### Power Theorem

**Theorem:**For all ,

*Proof:*

*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 and are in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their product is proportional to the

*power per sample*at time , and becomes proportional to the total

*energy*supplied (or absorbed) by the driving force. By the power theorem, 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 .

^{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*

*normalized DFT*(§6.10), the power theorem becomes simply

(Normalized DFT case)

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 and 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).

**Next Section:**

Rayleigh Energy Theorem (Parseval's Theorem)

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Correlation Theorem