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Simplified Method of Designing FIR Filter with Controllable Center Frequency

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Simplified Method of Designing FIR Filter with Controllable Center Frequencyof,Of,FIR,with

    Simplified Method of Designing FIR Filter

    with Controllable Center Frequency Trans.TianjinUniv,2010.16:262-266

    DOf10.1007/sl2209.0101154.3

    SimplifiedMethodofDesigningFIRFilter

    withControllableCenterFrequency

    HUANGXiangdong(黄翔东),CHUJinghui(褚晶辉),L0Wei(吕卫),

    CUIHaitao(崔海涛),WANGZhaohua(王兆华)

    (SchoolofElectronicsandInformationEngineering,TianjinUniversity,Tianjin300072,China)

    TianjinUniversityandSpringerVerlagBerlinHeidelberg2010

    Abstract:AnovelsimplifiedmethodispresentedtodesignFIRfilterwithcontrollablecenterfrequency.Theproper

    tiesoftransfercurvesforall

    phasefiltersareillustratedunder3windowingconditions.Bycombiningsingle.window allphasefilterdesignstepsanddoublephase

    shiftcombination,aseriesofdesignformulasforpointpassfilter,notch

    filter,bandpassfilterandband

    stopfilterarederived,thusthedesigncomputationcomplexityisgreatlyreduced.Ex

    perimentsprovethatthecenterfrequencyofallthefilterscanbeaccuratelyspecifiedatarbitrarypositionbyadjusting

    frequencyparametersand.

    Keywords:filterdesign;al1.-phase;doublephaseshiftcombination;centerfrequency

    FIRfiltersfindapplicationsinmanyfieldswhere

    onecertalnfrequencycomponentmustbecancelledfrom

    thesignalspectrumonsomeoccasions.Forexample,

    50

    Hzinterferenceinrecordingelectrocardiogrammustbe canceled.andnarrowbandinterferenceindirectse. quencespreadspectrumsystemsshouldalsobere

    moved.Sodesigningthefilterwithcontrollablecenter frequencyisofsignificance.

    forcriticalfrequencytuningofFIRfilters. whichcan

    adjustthepositionofnotchfrequencyarbitrarilywhile preservingthenatureofthefiltesuchasmaximallyflat andequiripple,butthewidthofthebandsofthefilter willbebroadenedl1.

    BasedontheconeeDtofal1.phasedigitalfilter. ing..',Ref.[8]presentedallphasedoublephase.shift

    combinationmethodtodesignFIRnotchfilter,which However,designingahigh.accuracycenterfre

    canachievearbitrarynotchfrequencyonlybyadjusting quencyfilterwithsmallripplesinpass--bandandstop.. bandisnoteasy.Fornotchfilter.Parks.McClellantech. niqueisatypicaliterativemethod,however, thestan

    dardParksMcClellanalgorithmcannotprovidesarisfac. toryresultsindesigningFIRnotchfilters.Zahradnikpre. sentedfastanalyticaldesignalgorithmsforFIRnotch filterThisanalyticalschemeconsistsofmaximallyflat (MF)methodandequiripple(ER)method.bothof whichcanguaranteethatthefrequencyresponsevalueat thenotchfrequencypointcanreach0.Butthenotchfre. quencycanonlybechosenamonglimitednumbersof casesbvfastanalyticalmethod.whichimpliesthatthe positionofnotchfrequencycannotbespecifiedarbIitrar

    ily,andthechosenrangefornotchfrequencycanonlybe widenedattheexpenseofincreasingfilterlength?_Zah.

    radniklaterrecognizedthedisadvantagesofthefastana. 1yricalprocedureandproposedtheanalyticalprocedure phaseshiftparameterAwithoutanyiterativeoperation. InRef.[8],thenotchfilterisformedbysynthesizingtwo conjugatenotchsub?filters,theDCiternmustbeconsid

    eredandsubtractedfromthefiltercoefficient.Inthispa

    per,afilterdesignmethodisproposedbysynthesizing twoconjugatepointpasssub-filters,thusremovingDC

    itemneednotbetakenintoaccount.Moreover,thepro

    posedmethodcandesignnotonlypointpassfilterbut

    alsonotchfilter,band.passfilterandband.stopfilter. whosecriticalfrequencypositioncanbeadjustedconven. ientlybysubstitutingtheexpectedparameterintoafixed formula

    1Principleofall-phasefiltering

    All-phasefilterdesignmethodisanewFIRfilter designschemepossessingconcurrentlythemeritsof Accepteddate:20100118.

