Unit5DigitalSignalProcessing5.1IntroductiontoDigitalSignalProcessing

5.2StructuresforDiscrete-TimeSystem

5.3FilterDesignTechniques5.1IntroductiontoDigitalSignalProcessingWordsandExpressionsBackgroundsTextWordsandExpressionsMoore’sLaw 摩爾定律GordonEarleMoore(January3,1929–March24,2023)wasanAmericanbusinessman,engineer,andthe

co-founder

andemerituschairmanof

IntelCorporation.Heproposed

Moore'slaw

whichmakestheobservationthatthe

numberoftransistors

inan

integratedcircuit

(IC)doublesabouteverytwoyears.WordsandExpressions5Moore’sLaw–NowandintheFutureMoore’sLawhasalwaysbeenaboutinnovationandwecanconfidentlysayinnovationcontinuesunabated.By

Dr.AnnKelleherExecutiveVicePresidentGeneralManagerofTechnologyDevelopmentWordsandExpressions6microprocessor n.微處理器transformation n.變換FourierTransformLaplaceTransformZTransformAmodern

64bit

x86-64

processor(AMDRyzen52600,Basedon

Zen+,2017)WordsandExpressionsconvolution n.卷積parameter n.參數preprocessing n.預處理7Discrete2DConvolutionAnimationconvolutionkernelWordsandExpressionsmodulation n.調制8CategorizationforsignalmodulationbasedondataandcarriertypesAlow-frequencymessagesignal(top)maybecarriedbyanAMorFMradiowave.WordsandExpressions9signalconditioning信號調理Signalconditioningisanelectroniccircuitthatmanipulatesasignalinawaythatpreparesitforthenextstageofprocessing.Signalconditionerisoneofthefundamentalbuildingblocksofmoderndataacquisition(akaDASorDAQsystem).compression n.壓縮compressionstandards:JPEG\MPEG\H.264WordsandExpressions10transmission n.傳輸communicationchannel通信信道A

communicationchannel

referseithertoaphysical

transmissionmedium

suchasawire,ortoa

logicalconnection

overa

multiplexed

mediumsuchasaradiochannelin

telecommunications

and

computernetworking.A

TOSLINK

opticalfibercablewithaclearjacket.Thesecablesareusedmainlyfordigitalaudioconnectionsbetweendevices.WordsandExpressions11postprocessing n.后處理receiver n.接收機facsimile n.傳真；復制本analog adj.模擬的chip n.芯片WordsandExpressionsanalogtodigital(A/D)conversion 模數轉換digitaltoanalog(D/A)conversion 數模轉換12An8-levelADCcodingscheme8-channel

CirrusLogic

CS4382digital-to-analogconverterasusedina

soundcard.WordsandExpressionsfastFouriertransform(FFT)algorithm 快速傅里葉變換multiratetechnique 多采樣率技術polyphasefilterimplementation 多相位濾波器實現13AnexampleFFTalgorithmstructure,usingadecompositionintohalf-sizeFFTsIn

digitalsignalprocessing,

upsampling,

expansion,and

interpolation（內插）

aretermsassociatedwiththeprocessof

resampling

ina

multiratedigitalsignalprocessing

system.Downsampling,

compression,and

decimation（抽取）

aretermsassociatedwiththeprocessof

resampling

ina

multi-ratedigitalsignalprocessing

system.

