Chapter3 FourierSeries 信号与系统教学课件

3FourierSeriesRepresentationofPeriodicSignals 3 FourierSeriesRepresentationofPeriodicSignal JeanBaptisteJosephFourier bornin1768 inFrance 1807 periodicsignalcouldberepresentedbysinusoidalseries 1829 Dirichletprovidedpreciseconditions 1960s CooleyandTukeydiscoveredfastFouriertransform 3FourierSeriesRepresentationofPeriodicSignals 3 2TheResponseofLTISystemstoComplexExponentials 1 ContinuoustimeLTIsystem systemfunction 3FourierSeriesRepresentationofPeriodicSignals 2 DiscretetimeLTIsystem systemfunction 3FourierSeriesRepresentationofPeriodicSignals 3 InputasacombinationofComplexExponentials ContinuoustimeLTIsystem DiscretetimeLTIsystem Example3 1 3FourierSeriesRepresentationofPeriodicSignals 3 3FourierSeriesRepresentationofContinuous timePeriodicSignals 1 GeneralForm 3 3 1LinearCombinationsofHarmonicallyRelatedComplexExponentials Thesetofharmonicallyrelatedcomplexexponentials Fundamentalperiod T commonperiod 3FourierSeriesRepresentationofPeriodicSignals So arbitraryperiodicsignalcanberepresentedas Fundamentalcomponents Secondharmoniccomponents Nthharmoniccomponents Fourierseries Example3 2 3FourierSeriesRepresentationofPeriodicSignals 2 RepresentationforRealSignal Realperiodicsignal x t x t Soa k a k Let A 3FourierSeriesRepresentationofPeriodicSignals Let A B 3FourierSeriesRepresentationofPeriodicSignals 3 3 2DeterminationoftheFourierSeriesRepresentationofaContinuous timePeriodicSignal Orthogonalfunctionset Determiningthecoefficientbyorthogonality Multiplytwosidesby 3FourierSeriesRepresentationofPeriodicSignals FourierSeriesRepresentation 3FourierSeriesRepresentationofPeriodicSignals Example3 33 5 3FourierSeriesRepresentationofPeriodicSignals 3 4ConvergenceoftheFourierSeries Approximationerror 1 Finiteseries 3FourierSeriesRepresentationofPeriodicSignals Condition1 AbsolutelyIntegrable 2 Dirichletcondition 3FourierSeriesRepresentationofPeriodicSignals Condition2 Finitenumberofmaximaandminimaduringasingleperiod 3FourierSeriesRepresentationofPeriodicSignals Condition3 Finitenumberofdiscontinuity 3FourierSeriesRepresentationofPeriodicSignals 1898 AlbertMichelson AnAmericanphysicistConstructedaharmonicanalyzerObservedtruncatedFourierseriesxN t lookedverymuchlikex t FoundastrangephenomenonJosiahGibbs AnAmericanmathematicalphysicistGivenoutamathematicalExplanation 3 Gibbsphenomenon 3FourierSeriesRepresentationofPeriodicSignals 3FourierSeriesRepresentationofPeriodicSignals Anycontinuity xN t1 x t1 Vicinityofdiscontinuity ripplespeakamplitudedoesnotseemtodecreaseDiscontinuity overshoot9 Gibbs sconclusion 3FourierSeriesRepresentationofPeriodicSignals 3 6 1LinearCombinationofHarmonicallyRelatedComplexExponentials Periodicsignalx n withperiodN x n x n N 3 6FourierSeriesRepresentationofDiscrete timePeriodicSignals Discrete timecomplexexponentialorthogonalsignalset 3FourierSeriesRepresentationofPeriodicSignals Propertyoforthogonalsignalset 3FourierSeriesRepresentationofPeriodicSignals 3 6 2DeterminationoftheFourierSeriesRepresentationofPeriodicSignals Fourierseriesofperiodicsignalx n Determinethecoefficientsakbyorthogonality 3FourierSeriesRepresentationofPeriodicSignals TheequationsofFourierseries akisperiodic 3FourierSeriesRepresentationofPeriodicSignals Example3 103 12 3FourierSeriesRepresentationofPeriodicSignals 1 Systemfunction 3 8FourierSeriesandLTISystem Continuoustimesystem Discrete timesystem 2 Frequencyresponse Continuoustimesystem Discrete timesystem 3FourierSeriesRepresentationofPeriodicSignals 3 Systemresponse Continuoustimesystem Discrete timesystem Example3 163 17 3FourierSeriesRepresentationofPeriodicSignals 3 9Filtering 3 9 1Frequency shapingfilters Example1 Equalizer 3FourierSeriesRepresentationofPeriodicSignals Example2 ImageFiltering Edgeenhancement 3FourierSeriesRepresentationofPeriodicSignals 3 9 2Frequency selectivefilters Severaltypeoffilter 1 Lowpassfilter 2 Highpassfilter 3 Bandpassfilter 3FourierSeriesRepresenta