7.2 IIR Digital Filter Design_ Bilinear Transation A.ppt

7 0Introduction DigitalfilterdesignistheprocessofderivingthetransferfunctionG z ofthefilter AfterG z hasbeenobtained thenextstepistorealizeitintheformofasuitablefilterstructure Then chooseacertaindigitalhardware bywritingaprogramwhichimplementsthealgorithmtothehardware adigitalfilterhasbeenmade 7 0Introduction Wefirstreviewsomeoftheissuesassociatedwiththefilterdesignproblem i e thefrequencyresponsespecificationoffilters andreviewtheprocessofanalogfilterdesign here wecallananalogfilterastheprototypeanalogfilter Infact anIIRdigitaltransferfunctionisoftenobtainedthroughtheconversionofaprototypeanalogtransferfunction 7 0Introduction FortheFIRdigitalfilterdesign thepopularwayistheapproximationbasedonthewindowfunction FortheIIRdigitalfilterdesign themainwaysarethebilineartransformationandtime domainapproximationmethods Inaddition weshallintroducesomeoftheM filesdesigningIIRandFIRdigitalfilters 一 IIR滤波器的优缺点 IIR数字滤波器的优点 可以利用模拟滤波器设计的结果 而模拟滤波器的设计有大量图表可查 方便简单 IIR数字滤波器的缺点 相位的非线性 将引起频率的色散 若须线性相位 则要采用全通网络进行相位校正 使滤波器设计变得复杂 成本也高 二 FIRDF优点 FIR滤波器在保证幅度特性满足技术要求的同时 很容易做到有严格的线性相位特性 设FIR滤波器单位冲激响应h n 长度为N 其系统函数H z 为 H z 是z 1的N 1次多项式 它在z平面上有N 1个零点 原点z 0是N 1阶重极点 因此 H z 永远稳定 稳定和线性相位特性是FIR滤波器突出的优点 三 为何要设计FIR滤波器 1 语音处理 图象处理以及数据传输要求线性相位 任意幅度 即要求信道具有线性相位特性 而FIR数字滤波器具有严格的线性相位 而且同时可以具有任意的幅度特性 2 另外FIR数字滤波器的单位抽样响应是有限长的 因而滤波器一定是稳定的只要经过一定的延时 任何非因果有限长序列都变成因果的有限序列 3 FIR可以用FFT算法来实现过滤信号 7 1PreliminaryConsiderations TherearetwomajorissuesthatneedtobeansweredbeforeonecandevelopthedigitaltransferfunctionG z Thefirstandforemostissueisthedevelopofareasonablefilterfrequencyresponsespecificationfromtherequirementsoftheoverallsysteminwhichthedigitalfilteristobeemployed 7 1PreliminaryConsiderations ThesecondissueistodeterminewhetheranFIRoranIIRdigitalfilteristobedesigned Inthissection weexaminethesetwoissues Inaddition wereviewthebasicapproachestothedesignofIIRandFIRdigitalfiltersanddeterminationofthefilterordertomeettheprescribedspecifications 7 1PreliminaryConsiderations Therearefourimportantparametersofananalogfilter 1 Themaximumpassbandattenuation pindB 2 Theminimumstopbandattenuation sindB 3 Thepassbandedge p 4 Thestopbandedge s 7 1PreliminaryConsiderations Forthedigitalfilters therearealsofourimportantparameterscorrespondingtotheanalogfilters Peakpassbandripple p 20log10 1 p dBMinimumstopbandattenuation s 20log10 s dB Passbandedge p Stopbandedge s 7 1PreliminaryConsiderations Determinetheircorrespondingpeakripplevalues pand s Example7 1Thepeakpassbandripple pandtheminimumstopbandattenuation sofadigitalfilterare respectively 0 1dBand35dB 7 1PreliminaryConsiderations Solution From p 20log10 1 p dB s 20log10 s dB 7 1PreliminaryConsiderations Thespecificationsforadigitallowpassfiltermaybegivenintermsofitsmagnituderesponse asinthefollowingfigure Normalizedform 一 巴特沃兹逼近 Butterworth 通带内最平坦滤波器 二 切比雪夫逼近 Chebyshev 简要介绍 通带等起伏 等纹波 特性滤波器 纹波系数 n 4 n 5 7 1PreliminaryConsiderations Thepassbandandstopbandedgefrequencies inmostapplications arespecifiedinHz