Eigenstructure
Starting with the defining equation for an eigenvector
and its
corresponding eigenvalue
,


We normalized the first element of



Equation (C.141) gives us two equations in two unknowns:
Substituting the first into the second to eliminate

![\begin{eqnarray*}
g+gc+c\eta_i &=& [gc+\eta_i(c-1)]\eta_i = gc\eta_i + \eta_i^2 ...
...{g\left(\frac{1+c}{1-c}\right)
- \frac{c^2(1-g)^2}{4(1-c)^2}}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4224.png)
As approaches
(no damping), we obtain

![\begin{eqnarray*}
\underline{e}_1&=&\left[\begin{array}{c} 1 \\ [2pt] \eta \end{...
...t{g\left(\frac{1+c}{1-c}\right)
- \frac{c^2(1-g)^2}{4(1-c)^2}}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4226.png)
They are linearly independent provided . In the undamped
case (
), this holds whenever
. The eigenvectors are
finite when
. Thus, the nominal range for
is the
interval
.
We can now use Eq.(C.142) to find the eigenvalues:
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



Eigenvalues in the Undamped Case
When , the eigenvalues reduce to


where




For
, the eigenvalues are real, corresponding to
exponential growth and/or decay. (The values
were
excluded above in deriving Eq.
(C.144).)
In summary, the coefficient in the digital waveguide oscillator
(
) and the frequency of sinusoidal oscillation
is simply

Next Section:
Summary
Previous Section:
State-Space Analysis