6.4 Basic IIR Digital Filter Structures Then consider H 2.ppt

Chapter6 DigitalFilterStructures 6 0Introduction Theconvolutionoperationinvolvesaddition multiplicationandtimeshift ordelay ForanIIRsystem i e anLTIsystemwithaninfinite lengthimpulseresponse thisapproachisnotpractical ThemostfundamentalcharacterizationofanLTIdigitalfilteristheconvolutionrepresentation 6 0Introduction ForacausalFIRsystem theconvolutionrepresentationisoftheform or However anIIRsystemcanbedescribedbyalinearconstant coefficientdifferenceequation isquitepractical 6 0Introduction Now theactualimplementationofadigitalfiltercouldbeeitherinsoftwareorhardwareformdependingonapplications InthisChapter weconsidertherealizationproblemofcausalIIRandFIRfiltersandoutlinerealizationmethodsbasedonboththetime domainandthetransform domainrepresentations 6 0Introduction Torealizeadigitalfilter thekeyisthatwemustfirstdescribethestructureofthedigitalfilterusingtheblockdiagramrepresentation calledthestructuralrepresentation Therearevariousformsofthestructuralrepresentationofadigitalfilter ThisistheemphasisinthisChapter 6 1BlockDiagramRepresentation 6 1 1BasicBuildingBlocksTheblockdiagramrepresentationconsistsofthesethreebasicoperations 1 Addition 2 Multiplication 3 Unitdelay 6 1BlockDiagramRepresentation Areviewofafirst orderLTIdigitalfilterConsidertheLTIdigitalsystem 6 1BlockDiagramRepresentation Fromthisreview weoftenneedtoexpressthedifferenceequationintotherecursiveformlikethis 6 1BlockDiagramRepresentation Thereareseveraladvantagesinrepresentingthedigitalfilterinblockdiagramform 1 Itiseasytowritedownthecomputationalalgorithmbyinspection 2 Itiseasytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandtheinput 3 Itiseasytomanipulateablockdiagramtoderiveother equivalent blockdiagramsyieldingdifferentcomputationalalgorithm 4 Itiseasytodeterminethehardwarerequirements 5 Itiseasiertodevelopblockdiagramrepresentationsfromthetransferfunctiondirectlyleadingtovarietyof equivalent representations 6 1BlockDiagramRepresentation 6 2EquivalentStructures Ourmainobjectiveinthischapteristodevelopvariousrealizationsofagiventransferfunction 6 2EquivalentStructures Inthiscoursewerestrictourattentiontoadiscussionofsomecommonlyusedstructures Underinfiniteprecisionarithmeticanygivenrealizationofadigitalfilterbehavesidenticallytoanyotherequivalentstructure 6 2EquivalentStructures However inpractice duetothefinitewordlengthlimitations aspecificrealizationbehavestotallydifferentlyfromitsotherequivalentrealizations Hence itisimportanttochooseastructurethathastheleastquantizationeffectswhenimplementedusingfiniteprecisionarithmetic 6 2EquivalentStructures Onewaytoarriveatsuchastructureistodeterminealargenumberofequivalentstructures analyzethefinitewordlengtheffectsineachcase andselecttheoneshowingtheleasteffects Incertaincases itispossibletodevelopastructurethatbyconstructionhastheleastquantizationeffects 6 2EquivalentStructures Here wereviewsomesimplerealizationsthatinmanyapplicationsarequiteadequate 6 3BasicFIRDigitalFilterStructures WefirstconsidertherealizationofFIRdigitalfilters AcausalFIRfilteroforderNischaracterizedbyatransferfunctionH z givenby Inthetime domaintheinput outputrelationoftheaboveFIRfilterisgivenby 6 3BasicFIRDigitalFilterStructures 6 3 1DirectFormsThedirectformstructureofanFIRfiltercanbeobtainedfromtheconvolutionsumofthefilter Ingeneral theconvolutionsumrequiresN 1multipliersandNtwo inputadders SupposethattheorderofanFIRfilterisN 4 thismeansthatthelengthoftheimpulseresponseisN 1 5 6 3BasicFIRDigitalFilterStructures Expandingtheconvolutionsumexpression Thedirectformstructureisshowninthefollowing 6 3BasicFIRDigitalFilterStructures Wecanobtainanotherdirectformstructurebythetransposeoperation 