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LectureNotesforEE261TheFourierTransformanditsApplicationsProf.BradOsgoodElectricalEngineeringDepartmentStanfordUniversityContents1FourierSeries11.1IntroductionandChoicestoMake................................11.2PeriodicPhenomena........................................21.3Periodicity:Definitions,Examples,andThingstoCome....................41.4ItAllAddsUp...........................................91.5Lostatc...............................................101.6Period,Frequencies,andSpectrum................................131.7TwoExamplesandaWarning...................................161.8TheMath,theMajesty,theEnd.................................211.9Orthogonality............................................261.10Appendix:TheCauchy-SchwarzInequalityanditsConsequences...............331.11Appendix:MoreontheComplexInnerProduct.........................361.12Appendix:BestL2ApproximationbyFiniteFourierSeries..................381.13FourierSeriesinAction......................................391.14NotesonConvergenceofFourierSeries..............................501.15Appendix:PointwiseConvergencevs.UniformConvergence..................581.16Appendix:StudyingPartialSumsviatheDirichletKernel:TheBuzzIsBack........591.17Appendix:TheComplexExponentialsAreaBasisforL2([0,1])................611.18Appendix:MoreontheGibbsPhenomenon...........................622FourierTransform652.1AFirstLookattheFourierTransform..............................652.2GettingtoKnowYourFourierTransform............................753Convolution953.1A∗isBorn.............................................953.2WhatisConvolution,Really?...................................993.3PropertiesofConvolution:It’saLotlikeMultiplication....................101iiCONTENTS3.4ConvolutioninActionI:ALittleBitonFiltering........................1023.5ConvolutioninActionII:DifferentialEquations.........................1063.6ConvolutioninActionIII:TheCentralLimitTheorem.....................1163.7TheCentralLimitTheorem:TheBellCurveTollsforThee..................1283.8Fouriertransformformulasunderdifferentnormalizations...................1303.9Appendix:TheMeanandStandardDeviationfortheSumofRandomVariables......1313.10MoreDetailsontheCentralLimitTheorem...........................1323.11Appendix:Heisenberg’sInequality................................1334DistributionsandTheirFourierTransforms1374.1TheDayofReckoning.......................................1374.2TheRightFunctionsforFourierTransforms:RapidlyDecreasingFunctions.........1424.3AVeryLittleonIntegrals.....................................1484.4Distributions............................................1524.5APhysicalAnalogyforDistributions...............................1644.6LimitsofDistributions.......................................1654.7TheFourierTransformofaTemperedDistribution.......................1684.8FluxionsFinis:TheEndofDifferentialCalculus........................1744.9ApproximationsofDistributions.................................1774.10TheGeneralizedFourierTransformIncludestheClassicalFourierTransform........1784.11OperationsonDistributionsandFourierTransforms......................1794.12Duality,ChangingSigns,EvennessandOddness........................1794.13AFunctionTimesaDistributionMakesSense..........................1824.14TheDerivativeTheorem......................................1854.15ShiftsandtheShiftTheorem...................................1864.16ScalingandtheStretchTheorem.................................1884.17ConvolutionsandtheConvolutionTheorem...........................1904.18δHardatWork...........................................1954.19Appendix:TheRiemann-Lebesguelemma............................2054.20Appendix:SmoothWindows...................................2064.21Appendix:1/xasaPrincipalValueDistribution........................2095III,Sampling,andInterpolation2115.1X-RayDiffraction:ThroughaGlassDarkly1..........................2115.2TheIIIDistribution........................................2125.3TheFourierTransformofIII....................................216CONTENTSiii5.4PeriodicDistributionsandFourierseries.............................2195.5SamplingSignals..........................................2235.6SamplingandInterpolationforBandlimitedSignals......................2255.7InterpolationaLittleMoreGenerally...............................2295.8FiniteSamplingforaBandlimitedPeriodicSignal.......................2315.9TroubleswithSampling......................................2365.10Appendix:HowSpecialisIII?...................................2465.11Appendix:Timelimitedvs.BandlimitedSignals.........................2486DiscreteFourierTransform2516.1FromContinuoustoDiscrete...................................2516.2TheDiscreteFourierTransform(DFT)..............................2546.3TwoGrids,ReciprocallyRelated.................................2596.4Appendix:Gauss’sProblem....................................2606.5GettingtoKnowYourDiscreteFourierTransform.......................2616.6Periodicity,Indexing,andReindexing...........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