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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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``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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA