofwaveform

Solution:applyinputfilterbeforesamplingtoremoveallunwantedinputs.AnyenergyremainingabovetheNyquistfrequencywillbemappedontolowerfrequencies.Ofcourse,onmoderndigitalequipment,thisfilteringistakencareofforyou.

3.Quantizationofamplitude(limitedsetofamplitudevalues)

Whenthesampledsignalisconvertedbackintorealtime,ofcourse,thereareonlyassignedvaluesspecifiedatthesamplepoints.Theoutputsignalwilljustbeflatuntilthenextsamplecomesalong.Theseflatspotswillbeperceivedbyalistenerasahigh-frequencysignal(abovetheNyquistfreq),butitwillbenoise.

Sotheredcurve(Fig5.3)belowmustbesmoothed(lowpassfiltered)intothegreencurvebelowinordertosoundright.Ofcourse,ifthesamplerateisthatofthecommercialCDstandard,thenthisnoisewillbeabovethelimitsofhumanhearing—soyourownearcanserveasthelowpassquantizationfilter.Indeed,sinceloudspeakersdonotnormallyproducesoundsabove20kH,theytoocanalsoserveasthislowpassoutputfilter.

Fig5.3Redcurveissmoothedintogreencurve

Technicalwordsandphrases

overview n.綜述；概觀

waveform n.波形

analog adj.模擬的

anti- pref.表示“反對(duì)，抵抗”之義

amplitude

n.振幅

processor n.處理器

irregularity n.不規(guī)則；無(wú)規(guī)律；不整齊

jag vt.使成鋸齒狀；使成缺口

artefact n.人工制品，加工品

interval

n.間隔；間距

alias n.別名，化名adv.別名叫；化名為

curve n.曲線

dot n.點(diǎn)，圓點(diǎn)

flat vt.使變平

perceive

vt.感覺(jué)；理解vi.感到，感知；認(rèn)識(shí)到

loudspeakern.喇叭，揚(yáng)聲器；擴(kuò)音器

alow-passfilter 低通濾波器

jumpyirregularities 不規(guī)則跳動(dòng)

inputwave 輸入波

Samplingtheorem 采樣定理

getmappedto 映射到

soundlike

被看做

bealiassedas 被混疊為

dottedcurve 虛線

bemappedonto 標(biāo)示到

quantizationofamplitude 振幅的量化

limitedset

極限設(shè)置

ADC(Analog-to-DigitalConverter) 模擬-數(shù)字轉(zhuǎn)換

DAC(Digital-AnalogConverter) 數(shù)字-模擬轉(zhuǎn)換

Nyquist奈奎斯特，美國(guó)物理學(xué)家。1917年獲得耶魯大學(xué)哲學(xué)博士學(xué)位。奈奎斯特為近代信息理論做出了突出貢獻(xiàn)。他總結(jié)的奈奎斯特采樣定理是信息論，特別是通信與信號(hào)處理學(xué)科中的一個(gè)重要基本結(jié)論。

5.1.1Exercises

1.PutthePhrasesintoEnglish

(1)數(shù)字信號(hào)處理系統(tǒng)；

(2)連續(xù)聲波；

(3)模擬低通濾波器；

(4)采樣值；

(5)音頻信號(hào)； (6)量化錯(cuò)誤。

2.PutthePhrasesintoChinese

(1)ananti-aliasingfilter;

(2)moderndigitalequipment;

(3)orderedsamplevalues;

(4)equally-spacedtimeintervals;

(5)samplepoints;

(6)lowpassquantizationfilter;

(7)limitsofhumanhearing;

(8)jumpyirregularities.

3.Translation

(1)Thesamplevalues,representingamplitudes,areencodedusingsomenumberofbitsthatdetermineshowaccuratelythesamplesaremeasured.

(2)WhenthesignalisconvertedbackintoanaudiosignalbytheDigital-AnalogConverter(DAC),therewillbejaggedirregularities(shownbelowonthispage)thatarequantizationerrors.

(3)Therightmostfigureismuchtoofast,sothesampledwavewillsoundlike(bealiasedas)theslowdottedcurve(whichisidenticaltotheleftmostcurve).

(4)Theoutputsignalwilljustbeflatuntilthenextsamplecomesalong.

(5)IfthesamplerateisthatofthecommercialCDstandard,thenthisnoisewillbeabovethelimitsofhumanhearing--soyourownearcanserveasthelowpassquantizationfilter.