alongwiththesamplingrateofthedigitalfilter Sinceallfilterdesigntechniquesaredevelopedintermsofnormalizedangularfrequencies pand s thespecifiedcriticalfrequenciesneedtobenormalizedbeforeaspecificfilteralgorithmcanbeapplied 7 1PreliminaryConsiderations LetFTdenotethesamplingfrequencyinHz andFpandFsdenotethepassbandandstopbandedgefrequenciesinHz Then 7 1PreliminaryConsiderations Example7 2Letthespecifiedpassbandandstopbandedgefrequenciesofadigitalhighpassfilteroperatingatasamplingrateof25kHzbe7kHzand3kHz respectively Determinethecorrespondingnormalizedangularbandedgefrequencies 7 1PreliminaryConsiderations 7 1 2SelectionoftheFilterTypeThesecondissueofinterestistheselectionofthedigitalfiltertype i e whetheranIIRoranFIRdigitalfilteristobeemployed TheobjectiveofdigitalfilterdesignistodevelopacausaltransferfunctionH z meetingthefrequencyresponsespecifications 7 1PreliminaryConsiderations ForFIRdigitalfilterdesign theFIRtransferfunctionisapolynomialofz 1 ForIIRdigitalfilterdesign theIIRtransferfunctionisarealrationalfunctionofz 1 7 1PreliminaryConsiderations ThereareseveraladvantagesinusinganFIRfilter sinceItcanbedesignedwithexactlinearphaseandthestructureisalwaysstablewithquantizedfiltercoefficients However inmostcases theorderNFIRofanFIRfilterisconsiderablyhigherthantheorderNIIRofanequivalentIIRfiltermeetingthesamemagnitudespecifications 7 1PreliminaryConsiderations Ingeneral theimplementationoftheFIRfilterrequiresapproximatelyNFIRmultiplicationsperoutputsample NFIR 4 Thenumberofmultiplications 5 7 1PreliminaryConsiderations WhereastheIIRfilterrequires2NIIR 1multiplicationsperoutputsample NIIR 3 Thenumberofmultiplications 7 7 1PreliminaryConsiderations Intheformercase iftheFIRfilterisdesignedwithalinearphase thenthenumberofmultiplicationsperoutputsamplereducestoapproximately NFIR 1 2 Ithasbeenshownthatformostpracticalfilterspecifications theratioNFIR NIIRistypicallyoforderoftensormoreand asaresult theIIRfilterusuallyiscomputationallymoreefficient 7 1PreliminaryConsiderations However ifthegroupdelayoftheIIRfilterisequalizedbycascadingitwithanallpassequalizer thenthesavingsincomputationmaynolongerbethatsignificant Inmanyapplications thelinearityofthephaseresponseofthedigitalfilterisnotanissue makingtheIIRfilterpreferablebecauseofthelowercomputationalrequirements 7 1PreliminaryConsiderations 7 1 3BasicapproachestodigitalfilterdesignInthecaseofIIRdigitalfilterdesign themostcommonpracticeistoconvertthedigitalfilterspecificationsintoanaloglowpassfilterspecifications todeterminetheanaloglowpassfiltertransferfunctionHa s meetingthesespecifications andthentotransformitintothedesireddigitalfiltertransferfunctionG z 7 1PreliminaryConsiderations Thisapproachhasbeenwidelyusedformanyreasons Analogapproximationtechniquesarehighlyadvanced Theyusuallyyieldclosedformsolutions Extensivetablesareavailableforanalogfilterdesign Manyapplicationsrequirethedigitalsimulationofanalogfilters 7 1PreliminaryConsiderations Inthesequel wedenoteananalogtransferfunctionas ThedigitaltransferfunctionderivedfromHa