Chapter-3-Fourier-Series-Representation-of-Periodic-Signals.ppt

Chapter3FourierSeriesRepresentationofPeriodicSignals 2012 2013 1 罗欣UESTC Contents ComplexExponentialsasEigenfunctionsofLTISystemsFourierSeriesrepresentationofperiodicsignals CT DT PropertiesofFourierSeriesApplications filtering 3 1Ahistoricalperspective JeanBaptisteJosephFourier bornin1768 inFrance 1807 periodicsignalcouldberepresentedbysinusoidalseries 1829 Dirichletprovidedpreciseconditions 1960s CooleyandTukeydiscoveredfastFouriertransform TwomainmostimportantcontributionofFourier Anyperiodicsignalscanberepresentedbyaseriesofharmonicallyrelatedsinusoids Anyaperiodicsignalscanberepresentedbyweightedintegralsofrelatedsinusoids FrancoisMarieCharlesFourier 1772 1837 Ideologist utopiansocialism 3 2TheResponseofLTISystemstoComplexExponentials Basicidea Torepresentsignalsaslinearcombinationofbasicsignals Thesetofbasicsignalscanbeusedtoconstructabroadandusefulclassofsignals TheresponseofanLTIsystemtoeachsignalshouldbesimple generalandinsightful LTI Previousfocus UnitsamplesandimpulsesFocusnow EigenfunctionsofallLTIsystemsEigenfunctionin samefunctionoutwitha gain Oneofchoices ThesetofcomplexexponentialsignalsSignalsofformestinCTSignalsofformzninDT 1 ContinuoustimeLTIsystem x t est y t H s est systemfunction eigenvalue eigenfunction 2 DiscretetimeLTIsystem x n zn y n H z zn systemfunction eigenvalue eigenfunction Example3 1 Wheninputsignal Hence Theimpulseresponseofthesystemis Then Then 3 InputasacombinationofComplexExponentials CT DT NowthetaskoffindingresponseofLTIsystemsistodetermineH sk orH nk FocusonrestrictedsetsofcomplexexponentialsCT s j purelyimaginaryi e signalsoftheformej tDT z ej i e signalsoftheformej n Note ForNow Howcanwerepresentasignalas sums ofcomplexexponentials CT DTFourierSeries PeriodicSignals andTransforms 3 3FourierSeriesRepresentationofContinuous timePeriodicSignals 1 GeneralForm 3 3 1LinearCombinationsofHarmonicallyRelatedComplexExponentials Thesetofharmonicallyrelatedcomplexexponentials Fundamentalperiod T commonperiod eigenfunction Alinearcombinationofharmonicallyrelatedcomplexexponentials Fourierseries periodicwithperiodT ak aretheFourier series coefficients k 0 constant directcurrentDCcomponent k 1firstharmonic fundamental components k 2secondharmoniccomponents k NtheNthharmoniccomponents eachharmoniccomponenthasdifferentamplitudeandfrequency spectrumofx t ak 0 First forsimpleperiodicsignalsconsistingofafewsinusoidalterms harmoniccomponentsk 1 3 harmoniccomponentsk 1 HowdowefindtheFouriercoefficients Example3 2 So wecanobtain where Euler srelation x1 t 1 2cos2 t x2 t cos4 t x3 t 2 3cos6 t x1 t x0 t x2 t x1 t x0 t x3 t x2 t x1 t x0 t 2 RepresentationforRealSignal Realperiodicsignal x t x t Therefore ak a kora k a k Let A B 3 3 2DeterminationoftheFSRepresentationofaCTPeriodicSignal Orthogonalfunctionset Here TdenotsintegeraloveranyintervaloflengthT oneperiod Determiningthecoefficientbyorthogonality Multiplytwosidesby Continuous timeFourierSeriespair Thesetofcomplexcoefficients ak areoftencalledspectralcoefficientsofx t theaveragevalueofx t Example3 3 because Example3 4 T1T1 fixed fixed ThespectralcharacteristicofaCTperiodicsignalx t DiscreteHarmonicConvergent Homework P250 3 23 3 Fouriermaintainedthat any periodicsignalcouldberepresentedbyaFourierseriesThetruthisthatFourierseriescanbeusedtorepresentanextremelylargeclassofperiodicsignalsThequestionisthatwhenaperiodicsignalx t doesinfacthaveaFourierseriesrepresentation 3 4ConvergenceoftheFourierSeries Approximationerror 1 Finiteseries Theenergyintheerroroveroneperiod oneusefulnotionforengineers If thentheseriesis convergent xN t x t 3 66 Tominimizetheenergyintheerror theparticularchoiceofthecoefficientsis Conclusion thetruncatedFourierseriesisthebestapproximationtox t usingonlyafinitenumeberofharmonicallyrelatedcomplexexponentialsunderMSEcriterion ak Afinite energyperiodicsignal Wedefine 2 Infiniteseries Conclusion