WordsandExpressionswavelet n.小波expansion n.展開sequence n.序列14MorletandMexicanhatAnexampleofthe2D

discretewavelettransform

thatisusedin

JPEG2000.WordsandExpressionsquantization n.量化biomedicalengineering生物醫學工程Biomedicalengineering

(BME)or

medicalengineering

istheapplicationofengineeringprinciplesanddesignconceptstomedicineandbiologyforhealthcarepurposes(e.g.,diagnosticortherapeutic).153-bitresolutionwitheightlevels.WordsandExpressionsone-dimensional adj.一維的multidimensional adj.多維的16A2-Dfilter(left)definedbyits1-Dprototypefunction(right)andaMcClellantransformation.WordsandExpressions17telecommunication n.電信microelectronictechnology微電子技術asubfieldofelectronicthestudyandmanufactureofverysmallelectronicdesignsandcomponentsBackgrounds18DigitalSignalProcessing/courses/res-6-008-digital-signal-processing-spring-2011/TextOutlineBasicsofSignalProcessingDiscrete-timeSignalProcessingMultidimensionalSignalProcessingDevelopmentsofSignalProcessing19BasicsofSignalProcessingSignalprocessingapplicationsspananimmensesetofdisciplinesthatincludeentertainment,communications,spaceexploration,medicine,archeology,geophysics,justtonameafew.信號處理應用涵蓋了一系列學科，包括娛樂、通信、太空探索、醫學、考古學、地球物理學等等。20BasicsofSignalProcessingBysomeestimates,evenwithimpendinglimitationsonMoore’sLaw,theprocessingcapabilityofbothspecial-purposesignalprocessingmicroprocessorsandpersonalcomputersislikelytoincreasebyseveralordersofmagnitude.據估計，即使摩爾定律即將受到限制，專用信號處理微處理器和個人計算機的處理能力也可能增加幾個數量級。21BasicsofSignalProcessingSignalprocessingdealswiththerepresentation,transformation,andmanipulationofsignalsandtheinformationthesignalscontain.信號處理關注的是信號及其所包含信息的表示、變換和處理。22BasicsofSignalProcessingIncommunicationsystem,itisgenerallynecessarytodopreprocessingsuchasmodulation,signalconditioning,andcompressionpriortotransmissionoveracommunicationschannel,andthentocarryoutpostprocessingatthereceivertorecoverafacsimileoftheoriginalsignal.在通信系統中，信號在一條信道上傳輸之前一般要做一些像調制、信號調理和壓縮等這樣的預處理，然后在接收機處進行后處理以恢復原始信號。23BasicsofSignalProcessingThesedevelopmentsintechnologyhavebeenreinforcedbymanyimportanttheoreticaldevelopments,suchasthefastFouriertransform(FFT)algorithm,parametricsignalmodeling,multiratetechniques,polyphasefilterimplementation,andnewwaysofrepresentingsignals,suchaswaveletexpansions.許多重要的理論發展加強了技術的發展，如快速傅立葉變換（FFT）算法、參數信號建模、多速率技術、多相濾波器實現，以及表示信號的新方法，如小波展開。24Discrete-timeSignalProcessingDiscrete-timesignalprocessingisbasedonprocessingofnumericsequencesindexedonintegervariablesratherthanfunctionsofacontinuousindependentvariable.離散時間信號處理是基于對以整數變量為索引的數字序列的處理，而不是連續自變量的函數。25Discrete-timeSignalProcessingAscomputerspeedshaveincreased,discrete-timeprocessingofcontinuous-timesignalsinrealtimehasbecomecommonplaceincommunicationsystems,radarandsonar,speechandvideocodingandenhancement,biomedicalengineering,andmanyotherareasofapplications.隨著計算機速度的提高，在通信系統、雷達和聲納、語音和視頻編碼與增強、生物醫學工程及許多其他應用領域，對連續時間信號進行實時離散時間處理已變得司空見慣。26MultidimensionalSignalProcessingManyimageprocessingapplicationsrequiretheuseoftwo-dimensionalsignalprocessingtechniques.Thisisthecaseinsuchareasasvideocoding,medicalimaging,enhancementandanalysisofaerialphotographs,analysisofsatelliteweatherphotos,andenhancementofvideotransmissionsfromlunaranddeep-spaceprobes.許多圖像處理應用需要使用二維信號處理技術。視頻編碼、醫學成像、航空圖像的增強和分析、衛星天氣圖像的分析以及月球和深空探測器視頻傳輸的增強等領域都是如此。27DevelopmentsofSignalProcessingAnotableexampleisintheareaoftelecommunications,wherediscrete-timesignalprocessingtechniques,microelectronictechnology,andfiberoptictransmissionhavecombinedtochangethenatureofcommunicationsystemsintrulyrevolutionaryways.電信領域就是一個明顯的例子，在該領域中，離散時間信號處理技術、微電子技術和光纖傳輸技術的結合，正以真正革命性的方式改變著通信系統的面貌。285.2StructuresforDiscrete-TimeSystemWordsandExpressionsBackgroundsTextWordsandExpressions30rational adj.有理的(eg.rationalfraction)Arationalfunctionisafunctionthatcanbewrittenasaquotientoftwopolynomialfunctions.Insymbols,thefunction: iscalledarationalfunction.systemfunction 系統函數transferfunction 傳遞函數WordsandExpressionsinterconnection n.互連constant n.常數delay n.延遲InterconnectionofLTIsystemsWordsandExpressionsformula n.公式expression 表達式equation 等式inequality 不等式LinearconstantcoefficientdifferenceequationWordsandExpressionsrecursive adj.遞歸的Incomputerprogramming,thetermrecursivedescribesafunctionormethodthatrepeatedlycalculatesasmallerpartofitselftoarriveatthefinalresult.Itissimilartoiteration,butinsteadofrepeatingasetofoperations,arecursivefunctionaccomplishesrepetitionbyreferringtoitselfinitsowndefinition.WordsandExpressionsinitial-restcondition 初始松弛條件Thedefinitionofinitialrestconditionforasystemingeneralisthatifx(t)=0fort<t0theny(t)=0fort<t0.

