Digital Filter Design LabRensselaer Polytechnic InstituteWord文档格式.docx
- 文档编号:19169016
- 上传时间:2023-01-04
- 格式:DOCX
- 页数:32
- 大小:514KB
Digital Filter Design LabRensselaer Polytechnic InstituteWord文档格式.docx
《Digital Filter Design LabRensselaer Polytechnic InstituteWord文档格式.docx》由会员分享,可在线阅读,更多相关《Digital Filter Design LabRensselaer Polytechnic InstituteWord文档格式.docx(32页珍藏版)》请在冰豆网上搜索。
NumberofSessions–4
INTRODUCTION
ThislabdemonstratestheuseofdigitalfiltersonaDSP(digitalsignalprocessor).Itconsistsoftwosteps,thefirstbeingthedesignofsomefiltersusingMATLABandLabVIEW,andthesecond,implementingthosefilters,aswellassomeothersontheDSPsystem.
ThefirstpartinvolvesthedesignofsomefourthorderandhigherfiltersusingMATLABandafilterdesigntoolkitwithinLabVIEW.Tohelpyoutowardsthisend,theAppendixprovidesalistofMATLABcommandsusefulforfilterdesign.FurtherdetailsmaybefoundintheMATLABhelppages.InstructionsforusingtheNationalInstrumentsLabVIEWDigitalFilterDesigntoolkitareincludedlaterinthelabprocedure.
ThesecondpartinvolvestheimplementationandanalysisofthesefiltersontheDSPboard.Twomethodsofimplementationwillbeevaluated.Inordertogainsomefamiliaritywithdigitalfilterdesign,youarerequiredtosolvethefollowingproblempriortothefirstlabsession:
Converttheanaloglow-passfilter:
toanequivalentdigitallow-passfilter.Thefiltershouldhaveacut-offfrequencyof143rad/sec.Usetheimpulseinvarianttransformationwithasamplingperiod(T)of7msec.YouranswershouldbeintheZdomain,andshouldincludeanassociatedblockdiagram.
Asinglefourthorderdigitalfiltermaybeimplementedtwodifferentwaysmathematically:
and
BlockdiagramsofthetwofilterformsareshowninFigure1andFigure2,howeveronlythesecondform(cascadeTypeII)isimplementedbythefiltertoolkit.ThefiltersarerunonaSpeedy-33DSPunderLabVIEW.Thereareafewitemstonotehere.OnlytheleftchanneliswiredintheBNC-1/8”ministereoplugcablesinthelab.TheA/Dinputusedfortheincominganalogsignalhasarange1.2Vp-pandtheD/Aoutputhasamaximumrangeofalittleover3.5Vp-p.AlthoughtheDSPsystemhasgoodlowfrequencyresponse,asanaudiosystemitpurposelyblocksDCtopreventdamagetocomponents.Thissystemwillattenuatefrequenciesbelowabout0.5Hz.AllfiltersimplementedbytheLabVIEWtoolhaveagainof3whenaunitygainwouldbeexpected.Thisisconsistentandshouldbetakenintoconsiderationwhenanalyzingdata.
y(k)
F+Gz-1+Hz-2
1+Iz-1+Jz-2
FIGURE1.ParallelFilter(TypeI).
H
FIGURE2.CascadeFilter(TypeII).
BACKGROUND
TheApproximationProblem
Theapproximationproblemisoneoffindingamatchbetweentheidealizedfrequencyresponsedesired,andthevariousresponsespossible.
Theideallow-passfilterresponseisasshowninFigure3a.Thefilterhasagainequalto1for|f|<
fc,andagainequalto0for|f|>
fc.Thisresponseispracticallyandtheoreticallyunrealizable.ConsidertheinverseFouriertransformofthisfilter.Itisasincpulsecenteredatt=0(Figure3b),whichisanon-causaloutput.AtimedelaycanbeaddedtothefilterandtheresponseisnowasinFigure3c.Foralargeenoughdelay,h(t)willbenegligiblefort<
0,andcanbeapproximatedbyarealizablefilter.
FIGURE3a.IdealLow-Pass.
FIGURE3b.ImpulseResponseofaLow-Passfilter.
FIGURE3c.DelayedResponseofafilter.
Therearethreemaintypesoflow-passfilterapproximations.TheyaretheButterworth(ormaximallyflat),theChebyshev(twoversions),andtheellipticapproximations.
TheButterworthlow-passfilterofordernhasanamplituderatiogivenby
Thisfilter,whoseBodeplotisshowninFigure4,hasagainwhichdecreasesmonotonicallyfromunityatf=0Hz.Asn(thefilterorder)isincreased,therateoffalloffofthefilteratitscutofffrequencyisincreased.Thisisnotwithoutapenalty,becauseasthefilterdegreeincreases,thephaseshiftgetsworse,andtheimpulseresponsedoesnotfollowthesincpulseveryclosely.
TheChebyshevfilterTypeI(butnottobeconfusedwithparallelTypeIimplementation)hasarippleinthepass-band,anddecreasesmonotonicallyinthestop-band.TheTypeIIfilterreversesthesebands.AtypicalfrequencyresponseisshowninFigure5.Thisfilterhastheadvantageofafasterrateoffalloff,andamorelinearphaseshift.Inthepass-band,themagnitudeofthefrequencyresponsefluctuatesbetween1and1/(1+e2)1/2.Foralargere,therippleislargerbutthefalloffisfaster.Thereisadesigntrade-offbetweentheripplesizeandthefalloffforagivenfilterorder.
