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.E.3 [90].

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

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

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

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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