5.1.2參考譯文

數(shù)字信號(hào)處理概述

1.波形的數(shù)字表示

數(shù)字信號(hào)處理系統(tǒng)將連續(xù)聲波作為輸入信號(hào)，把這些信號(hào)送入模擬低通濾波器(抗混疊濾波器)以消除所有高于采樣率一半以上的頻率(見(jiàn)奈奎斯特采樣定理)。這種模擬-數(shù)字轉(zhuǎn)換器(ADC)在間距相等的時(shí)間間隔內(nèi)過(guò)濾和采樣波的振幅，并在計(jì)算機(jī)中的連續(xù)內(nèi)存位置生成一個(gè)有序樣本值的簡(jiǎn)單列表。采樣值即振幅，由一些決定如何準(zhǔn)確測(cè)量樣品的位數(shù)編碼而成。處理器是一臺(tái)適用于采樣波數(shù)值運(yùn)算的電腦。

如圖5.1所示，處理器有低通濾波信號(hào)的功能，從而消除了輸入波的不規(guī)則跳動(dòng)。當(dāng)信號(hào)被數(shù)字-模擬轉(zhuǎn)換器(DAC)轉(zhuǎn)換成音頻信號(hào)后，會(huì)有鋸齒狀的不規(guī)則波形即量化錯(cuò)誤。當(dāng)然，這些都將基于之前提到的奈奎斯特頻率，所以新的模擬信號(hào)(實(shí)時(shí))在輸出端需要另一個(gè)模擬低通濾波器，因?yàn)樗懈哂谀慰固仡l率的都是嘈雜現(xiàn)象。

2.采樣定理

奈奎斯特頻率?=?采樣頻率/2

混疊問(wèn)題(奈奎斯特頻率以上的頻率映射到較低的頻率)。三種波形如圖5.2所示。圖中最左邊的波形每個(gè)周期有8個(gè)樣本(即頻率為1/8的采樣率(srate))。中間波形的頻率是最高的，可以通過(guò)這個(gè)采樣率來(lái)表示。最右邊的波形頻率特別高，所以采樣波將被看做(被混疊為)緩慢的虛線(這與最左邊的曲線相同)。

解決：在采樣以前使用輸入濾波器消除所有不需要的輸入。任何高于奈奎斯特頻率的能量將被映射到較低的頻率。當(dāng)然，在現(xiàn)代的數(shù)字設(shè)備上，濾波器會(huì)自動(dòng)完成這些工作。

3.振幅的量化(振幅值的極限設(shè)置)

當(dāng)然，當(dāng)采樣信號(hào)被轉(zhuǎn)化回實(shí)時(shí)信號(hào)時(shí)，只在采樣點(diǎn)分配指定值。輸出信號(hào)將較為平緩直到下一個(gè)采樣進(jìn)來(lái)。這些平緩的信號(hào)將被收聽(tīng)者視為一個(gè)高頻信號(hào)(高于奈奎斯特頻率)，但它也可能是噪音。因此，為了使收聽(tīng)者收到正常信號(hào)，圖5.3中的紅色曲線必須平滑過(guò)渡到(低通濾波)綠色曲線。當(dāng)然，如果采樣率是商業(yè)CD標(biāo)準(zhǔn)，那么這種噪音會(huì)高于人類聽(tīng)覺(jué)的限制，所以你自己的耳朵可以作為低通量化濾波器。事實(shí)上，由于揚(yáng)聲器不再產(chǎn)生20kHz以上的聲音，它們也能作為低通輸出濾波器。

5.2ReadingMaterials

5.2.1TheDiscrete-timeFourierTransform

Inmathematics,thediscrete-timeFouriertransform(DTFT)(Fig5.4)isoneofthespecificformsofFourieranalysis.Assuch,ittransformsonefunctionintoanother,whichiscalledthefrequencydomainrepresentation,orsimplythe“DTFT”,oftheoriginalfunction(whichisoftenafunctioninthetime-domain).ButtheDTFTrequiresaninputfunctionthatisdiscrete.Suchinputsareoftencreatedbysamplingacontinuousfunction,likeaperson’svoice.Fig5.4Fouriertransforms

TheDTFTfrequency-domainrepresentationisalwaysaperiodicfunction.Sinceoneperiodofthefunctioncontainsalloftheuniqueinformation,itissometimesconvenienttosaythattheDTFTisatransformtoa“finite”frequency-domain(thelengthofoneperiod),ratherthantotheentirerealline.ItisPontryagindualtotheFourierseries,whichtransformsfromaperiodicdomaintoadiscretedomain.

