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