    SupportedbyNationalNaturalScienceFoundationofChina(No60802048)andNewTeache

    l'PhDProgramsFoundationofMinistryofEducationof China(No.200700056lO5)

    HUANGXiangdong,bornin1979,male,Dr,associateProf. CorrespondencetoHUANGXiangdong,E-mail:xdhuang@Ou.educn HUANGXiangdongelQl:SimpliftedMethodofDesigningFIRFilterwithControllableCen

    terFrequency

    /J. conventionalwindowingandfrequencysampling

    Similartothefrequencysamplingmethod,theconstruc-

    tionofall?phasefilterneedstospecifyanN-lengthfre-- quencyvectorHsatisfying

    ()=(N-k)1,2,,J7v1(1)

    Besidesthis,anN-lengthfrontwindowfandanN- lengthbackwindowbshouldalsobespecified.Letbe rectanglewindow.Sotheallphasefiltercanbeclassified into3cases:no-windowcase(iff==R?),singlewin-

    dowcase(iff=RN,b?RNor?RN,西=Randdual-

    windowcase(iff=b?R.ThenanallphaseFIRfilter

    gcanbeconstructedbythefollowing3steps.

    Step1ImplementinversediscreteFouriertrans- form(IDFT)ontoconstructh.Thenextendthedefin. ingrangeofhtoforma(21)-lengthvectorh [h(一?+1),,h(0),,h(AL1)].

    Step2Implementconvolutionbetweenfandb(in reverseformat)togeneratetheconvolutionwindowWe, thennormalizeWe.

    Step3Multiplythecorrespondingelementsofh byctogeneratethecoefficientvectorg. FromStep1andEq.(1),

    h(r1)=h(r1)=h(N-n)

    =

    0,1,2,,?1(2)

    FromStep2,

    ImplementingFouriertransformonEq.(3), (j=F(jB0co)(5)

    FromStep3,

    g()=we(n)()?【-N+I,N1](6)

    CombiningEqs.(2)(6).thefrequencytransferfunc-

    tionG(j?)ofall?phaseFIRfiltercanbededucedas G(j=?g(n)e..=

    ?wan)(,z)e-jn~o=

    [-1

    N-I,?ejn~o=

    ?()?Wc()e_V=oN+I

    We[j(co-2kn/N)](7)

    Eq.(7)indicatesthatthefrequencytransferfunction ofthefilterequalsthediscreteconvolutionbetweenfre. quencysamplingvaluesandthespectrumfunctionof convolutionwindow.BasedonEq.(3).thispropertyre

    latesthetransfercurveofthefiltertothefrequencysam

    piingpointsofH.Nowwegiveanexample:LetN=16 andsetthefrequencyvectorHasfollows:

    H=[00010000000001oo]T(8)

    Fig.1givesthepointpasstransfercurvesforno

    windowcase,singlewindowcaseanddualwindowcase

    respectively(Whdenotesthehanningwindowwithlength N).Fig.1(a)andFig.1(b)showthatthetransfercurves ofno--windowandsingle--windowall?-phasefilterpass throughthefrequencysamplingpointssetbyH.which doesnotsuitthedua1.windowcase.Thereasonforthisis theoreticallyprovedinRef.[6].Moreover,itcanalsobe foundthattheripplesinthepassbandofsinglewindow

    transfercurvearemuchsmallerthanthoseofnowindow

    transfercurve.Sointhenextsection,thesinglewindow

    caseischosentodesignnotchfilterwhosecenterfre? quencycanbespecifiedarbitrarily.

    :

    l0

0.5

    1.0

    

    o5

    1.0

    _.

    o.5

    0123456789l0ll123l4l5l6

    /(2~c/16Far{?s1

    (aJNowindowrase:b=R

    0123456789101112l3l4l5l6

    co/(2n/16ra(1.s1

    (b)Singlewindow(asP:'.'h,b=R,

    0123456789l0l】】2l3l4516

    1(2n/16rad?1

    c)1)ualwindm~ease:f_-6=..t

    Fig.1Notchtransfercurvesfor3windowingcases 2Designofsimplifiedfilterdesignmethod 2.1Designofpoint?passfilterwithcontrollable centerfrequency

    Inthisdesignscheme,twoconjugatesub-filtersneed tObeconstructed.Andinversephaseshiftoperationsare requiredonthesetwofilters.Thensimplysummingup thesetwophase??shiftfilterswillgeneratethepoint?-pass

    filterwithcontrollablecenterfrequency. Toachievethis,thevectorinEq.(8)canbede

    tachedintotwovectorsH1andH2satisfying

    263

    ?