Definition

Givenadiscretesetofrealorcomplexnumbers:(integers),thediscrete-timeFouriertransform(orDTFT)ofisusuallywritten:

(5-2-1)

Relationshiptosampling

Oftenthesequencerepresentsthevalues(akasamples)ofacontinuous-timefunction,,atdiscretemomentsintime:,whereTisthesamplinginterval(inseconds),and

1/T?=?fsisthesamplingrate(samplespersecond).ThentheDTFTprovidesanapproximationofthecontinuous-timeFouriertransform:

(5-2-2)

Tounderstandthis,considerthePoissonsummationformula,whichindicatesthataperiodicsummationoffunctionX(f)canbeconstructedfromthesamplesoffunctionx(t)Theresultis:

(5-2-3)

Theright-handsidesofEq.2andEq.1areidenticalwiththeseassociations:

(5-2-4)

(5-2-5)

XT(f)comprisesexactcopiesofX(f)thatareshiftedbymultiplesof?sandcombinedbyaddition.Forsufficientlylarge?s,thek=0termcanbeobservedintheregion[??s/2,?s/2]withlittleornodistortion(aliasing)fromtheotherterms.

Periodicity

Samplingx(t)causesitsspectrum(DTFT)tobecomeperiodic.Intermsofordinaryfrequency,f(cyclespersecond),theperiodisthesamplerate,fs.Intermsofnormalizedfrequency,f/fs(cyclespersample),theperiodis1.Andintermsof(radianspersample),theperiodis2π,whichalsofollowsdirectlyfromtheperiodicityof.Thatis:

(5-2-6)

wherebothnandkarearbitraryintegers.Therefore:

(5-2-7)

ThepopularalternatenotationX(eiω)fortheDTFTX(ω):

1.highlightstheperiodicityproperty.

2.HelpsdistinguishbetweentheDTFTandunderlyingFouriertransformofx(t);thatis,X(f)(orX(ω)).

3.emphasizestherelationshipoftheDTFTtotheZ-transform.

However,itsrelevanceisobscuredwhentheDTFTisformedbythefrequencydomainmethod(superposition),asdiscussedabove.Sothenotationisalsocommonlyused,asinthetabletofollow.

Inversetransform

Thefollowinginversetransformsrecoverthediscrete-timesequence:

(5-2-8)

TheintegralsspanonefullperiodoftheDTFT,whichmeansthatthex[n]samplesarealsothecoefficientsofaFourierseriesexpansionoftheDTFT.Infinitelimitsofintegrationchangethetransformintoacontinuous-timeFouriertransform[inverse],whichproducesasequenceofDiracimpulses.Thatis:

(5-2-9)

FIRfiltersarefiltershavingatransferfunctionofapolynomialinz-andisanall-zerofilterinthesensethatthezeroesinthez-planedeterminethefrequencyresponsemagnitudecharacteristic.TheztransformofaN-pointFIRfilterisgivenby

(5-2-10)

FIRfiltersareparticularlyusefulforapplicationswhereexactlinearphaseresponseisrequired.TheFIRfilterisgenerallyimplementedinanon-recursivewaywhichguaranteesastablefilter.FIRfilterdesignessentiallyconsistsoftwoparts:

(i)approximationproblem.

(ii)realizationproblem.

TheWindowMethod

Inthismethod,thedesiredfrequencyresponsespecificationHd(ω),correspondingunitsampleresponsehd(n)isdeterminedusingthefollowingrelation:

-∞≤n≤∞

(5-2-11)

Where

(5-2-12)

Ingeneral,unitsampleresponsehd(n)obtainedfromtheaboverelationisinfiniteinduration,soitmustbetruncatedatsomepointsayn?=M-1toyieldanFIRfilteroflengthM(i.e.0toM-1).Thistruncationofhd(n)tolengthM-1issameasmultiplyinghd(n)bytherectangularwindowdefinedas:

w(n)=10≤n≤M-1

0otherwise

(5-2-13)

ThustheunitsampleresponseoftheFIRfilterbecomes:

h(n)?=?hd(n)w(n)?

