离散时间信号处理

离散时间信号处理离散时间信号处理第1页Introduction离散时间信号处理第2页3DigitalSignalProcessing(DSP)isusedinawidevarietyofapplications.Telephone&telegramradarAudiosignalprocessingMultimediasystemImageprocessingMobiletelephoneCommunicationsystemdigitalTVIntroduction离散时间信号处理第3页4IntroductionSignalsAsignalcanbedefinedasafunctionthatconveysinformation.Signalsarepresentedmathematicallyasfunctionsofoneormoreindependentvariables.forexample:aspeechsignalwouldberepresentedmathematicallyasafunctionofonetimevariable------f(t);-----One-dimensional(1-D)signal一维信号apicturewouldberepresentedmathematicallyasabrightnessfunctionoftwospatialvariables------f(x,y).------Two-dimensional(2-D)signal二维信号acolorvideosignal(aRGBtelevisionsignal)isa3-Dsignal.

-----Multidimensional(M-D)signal多维信号离散时间信号处理第4页5IntroductionWhatisDigitalSignalProcessing?DigitalSignalProcessingisthesciencetoprocesssignalsbydigitalmeans.

Thisincludesawidevarietyofgoals:filtering,transformation,recognition,enhancement,compression,andmuchmore.DSPisoneofthemostpowerfultechnologiesthatwillshapescienceandengineeringinthetwenty-firstcentury.Supposeweattachananalog-to-digitalconvertertoacomputer,andthenuseittoacquireachunkofrealworlddata.离散时间信号处理第5页6Introduction简单说，数字信号处理是利用计算机或专用处理设备，以数值计算方法对信号进行采集、变换、综合、估值与识别等加工处理，借以到达提取信息和便于应用目标。离散时间信号处理第6页7IntroductionDigitalsignalprocessingincludestwomeanings:Processinganalogsignalsinadigitalway.Processingdigitalsignals.Advantages:Highreliability（可靠性）Highagility(灵活性好，易于实现系统性能）Highprecision（高精度）Lowcost（成本低）…离散时间信号处理第7页8IntroductionMaintools:Discrete-timesignalrepresentations.Discretetransformsandtheirfastalgorithms(ztransform,DFT&FFT).Designandimplementationofdigitalfilter(IIR（无限长单位冲激响应）&FIR（有限长单位冲激响应）filter).Multiratesystems（多率系统）,filterbanks（滤波器组）,andwavelets（小波）.Implementationofdigitalsignalprocessingsystems.离散时间信号处理第8页9References程佩青，数字信号处理教程，清华大学出版社胡广书，数字信号处理——理论、算法与实现，清华大学出版社高西全，丁玉美，阔永红，数字信号处理——原理、实现及应用，电子工业出版社…离散时间信号处理第9页Chapter2Discrete-TimeSignalsandSystem离散时间信号处理第10页112.1Discrete-TimeSignals:SequencesTheindependentvariableofasignalmaybeeithercontinuousordiscrete.Continuous-timesignalsarethosethataredefinedatcontinuoustimes.Discrete-timesignalsarethosethataredefinedatdiscretetimes.tAmplitudetAmplitudeContinue-timesignalDiscrete-timesignal离散时间信号处理第11页122.1Discrete-timesignals—notationsAdiscrete-timesignalcanberepresentedas

whereTistimeintervalbetweensamples.Eachsampleofsequencex(nT)isdeterminedbytheamplitudeofsignalatinstantnT.Forexample

whereTis0.03.Anothernotationisasequenceofnumbers.Forexample,thesequencexcanberepresentedas

whereZisthesetofintegernumbers（整数集）,andx(n)isreferredtoasthe“nthsample”ofthesequence.ForexampleAconvenientnotationforthesequencexjustisx(n).离散时间信号处理第12页132.1Discrete-timesignals—graphDiscrete-timesignalsareoftendepictedgraphically.x(n)orx(nT)nornT离散时间信号处理第13页14unitsamplesequence（单位抽样序列）Thedefinitionoftheunitimpulseδ(n)n01离散时间信号处理第14页15delayedunitsamplesequence（延时单位抽样序列）Thedefinitionofthedelayedunitsamplesequenceδ(n–m)n01m离散时间信号处理第15页16unitstepsequence

（单位阶跃序列）Thedefinitionoftheunitstepu(n)n01——u(n)后向差分离散时间信号处理第16页17cosinefunctionThedefinitionofthecosinefunctionis,whoseangularfrequency（角频率）isωrad/sample.n0离散时间信号处理第17页18ExponentialSequence