s isdenotedby 7 1PreliminaryConsiderations ThebasicideabehindtheconversionofananalogprototypetransferfunctionHa s intoadigitalIIRtransferfunctionG z istoapplyamappingfromthes domaintothez domainsothattheessentialpropertiesoftheanalogfrequencyresponsearepreserved 7 1PreliminaryConsiderations ThusmappingfunctionshouldbesuchthatImaginary j axisinthes planebemappedontotheunitcircleofthez plane Astableanalogtransferfunctionbemappedintoastabledigitaltransferfunction 7 1PreliminaryConsiderations Tothisend themostwidelyusedtransformationisthebilineartransformationdescribedinSection7 2 UnlikeIIRfilterdesign FIRfilterdesignisbasedonadirectapproximationofthespecifiedmagnituderesponse withtheoftenaddedrequirementthatthephasebelinear 7 1PreliminaryConsiderations ThedesignofanFIRfilteroforderNmaybeaccomplishedbyfindingeitherthelength N 1 impulseresponsesamples h n orthe N 1 samplesofitsfrequencyresponseH ej 7 1PreliminaryConsiderations ThreecommonlyusedapproachestoFIRfilterdesign 1 WindowedFourierseriesapproach 2 Frequencysamplingapproach 3 Computer basedoptimizationmethods Impulseinvariancetransformation Thedigitalfilterimpulseresponsetolooksimilartothatofafrequency selectiveanalogfilter Sampleha t atsomesamplingintervalTtoobtainh n h n ha nT Sincez ejwontheunitcircleands j ontheimaginaryaxis wehavethefollowingtransformationfromthes planetothez plane z esT Frequency domainaliasingformula Giventhedigitallowpassfilterspecificationswp ws RpandAs wewanttodetermineH z byfirstdesigninganequivalentanalogfilterandthenmappingitintothedesireddigitalfilter DesignProcedure 1 ChooseTanddeterminetheanalogfrequencies p wp T s ws T2 DesignananalogfilterHa s 3 Usingpartialfractionexpansion expandHa s into4 Nowtransformanalogpoles pk intodigitalpoles epkT toobtainthedigitalfilter S平面上的极点S Si变换到Z平面上是极点 而Ha s 与H Z 中部分分式所对应的系数不变 但要注意 这种Ha s 到H Z 的对应变换关系 只有将Ha s 表达为部分分式形式才成立 三 设计流程的公式推导1 设计步骤 冲激响应不变法设计数字滤波器的思路为 1 先根据要求 设计出中间模拟滤波器系统函数 2 然后经下列变换设计出H z Ha s ha t h n H z 即 Ha s 求ha t L 1 Ha s ha t 抽样 h n ha t t nT Ha nT 会导致频谱中幅度变小 Th n Tha t t nT 把幅度加大 让它频谱幅度一样 H z Z Th n 可见整个过程很复杂 3 设计公式推导 四 模拟滤波器与数字滤波器的变换关系 五 数字滤波器的频率响应 数字滤波器的频率响应 与抽样周期T成反比 当抽样频率很高时 将产生很高的增益 为稳定增益 令h n Tha nT 则 冲激响应不变法 例 I 冲激响应不变法 例 II AdvantagesofImpulseInvarianceMapping Itisastabledesignandthefrequencies andwarelinearlyrelated DisadvantageWeshouldexpectsomealiasingoftheanalogfrequencyresponse andinsomecasesthisaliasingisintolerable Consequently thisdesignmethodisusefulonlywhentheanalogfilterisessentiallyband limitedtoalowpassorbandpassfilterinwhichtherearenooscillationsinthestopband 冲激不变法设计IIRDF的优缺点 1 冲激不变法使得数字滤波器的冲激响应完全模仿模拟滤波器的冲激响应 也就是时域逼近良好 2 模拟频率 和数字频率w之间呈线性关系 w T如 一个线性相位的模拟滤波器 例贝塞尔滤波器 可以映射成一个线性相位的数字滤波器 3 缺点 由于有频率混叠效应 所以冲激响应不变法只适用于限带的模拟滤波器 八 冲激不变法应用的局限性 由于具有频率的混叠效应 所以高通和带阻滤波器不宜采用冲激不变法 因为它们高频部分不衰减 将完全混淆在低频中 从而使整个频响面目全非 若要对高通和带阻实行冲激不变法 则必须先对高通和带阻滤波器加一保护滤波器 滤掉高于折叠频率以上的频带 它会增加设计的复杂性和滤波器的阶数 因而只有在一定要追求频率线性关系或保持网络瞬态响应不变时才使用 对于带通和低通滤波器 需充分限带 若阻带衰减越大 则混叠效应越小 数字滤波器的频率响应 数字滤波器的频率响应 与抽样周期T成反比 当抽样频率很高时 将产生很高的增益 为稳定增益 