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Normalized DFT Power Theorem

Note that the power theorem would be more elegant if the DFT were defined as the coefficient of projection onto the normalized DFT sinusoids

$\displaystyle \tilde{s}_k(n) \isdef \frac{s_k(n)}{\sqrt{N}}.

That is, for the normalized DFT6.10), the power theorem becomes simply

$\displaystyle \left<x,y\right> = \langle \tilde{X},\tilde{Y}\rangle$   (Normalized DFT case)$\displaystyle . \protect$

We see that the power theorem expresses the invariance of the inner product between two signals in the time and frequency domains. If we think of the inner product geometrically, as in Chapter 5, then this result is expected, because $ x$ and $ \tilde{X}$ are merely coordinates of the same geometric object (a signal) relative to two different sets of basis signals (the shifted impulses and the normalized DFT sinusoids).

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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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University