（实指数序列）Thedefinitionoftherealexponentialfunctionis

n0αn离散时间信号处理第18页19unitramp（单位斜坡序列）Thedefinitionoftheunitrampr(n)n0离散时间信号处理第19页202.1Discrete-timesignalsAnarbitrarysequencecanbeexpressedasasumofscaled,delayedunitimpulses.Theunitstepu(n)canbeexpressedasAndtheunitrampr(n)canbeexpressedas离散时间信号处理第20页21Example:generatethesignalwithimpulsesequence-3-2-1012345x(n)nn00n0n离散时间信号处理第21页22PeriodicsequenceAsequencex(n)isdefinedtobeperiodicifandonlyifthereisaninteger

N≠0suchthatx(n)=x(n

+

N)foralln.Insuchacase,Niscalledtheperiodofthesequence.Note,notalldiscretecosinefunctionsareperiodic.If2π/ωisaninteger（整数）

orarationalnumber（有理数),thissequencewillbeperiodic;If2π/ωisanirrationalnumber（无理数）,thiscosinefunctionwillnotbeperiodicatall.离散时间信号处理第22页23ExampleDeterminewhetherfollowingdiscretesignalisperiodicornot.Ifitis,calculatetheperiodofthesignal.Solution:ifthesignalisperiodic,thenwehaveThensosystemisperiodic,itsperiodis40(whenk=17).离散时间信号处理第23页242.2Discrete-timesystemsDefinition:

Asystemisdefinedmathematicallyasauniquetransformationoroperatorthatmapsaninputsequencex(n)intoanoutputsequencey(n).Thiscanbedenotedas

y(n)=T{x(n)} whereT{•}expressesadiscrete-timesystem.Discrete-timesystemy(n)x(n)ExcitationResponseT{•}离散时间信号处理第24页252.2.1MemorylessSystemsAsystemisreferredtoasmemorylessiftheoutputy(n)ateveryvalueofndependsonlyontheinputx(n)atthesamevalueofn.离散时间信号处理第25页262.2.2LinearitySystemsLinearity（线性）

Ify1(n)andy2(n)aretheresponseswhenx1(n)andx2(n)aretheinputsrespectively,thenasystemislinearifandonlyif

T{a

x(n)}=a

T{x(n)}andT{

x1(n)+

x2(n)}=T{x1(n)}+T{x2(n)} foranyconstantsaandb.

Example:y(n)=T{x(n)}=3x(n)+4

T{a

x(n)}=3a

x(n)+4

aT{x(n)}=3a

x(n)+4aT{a

x(n)}≠aT{x(n)}

Soitisnotalinearitysystem.离散时间信号处理第26页272.2.3Time-InvariantSystemsTimeinvariance（时不变性）

Adiscrete-timesystemistimeinvariantifandonlyif,foranyinputsequencex

(n)andintegern0,thenT{

x(n-n0)}=y(n-n0)withy

(n)=T{

x(n)}. Note:anothernameoftimeinvarianceisshiftinvariance（移不变性）.

Example:

y(n)=3x(n)+4T[x(n-n0)]=3x[(n-n0)]+4y(n-n0)=3x[(n-n0)]+4y(n-n0)=T[x(n-n0)]Sothissystemistimeinvariance.离散时间信号处理第27页282.2.4CausalityCausality（因果性）

Adiscrete-timesystemiscausalifandonlyif,whenx1(n)=x2(n)forn<n0,then

T{x1(n)}=T{x2(n)},forn<n0Acausalsystemisoneforwhichtheoutputatinstantndoesnotdependonanyinputoccurringafter

n.Usually,inthecaseofadiscrete-timesignal,anoncausalsystemisnotimplementableinrealtime.However,insomecasesadiscretesignaldoesnotconsistoftimesamples,anoncausalsystemcanbeeasilyimplemented.离散时间信号处理第28页292.3LinearTime-InvariantSystemAninputsequencex(n)canbeexpressedasasumofscaled,shiftedunitimpulses.Theoutputcanbeexpressedas离散时间信号处理第29页30ImpulseresponsesandconvolutionsumsIfthesystemislinear,wecanobtainSincex(k)intheaboveequationisjustaconstant（常数）,theoutputis wherewedefine离散时间信号处理第30页31Impulseresponsesandconvolutionsumsh(n)=T{δ(n)}isreferredtoastheimpulseresponse（冲激响应）ofthesystem.Theequation iscalledaconvolutionsumoradiscrete-timeconvolution（卷积和，又称离散卷积和、线性卷积和）.

