Power Theorem for the DTFT
The inner product of two spectra may be defined as
Note that the frequency-domain inner product includes a normalization factor while the time-domain definition does not.
That is, the inner product of two signals in the time domain equals the inner product of their respective spectra (a complex scalar in general).
When we consider the inner product of a signal with itself, we have the special case known as the energy theorem (or Rayleigh's energy theorem):
where denotes the norm induced by the inner product. It is always real.