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Chapters

Physical Audio Signal Processing
    The Digital Waveguide Oscillator
      
             Eigenstructure

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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.$ \,$(J.17),

$\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 $ 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.$ \,$(J.13), provided it is known that the poles form a complex-conjugate pair.

To obtain a desired frequency of oscillation, we must solve