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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.143)

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.144).) 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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