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A Musical Time-Varying Filter Example

Note, however, that a gain $ g$ may vary with time independently of $ x$ to yield a linear time-varying filter. In this case, linearity may be demonstrated by verifying

$\displaystyle g(n) \left[ \alpha \cdot x_1(n) + \beta \cdot x_2(n)\right]
= \alpha \cdot [g(n)\cdot x_1(n)] + \beta\cdot[g(n)\cdot x_2(n)]
$

to show that both scaling and superposition hold. A simple example of a linear time-varying filter is a tremolo function, which can be written as a time-varying gain, $ y(n)=g(n)x(n)$ . For example, $ g(n) = 1
+ \cos[2\pi (4)nT]$ would give a maximally deep tremolo with 4 swells per second.


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-04-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA