Power Theorem

**Theorem: **For all
,

*Proof: *

As mentioned in §5.8, physical *power* is
*energy per unit time*.^{7.20} 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
.^{7.21}

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University