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

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Power Theorem for the DTFT

The inner product of two signals is defined in the time domain by

$\displaystyle \left<x,y\right> \isdef \sum_{n=-\infty}^{\infty} x_n\overline{y}_n. $

The inner product of two spectra may be defined as

$\displaystyle \left<X,Y\right> \isdef \frac{1}{2\pi}\int_{-\pi}^\pi X(\omega)\overline{Y(\omega)} d\omega. $

Note that the frequency-domain inner product includes a normalization factor while the time-domain definition does not.

Using inner-product notation, the power theorem (or Parseval's theorem [186]) for DTFTs can be stated as follows:

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

That is, the inner product of two signals in the time domain equals the inner product of their respective spectra (a complex scalar in general).

When we consider the inner product of a signal with itself, we have the special case known as the energy theorem (or Rayleigh's energy theorem):

$\displaystyle \Vert x \Vert^2 = \left<x,x\right> = \left<X,X\right> = \Vert X \Vert^2 $

where $ \Vert\,\cdot\,\Vert$ denotes the $ L2$ norm induced by the inner product. It is always real.



Proof:

\begin{eqnarray*} \left<x,y\right> &\isdef & \sum_{n=-\infty}^{\infty} x(n)\over... ...} X(\omega)\overline{Y(\omega)} d\omega \isdef \left<X,Y\right> \end{eqnarray*}

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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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