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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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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