Correlation Theorem

Theorem: For all ,

where the correlation operation `' was defined in §7.2.5.

Proof:

The last step follows from the convolution theorem and the result from §7.4.2. Also, the summation range in the second line is equivalent to the range because all indexing is modulo .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.