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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Rayleigh Energy Theorem (Parseval's Theorem)


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Rayleigh Energy Theorem (Parseval's Theorem)



Theorem: For any $ x\in{\bf C}^N$,

$\displaystyle \zbox {\left\Vert\,x\,\right\Vert^2 = \frac{1}{N}\left\Vert\,X\,\right\Vert^2.} $

I.e.,

$\displaystyle \zbox {\sum_{n=0}^{N-1}\left\vert x(n)\right\vert^2 = \frac{1}{N}\sum_{k=0}^{N-1}\left\vert X(k)\right\vert^2.} $



Proof: This is a special case of the power theorem.

Note that again the relationship would be cleaner ( $ \left\Vert\,x\,\right\Vert = \vert\vert\,\tilde{X}\,\vert\vert $) if we were using the normalized DFT.

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Next: Stretch Theorem (Repeat Theorem)
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (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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