### Power Theorem

Theorem: For all ,

Proof:

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 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 DFTPower 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

That is, for the normalized DFT6.10), 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).

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