Starting with the defining equation for an eigenvector
and its
corresponding eigenvalue
,
we get
Equation (G.23) gives us two equations in two unknowns:
Thus, we have found both eigenvectors
They are linearly independent provided
and finite provided
.
We can now use Eq.(G.24) to find the eigenvalues:
Assuming
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 see that the coefficient range (-1,1) corresponds to frequencies in the range
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.