令h n Tha nT 则 例子1 Ha j w H ejw 由于模拟滤波器不是充分限带 所以数字滤波器产生很大的频谱混叠失真 例子2 设低通DF的3dB带宽频率wc 0 2 止带频率ws 0 4 在w ws处的止带衰减20lg H ejws 15dB 试用脉冲响应不变法 冲激不变法 设计一个Butterworth低通DF 设采样频率fs 20kHz 解 设计分为4步 1 将数字滤波器的设计指标转变为模拟滤波器的设计指标 因为 fs 20kHz 则采样间隔为T 1 fs 1 20kHz 对于冲激不变法 频率变换是线性的 2 设计Ha s 7 2IIRDigitalFilterDesign BilinearTransformationMethod AnumberoftransformationshasbeenproposedtoconvertananalogtransferfunctionHa s intoadigitaltransferfunctionG z sothatessentialpropertiesoftheanalogtransferfunctioninthes domainarepreservedforthedigitaltransferfunctioninthez domain Ofthese thebilineartransformationismorecommonlyused Abovetransformationmapsasinglepointinthes planetoauniquepointinthez planeandvice versa 7 2IIRDigitalFilterDesign BilinearTransformationMethod 7 2 1ThebilineartransformationmethodThebilineartransformationfromthes planetothez planeisgivenby Tisthesamplinginterval 7 2IIRDigitalFilterDesign BilinearTransformationMethod RelationbetweenG z andHa s isthengivenby Asaresult theparameterThasnoeffectonG z andT 2ischosenforconvenience 7 2IIRDigitalFilterDesign BilinearTransformationMethod Digitalfilterdesignconsistsof3steps 1 DevelopthespecificationsofHa s byapplyingtheinversebilineartransformationtospecificationsofG z 2 DesignHa s 3 DetermineG z byapplyingbilineartransformationtoHa s 7 2IIRDigitalFilterDesign BilinearTransformationMethod Mappingofs planeintothez plane 7 2IIRDigitalFilterDesign BilinearTransformationMethod Letz ej withT 2ands j wehave or tan 2 7 2IIRDigitalFilterDesign BilinearTransformationMethod Mappingishighlynonlinearbutone to onemapping Completenegativeimaginaryaxisinthes planefrom to 0ismappedintothelowerhalfoftheunitcircleinthez planefromz 1toz 1 7 2IIRDigitalFilterDesign BilinearTransformationMethod Completepositiveimaginaryaxisinthes planefrom 0to ismappedintotheupperhalfoftheunitcircleinthez planefromz 1toz 1 7 2IIRDigitalFilterDesign BilinearTransformationMethod Nonlinearmappingintroducesadistortioninthefrequencyaxiscalledfrequencywarping Effectofwarpingisshownright 7 2IIRDigitalFilterDesign BilinearTransformationMethod StepsinthedesignofanIIRdigitalfilter 1 Prewarp p s tofindtheiranalogequivalents p s 2 DesigntheanalogfilterHa s meetingthespecifications p sand p s 3 DesignthedigitalfilterG z byapplyingbilineartransformationtoHa s 7 2IIRDigitalFilterDesign BilinearTransformationMethod Transformationcanbeusedonlytodesigndigitalfilterswithprescribedmagnituderesponsewithpiecewiseconstantvalues Transformationdoesnotpreservephaseresponseofanalogfilter 二 性能分析 1 解决了冲激不变法的混叠失真问题 2 它是一种简单的代数关系 只须将上述关系代入AF的Ha s 中 对直接 级联 并联结构都适用 即可求出DF的H z 设计十分方便 4 双线性变换法不适用于设计 1 设计线性相位的DF 2 它要求AF的幅频响应是分段常数型 即幅度变换是线性的 一般低通 高通 带通 带阻型滤波器的频率响应特性都是分段常数 即模拟角频率与数字角频率存在非线性关系 所以双线性变换避免了混叠失真 却又带来了非线性的频率失真 3 由于双线性变换中 5 同时 看出双线性变换 1 在零频附近 模拟角频率与数字角频率变换关系接近线性关系 2 又要求AF的幅频响应是分段常数型 即幅度变换是线性的所以称之为双线性变换 频率升高时 非线性失真严重 6 对于分段常数型AF滤波器 经双线性变换后 仍得到幅频特性为分段常数的DF 但在各个分段边缘的临界频率点产生畸变 这种频率的畸变 可通过频率预畸变加以校正 7 2IIRDigitalFilterDesign BilinearTransformationMethod Example7 7 Consider c 0 5 2 pi example7 