文档介绍：MATHEMATICS OF THE DISCRETE FOURIER TRANSFORM (DFT) WITH AUDIO APPLICATIONS SECOND EDITION
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MATHEMATICS OF THE
DISCRETE FOURIER TRANSFORM (DFT)
WITH AUDIO APPLICATIONS
SECOND EDITION
JULIUS O. SMITH III
Center puter
Research in Music and
Acoustics (CCRMA)
Department of Music ,
Stanford University ,
Stanford, California
94305 USA
● Preface
❍ Chapter Outline
❍ Acknowledgments
❍ Errata
● Introduction to the DFT
❍ DFT Definition
❍ Inverse DFT
❍ Mathematics of the DFT
❍ DFT Math Outline
● Complex Numbers
❍ Factoring a Polynomial
❍ The Quadratic Formula
❍ Complex Roots
❍ Fundamental Theorem of Algebra
http://ccrma-/%7Ejos/r320/ (1 of 9) [2007-5-27 23:45:15]
MATHEMATICS OF THE DISCRETE FOURIER TRANSFORM (DFT) WITH AUDIO APPLICATIONS SECOND EDITION
❍ Complex Basics
❍ plex Plane
❍ More Notation and Terminology
❍ Elementary Relationships
❍ Euler's Identity
❍ De Moivre's Theorem
❍ Conclusion
❍ Complex_Number Problems
● Proof of Euler's Identity
❍ Euler's Identity
❍ Positive Integer Exponents
❍ Properties of Exponents
❍ The Exponent Zero
❍ Negative Exponents
❍ Rational Exponents
❍ Real Exponents
❍ A First Look at Taylor Series
❍ Imaginary Exponents
❍ Derivatives of f(x)=a^x
❍ Back to e
❍ e^(j theta)
❍ Back to Mth Roots
❍ Roots of Unity
❍ Direct Proof of De Moivre's Theorem
❍ Euler_Identity Problems
● Sinusoids and Exponentials
❍ Sinusoids
■ Example Sinusoids
■ Why Sinusoids are Important
■ In-Phase & Quadrature ponents
■ Sinusoids at the Same Frequency
■ Constructive and Destructive Interference
■ Sinusoid Magnitude Spectra
❍ Exponentials
■ Why Exponentials are Important
■ Audio Decay Time (T60)
❍ Complex Sinusoids
http://ccrma-/%7Ejos/r320/ (2 of 9) [2007-5-27 23:45:15]
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