The Digital Waveguide Oscillator

Let's use state-space analysis to determine the frequency of oscillation of the following system:

The second-order digital waveguide oscillator.

Note the assignments of unit-delay *outputs* to state variables
and
.

We have

and

In matrix form, the state transition can be written as

or, in vector notation,

The poles of the system are given by the eigenvalues of , which are the roots of its characteristic polynomial. That is, we solve

for , , or, for our problem,

Using the quadratic formula, the two solutions are found to be

Defining , we obtain the simple formula

It is now clear that the system is a real sinusoidal oscillator for , oscillating at normalized radian frequency .

We determined the frequency of oscillation
from the
eigenvalues
of
. To study this system further, we
can *diagonalize*
. For that we need the eigenvectors as well
as the eigenvalues.

Download StateSpace.pdf

Download StateSpace_2up.pdf

Download StateSpace_4up.pdf

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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