Power Theorem

**Theorem: **For all
,

*Proof: *

Note that the power theorem would be more elegant
if the DFT were defined as the coefficient of
projection onto the *normalized DFT sinusoid*
. That, for the *normalized DFT* (defined in
§6.10), the power theorem becomes simply

(Normalized DFT case)

The power theorem is also sometimes called *Parseval's theorem*
[51].

In the field of physics, much progress has been made by formulating
physical laws such that they are *invariant with respect to
changes of coordinates*. The inner product is an example of a
coordinate-invariant relationship between two vectors and in
. In the present setting, as developed in Chapter 5, we are
concerned only with *two* different coordinate systems,
corresponding to time-domain signals and their spectra.

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