Fordiscretetimesystems,itbasicallysaysthat

theinputiszerouptoacertainpoint(orx[n]=0forn<n0),thentheoutputshouldbezerouptothesamepoint(ory[n]=0forn<n0).recurrence n.重現WordsandExpressionsblockdiagram 方框圖cascade n.級聯canonicformimplementation 規范形式實現Cascadestructurefora6th–ordersystemwithadirectformIIrealizationofeach2nd-ordersubsystemCanonicformimplementationofNth–orderLTIsystemBackgrounds36UCBerkeleyEE123.DigitalSignalProcessing/Courses/EE123/TextOutline37Representationsfordiscrete-timesystemDifferenceEquationImplementationBasicElementsBlockDiagramsRepresentationsfordiscrete-timesystemAnLTIsystemwitharationalsystemfunctionhasthepropertythattheinputandoutputsequencessatisfyalinearconstant-coefficientdifferenceequation.具有有理系統函數的線性時不變系統有這樣的性質，即其輸入和輸出序列滿足線性常系數差分方程。38

39Representationsfordiscrete-timesystemSystemsdescribedbylinearconstant-coefficientdifferenceequationscanberepresentedbystructuresconsistingofaninterconnectionofthebasicoperationsofaddition,multiplicationbyaconstant,anddelay,theexactimplementationofwhichisdictatedbythetechnologytobeused.由線性常系數差分方程描述的系統能夠用由加法、常數乘法和延遲的基本運算互連而成的結構來表示，它的真正實現則取決于所用的技術。40Representationsfordiscrete-timesystemBasicElementsThebasicelementsrequiredfortheimplementationofanLTIdiscrete-timesystemsareadders,multipliers,andmemoryforstoringdelayedsequencevaluesandcoefficients.實現線性時不變離散時間系統所需的基本單元是加法器、乘法器和存儲延遲序列值和系數的存貯器。41BasicElements42Figure5.1Blockdiagramsymbols.Additionoftwosequences.Multiplicationofasequencebyaconstant.Unitdelay.BlockDiagram43

