mathematics of the discrete fourier transform

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1、MathematicsoftheDiscreteFourierTransform(DFT)JuliusO.SmithIII(jos@ccrma.stanford.edu)CenterforComputerResearchinMusicandAcoustics(CCRMA)DepartmentofMusic,StanfordUniversityStanford,California94305March15,2002PageiiDRAFTof“MathematicsoftheDiscreteFourierTransform(DFT),”

2、byJ.O.Smith,CCRMA,Stanford,Winter2002.ThelatestdraftandlinkedHTMLversionareavailableon-lineathttp://www-ccrma.stanford.edu/~jos/mdft/.Contents1IntroductiontotheDFT11.1DFTDefinition........................11.2MathematicsoftheDFT..................31.3DFTMathOutline.......

3、...............62ComplexNumbers72.1FactoringaPolynomial....................72.2TheQuadraticFormula...................82.3ComplexRoots........................92.4FundamentalTheoremofAlgebra..............112.5ComplexBasics........................112.5.1TheComplexPlane...

4、..............132.5.2MoreNotationandTerminology..........142.5.3ElementaryRelationships..............152.5.4Euler’sFormula....................152.5.5DeMoivre’sTheorem................172.6NumericalToolsinMatlab.................172.7NumericalToolsinMathematica...........

5、...233ProofofEulersIdentity273.1Euler’sTheorem.......................273.1.1PositiveIntegerExponents.............273.1.2PropertiesofExponents...............283.1.3TheExponentZero..................283.1.4NegativeExponents.................283.1.5RationalExponents........

6、.........293.1.6RealExponents....................303.1.7AFirstLookatTaylorSeries............313.1.8ImaginaryExponents................32iiiPageivCONTENTS3.1.9Derivativesoff(x)=ax...............323.1.10Backtoe.......................333.1.11SidebaronMathematica...........

7、....343.1.12Backtoejθ......................343.2InformalDerivationofTaylorSeries............363.3TaylorSerieswithRemainder................373.4FormalStatementofTaylor’sTheorem...........393.5WeierstrassApproximationTheorem............403.6DifferentiabilityofAudioSignals

8、..............404Logarithms,Decibels,andNumberSystems414.1Logarithms..........................414.1.1ChangingtheBase.