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# Simplified Method of Designing FIR Filter with Controllable Center Frequency

By Anne Hicks,2014-02-18 20:38
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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

frequencyparametersand.

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

FIRfiltersfindapplicationsinmanyfieldswhere

onecertalnfrequencycomponentmustbecancelledfrom

thesignalspectrumonsomeoccasions.Forexample,

50

moved.Sodesigningthefilterwithcontrollablecenter frequencyisofsignificance.

forcriticalfrequencytuningofFIRfilters. whichcan

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

combinationmethodtodesignFIRnotchfilter,which However,designingahigh.accuracycenterfre

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

ily,andthechosenrangefornotchfrequencycanonlybe widenedattheexpenseofincreasingfilterlength?_Zah.

eredandsubtractedfromthefiltercoefficient.Inthispa

per,afilterdesignmethodisproposedbysynthesizing twoconjugatepointpasssub-filters,thusremovingDC

itemneednotbetakenintoaccount.Moreover,thepro

posedmethodcandesignnotonlypointpassfilterbut

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

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&lt;'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

metricwith=.However.itshouldbenotedthatthe

point-passfrequencypointscanonlyfallattheposition

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/

whicharespecifiedbythefrequencyshiftingparame

(15)ter.

UsingEq.(12)Eq.(15),

g()=()n?[-N+1,1](16)

Fig.2showsthetransfercurvesofthesetwosub

filters.

(lJSubfilterg:

}?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.