Damping and Tuning Parameters

Free Books Physical Audio Signal Processing

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

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

$\begin{eqnarray*} e^{-2T/\tau} &=& \left\vert{\lambda_i}\right\vert^2 = c^2\left... ...left[g(1-c^2) - c^2\left(\frac{1-g}{2}\right)^2\right]\\ &=& g \end{eqnarray*}$

Therefore, given a desired decay time-constant $\tau$ (and the sampling interval

), we may compute the damping parameter

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

$\begin{eqnarray*} \theta = \omega T &=& \tan^{-1}\left[\frac{\sqrt{g(1-c^2) - [... ...,\tan^2{\theta} &=& \frac{g(1-c^2) - [c(1-g)/2]^2}{[c(1+g)/2]^2} \end{eqnarray*}$

for