Power Theorem for the DTFT

The *inner product* of two *signals* may be defined in the
time domain by [264]

(3.25) |

The inner product of two

(3.26) |

Note that the frequency-domain inner product includes a normalization factor while the time-domain definition does not.

Using inner-product notation, the *power theorem* (or
*Parseval's theorem* [202]) for DTFTs can
be stated as follows:

(3.27) |

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*):

(3.28) |

where denotes the norm induced by the inner product. It is always real.

*Proof: *
Using the previous correlation theorem:

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