NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Fourier Theorems for the DFT

This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem,correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, andstatistical signal processing.

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Subsections

• The DFT and its Inverse
  • Notation and Terminology
  • Modulo Indexing, Periodic Extension
• Signal Operators
  • Flip Operator
  • Shift Operator
  • Convolution
    • Graphical Convolution
    • Polynomial Multiplication
    • Multiplication of Decimal Numbers
  • Correlation
  • Stretch Operator
  • Zero Padding
  • Repeat Operator
  • Decimation Operator
  • Alias Operator
• Even and Odd Functions
• The Fourier Theorems
  • Linearity
  • Conjugation and Reversal
  • Symmetry
  • Shift Theorem
    • Linear Phase Terms
    • Application of the Shift Theorem to FFT Windows
  • Convolution Theorem
  • Dual of the Convolution Theorem
  • Correlation Theorem
  • Power Theorem
  • Rayleigh Energy Theorem (Parseval's Theorem)
  • Stretch Theorem (Repeat Theorem)
  • Decimation Theorem (Aliasing Theorem)
  • Zero Padding Theorem
  • Bandlimited Interpolation in Time
• Conclusions
• Acknowledgement
• Appendix A: Linear Time-Invariant Filters and Convolution
  • LTI Filters and the Convolution Theorem
• Appendix B: Introductory Statistical Signal Processing
  • Cross-Correlation
  • Applications of Cross-Correlation
  • Autocorrelation
  • Coherence
• Appendix C: Mathematica/Matlab Examples
• Appendix D: The Similarity Theorem

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