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Eigenvalues in the Undamped Case

When $ g=1$ , the eigenvalues reduce to

$\displaystyle {\lambda_i}= c\pm j\sqrt{1-c^2}
$

Assuming $ \left\vert c\right\vert<1$ , the eigenvalues can be expressed as

$\displaystyle {\lambda_i}= c\pm j\sqrt{1-c^2} = \cos(\theta) \pm j\sin(\theta) = e^{\pm j\theta} \protect$ (C.140)

where $ \theta=\omega T$ denotes the angular advance per sample of the oscillator. Since $ c\in(-1,1)$ corresponds to the range $ \theta\in(-\pi,\pi)$ , we see that $ c$ in this range can produce oscillation at any digital frequency.

For $ \left\vert c\right\vert>1$ , the eigenvalues are real, corresponding to exponential growth and/or decay. (The values $ c=\pm 1$ were excluded above in deriving Eq.$ \,$ (C.141).)

In summary, the coefficient $ c$ in the digital waveguide oscillator ($ g=1$ ) and the frequency of sinusoidal oscillation $ \omega $ is simply

$\displaystyle \fbox{$\displaystyle c= \cos(\omega T)$}.
$


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA