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Damping and Tuning Parameters

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

$\displaystyle {\lambda_i}= e^{\frac{T}{\tau}} e^{\pm j\omega T}

where $ T$ denotes the sampling interval, $ \tau $ is the time constant of decay, and $ \omega $ is the frequency of oscillation in radians per second. The eigenvalues are presumed to be complex, which requires, from Eq.$ \,$(C.144),

$\displaystyle g(1-c^2) \geq\frac{c^2(1-g)^2}{4} \,\,\Rightarrow\,\,c^2 \leq \frac{4g}{(1+g)^2}

To obtain a specific decay time-constant $ \tau $, we must have
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
Therefore, given a desired decay time-constant $ \tau $ (and the sampling interval $ T$), we may compute the damping parameter $ g$ for the digital waveguide resonator as

$\displaystyle \zbox {g = e^{-2T/\tau}.}

Note that this conclusion follows directly from the determinant analysis of Eq.$ \,$(C.140), provided it is known that the poles form a complex-conjugate pair. To obtain a desired frequency of oscillation, we must solve
\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}
for $ c$, which yields

$\displaystyle \zbox {
c= \sqrt{\frac{1}{1 + \frac{\tan^2(\theta)(1+g)^2+(1-g)^2}{4g}}}
\approx 1 - \frac{\tan^2(\theta)(1+g)^2 + (1-g)^2}{8g}.

Note that this reduces to $ c=\cos(\theta)$ when $ g=1$ (undamped case).
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Eigenvalues in the Undamped Case
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Miscellaneous Properties