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)}...
...^{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.7.19 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$.7.20

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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Rayleigh Energy Theorem (Parseval's Theorem)
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Correlation Theorem