6 3BasicFIRDigitalFilterStructures 6 3 2CascadeFormAhigher orderFIRtransferfunctioncanalsoberealizedasacascadeofFIRsectionswitheachsectioncharacterizedbyeitherafirst orderorasecond orderrealcoefficienttransferfunction ThewayistofactorthetransferfunctionoftheFIRfilter 6 3BasicFIRDigitalFilterStructures whereK N 2ifNiseven andK N 1 2ifNisoddandH z mustcontainafirst ordersection If 2k 0 thenthesectionisafirst ordersection Infact afirst ordersectioncanbeviewedasasecond ordersectionwith 2k 0 Let ssee 6 3BasicFIRDigitalFilterStructures Asecond ordersection 6 3BasicFIRDigitalFilterStructures A6 orderFIRfilterincascadeform 6 3BasicFIRDigitalFilterStructures 6 3 4Linear PhaseFIRStructuresRemembertheconditionsforwhichanFIRdigitalfilterhaslinearphase Foralinear phaseFIRdigitalfilter itsimpulseresponsesatisfiesthesymmetryorantisymmetryproperty 6 3BasicFIRDigitalFilterStructures Thesymmetry orantisymmetry propertyofalinear phaseFIRfiltercanbeexploitedtoreducethenumberofmultipliersintoalmosthalfofthatinthedirectformimplementations Consideralength 7Type1FIRtransferfunctionwithasymmetricimpulseresponse 6 3BasicFIRDigitalFilterStructures Thetransferfunctionisobtainedfrom 6 3BasicFIRDigitalFilterStructures Fromtheexpression weobtaintherealizationshownbelow 6 3BasicFIRDigitalFilterStructures Foralength 8Type2FIRfilter itstransferfunctioncanbeexpressedas 6 3BasicFIRDigitalFilterStructures Fromthetwoexamples weseethattheType1linear phasestructureforalength 7FIRfilterrequires4multipliers whereasadirectformrealizationrequires7multipliers andtheType2linear phasestructureforalength 8FIRfilterrequires4multipliers whereasadirectformrealizationrequires8multipliers 6 3BasicFIRDigitalFilterStructures SimilarsavingsoccursintherealizationofType3andType4linear phaseFIRfilterswithantisymmetricimpulseresponses FIR级联型结构特点 由于这种结构所需的系数比直接型多 所需乘法运算也比直接型多 很少用 由于这种结构的每一节控制一对零点 因而只能在需要控制传输零点时用 其他常用结构 1 频率抽样型结构2 快速卷积结构3 格型滤波器 6 4BasicIIRDigitalFilterStructures IIRdigitalfilterisanotherimportantdigitalfilterthatwemainlydiscussinthiscourse Thecharacteristicsareinfinite lengthimpulseresponseandnonlinear phase ThetransferfunctionofacausalIIRdigitalfilterisoftenintheform 6 4BasicIIRDigitalFilterStructures 6 4 1DirectFormsFromrelationshipbetweenthedifferenceequationandthetransferfunction thedifferenceequationcanbewrittenas 6 4BasicIIRDigitalFilterStructures Inordertodevelopthedirectform wecanviewthetransferfunctionasaproductoftwotransferfunctions 6 4BasicIIRDigitalFilterStructures Forexample letsconsiderforsimplicitya3 orderIIRfilterwithatransferfunction 6 4BasicIIRDigitalFilterStructures WefirstconsiderthestructureofH1 z 6 4BasicIIRDigitalFilterStructures ThenconsiderH2 z 6 4BasicIIRDigitalFilterStructures ThenconsiderH2 z 6 4BasicIIRDigitalFilterStructures ThencascadethestructuresofH1 z andH2 z weobtainthedirectformIstructureoftheIIRdigitalfilter DirectformI 6 4BasicIIRDigitalFilterStructures ExchangingtheorderofH1 z andH2 z andreducing weobtainthedirectformIIstructureofH z 6 4BasicIIRDigitalFilterStructures ExchangingtheorderofH1 z andH2 z andreducing weobtainthedirectformIIstructureofH z DirectformII 6 4BasicIIRDigitalFilterStructures Thetransposeofthisstructureissketchedinthefollowingfigure directformI结构缺点 需要2N个延迟器 z 1 太多 系数ai bi对滤波器性能的控制不直接 对极 零点的控制难 一个ai bi的改变会影响系统的零点或极点分布 对字长变化敏感 对ai bi的准确度要求严格 易不稳定 阶数高时 上述影响更大 directformII优缺点 优点 延迟线减少一半 为N个 可节省寄存器或存储单元 缺点 同直接型 通常在实际中很少采用上述两种结构实现高阶系统 而是把高阶变成一系列不同组合的低阶系统 一 二阶 来实现 6 4BasicIIRDigitalFilterStructures 6 4 2CascadeRealizationsByexpressingthenumeratoranddenominatorpolynomialsofthetransferfunctionH z asaproductofpolynomialsoflowerdegree adigitalfilterisoftenasacascadeoflow orderfiltersections