Damping and Tuning Parameters
The tuning and damping of the resonator impulse response are governed by the relation






To obtain a specific decay time-constant , we must have
![\begin{eqnarray*}
e^{-2T/\tau} &=& \left\vert{\lambda_i}\right\vert^2 = c^2\left...
...left[g(1-c^2) - c^2\left(\frac{1-g}{2}\right)^2\right]\\
&=& g
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4234.png)
Therefore, given a desired decay time-constant (and the
sampling interval
), we may compute the damping parameter
for
the digital waveguide resonator as


To obtain a desired frequency of oscillation, we must solve
![\begin{eqnarray*}
\theta = \omega T
&=& \tan^{-1}\left[\frac{\sqrt{g(1-c^2) - [...
...,\tan^2{\theta} &=& \frac{g(1-c^2) - [c(1-g)/2]^2}{[c(1+g)/2]^2}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4236.png)
for , which yields



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Eigenvalues in the Undamped Case
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Miscellaneous Properties