=

hd(n)0≤n≤M-1

=?0otherwise

?(5-2-14)

Now,themultiplicationofthewindowfunctionw(n)withhd(n)isequivalenttoconvolutionofHd(ω)withW(ω),whereW(ω)isthefrequencydomainrepresentationofthewindowfunction:

(5-2-15)

ThustheconvolutionofHd(ω)withW(ω)yieldsthefrequencyresponseofthetruncatedFIRfilter:

(5-2-16)

Thefrequencyresponsecanalsobeobtainedusingthefollowingrelation:

(5-2-17)

Butdirecttruncationofhd(n)toMtermstoobtainh(n)leadstotheGibbsphenomenoneffectwhichmanifestsitselfasafixedpercentageovershootandripplebeforeandafteranapproximateddiscontinuityinthefrequencyresponseduetothenon-uniformconvergenceofthefourierseriesatadiscontinuity.Thusthefrequencyresponseobtainedbyusing(8)containsripplesinthefrequencydomain.

Inordertoreducetheripples,insteadofmultiplyinghd(n)witharectangularwindoww(n),hd(n)ismultipliedwithawindowfunctionthatcontainsataperanddecaystowardzerogradually,insteadofabruptlyasitoccursinarectangularwindow.Asmultiplicationofsequenceshd(n)andw(n)intimedomainisequivalenttoconvolutionofHd(ω)andW(ω)inthefrequencydomain,ithastheeffectofsmoothingHd(ω).

TheseveraleffectsofwindowingtheFouriercoefficientsofthefilterontheresultofthefrequencyresponseofthefilterareasfollows:

(i)AmajoreffectisthatdiscontinuitiesinH(ω)becometransitionbandsbetweenvaluesoneithersideofthediscontinuity.

(ii)Thewidthofthetransitionbandsdependsonthewidthofthemainlobeofthefrequencyresponseofthewindowfunction,w(n)i.e.W(ω).

(iii)Sincethefilterfrequencyresponseisobtainedviaaconvolutionrelation,itisclearthattheresultingfiltersareneveroptimalinanysense.

(iv)AsM(thelengthofthewindowfunction)increases,themainlobewidthofW(ω)isreducedwhichreducesthewidthofthetransitionband,butthisalsointroducesmorerippleinthefrequencyresponse.

(v)Thewindowfunctioneliminatestheringingeffectsatthebandedgeanddoesresultinlowersidelobesattheexpenseofanincreaseinthewidthofthetransitionbandofthefilter.

TheFrequencySamplingTechnique

Inthismethod,thedesiredfrequencyresponseisprovidedasinthepreviousmethod.NowthegivenfrequencyresponseissampledatasetofequallyspacedfrequenciestoobtainNsamples.Thus,samplingthecontinuousfrequencyresponseHd(ω)atNpointsessentiallygivesustheN-pointDFTofHd(2pnk/N).ThusbyusingtheIDFTformula,thefiltercoefficientscanbecalculatedusingthefollowingformula:

(5-2-18)

NowusingtheaboveN-pointfilterresponse,thecontinuousfrequencyresponseiscalculatedasaninterpolationofthesampledfrequencyresponse.Theapproximationerrorwouldthenbeexactlyzeroatthesamplingfrequenciesandwouldbefiniteinfrequenciesbetweenthem.Thesmootherthefrequencyresponsebeingapproximated,thesmallerwillbetheerrorofinterpolationbetweenthesamplepoints.

Onewaytoreducetheerroristoincreasethenumberoffrequencysamples[Rab75].Theotherwaytoimprovethequalityofapproximationistomakeanumberoffrequencysamplesspecifiedasunconstrainedvariables.Thevaluesoftheseunconstrainedvariablesaregenerallyoptimizedbycomputertominimizesomesimplefunctionoftheapproximationerrore.g.onemightchooseasunconstrainedvariablesthefrequencysamplesthatlieinatransitionbandbetweentwofrequencybandsinwhichthefrequencyresponseisspecifiede.g.inthebandbetweenthepassbandandthestopbandofalowpassfilter.

Therearetwodifferentsetoffrequenciesthatcanbeusedfortakingthesamples.Onesetoffrequencysamplesareatfk=k/Nwherek=0,1,…,N-1.Theothersetofuniformlyspacedfrequencysamplescanbetakenatfk=(k+?)/Nfork=0,1,…,N-1.

