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Phase-Preserving Restrikes

An interesting third (nonlinear) input possibility is to add the input to the instantaneous magnitude of $ z(n)$. This mode is useful for ``restriking'' a decaying filter oscillation in a phase-preserving manner. The result is that the amplitude of a decaying oscillation can be jumped without altering its phase at all.

Adding a real signal $ u(n)$ to the magnitude of $ z(n)$ is equivalent to scaling $ z(n)$ by $ g(n) = u(n)/r(n) + 1$, where $ r(n)=\vert z(n)\vert$, to get

$\displaystyle \left[\frac{u(n)}{r(n)}+1\right]z(n) = [u(n)+r(n)]e^{j\theta(n)}.

Since amplitude jumps can be perceived as ``clicks'' when they are too fast, it is useful to delay amplitude jumps until the next zero-crossing of the output $ y(n)$. Waiting for a zero-crossing in $ y(n)$ is equivalent to waiting for the phase of $ z(n)$ to reach either 0 or $ \pi$.

Waiting for phase 0 to scale the state $ z(n)=x(n)$ by some gain factor $ g$ is equivalent to adding an input impulse with amplitude $ u(n)=(g-1)\vert z(n)\vert = (g-1)x(n)$ to $ x(n)$ at that time. Therefore, another approach to achieving smooth filter restriking is to delay them until the next positive-going zero-crossing in the output signal $ y(n)$.

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Download smac03maxjos.pdf

``Methods for Synthesizing Very High Q Parametrically Well Behaved Two Pole Filters'', by Max Mathews and Julius Smith, Proceedings of the Stockholm Musical Acoustics Conference (SMAC-03), pp. 405-408, August 6-9, 2003.
Copyright © 2008-03-12 by Max Mathews and Julius Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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