7 m 7 2IIRDigitalFilterDesign BilinearTransformationMethod Applyingbilineartransformationtotheabovewegetthetransferfunctionofafirst orderdigitallowpassButterworthfilter Rearrangingtermsweget 7 2IIRDigitalFilterDesign BilinearTransformationMethod example7 7 m c 0 5 2 pi 7 2IIRDigitalFilterDesign BilinearTransformationMethod 7 2 2DesignofDigitalIIRNotchFiltersWeconsidernextthedesignofasecondIIRnotchfilterasanexampleoftheapplicationofthebilineartransformationmethod Asecond orderanalognotchfilterhasatransferfunctiongivenby 7 2IIRDigitalFilterDesign BilinearTransformationMethod Fromthetransferfunction itsmagnituderesponseisgivenby The3 dBnotchbandwidth 2 1isequaltoB 7 2IIRDigitalFilterDesign BilinearTransformationMethod Themagnituderesponseandthezero polediagramareplottedas 7 2IIRDigitalFilterDesign BilinearTransformationMethod ApplyingabilineartransformationtoHa s where 7 2IIRDigitalFilterDesign BilinearTransformationMethod Itisasimpleexercisetoshowthatthenotchfrequency oandthe3 dBnotchbandwidthBwofthedigitalnotchfilterarerelatedtotheconstants and through 7 2IIRDigitalFilterDesign BilinearTransformationMethod Example7 8 Designasecond orderdigitalnotchfilteroperatingatasamplingrateof400Hzwithanotchfrequencyat60Hz 3 dBnotchbandwidthof6Hz 7 2IIRDigitalFilterDesign BilinearTransformationMethod Example7 8 Designasecond orderdigitalnotchfilteroperatingatasamplingrateof400Hzwithanotchfrequencyat60Hz 3 dBnotchbandwidthof6Hz Thenotchfrequency 0 2 60 400 0 3 The3 dBbandwidthBw 2 6 400 0 03 Fromtheabovevaluesweget 0 90993 0 587785 7 2IIRDigitalFilterDesign BilinearTransformationMethod Thegainandphaseresponsesareshownbelow Thus 7 2IIRDigitalFilterDesign BilinearTransformationMethod B 2 pi 6Hz 400Hz o 2pi 60Hz 400Hz FT 400Hz 7 3DesignofLowpassIIRDigitalFilter Example DesignalowpassButterworthdigitalfilterwith p 0 25 s 0 55 p 0 5dB and s 15dB Solution Wefirstdeterminetheparameters andA Thus 2 0 1220185 A2 31 622777If G ej0 1 thisimplies20log10 G ej0 25 0 520log10 G ej0 55 15 7 3DesignofLowpassIIRDigitalFilter SecondwedeterminethecorrespondingspecificationsoftheanaloglowpassButterworthfilter Byprewarpingweget p tan p 2 tan 0 25 2 0 4142136 s tan s 2 tan 0 55 2 1 1708496Thepassbandandthestopbandattenuationsarestill p 0 5dB and s 15dB 7 3DesignofLowpassIIRDigitalFilter Determinetheparameterskandk1andtheorderN Theinversetransitionratiois1 k s p 2 8266809Theinversediscriminationratiois1 k1 A2 1 15 841979ThusN log10 1 k1 log10 1 k 2 6586997ChooseN 3 7 3DesignofLowpassIIRDigitalFilter Todetermine cweuse Ha j p 2 1 1 p c 2N 1 1 2 Wethenget c 1 419915 p 0 5881483rd orderlowpassButterworthtransferfunctionfor c 1isHan s 1 s 1 s2 s 1 Denormalizingtoget c 0 588148wearriveatHa s Han s 0 588148 7 3DesignofLowpassIIRDigitalFilter ApplyingbilineartransformationtoHa s wegetthedesireddigitaltransferfunction MagnitudeandgainresponsesofG z shownbelow 7 6FIRFilterDesignBasedonWindowedFourierSeries Wenowturnourattentiontothedesignofreal coefficientFIRfilters Thesefiltersaredescribedbyatransferfunctionthatisapolynomialinz 