State-Space Analysis Example:
The Digital Waveguide Oscillator
Note the assignments of unit-delay outputs to state variables and . From the diagram, we see that
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 ).
We normalized the first element of to 1 since is an eigenvector whenever is. (If there is a missing solution because its first element happens to be zero, we can repeat the analysis normalizing the second element to 1 instead.)
Equation (G.23) gives us two equations in two unknowns:
Substituting the first into the second to eliminate , we get
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:
and so this is the range of corresponding to sinusoidal oscillation. For , the eigenvalues are real, corresponding to exponential growth and decay. The values yield a repeated root (dc or oscillation).
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
Recalling that , the output signal from any diagonal state-space model is a linear combination of the modal signals. The two immediate outputs and in Fig.G.3 are given in terms of the modal signals and as
The output signal from the first state variable is
The initial condition corresponds to modal initial state