Thisequationindicatesthatalineartime-invariantsystemiscompletelycharacterizedbyitsunitimpulseresponseh(n).(1.37)离散时间信号处理第31页32ImpulseresponsesandconvolutionsumsTheconvolutionsumcanalsobewrittenasAshorthandnotationfortheconvolutionis

and

where“*”representstheconvolutionsum.离散时间信号处理第32页33Example Computethelinearconvolutiony(n)=x(n)*h(n),and Solution:离散时间信号处理第33页34Cont.离散时间信号处理第34页35Cont. SolutionII:离散时间信号处理第35页362.4PropertiesofLinearTime-InvariantSystemSupposenowthattherearetwolineartime-invariantsystemwhichareincascade.Thatistosay,theoutputofasystemwithimpulseresponseh1(n)istheexcitationforasystemwithimpulseresponseh2(n).h1(n)h2(n)y(n)x(n)离散时间信号处理第36页37Systemsincascade Theoutputofthefirstsystemh1(n)is

Andtheoutputofthesecondsystemh2(n)is离散时间信号处理第37页38Systemsincascade Byexchangethesummingsequence,wegetThenwecanexpressitsoutputas离散时间信号处理第38页39SystemsincascadeConclusion:Twolineartime-invariantsystemsincascadeformalineartime-invariantsystemwithanimpulseresponsewhichistheconvolutionsumofthetwoimpulseresponses.h(n)=h1(n)*h2(n)y(t)x(n)h1(n)h2(n)y(n)x(n)离散时间信号处理第39页40*Discrete-timesystems—stability（稳定性）Asystemisreferredtoasbounded-inputbounded-output(BIBO)stableif,foreveryinputlimitedinamplitude,theoutputsignalisalsolimitedinamplitude.（有界输入产生有界输出）Ifx(n)isbounded,i.e.,|x(n)|≤

xmax

<∞foralln,thenLinearshiftinvariantsystemsarestableifandonlyif（充要条件）离散时间信号处理第40页41ExampleCharacterizefollowingsystemasbeingeitherlinearornonlinear,timeinvariantortimevarying,causalornoncausal,stableornotstable.Solution:For

Soitisnotalinearsystem.离散时间信号处理第41页42Cont.Forsoitistimevarying;Becausei.e.theoutputforacertaintimet=nofthissystemnotonlydependsontheinputattimet=n,

butalsodependsonthetimeaftern(i.e.t=n+2).Sothesystemisnoncausal;离散时间信号处理第42页43Cont.foraspecialboundedinputwehavethentheoutputforsystemi.e.system’soutputisunbounded.Sothesystemisunstable.Thereforethesystemisnonlinear,timeinvariant,noncausalandunstable.

离散时间信号处理第43页442.5LinearConstant-CoefficientDifferenceEquationsTheinputx(n)andtheoutputy(n)ofasystemdescribedbyalineardifferenceequation（线性差分方程）aregenerallyrelatedby线性：指方程中各y(n-i)和x(n-l)项都只有一次幂，且不存在它们相乘项。输出序列y(n)变量序号最高值与最低值之差N称为差分方程阶数。SystemtypeDescriptionofthesystemthecontinuous-timesystemthedifferentialequation（微分方程）thediscrete-timesystemthedifferenceequation（差分方程）(2.1)离散时间信号处理第44页45IIRandFIRfiltersEquation(2.1)canberewritten,withoutlossofgenerality,consideringthata0=1,yieldingSotheoutputy(n)isdependentbothonsamplesoftheinput

x(n),x(n-1),…,x(n-M),andonprevioussamplesoftheoutput

y(n-1),y(n-2),…,y(n-N).(2.2)离散时间信号处理第45页46IIRandFIRfiltersSinceinordertocomputetheoutput,weneedthepastsamplesoftheoutputitself,wesaythatthesystemisrecursive（递归）.Whena1=a2=…=aN=0,thentheoutputatsamplendependsonlyonvaluesoftheinputsignal.Insuchcase,thesystemiscallednonrecursive（非递归）.Itis(2.3)离散时间信号处理第46页47IIRandFIRfiltersIfwecomparetheaboveequationwithequationweseethatthesysteminequation(2.3)hasafinite-durationimpulseresponse.Suchdiscrete-timesystemareoftenreferredtoasfinite-durationimpulse-response(FIR)

（有限长单位冲激响应）filters.Incontrast,wheny(n)dependsonitspastvalues,asshowninequation(2.2),theimpulseresponseofthesystemmightnotbezerowhenn→∞.Therefore,recursivedigitalsystemareoftenreferredtoasinfinite-durationimpulse-response(IIR)（无限长单位冲激响应）filters.离散时间信号处理第47页48ReviewAsequencex(n)isdefinedtobeperiodicifandonlyifthereisaninteger

N≠0suchthatx(n)=x(n

+

N)foralln.Insuchacase,Niscalledtheperiodofthesequence.Note,notalldiscretecosinefunctionsareperiodic.If2π/ωisaninteger（整数）

orarationalnumber（有理数),thissequencewillbeperiodic;If2π/ωisanirrationalnumber（无理数）,thiscosinefunctionwillnotbeperiodicatall.离散时间信号处理第48页49ReviewThecharacteristicsofthediscrete-timesystemy(n)=H{x(n)}:Linearity:Ify1(n)=H{x1(n)},y2(n)=H{x2(n)},thenH{ax(n)}=aH{x(n)}and

H{x1(n)+x2(n)}=H{x1(n)}+H{x2(n)}foranyconstantsaandb.timeinvariance:

Ify