Ablockdiagramcanberearrangedormodifiedinavarietyofwayswithoutchangingtheoverallsystemfunction.Eachappropriaterearrangementrepresentsadifferentcomputationalalgorithmforimplementingthesamesystem.在不改變整個系統函數的情況下，可以各種方式重新排列或修改框圖。每個適當的重排表示用于實現相同系統的不同計算算法。44BlockDiagramSinceeachofthetwosystemsisanLTIsystem(assuminginitial-restconditionsforthedelayregisters),theorderinwhichthetwosystemsarecascadedcanbereversed,asshowninFigure5.3,withoutaffectingtheoverallsystemfunction.由于兩個系統都是線性時不變系統（假設延遲寄存器初始松弛），那么兩個系統在級聯中的次序就可以交換成如圖5.3所示，而不會影響總的系統函數。45BlockDiagram46Figure5.3Rearrangementofblockdiagramofmiddlefigure.BlockDiagramConsequently,thetwocanbecollapsedintoonechain,asindicatedinFigure5.4.Animplementationwiththeminimumnumberofdelayelementsiscommonlyreferredtoasacanonicformimplementation.因此，如圖5.4所示，兩者可以折疊成一條鏈。具有最小數量延遲單元的實現通常被稱為規范形式實現。47BlockDiagramBlockDiagram48Figure5.4ThedirectformIIorcanonicdirectformimplementation.5.3FilterDesignTechniquesWordsandExpressionsBackgroundsTextWordsandExpressionsbilineartransformation 雙線性變換法Thebilineartransformisaspecialcaseofaconformalmapping,oftenusedtoconvertatransferfunction

Ha(s)ofalinear,time-invariant(LTI)filterinthecontinuous-timedomain(oftencalledananalogfilter)toatransferfunctionHd(z)ofalinear,shift-invariantfilterinthediscrete-timedomain.

51WordsandExpressionsaliasing n.混疊WordsandExpressionspassband n.通帶stopband n.阻帶transition n.過渡帶tolerance n.容限causal adj.因果的Lowpassfiltertolerancescheme.Inordertobeimplementable,anytime-dependentfilter(operatinginrealtime)mustbe

causal:thefilterresponseonlydependsonthecurrentandpastinputs.Backgrounds53CMUDigitalSignalProcessing/~ee791/TextOutline54TypesofDigitalFiltersInfiniteImpulseResponse(IIR)filterFiniteImpulseResponse(FIR)filterFilterDesignTechniquesIIR:impulseinvariance,bilineartransformationFIR:windowfunction,optimumapproximationEconomicConcernsTypesofDigitalFiltersFiltersareaparticularlyimportantclassofLTIsystems.Strictlyspeaking,thetermfrequency-selectivefiltersuggestsasystemthatpassescertainfrequencycomponentsofaninputsignalandtotallyrejectsallothers,butinabroadercontext,anysystemthatmodifiescertainfrequenciesrelativetoothersisallcalledfilter.濾波器是LTI系統中一類特別重要的系統。嚴格來說，選頻濾波器指的是讓輸入信號的某些頻率分量通過并完全抑制所有其他頻率分量的系統，但在更廣泛的定義下，任何相對于其他頻率分量修改特定頻率分量的系統都被稱為濾波器。55TypesofDigitalFiltersThedesignofdiscrete-timefilterscorrespondstodeterminingtheparametersofatransferfunctionordifferenceequationthatapproximatesadesiredimpulseresponseorfrequencyresponsewithinspecifiedtolerances.離散時間濾波器的設計即確定其傳遞函數或差分方程的參數，由此得到近似于規定容限內的期望系統的脈沖響應或頻率響應。56TypesofDigitalFiltersDiscrete-timesystemsimplementedwithdifferenceequationsfallintotwobasiccategories:infiniteimpulseresponse(IIR)systemsandfiniteimpulseresponse(FIR)systems.用差分方程實現的離散時間系統分為兩類：無限脈沖響應（IIR，也譯為無限沖激響應）系統和有限脈沖響應（FIR，也譯為有限沖激響應）系統。57TypesofDigitalFiltersDesigningIIRfiltersimpliesobtaininganapproximatingtransferfunctionthatisrationalfunctionofz,whereasdesigningFIRfiltersimpliespolynomialapproximation.設計IIR濾波器意味著獲得近似的傳遞函數，該傳遞函數是z變量的有理函數，而設計FIR濾波器則意味著多項式逼近。58FilterdesigntechniquesWhendiscrete-timefiltersfirstcameintocommonuse,theirdesignwerebasedonmappingwell-formulatedandwell-understoodcontinuous-timefilterdesignstodiscrete-timedesignsthroughtechniquessuchasimpulseinvarianceandthebilineartransformation.當離散時間濾波器首次普遍使用時，它們的設計是通過脈沖（沖激）響應不變法和雙線性變換法等技術將精心設計且易于理解的連續時間濾波器設計映射到離散時間濾波器的設計。59FilterdesigntechniquesTheuseofthisproceduremaybemotivatedbyadesiretomaintaintheshapeoftheimpulseresponseorbytheknowledgethatifthecontinuous-timefilterisbandlimited,consequentlythediscrete-timefilterfrequencyresponsewillcloselyapproximatethecontinuous-timefrequencyresponse.使用本方法的出發點可能是希望保持沖激響應的形狀，或者若已知連續時間濾波器是帶限的，那么離散時間濾波器的頻率響應將非常接近連續時間濾波器的頻率響應。60Filterdesigntechniques

