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Power Theorem



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

$\displaystyle \zbox {\left<x,y\right> = \frac{1}{N}\left<X,Y\right>.}
$



Proof:

\begin{eqnarray*}
\left<x,y\right> &\isdef & \sum_{n=0}^{N-1}x(n)\overline{y(n)}
= (y\star x)_0
= \hbox{\sc DFT}_0^{-1}(\overline{Y}\cdot X) \\
&=& \frac{1}{N} \sum_{k=0}^{N-1}X(k)\overline{Y(k)}
\isdef \frac{1}{N} \left<X,Y\right>.
\end{eqnarray*}

As mentioned in §5.8, physical power is energy per unit time.7.20 For example, when a force produces a motion, the power delivered is given by the force times the velocity of the motion. Therefore, if $ x(n)$ and $ y(n)$ are in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their product $ x(n)y(n)\isdeftext
f(n)v(n)$ is proportional to the power per sample at time $ n$ , and $ \left<f,v\right>$ becomes proportional to the total energy supplied (or absorbed) by the driving force. By the power theorem, $ {F(k)}\overline{V(k)}/N$ can be interpreted as the energy per bin in the DFT, or spectral power, i.e., the energy associated with a spectral band of width $ 2\pi/N$ .7.21



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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