数字滤波器设计英文文献Word文档格式.docx

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Thedesignprocessofadigitalfilterislongandtediousifdonebyhand.Withtheaidof

computerprogramsperformingfilterdesignalgorithms,designingandoptimizingfilters

canbedonerelativelyquickly.ThispaperdiscussestheuseofMatlab,amathematical

softwarepackage,todesign,manipulate,andanalyzedigitalfilters.

ThedesignoptionsinMatlaballowtheusertoeithercreateacodefordesigningfilters

thatcallsbuilt-infunctions,ortodesignfiltersinSptool,agraphicaluserinterface.Each

ofthesemethodsareexaminedinthispaper.Thestrengthsandweaknessesofeachare

detailedinthefollowingdiscussion.

ThispaperconcludeswithadiscussionofhowthedatagivenbyMatlabforvariousfilters

canbeusedtoimplementfiltersonrealdigitalsignalprocessors.Matlabprovidesallthe

informationnecessaryforbuildingahardwarereplicaofthefilterdesignedinsoftware.

TABLEOFCONTENTS

1.Abstract……………………………………………………………..4

2.Introduction.………………………………………………………..4

3.LowpassFilterDesign………………………………………………7

4.HighpassandBandpassFilterDesign………………………………11

5.Sptool……………………………………………………………….13

6.FutureDirections……………………………………………………16

7.Acknowledgments…………………………………………………..16

8.References…………………………………………………………..16

9.Appendix…………………………………………………………….17

Abstract

Matlabprovidesdifferentoptionsfordigitalfilterdesign,whichincludefunctioncallsto

filteralgorithmsandagraphicaluserinterfacecalledSptool.Avarietyoffilterdesign

algorithmsareavailableinMatlabforbothIIRandFIRfilters.Thispaperdiscussesthe

differentoptionsinMatlabandgivesexamplesoflowpass,highpass,andbandpassfilter

designs.

ResultsshowthatthegraphicaluserinterfaceSptoolisaquickerandsimpleroptionthan

theoptionofmakingfunctioncallstothefilteralgorithms.Sptoolhasamoreuser-

friendlyenvironmentsincethespectrumofthefilterisimmediatelydisplayedtotheuser,

andtheusercanquicklyzoominandexamineparticularareasofinterestinthespectrum

（i.e.thepassband）.However,theshortcomingofSptoolisthatitonlydisplaysthe

magnituderesponseofthefilter,notthephaseresponse.

Introduction

Akeyelementinprocessingdigitalsignalsisthefilter.Filtersperformdirect

manipulationsonthespectraofsignals.Tocompletelydescribedigitalfilters,threebasic

elements（orbuildingblocks）areneeded:

anadder,amultiplier,andadelayelement.The

adderhastwoinputsandoneoutput,anditsimplyaddsthetwoinputstogether.The

multiplierisagainelement,anditmultipliestheinputsignalbyaconstant.Thedelay

elementdelaystheincomingsignalbyonesample.Digitalfilterscanbeimplemented

usingeitherablockdiagramorasignalflowgraph.Figure1showsthethreebasic

elementsinblockdiagramform,andFigure2showstheminsignalflowgraphform.

Withthebasicbuildingblocksathand,thetwodifferentfilterstructurescaneasilybe

implemented.ThesetwostructuresareInfiniteImpulseResponse（IIR）andFinite

ImpulseResponse（FIR）,dependingontheformofthesystem’sresponsetoaunitpulse

input.IIRfiltersarecommonlyimplementedusingafeedback（recursive）structure,while

FIRfiltersusuallyrequirenofeedback（non-recursive）.

InthedesignofIIRfilters,acommonlyusedapproachiscalledthebilinear

transformation.Thisdesignbeginswiththetransferfunctionofananalogfilter,then

performsamappingfromthes-domaintothez-domain.Usingdifferentialequations,it

canbeshown（Proakis677）thatthemappingfromthes-planetothez-planeis

ThismappingresultsinageneralformforanIIRfilterwithanarbitrarynumberofpoles

andzeros.Thesystemresponseandthedifferenceequationforthisfilterisasfollows:

Thissystemresponsecanbeeasilyrealizedusingasignalflowgraph

AnFIRfilterhasadifferenceequationof

Bytakingthez-transform,thesystemresponseis

TherealizationofanFIRfilterusingasignalflowgraphisstraightforward.

MatlabhasseveraldesignalgorithmsthatcanbeusedtocreateandanalyzebothIIRand

FIRdigitalfilters.TheIIRfiltersthatcanbecreatedinMatlabareButterworth,

Chebyshevtype1and2,andelliptic.TheFIRfilteralgorithmsinMatlabareequiripple,

leastsquares,andKaiserwindow.TheMatlabcoderequiredtoimplementthesefilters

involvesbilineartransformationsandfunctioncallstoanalogprototypefilters.The

followingsectionsgiveexamplesofMatlabimplementationoftheIIRfilterslistedabove.

