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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Power Theorem

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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... ...d\omega = \ensuremath{\int_{-\infty}^{\infty}}X(2\pi f)\overline{Y(2\pi f)}df. $

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}}X(\omega)\overline{Y(\omega)}e^{j\omega t}d\omega\right\vert _{t=0}. $

By the convolution theorem (§2.4.6) and flip theorem (§2.4.7),

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

which at $ t=0$ gives

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

Thus,

$\displaystyle \zbox {\left<x,y\right> \longleftrightarrow \left<X,Y\right>.} $

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Previous: Flip Theorems
Next: The Continuous-Time Impulse
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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