Thesecondsetgivesustheadditionalflexibilitytospecifythedesiredfrequencyresponseatasecondpossiblesetoffrequencies.Thusagivenbandedgefrequencymaybeclosertotype-IIfrequencysamplingpointthattotype-Iinwhichcaseatype-IIdesignwouldbeusedinoptimizationprocedure.

Meritsoffrequencysamplingtechnique

(i)Unlikethewindowmethod,thistechniquecanbeusedforanygivenmagnituderesponse.

(ii)Thismethodisusefulforthedesignofnon-prototypefilterswherethedesiredmagnituderesponsecantakeanyirregularshape.

Therearesomedisadvantageswiththismethodi.ethefrequencyresponseobtainedbyinterpolationisequaltothedesiredfrequencyresponseonlyatthesampledpoints.Attheotherpoints,therewillbeafiniteerrorpresent.

OptimalFilterDesignMethods

Manymethodsarepresentunderthiscategory.Thebasicideaineachmethodistodesignthefiltercoefficientsagainandagainuntilaparticularerrorisminimized.Thevariousmethodsareasfollows:

(i)Leastsquarederrorfrequencydomaindesign.

(ii)NonlinearequationsolutionformaximalrippleFIRfilters.

(iii)PolynomialinterpolationsolutionformaximalrippleFIRfilters.

Leastsquarederrorfrequencydomaindesign

Asseeninthepreviousmethodoffrequencysamplingtechniquethereisnoconstraintontheresponsebetweenthesamplepoints,andpoorresultsmaybeobtained.

Thefrequencysamplingtechniqueismoreofaninterpolationmethodratherthananapproximationmethod.Thismethodcontrolstheresponsebetweenthesamplepointsbyconsideringanumberofsamplepointslargerthantheorderofthefilter.Thepurposeofmostfiltersistoseparatedesiredsignalsfromundesiredsignalsornoise.Astheenergyofthesignalisrelatedtothesquareofthesignal,asquarederrorapproximationcriterionisappropriatetooptimizethedesignoftheFIRfilters.

ThefrequencyresponseoftheFIRfilterisgivenby(5-2-19)foraN-pointFIRfilter.Anerrorfunctionisdefinedasfollows:

(5-2-19)

WhereandHd(ωk)areLsamplesofthedesiredresponse,whichistheerrormeasureasasumofthesquareddifferencesbetweentheactualanddesiredfrequencyresponseoverasetofLfrequencysamples.Themethodconsistsofthefollowingsteps:

(i)First‘L’samplesfromthecontinuousfrequencyresponsearetaken,whereL>N(lengthoftheimpulseresponseoffiltertobedesigned).

(ii)Thenusingthefollowingformula:

(5-2-20)

theL-pointfilterimpulseresponseiscalculated.

(iii)ThentheobtainedfilterimpulseresponseissymmetricallytruncatedtodesiredlengthN.

(iv)Thenthefrequencyresponseiscalculatedusingthefollowingrelation:

(5-2-21)

(v)Themagnitudeofthefrequencyresponseatthesefrequencypointsforwillnotbeequaltothedesiredones,buttheoverallleastsquareerrorwillbereducedeffectivelythiswillreducetherippleinthefilterresponse.

Tofurtherreducetherippleandovershootnearthebandedges,atransitionregionwillbedefinedwithalineartransferfunction.ThentheLfrequencysamplesaretakenatusingwhichthefirstNsamplesofthefilterarecalculatedusingtheabovemethod.Usingthismethod,reducestherippleintheinterpolatedfrequencyresponse.

NonlinearEquationsolutionformaximalrippleFIRfilters

TherealpartofthefrequencyresponseofthedesignedFIRfiltercanbewrittenaswherelimitsofsummationanda(n)varyaccordingtothetypeofthefilter.ThenumberoffrequenciesatwhichH(ω)couldattainanextremumisstrictlyafunctionofthetypeofthelinearphasefilteri.e.whetherlengthNoffilterisoddorevenorfilterissymmetricoranti-symmetric.

Ateachextremum,thevalueofH(ω)ispredeterminedbyacombinationoftheweightingfunctionW(ω),thedesiredfrequencyresponse,andaquantitythatrepresentsthepeakerrorofapproximationdistributingthefrequenciesatwhichH(ω)attainsanextremalvalueamongthedifferentfrequencybandsoverwhichadesiredresponsewasbeingapproximated.Sincethesefiltershavethemaximumnumberofripples,theyarecalledmaximalripplefilters.

