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**Search Mathematics of the DFT**

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In this section the main Fourier theorems are stated and proved. It is no small matter how simple these theorems are in the DFT case relative to the other three cases (DTFT, Fourier transform, and Fourier series, as defined in Appendix B). When infinite summations or integrals are involved, the conditions for the existence of the Fourier transform can be quite difficult to characterize mathematically. Mathematicians have expended a considerable effort on such questions. By focusing primarily on the DFT case, we are able to study the essential concepts conveyed by the Fourier theorems without getting involved with mathematical difficulties.

- Linearity
- Conjugation and Reversal
- Symmetry
- Shift Theorem
- Linear Phase Terms
- Linear Phase Signals
- Zero Phase Signals
- 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)
- Downsampling Theorem (Aliasing Theorem)

- Zero Padding Theorem (Spectral Interpolation)
- Interpolation Theorems

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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