**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 theoremsfor 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.

Subsections

- The DFT and its Inverse
- Signal Operators
- Even and Odd Functions
- The Fourier Theorems

- Linearity
- Conjugation and Reversal
- Symmetry
- Shift Theorem
- 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
- Appendix B: Introductory Statistical Signal Processing
- Appendix C: Mathematica/Matlab Examples
- Appendix D: The Similarity Theorem