ThisequationDoesnotimplythatthesignalx t anditsFourierseriesrepresentationareequalateveryvalueoft Whatdoesitsayisthatthereisnoenergyintheirdifferent Condition1 AbsolutelyIntegrable 3 Dirichletcondition 0 t 1 Condition2 Finitenumberofmaximaandminimaduringasingleperiod 0 t 1 Condition3 Finitenumberofdiscontinuity Dirichletconditionsaremetforthesignalswewillencounterintherealworld Then TheFourierseries x t atpointswherex t iscontinuous TheFourierseries midpoint atpointsofdiscontinuityConclusion Thetwosignalsdifferonlyatisolatedpoints andbehaveidenticallyunderconvolution HowtheFourierseriesconvergesforaperiodicsignalwithdiscontinuities 1898 AlbertMichelson anAmericanphysicist usedhisharmonicanalyzertoobservetruncatedFourierseriesapproximationforthesquarewave 3 Gibbsphenomenon JosiahGibbs AnAmericanmathematicalphysicistGivenoutamathematicalExplanation WhenN Anycontinuity say t t1 xN t1 x t1 Vicinityofdiscontinuity highfrequencyripplesandovershoot asNincreases theripplesbecomecompressedtowardthediscontinuity peakamplitudedoesnotseemtodecrease tendstobe9 Gibbs sconclusion 3 5PropertiesofCTFS 3 5 1Linearity T commonperiod 3 5 2TimeShifting 3 5 3TimeReversal 3 5 6ConjugationandConjugateSymmetry ifx t isreal x t x t thenak a k orak a kSymmetryproperties Re ak Re a k Im ak Im a k ak a k ak a k Note ifx t iseven thatisx t x t thenak a kifx t isodd thenak a kifx t isreal thenak a kifx t isrealandeven thenak a k akakisrealandeven ifx t isrealandodd thenak a k akakispurelyimaginaryandodd 3 5 4TimeScaling 3 5 5Multiplication ConvolutionSum T commonperiod 3 5 7Parseval sRelation 3 46 Table3 1 3 5 8Differentiation Example3 6 T1T1 Example3 7 theaveragevalueofx t Example3 8 Ifx t ak theng t ck PeriodicImpulsetrain important samplefunction Allcomponentshave 1 thesameamplitude 2 thesamephase Homework P251 3 53 83 22fig a 3 40 a d 3 43 3 6FourierSeriesRepresentationofDiscrete timePeriodicSignals 3 6 1LinearCombinationofHarmonicallyRelatedComplexExponentials Periodicsignalx n withperiodN x n x n N Discrete timecomplexexponentialorthogonalsignalset Propertyoforthogonalsignalset discrete timeFourierSeries ak Fourierseriescoefficients 3 6 2DeterminationoftheFourierSeriesRepresentationofPeriodicSignals Solution1 NequationsforNunknowns a0 a1 aN 1 Determinethecoefficientsakbyorthogonality Solution2 Discrete timeFourierseriespair ak theFouriercoefficientsorthespectralcoefficientsofx n ak ak N WeonlyuseNconsecutivevaluesofakinthesynthesisequation Synthesisequation Analysisequation Example1 sumofapairsinusoids periodicwithperiodN 16 Example3 12 and Supposetogetspectrumofx n Note theenvelopeshapeofakisas Note TheFourierSeriesrepresentationofadiscrete timeperiodicsignalisafiniteseries Therearenomathematicalissuesofconvergence ak thespectralcoefficientsofx n isperiodicwithN ThefeaturesoftheFSakofperiodicdiscrete timesignalDiscreteHarmonicPeriodic 3 7PropertiesofDiscrete TimeFourierSeries Seepage221Table3 2 CT DT Systemfunction Systemfunction Recall 3 8FourierSeriesandLTISystem FrequencyresponseofanLTIsystem CTFrequencyResponse DTFrequencyResponse s j z ej t CT DT FourierSeriesandLTISystem Example3 16 and Example3 2 calculatetheFourierSeriescoefficientsoftheoutputy t where Solution Accordingto Hence Where Homework P253 3 133 15P262 3 343 35 3 9Filtering filteringfrequency shapingfiltersfrequency selectivefilters 3 9 1Frequency shapingfilters Example1 Equalizer inhigh fidelityaudiosystems BychoiceofH j orH ej asafunctionof wecanshapethefrequencycompositionoftheoutput Foraudiosignals theamplitudeismuchmoreimportantthanthephase Demo Filteringeffectsonaudiosignals Example2 ImageFiltering Edgeenhancement 3 9 2Frequency selectivefilters Severaltypeoffilter 1 Lowpassfilter 2 Highpassfilter 3 Bandpassfilter threetypesofidealfilter Continuoustime LPF HPF BPF Symmetric Note Onlyshowamplitudehere H 1and H 0fortheidealfiltersinthepassbands noneedforthephaseplot c cutofffrequency c1 lowercutofffrequency c2 uppercutofffr