差分方程与滤波.ppt

Chapter3 4 Discrete TimeSignalsandSystems MainContents ImportanttypesofsignalsandtheiroperationsLinearandshift invariantsystemEasiertoanalyzeandimplementTheconvolutionanddifferenceequationrepresentationsRepresentationsandimplementationofsignalandsystemsusingMATLAB Discrete timesignals Analoganddiscretesignalsanalogsignaltrepresentsanyphysicalquantity timeinsec Discretesignal discrete timesignalNisintegervalued representsdiscreteinstancesintimes Discrete timesignal InMatlab afinite durationsequencerepresentationrequirestwovectors andeachforxandn Example Question whetherornotanarbitraryinfinite durationsequencecanberepresentedinMATLAB Typesofsequences Elementarysequenceforanalysispurposes1 UnitsamplesequenceRepresentationinMATLAB Function x n impseq n0 n1 n2 A n n1 n2 x zeros 1 n2 n1 1 x n0 n1 1 1 B n n1 n2 x n n0 0 stem n x ro ProgramP1 1 2 Unitstepsequence A n n1 n2 x zeros 1 n2 n2 1 x n0 n1 1 end 1 B n n1 n2 x n n0 0 3 Real valuedexponentialsequence ForExample n 0 10 x 0 9 n stem n x ro MATLAB中数组 矩阵基本运算符 运算符号举例加法 a b 12 34 1 2 3减法 a b 12 34 1 2 3乘法 a b 1 2 3 1 2 3 4 a b 1 2 3 4 3 8 逆乘 左乘 ax b x a b inv a b右乘 xa b x b a b inv a 乘幂 方阵的幂 a 2 a a元素的幂 a 2x 3 4 Complex valuedexponentialsequence Attenuation 衰减因子frequencyinradians ForExample n 0 10 x exp 2 3j n 5 Sinusoidalsequence PhaseinradiansForExample n 0 10 x 3 cos 0 1 pi n pi 3 2 sin 0 5 pi n 6 Randomsequence Rand 1 N GeneratealengthNrandomsequencewhoseelementsareuniformlydistributedbetween 0 1 Randn 1 N GeneratealengthNGaussianrandomsequencewithmean0andvariance1 en 0 1 7 Periodicsequence Asequencex n isperiodicifx n x n N ThesmallestintegerNiscalledthefundamentalperiodForexampleA xtilde x x x x B xtilde x ones 1 P xtilde xtilde xtilde xtilde transposition Operationsonsequence 1 SignaladditionSample by sampleaddition x1 n x2 n x1 n x2 n 3 Scaling a x n ax n 5 folding y n x n k m n k y x 4 Shifting y n x n y fliplr x n fliplr n 6 Samplesummationss sum x n1 n2 7 Sampleproductionsp prod x n1 n2 8 Signalenergyse sum x conj x orse sum abs x 2 9 Signalpower Someusefulresults UnitsamplesynthesisAnyarbitrarysequencecanbesynthesizedasaweightedsumofdelayedandscaledunitsamplesequence EvenandoddsynthesisEven symmetric xe n xe n Odd antisymmetric xo n xo n Anyarbitraryreal valuedsequencecanbedecomposedintoitsevenandoddcomponent x n xe n xo n Thegeometricseries Aone sideexponentialsequenceoftheform an n 0 whereaisanarbitraryconstant iscalledageometricseries Expressionforthesumofanyfinitenumberoftermsoftheseries Correlationsofsequences Itisameasureofthedegreetowhichtwosequencesaresimilar Giventworeal valuedsequencesx n andy n offiniteenergy CrosscorrelationAutocorrelation Theindexliscalledtheshiftorlagparameter Thespecialcase y n x n DiscreteSystems Mathematically anoperationT y n T x n x n excitation inputsignaly n response outputsignalClassificationLinearsystemsNonlinearsystems LinearoperationL IffL satisfiestheprincipleofsuperpositionTheoutputy n ofalinearsystemtoanarbitraryinputx n iscalledimpulseresponse andisdenotedbyh n k h n k thetime varyingimpulseresponse Lineartime invariant LTI system Alinearsysteminwhichaninput outputpairisinvarianttoashiftnintimeiscalledalineartimes invariantsystemy n L x n y n k L x n k TheoutputofaLTIsystemiscallalinearconvolutionsumAnLTIsystemiscompletelycharacterizedinthetimedomainbytheimpulseresponseh n PropertiesoftheLTIsystem StabilityAsystemissaidtobebounded inputbounded output BIBO stableifeveryboundedinputproducesaboundedoutput Condition absolutelysummableToavoidbuildingharmfulsystemsortoavoidburnoutorsaturationinsystemoperation PropertiesoftheLTIsystem CausalityAsystemissaidtobecausaliftheoutputatindexn0dependsonlyontheinputuptoandincludingtheindexn0TheoutputdoesnotdependonthefuturevaluesoftheinputCondition h n 0 n 0Suchasequenceistermedacausalsequence Tomakesurethatsystemscanbebuilt Convolution ConvolutioncanbeevaluatedinmanydifferentwaysIfthesequencesaremathematicalfunctions thenwecananalyticallyevaluatex n h n forallntoobtainafunctionalformofy n Graphicalinterpretation folded and shiftedversion Functionformofconvolution Threedifferentconditionsunderwhichu n k canbeevaluated Case1 n 9 completelyoverlaps Folded and shifted Signalsx x 1 x 2 x 3 x 4 x 5 SystemImpulseResponse h h 1 h 2 h 3 h 4 y conv x h y 1 x 1 h 1 y 2 x 1 h 2 x 2 h 1 y 3 x 1 h 3 x 2 h 2 x 3 h 1 x 1 x 2 x 3 x 4 x 5 h 4 h 3 h 2 h 1 Notethattheresultingsequencey n hasalongerlengththanboththex n andh n sequence Sequencecorrelationsrevisited Thecorrelationcanbecomputedusingtheconvfunctionifsequencesareoffiniteduration Example2 8ThemeaningofthecrosscorrelationThisapproachcanbeusedinapplicationslikeradarsignalprocessinginidentifyingandlocalizingtargets DifferenceEquation AnLTIdiscretesystemcanalsobedescribedbyalinearconstantcoefficientdifferenceequationoftheformIfaN 0 thenthedifferenceequationisoforderNItdescribesarecursiveapproachforcomputingthecurrentoutput giventheinputvaluesandpreviouslycomputedoutputvalues Solutionofdifferenceequation y n yH n yP n Homogeneouspart yH n Particularpart yP n AnalyticalapproachusingZ transformwillbediscussedinthefuntureNumericalsolutionwithMatlaby filter b a x Zero inputandZero stateresponse InDSPthedifferenceequationisgenerallysolvedforwardintimefromn 0 Thereforeinitialconditionsonx n andy n arenecessarytodeterminetheoutputforn 0 Subjecttotheinitialconditions Solution Zero inputandZero stateresponse yZI n zero inputsolutionAsolutionduetotheinitialconditionsaloneyZS n zero statesolutionAsolutionduetoinputx n alone Digitalfilter Discrete timeLTIsystemsarealsocalleddigitalfilter Classifica