Mathematics Of The Discrete Fourier Transform, Julius Smith.pdf

文档介绍：Mathematics of the Discrete Fourier Transform (DFT)
Julius O. Smith III ()
Center puter Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University
Stanford, California 94305
August 11, 2002
Page ii
Contents
1 Introduction to the DFT 1
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plex Numbers 7
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plexPlane..................... 13
MoreNotationandTerminology.............. 15
ElementaryRelationships.................. 15
Euler’s Formula . ....................... 16
De Moivre’s Theorem . ................... 17
Numerical Tools in Matlab . . ................... 17
Numerical Tools in Mathematica . . . ............... 24
3 Proof of Euler’s Identity 27
Euler’s Theorem ........................... 27
PositiveIntegerExponents................. 27
PropertiesofExponents................... 28
The Exponent Zero . . . ................... 28
NegativeExponents..................... 28
RationalExponents..................... 29
RealExponents........................ 30
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Page iv CONTENTS
A First Look at Taylor Series . . . ............. 31
Imaginary Exponents . . . ................. 32
Derivatives of f(x)=ax ................... 32
Back to e ........................... 33
................... 34
Back to ejθ.......................... 34
Informal Derivation of Taylor Series . . . ............. 36
Taylor Series with Remainder . . . ................. 38
’sTheorem............... 39