1andthereforerequiredifferentapproachesfortheirdesign AdirectandstraightforwardmethodisbasedontruncatingtheFourierseriesrepresentationoftheprescribedfrequencyresponse 7 6FIRFilterDesignBasedonWindowedFourierSeries Thesecondmethodisbasedontheobservationthatforalength NFIRdigitalfilter NdistinctequallyspacedfrequencysamplesofitsfrequencyresponseconstitutetheN pointDFTofitsimpulseresponse andhence theimpulseresponsesequencecanbereadilycomputedbyapplyinganinverseDFTtothesefrequencysamples 7 6FIRFilterDesignBasedonWindowedFourierSeries 7 6 1LeastIntegral SquaredErrorDesignofFIRFiltersLetHd ej denotethedesiredfrequencyresponsefunction SinceHd ej isaperiodicfunctionof withaperiod2 itcanbeexpressedasaFourierseries 7 6FIRFilterDesignBasedonWindowedFourierSeries WheretheFourierseriescoefficients hd n arepreciselythecorrespondingimpulseresponsesamplesandaregivenby Thus givenafrequencyresponsespecificationHd ej wecancomputehd n usingtheaboveequation 7 6FIRFilterDesignBasedonWindowedFourierSeries However formostpracticalapplications thedesiredfrequencyresponseispiecewiseconstantwithsharptransitionsbetweenbands inwhichcase thecorrespondingimpulseresponsesequence hd n isofinfinitelengthandnoncausal 7 6FIRFilterDesignBasedonWindowedFourierSeries Ourobjectiveistofindafinite durationimpulseresponsesequence ht n oflength2M 1whoseDTFTHt ej approximatesthedesiredDTFTHd ej insomesense Onecommonlyusedapproximationcriterionistominimizetheintegral squarederror 7 6FIRFilterDesignBasedonWindowedFourierSeries UsingParseval srelationwehave Thismeansthattheintegral squarederrorisminimumwhenht n hd n for M n M 7 6FIRFilterDesignBasedonWindowedFourierSeries Orinotherword thebestfinite lengthapproximationtotheidealinfinite lengthimpulseresponseinthemean squareerrorsenseissimplyobtainedbytruncation AcausalFIRfilterwithanimpulseresponseh n canbederivedfromht n bydelayingthelattersequencebyMsamples 7 6FIRFilterDesignBasedonWindowedFourierSeries 7 6 2ImpulseResponseofIdealFiltersHere weonlyreviewtheimpulseresponsehLP n oftheideallowpassfilter Assumethatthereisanideallowpassfilterwithzero phase itsfrequencyresponseisdepictedbelow Itsimpulseresponseisgivenby 7 6FIRFilterDesignBasedonWindowedFourierSeries TheplotoftheimpulseresponsehLP n isdepictedinthefollowingfigure ItcanbeseenthathLP n isnoncausalandofinfinite lengthsothatitcannotberealized 7 6FIRFilterDesignBasedonWindowedFourierSeries Note Ourobjectiveistofinda2M 1 lengthimpulseresponseh n definedintherangeof0 n 2M Tothisend shifthLP n totherightbyMpointsandtruncatehLP n byalength2M 1rectangularwindowfunctionw n definedby 7 6FIRFilterDesignBasedonWindowedFourierSeries Now wefirstshifthLP n totherightbyMpoints thentheimpulseresponsehLP n comesto Suchanideallowpassfilterwillnolongerhavezero phasebuthavealinearphase 7 6FIRFilterDesignBasedonWindowedFourierSeries TheplotoftheshiftedimpulseresponseforM 10 7 6FIRFilterDesignBasedonWindowedFourierSeries Thentruncatetheshiftedimpulseresponsebytherectangularwindowfunction