61FilterdesigntechniquesThedesignoffiltersinvolvesthefollowingstages,thespecificationofthedesiredpropertiesofthesystem,theapproximationofthespecificationsusingacausaldiscrete-timesystem,andtherealizationofthesystem.濾波器的設計涉及以下步驟：給定系統所要求特性的技術指標、用因果離散時間系統逼近這些技術指標及實現該系統。62Filterdesigntechniques63Figure5.5Lowpassfiltertolerancescheme.FilterdesigntechniquesManyofthefiltersusinginpracticearespecifiedbyatoleranceschemewithnoconstraintsonthephaseresponse

otherthanthoseimposedimplicitlybyrequirementsofstabilityandcausality.在實踐中使用的許多濾波器是由容限方案指定的，除了穩定性和因果關系要求隱含施加的約束之外，對相位響應沒有任何約束。64FilterdesigntechniquesForthedesignofFIRfilters,windowfunctionmethodandoptimumapproximationmethodareapplied.ThechoicebetweenanFIRfilterandanIIRfilterdependsontheimportancetothedesignproblemoftheadvantagesofeachtype.IIRfilters,forexample,havetheadvantagethatavarietyoffrequency-selectivefilterscanbedesignedusingclosed-formdesignformulas.FIR濾波器的設計采用了窗函數法和最優逼近法。FIR濾波器和IIR濾波器之間的選擇取決于每種類型的優點對設計問題的重要性。例如，IIR濾波器的優點是，可以使用閉合形式的設計公式來設計各種頻率選擇性濾波器。65FilterdesigntechniquesAlthoughthewindowmethodisstraightforwardtoapply,someiterationmaybenecessarytomeetaprescribedspecification.TheParks-McClellanalgorithmleadstolowerorderfiltersthanthewindowmethoddoes,andbothmethodscanbeimplementedonapersonalcomputeroraworkstation.盡管窗函數法應用起來很簡單，但為滿足規定的技術指標，可能需要進行一些迭代。Parks-McClellan算法比窗函數法設計的濾波器階次低，并且這兩種方法都可以在個人計算機或工作站上實現。66EconomicConcernsClearly,thefinalchoicewillmostoftenbemadebyengineeringjudgementonsuchquestionsastheformulationofthespecifications,themethodofimplementationofthefilter,andthecomputationalfacilitiesandsof