《信息工程专业英语》课件第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.综述；概观

waveform n.波形

analog adj.模拟的

anti- pref.表示“反对，抵抗”之义

amplitude

n.振幅

processor n.处理器

irregularity n.不规则；无规律；不整齐

jag vt.使成锯齿状；使成缺口

artefact n.人工制品，加工品

interval

n.间隔；间距

alias n.别名，化名adv.别名叫；化名为

curve n.曲线

dot n.点，圆点

flat vt.使变平

perceive

vt.感觉；理解vi.感到，感知；认识到

loudspeakern.喇叭，扬声器；扩音器

alow-passfilter 低通滤波器

jumpyirregularities 不规则跳动

inputwave 输入波

Samplingtheorem 采样定理

getmappedto 映射到

soundlike

被看做

bealiassedas 被混叠为

dottedcurve 虚线

bemappedonto 标示到

quantizationofamplitude 振幅的量化

limitedset

极限设置

ADC(Analog-to-DigitalConverter) 模拟-数字转换

DAC(Digital-AnalogConverter) 数字-模拟转换

Nyquist奈奎斯特，美国物理学家。1917年获得耶鲁大学哲学博士学位。奈奎斯特为近代信息理论做出了突出贡献。他总结的奈奎斯特采样定理是信息论，特别是通信与信号处理学科中的一个重要基本结论。

5.1.1Exercises

1.PutthePhrasesintoEnglish

(1)数字信号处理系统；

(2)连续声波；

(3)模拟低通滤波器；

(4)采样值；

(5)音频信号； (6)量化错误。

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.

5.1.2参考译文

数字信号处理概述

1.波形的数字表示

数字信号处理系统将连续声波作为输入信号，把这些信号送入模拟低通滤波器(抗混叠滤波器)以消除所有高于采样率一半以上的频率(见奈奎斯特采样定理)。这种模拟-数字转换器(ADC)在间距相等的时间间隔内过滤和采样波的振幅，并在计算机中的连续内存位置生成一个有序样本值的简单列表。采样值即振幅，由一些决定如何准确测量样品的位数编码而成。处理器是一台适用于采样波数值运算的电脑。

如图5.1所示，处理器有低通滤波信号的功能，从而消除了输入波的不规则跳动。当信号被数字-模拟转换器(DAC)转换成音频信号后，会有锯齿状的不规则波形即量化错误。当然，这些都将基于之前提到的奈奎斯特频率，所以新的模拟信号(实时)在输出端需要另一个模拟低通滤波器，因为所有高于奈奎斯特频率的都是嘈杂现象。

2.采样定理

奈奎斯特频率 = 采样频率/2

混叠问题(奈奎斯特频率以上的频率映射到较低的频率)。三种波形如图5.2所示。图中最左边的波形每个周期有8个样本(即频率为1/8的采样率(srate))。中间波形的频率是最高的，可以通过这个采样率来表示。最右边的波形频率特别高，所以采样波将被看做(被混叠为)缓慢的虚线(这与最左边的曲线相同)。

解决：在采样以前使用输入滤波器消除所有不需要的输入。任何高于奈奎斯特频率的能量将被映射到较低的频率。当然，在现代的数字设备上，滤波器会自动完成这些工作。

3.振幅的量化(振幅值的极限设置)

当然，当采样信号被转化回实时信号时，只在采样点分配指定值。输出信号将较为平缓直到下一个采样进来。这些平缓的信号将被收听者视为一个高频信号(高于奈奎斯特频率)，但它也可能是噪音。因此，为了使收听者收到正常信号，图5.3中的红色曲线必须平滑过渡到(低通滤波)绿色曲线。当然，如果采样率是商业CD标准，那么这种噪音会高于人类听觉的限制，所以你自己的耳朵可以作为低通量化滤波器。事实上，由于扬声器不再产生20kHz以上的声音，它们也能作为低通输出滤波器。

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)thatareshiftedbymultiplesofƒsandcombinedbyaddition.Forsufficientlylargeƒs,thek=0termcanbeobservedintheregion[−ƒs/2,ƒs/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+½)/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