LowpassFilterDesign

UsingMatlab,alowpassdigitalfilterisdesignedusingvariousanalogprototypes:

Chebyshev,Butterworth,andElliptic.Theoptimumfiltertypeischosenonthebasisof

implementationcomplexity,magnituderesponse,andphaseresponse.Thedesign

specificationsforthefilterareasfollows:

•Cutofffrequency=1000Hz

•Samplefrequency=8000Hz

•Passbandripple=0.5dB

•Stopbandattn.=60dB

•Transitionband=100Hz

MatlabCode（Chebyshev）:

%LowpassdigitalfilterwithChebyshev-Ianalogprototype

%

%DigitalFilterSpecifications:

wp=0.125*2*pi;

%digitalpassbandfrequencyinHz（normalized）

ws=0.1375*2*pi;

%digitalstopbandfrequencyinHz（normalized）

Rp=0.5;

%passbandrippleindB

As=20;

%stopbandattenuationindB

%AnalogPrototypeSpecifications:

Fs=1;

T=1/Fs;

OmegaP=（2/T）*tan（wp/2）;

%prewarpprototypepassbandfrequency

OmegaS=（2/T）*tan（ws/2）;

%prewarpprototypestopbandfrequency

%AnalogChebyshev-1PrototypeFilterCalculation:

[c,d]=chb1（OmegaP,OmegaS,Rp,As）;

%BilinearTransformation:

[b,a]=bilinear（cs,ds,Fs）;

[db,mag,pha,grd,w]=freqz（b,a）;

plot（w*8000/2/pi,db）;

xlabel（'

frequency（Hz）'

）;

ylabel（'

decibels'

title（'

Magnitudein

dB'

ThisexactcodeisalsousedfortheellipticandButterworthdesigns.Theonlychangeis

inthefiltercalculationsofeachtype.Insteadofcallingchb1（）,theellipticfilterdesign

callsafunction“elliptic（）”andtheButterworthdesigncallsafunction

“butterworth（）”.SeetheappendixfortheMatlabcodeofthefunctionchb1（）.

Thefollowingfiguresshowthemagnitudeandphaseresponsesofeachtypeoffilter.

MagnitudeResponseofChebyshevFilter

PhaseofChebyshevFilter

MagnitudeResponseofEllipticFilter

PhaseofEllipticFilter

MagnitudeResponseofButterworthFilter

PhaseofButterworthFilter

TheMatlabcodeoutputsthefilterorderandthefiltercoefficients.Forthisexample,the

Chebyshevfilterorderwasnine.Theellipticfilterhadanorderoffive,andthe

Butterworthfilterorderwasthirty-two.

Severalconclusionscanbedrawnabouttheselow-passfilterdesignsfromthissimple

example.First,ingeneral,foragivensetofdesignconstraints,theellipticfilterdesign

algorithmwillresultinthesimplestfilter（intermsofcomplexity）.Themostcomplex

filteristheButterworthfilterwithanorderofthirty-two.Intermsofpassbandripple,the

Butterworthfiltergivestheoptimumresponse.Inthepassband,thereisalmostnoripple

（monotonic）.TheellipticandChebyshevfiltersbothhavemuchmorerippleinthe

passband.So,thereisatradeoffbetweenthesethreedifferenttypesoffilters.Intermsof

magnituderesponseandcomplexity,theellipticrippleismostlikelytheoptimumchoice.

However,theellipticripplehasaphaseresponsethatismorenonlinearthanthe

ChebyshevandButterworthfilters.Therefore,ifasharpcutoffandrelativelylow

complexityisrequired,thechoicewouldbetheellipticfilter.Ifthephaseresponsewould

needtobemorelinear,aChebyshevorButterworthfiltershouldbechosenoverthe

ellipticfilter.

HighpassandBandpassFilterDesign

Matlabprovidesfunctionsforimplementinglowpass-to-highpassandlowpass-to-bandpass

conversions.Byprovidingafilterorder,thepassbandripple,andthe3dBcutoff

frequencytothefunctioncheby1（）,ahighpassfiltercanbedesigned.Thefilterorderis

foundusingthefunctionchebord（）.ForaButterworthprototype,thefunctionsare

butter（）andbuttord（）.Fortheellipticprototype,thefunctionsareellip（）and

ellipord（）.

ThefollowingMatlabcodeisusedtodesignaChebyshevhighpassdigitalfilterwitha

passbandat1100Hzanda100Hztransitionband.

%HighpassChebyshevDigitalFilter

ws=0.125*2*pi;

%digitalstopbandfrequencyinrad/s

wp=0.1375*2*pi;

%digitalpassbandfrequencyinrad/s

[N,wn]=cheb1ord（wp/pi,ws/pi,Rp,As）;

[b,a]=cheby1（N,Rp,wn,'

high'

[db,mag,pha,grd,w]=freqz_m（b,a）;

axis（[8001200-221]）;

Thefollowingfigureshowsthemagnituderesponseofthehighpassfilter.

Bandpassfiltersarefoundusingthesesametwofunctions.However,withbandpass

filters,thepassbandandstopbandfrequencies（wpandws）aretwo-elementvectorssince

therearetwopassbandfrequenciesandtwostopbandfrequencies.TheMatlabcode

belowshowsthedesignofanellipticdigitalbandpass