State-Space Analysis Example:

The Digital Waveguide Oscillator

As an example of state-space analysis, we will use it to determine the frequency of oscillation of the system of Fig.G.3 [90].

Note the assignments of unit-delay outputs to state variables and . From the diagram, we see that

and

In matrix form, the state time-update can be written

or, in vector notation,

We have two natural choices of output, and :

A basic fact from linear algebra is that the *determinant* of a
matrix is equal to the *product of its eigenvalues*. As a quick
check, we find that the determinant of
is

Since an undriven sinusoidal oscillator must not lose energy, and since every lossless state-space system has unit-modulus eigenvalues (consider the modal representation), we expect .

Note that . If we diagonalize this system to obtain , where diag , and is the matrix of eigenvectors of , then we have

where denotes the state vector in these new ``modal coordinates''. Since is diagonal, the modes are decoupled, and we can write

If this system is to generate a real sampled sinusoid at radian frequency , the eigenvalues and must be of the form

(in either order) where is real, and denotes the sampling interval in seconds.

Thus, we can determine the frequency of oscillation (and verify that the system actually oscillates) by determining the eigenvalues of . Note that, as a prerequisite, it will also be necessary to find two linearly independent eigenvectors of (columns of ).

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