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

We normalized the first element of



Equation (G.23) gives us two equations in two unknowns:
Substituting the first into the second to eliminate

![\begin{eqnarray*}
1+c+c\eta_i &=& [c+\eta_i (c-1)]\eta_i = c\eta_i + \eta_i ^2 (...
...-1)\\
\,\,\Rightarrow\,\,\eta_i &=& \pm \sqrt{\frac{c+1}{c-1}}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img2261.png)
Thus, we have found both eigenvectors
![\begin{eqnarray*}
\underline{e}_1&=&\left[\begin{array}{c} 1 \\ [2pt] \eta \end{...
...ght], \quad \hbox{where}\\
\eta&\isdef &\sqrt{\frac{c+1}{c-1}}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img2262.png)
They are linearly independent provided
and finite provided
.
We can now use Eq.(G.24) to find the eigenvalues:


and so this is the range of




Let us henceforth assume
. In this range
is real, and we have
,
. Thus, the eigenvalues can be expressed as follows:

Equating to
, we obtain
, or
, where
denotes the sampling rate. Thus the
relationship between the coefficient
in the digital waveguide
oscillator and the frequency of sinusoidal oscillation
is
expressed succinctly as


We have now shown that the system of Fig.G.3 oscillates
sinusoidally at any desired digital frequency rad/sec by simply
setting
, where
denotes the sampling interval.
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Choice of Output Signal and Initial Conditions
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Jordan Canonical Form