    +

    ?

    

    i:

    )?,??厂=?一C=\^

    C

    TransactionsofTianjinUniversityVo1.16No.42010

    .()=

    H()=

    0??一1<'J,m—一一Il0others2

    =?一0??一N1)0others',2

    (9)

    (10)

    ForthecaseofEq.(8),N=l6,m=3,accordingto Eq.(9)andEq.(10),H1andareexpressedas l

    [000100000000000(T

    (11)

    H2=[0000000000000100]

    Notethatthevalue1inHlisinthevector'sformer

    halfpart,whichguaranteesthatthepointpassfrequency ofHlcanbelimitedintherange??(0,7c).Thepoint- passfrequencyof//2canbelimitedintherange?(7

    2n).Obviously,andt/2satisfy (:()k=1,2,,JV_1(12)

    SotheIDFTofillisconjugatetothatofH2, hl()=h2()(13)

NowH1and/42areutilizedtodesigntwoallphase

    FIRsubfiltersgandaccordingtothe3stepsde

    scribedabove.Let=exp(-j2n/N),f=Whandb

    ?,

    _wc(?==

    wc()

    N

    .

    2n

    mn

    _

    

    w

    e

    _

    (n)eJU

    (

    -

    m"

    nE[N+I,N-1](14)

    CombiningEq.(13)andEq.(14)wecanalsoob

    taintheothersubfiltergas

    g;(,2)=wc(n)h2(n)=wc()e

    n?【一?+l,N-1

    ?

    InFig.2,Ic;Oco)Iandl(j)Ipassthroughthe

    frequencysamplingpointatCOl3x2n/16radsand

    092=13x2n/16rad'srespectively.Andtheyaresym-

    metricwith=.However.itshouldbenotedthatthe

    point-passfrequencypointscanonlyfallattheposition

    ofintegertimesofAco=2n/16rad?s(.Inordertomove thecenterfrequencyarbitrarily,thephaseshiftoperation introducedinRef.[8]canbeimplementedonthesetwo

    subfilters.Soaphaseshiftvectorl,1[121(一?+1),,

    v1(0),,v1(1)]needstobespecifiedas v1()==e,一?+1?n?N(1(17)

    Thenanewsub-filterglcanbeobtainedthrough

    multiplyingVl()byg(n)as

    lmH

    g.():g():.

    e--j2Anx/N

    :e

    '(18)

    Similarly,anotherphaseshiftvectorY2=v2(

    ?+1),,v2(0),,v2(1)]canalsobespecifiedas 1,,()==e,Jv+1???一1(19)

    Soanewsub--filterg2canbeobtainedthroughmultiply?-

    ingv2()byg()as

    g2()=g()v2()=g()e=

    2x

    :ej2:e!?:一箐(

    NN

    (20)

    Fig.3givesthetransfercurvesofphaseshiftsub

    filtersglandg2.IncontrasttoFig.2,thepointpassfre-

    quencyvaluesof1Gl(j)land]G2(jo))lare(Ol2.7X2n/

    16rad.s(and?2=13.3x2n/16rad?s(,respectively, ,,

    whicharespecifiedbythefrequencyshiftingparame

    (15)ter.

UsingEq.(12)Eq.(15),

    g()=()n?[-N+1,1](16)

    Fig.2showsthetransfercurvesofthesetwosub

    filters.

    09/(2n/I6rad?s'l

    (lJSubfilterg:

    /(2nil6Iad?s)

    }?Subfilterg

    Fig.2Transfercurvesoftwosubfilters(N=16,m:3)

    264

    mf(2n/16ra(1?s1

    ?Subfiherg

    031(2hil6r?s'.1

    m)lI_fill,',g

    6

    Fig.3Transfercurvesoftwophaseshiftsubfilters (N=16,m=3,=0.3)

    However.Eq.(18)andEq.(20)showthatgl()

    andg2()arenotrealvaluedfiltercoefficients.Simply

    HUANGXiangdongetal:SimplifiedMethodofDesigningFIRFilterwithControllableCent

    erFrequency

    summingupgl(,z)andg2()cangeneratetherealusefulonmanyoccasions

    valuedfiltercoefficientg(n)as ,="]=

    cos

    [()-N+I,N-1](2-?J?,J

    Fig.4givesthetransfercurveoftheultimatepoint..

    passfilter.

    /(2x/16rad?s

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