The Fourier Transform and its Applications

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1、LectureNotesforEE261TheFourierTransformanditsApplicationsProf.BradOsgoodElectricalEngineeringDepartmentStanfordUniversityContents1FourierSeries11.1IntroductionandChoicestoMake................................11.2PeriodicPhenomena........................................

2、21.3Periodicity:Definitions,Examples,andThingstoCome....................41.4ItAllAddsUp...........................................91.5Lostatc...............................................101.6Period,Frequencies,andSpectrum................................121.7TwoExampl

3、esandaWarning...................................161.8TheMath,theMajesty,theEnd.................................211.9Orthogonality............................................261.10Appendix:TheCauchy-SchwarzInequalityanditsConsequences...............331.11Appendix:Moreo

4、ntheComplexInnerProduct.........................361.12Appendix:BestL2ApproximationbyFiniteFourierSeries..................381.13FourierSeriesinAction......................................391.14NotesonConvergenceofFourierSeries..............................501.15Appendi

5、x:PointwiseConvergencevs.UniformConvergence..................581.16Appendix:StudyingPartialSumsviatheDirichletKernel:TheBuzzIsBack........591.17Appendix:TheComplexExponentialsAreaBasisforL2([0,1])................611.18Appendix:MoreontheGibbsPhenomenon.................

6、..........622FourierTransform652.1AFirstLookattheFourierTransform..............................652.2Appendix:ChasetheConstant..................................752.3Examples..............................................752.4HowDoestheGraphoff(ax)ComparewiththeGraphoff(

7、x)?..............782.5GettingtoKnowYourFourierTransform............................793Convolution91iiCONTENTS3.1A∗isBorn.............................................913.2WhatisConvolution,Really?...................................953.3PropertiesofConvolution:It’saLotl

8、ikeMultiplication....................973.4ForWhomtheBellCurveTolls..................................993.5Appendix:Evaluation