Forexample 6 4BasicIIRDigitalFilterStructures Thereare9cascaderealizationsobtainedbydifferentpole zeropairingsareshownbelow 6 4BasicIIRDigitalFilterStructures Therearealso9cascaderealizationsobtainedbydifferentorderingofsectionsareshownbelow 6 4BasicIIRDigitalFilterStructures basedonpole zero pairingsandordering Therearealtogetheratotalof36differentcascaderealizationsof 6 4BasicIIRDigitalFilterStructures Usually thepolynomialsarefactoredintoaproductoffirst orderandsecond orderpolynomials Intheabove forafirst orderfactor 6 4BasicIIRDigitalFilterStructures Byfactoringthenumeratorandthedenominatorpolynomials Nowletsconsiderthethird ordertransferfunction 6 4BasicIIRDigitalFilterStructures Letsfirstconsiderthesection1and2 6 4BasicIIRDigitalFilterStructures Onepossiblecascaderealizationisgivenby First ordersection Second ordersection 级联型结构的优缺点 优点 简化实现 用一个二阶节 通过变换系数就可实现整个系统 极 零点可单独控制 调整 调整 可单独调整第对零点 调整 可单独调整第对极点 各二阶节零 极点的搭配可互换位置 优化组合以减小运算误差 可流水线操作 缺点 二阶节电平难控制 电平大易导致溢出 电平小则使信噪比减小 6 4BasicIIRDigitalFilterStructures 6 4 3ParallelRealizationsAnIIRtransferfunctioncanberealizedinaparallelformbymakinguseofthepartial fractionexpansionofthetransferfunction Apartial fractionexpansionofthetransferfunctionintheformofthefollowingequationleadstotheparallelformI 6 4BasicIIRDigitalFilterStructures ParallelformI thedegreeofthenumeratorpolynomialislessthanthedegreeofthedenominatorpolynomial ParallelformI TheparallelrealizationofIIRdigitalfilterconsistsofseveral2 orderbasicsectionswithreal coefficients 6 4BasicIIRDigitalFilterStructures Thetransferfunctionmaybeexpressedintheform ParallelformII ParallelformII thedegreeofthenumeratorpolynomialisequaltothedegreeofthedenominatorpolynomial 6 4BasicIIRDigitalFilterStructures Foreachsection thedirectformIrealizationsmaybe ParallelformI 6 4BasicIIRDigitalFilterStructures Foreachsection thedirectformIIrealizationsmaybe ParallelformII 6 4BasicIIRDigitalFilterStructures Dividingthenumeratoranddenominatorpolynomialsbyz2weobtainthedirectformexpressionofH z Example6 3Wedevelopdifferentrealizationsofthethird orderIIRtransferfunction 6 4BasicIIRDigitalFilterStructures ByfactoringH z wehave DirectformII ThedirectformIIrealizationisshownas 6 4BasicIIRDigitalFilterStructures Cascaderealization Onepossiblecascaderealizationisshownas 6 4BasicIIRDigitalFilterStructures Takingthepartial fractionexpansionofthetransferfunctioninz 1andguaranteeingthecoefficientstobereal ParallelformI 6 4BasicIIRDigitalFilterStructures ThecorrespondingparallelformIIrealizationisshownontheright TheexpansionofH z maybeintheform 并联型结构是用的加法器 乘法器 延时单元基本与级联结构相同 它只能独立的调整各极点的位置 不能单独调整零点的位置 但并联结构的误差比级联结构的运算误差小 6 5RealizationUsingMATLAB AlthoughtherealizationsofFIRandIIRdigitalfilterscanbeunderstoodeasily itisverydifficulttoobtainthecorrespondingexpressionsofeitherthecascaderealizationortheparallelrealization BytheuseofMATLAB thisproblemwillbecomeeasier 6 5RealizationUsingMATLAB TheM filetoobtainthecascaderealization zp2sosUsage sos zp2sos z p k Thestatementgeneratesamatrixsoscontainingthecoefficientsofeachsecond ordersectionoftheequivalenttransferfunctionH z determinedfromitspole zeroform 6 5RealizationUsingMATLAB sosisanL 6matrixoftheform Listhenumberofsections 6 5RealizationUsingMATLAB Inordertoobtainthepole zeromodelofthefilter anotherM filecanbeusedtodothis z p k tf2zp num den Example6 4Wedevelopthesecond orderfactorsofthesixth orderFIRtransferfunction 6 5RealizationUsingMATLAB num input Numeratorcoefficientvector den input Denominatorcoefficientvector z p k tf2zp num den sos zp2sos z p k Runprogram6 1andinputthevectors 50 428 0213 897 426 0931 and 1000000 theresultis sos 50 4000 35 280016 80001 0000001 00000 87500 25001 0000001