Thismethodisasfollows:

1.AteachoftheNeunknownexternalfrequencies,E(ω)attainsthemaximumvalueofeitherandE(ω)orequivalentlyH(ω)haszeroderivative.ThustwoNeequationsoftheform

areobtained.

(5-2-22)

(5-2-23)

Theseequationsrepresentasetof2NenonlinearequationsintwoNeunknowns,NeimpulseresponsecoefficientsandNefrequenciesatwhichH(ω)obtainstheextremalvalue.ThesetoftwoNeequationsmaybesolvediterativelyusingnonlinearoptimizationprocedure.

Animportantthingtonoteisthatherethepeakerror()isafixedquantityandisnotminimizedbytheoptimizationscheme.ThustheshapeofH(ω)ispostulatedaprioriandonlythefrequenciesatwhichH(ω)attainstheextremalvaluesareunknown.

Thedisadvantageofthismethodisthatthedesignprocedurehasnowayofspecifyingbandedgesforthedifferentfrequencybandsofthefilter.Thustheoptimizationalgorithmisfreetoselectexactlywherethebandswilllie.

PolynomialInterpolationSolutionforMaximalRippleFIRfilters

ThisalgorithmisbasicallyaniterativetechniqueforproducingapolynomialH(ω)thathasextremaofdesiredvalues.ThealgorithmbeginsbymakinganinitialestimateofthefrequenciesatwhichtheextremainH(ω)willoccurandthenusesthewell-knownLagrangeinterpolationformulatoobtainapolynomialthatalternativelygoesthroughthemaximumallowableripplevaluesatthesefrequencies.Ithasbeenexperimentallyfoundthattheinitialguessofextremalfrequenciesdoesnotaffecttheultimateconvergenceofthealgorithmbutinsteadaffectsthenumberofiterationsrequiredtoachievethedesiredresult.

Letusconsiderthecaseofdesignofalowpassfilterusingtheabovemethod.

TheFig5.5showstheresponseofalowpassfilterwithN=11.Thenumberofextremalfrequenciesi.e.thefrequencieswhereripplesoccurare6inthiscase.Theyaredividedinto3passbandextremaand3stopbandextrema.ThefilleddotsindicatetheinitialguessastotheextremalfrequenciesofH(ω).ThesolidlineistheinitialLagrangepolynomialobtainedbychoosingpolynomialcoefficientssothatthevaluesofthepolynomialattheguessedsetoffrequenciesareidenticaltotheassignedextremevalues.

Butthispolynomialhasextremathatexceedsthespecifiedmaximavalues.ThenextstageofthealgorithmistolocatethefrequenciesatwhichtheextremaofthefirstLagrangeinterpolationoccur.Thesefrequenciesarenowusedasthenewfrequenciesforwhichtheextremaofthefilterresponseoccur.ThissecondsetoffrequenciesareindicatedbyopendotsinFig5.5.Nowsimilarlythenewsetoffrequenciesaretakenasthosefrequencieswherethemaximumexceedsthespecifiedmaxima.Thusthemethodiscompletelyiterativeinnature.Fig5.5Iterativesolutionforamaximumripplelowpassfilter

IIRfilterdesign

Typicalfrequency-selectivefiltershavetheclosedformformulas,butarbitraryfiltershaven’ttheclosedformformulasindesign.Inthiscase,weapplythecomputer-aideddesigntechniquestodesignthedesiredfilter.

MostalgorithmicdesignproceduresforIIRfilterstakethefollowingform:

1.H(z)isassumedtoberationalfunction.Itcanberepresentedasaratioofpolynomialinz(orz-1),asaproductofnumeratoranddenominatorfactors(zerosandpoles),orasaproductofsecond-orderfactors.

2.TheordersofthenumeratoranddenominatorofH(z)arefixed.

3.Anidealdesiredfrequencyresponseandacorrespondingapproximationerrorcriterionischosen.

4.Byasuitableoptimizationalgorithm,thefreeparameters(numeratoranddenominatorcoefficients,zeroandpoles,etc)arevariedinasystematicwaytominimizetheapproximationerroraccordingtotheassumederrorcriterion.

5.Thesetofparametersthatminimizestheapproximationerrordeterminesthesystemfunctionofthedesiredsystem.

Deczky’sMethod

InDeczky’smethod,thesystemfunctionofthefilterisrepre