Fourier Transforms for Continuous/Discrete Time/Frequency

Fourier Theorems for the DTFT

Convolution Theorem for the DTFT

**Search Spectral Audio Signal Processing**

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The *convolution* of discrete-time signals and is defined
as

The *convolution theorem* is then

*Proof: *The result follows immediately from interchanging the order
of summations associated with the convolution and DTFT:

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.

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