Theellipticfilterallowsripplesinboththepassandstop-bands,asshowninFigure6.Thishasthefastestfalloffrateofthethreefiltertypesbuthasalargephaseshift.Thisfilteragainhasatradeoffbetweenripplesizeandfall-offrate.Forfurtherdetailsonanalogfiltertypesseereference[3].
Band-passFilterDesign
Aband-passfilterwithcenterpass-bandfrequencyw0canbederivedfromalow-passfilterbyusingthelow-passtoband-passtransformation.
Apole-zeropatternandfrequencyresponsecurveforatypicallow-passfilterisshowninFigure7a.Tomakeaband-passfilteryoumighttrytomakethesubstitutionss-jw0tomovethepolesuptojw0.Thiswouldnotworkbecauseanycircuitbuiltwithrealelementsmusthaveallcomplexpolesandzerosincomplexconjugatepairs.
AsubstitutionthatdoesworkisthereplacementoftheLaplacedomainvariablesinthelow-passfilterH(s)by
wheresbistheLaplacevariableofthetransformedband-passfilter.Then,forfrequenciesofoperationclosetothecenterfrequencyw0(i.e.sbisapproxequaltojw0),thetransformedlow-passfilterbecomes
whereDisthedeviationfromw0.Thus,theshapeandbandwidthofthelow-passfilterarepreserved.ThistransformationleadstocomplexconjugatepolesandzerosasshowninFigure7b,andisthereforerealizable.
FIGURE4a.ButterworthLow-PassFilter(1stOrder).
FIGURE4b.ButterworthLow-PassFilter(2ndOrder).
FIGURE4c.ButterworthLow-PassFilter(3rdOrder).
FIGURE5.ChebyshevTypeILow-PassFilter(passbandripple).
FIGURE6.EllipticLow-PassFilter(passandstopbandripple).
FIGURE7a.Low-PassFilter.
FIGURE7b.TransformedBand-PassFilter.
DISCRETETIMESYSTEMS
Adiscretesignalisanorderedsequenceofnumbers.Ifyousampledacontinuoussignalx(t)everyTseconds,youroutputwouldbeafunction(x(kT):
k=0,1,2,...),whichisadiscretesignal(i.e.,aseriesofvaluesoccurringeveryTseconds).Adiscretesystemisoneinwhichallthevariablesarediscretesignals.
Adiscretesystemisanalogoustoacontinuoussysteminmanyways.Theoutputofthesystematanyfuturetimeisknownifyouknowthesystem'
spresentstateandtheinput.
Astatevariableequationcanbewrittenasy(kT)=S[q0:
x(kT)]k≥k0,wherex(kT)istheinput,q0istheinitialstateatk=k0,andy(kT)istheoutput.Afixed,lineardiscretesystemwillobeytheprinciplesofdecomposability,superposition,andtimeinvariance(seereference[2]).Discretesystemsaredescribedbydifferenceequationsinthesamewaythatcontinuoussystemsaredescribedbydifferentialequations.Theblockdiagramelementsofadiscretesystemareunitdelayelementsandscalars.
a.Time-DelayElement.b.Scalar.
FIGURE8.SimulationDiagramElements.
Allsystemsinvolvingdigitalcomputersforsignalprocessingarediscretetimesystems.Toworkonasignal,itmustfirstbecodedintosomebinaryrepresentation.Thisanalogtodigitalconversiontakessomefiniteamountoftime.Therefore,thereissomemaximumsamplingfrequency.Ifthesignalistobeprocessedinrealtime,theamountoftimetakentoperformcalculationsandoutputresultsmustbeaddedtothisconversiontime.Thisreducesthemaximumpossiblesamplingfrequency.Tosimplifythemanipulationofcontinuoussystems,theLaplacetransformisused.AnanalogoustoolforthediscretesystemistheZ-transform.TheZ-transformofv(k)isdefinedastheinfinitesummation
whichissometimesexpressedasV(z)=Z{v(k)},orbythetransformpairv(k)V(z).TheZ-transformconvertsadifficulttosolvefinitedifferenceequationintoaneasytosolvealgebraicequationinz.
Therearemanytechniquesfordesigningdigitalfilters.Themethodusedinthislabistodesignananalogfilterforthedesiredresponse,andthentotransformtheH(s)intoanH(z)byoneofthreemethods.
Thefirsttwomethodsusedareimpulseandstepinvariance.Thesetwotechniquessettheresponseofthedigitalfiltertoaparticularinputtobeequaltotheresponseoftheanalogfiltertothesameinput.
Togettheimpulseinvariantfilter,itisnecessarytoobtainthetimedomainimpulseresponseh(t)ofthedesiredanalogfilter.Thisisthensampledgivingthevaluesh(0),h
(1),...etc.ThecorrespondingZ-transformoftheimpulseinvariantdigitalfilteristhus
Alternately,ifH(s)isthetransferfunctionoftheanalogfilter,then
AtthispointitshouldbenotedthattheDCorzerofrequencygainsofH(s)andH(z)willnotbethesame.ThusascalingfactorisneededforH(z),
ForH(s),
However,
Therefore,H(z)mustbemultipliedbyTpriortoitsimplementation.
Withthestepinvarianttransformation,theoutputofthedigitalfilteristobeequaltothesampledoutputsofthecorrespondinganalogfilter.AnexampleofthisisthestepinvariantButterworthsecondorderlow-passfiltershowninFigure9.Ascanbeseen,thedigitalresponseisidenticaltotheanalogresponseeveryTseconds.Thistechniqueguaranteestheoutputforastepinput,butinturnsaysnothingabouttheimpulseresponseofthedigitalfilter.
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- Digital Filter Design Lab Rensselaer Polytechnic Institute
链接地址:https://www.bdocx.com/doc/19169016.html