Modulation Theorem (Shift Theorem Dual)

The Fourier dual of the shift theorem is often called the modulation theorem:

This is proved in the same way as the shift theorem above by starting with the inverse Fourier transform of the right-hand side:

or,

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About the Author: 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.