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Power Theorem

The power theorem for Fourier transforms states that the inner product of two signals in the time domain equals their inner product in the frequency domain.

The inner product of two spectra $ X(\omega)$ and $ Y(\omega)$ may be defined as

$\displaystyle \left<X,Y\right> \isdef \frac{1}{2\pi} \ensuremath{\int_{-\infty}^{\infty}}X(\omega)\overline{Y(\omega)}d\omega = \ensuremath{\int_{-\infty}^{\infty}}X(2\pi f)\overline{Y(2\pi f)}df.$ (B.21)

This expression can be interpreted as the inverse Fourier transform of $ X\cdot\overline{Y}$ evaluated at $ t=0$ :

$\displaystyle \left<X,Y\right> \isdef \frac{1}{2\pi} \left.\ensuremath{\int_{-\infty}^{\infty}}X(\omega)\overline{Y(\omega)}e^{j\omega t}d\omega\right\vert _{t=0}.$ (B.22)

By the convolution theoremB.7) and flip theoremB.8),

$\displaystyle X\cdot \overline{Y}\;\longleftrightarrow\;x\ast \hbox{\sc Flip}(\overline{y}),$ (B.23)

which at $ t=0$ gives

$\displaystyle (x\ast \hbox{\sc Flip}(\overline{y}))(0) = \left.\ensuremath{\int_{-\infty}^{\infty}}x(\tau)\overline{y(\tau-t)}d\tau\right\vert _{t=0} = \ensuremath{\int_{-\infty}^{\infty}}x(\tau)\overline{y(\tau)}d\tau \isdef \left<x,y\right>$ (B.24)

Thus,

$\displaystyle \zbox {\left<x,y\right> \;\longleftrightarrow\;\left<X,Y\right>.}$ (B.25)


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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