Summary

A representation of an arbitrary linear time-varying digital filter has been constructed which characterizes such a filter as having the ability to generate an arbitrary output in response to each basis function in the signal space. The representation was obtained by casting the filter in moving-average form as a matrix, and studying its response to individual orthogonal basis functions which were chosen here to be complex sinusoids. The overall conclusion is that time-varying filters may be used to convert from a set of orthogonal signals (such as tones at distinct frequencies) to a set of unconstrained waveforms in a one-to-one fashion. Linear combinations of these orthogonal signals are then transformed by the LTV filter to the same linear combination of the transformed basis signals.

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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.