## 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 ).

### Finding the Eigenstructure of A

Starting with the defining equation for an eigenvector and its corresponding eigenvalue ,

we get

 (G.23)

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:

 (G.24) (G.25)

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:

Assuming , the eigenvalues are

 (G.26)

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

We see that the coefficient range (-1,1) corresponds to frequencies in the range , and that's the complete set of available digital frequencies.

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.

### Choice of Output Signal and Initial Conditions

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

For this initialization, the output from the first state variable is simply

A similar derivation can be carried out to show that the output is proportional to , i.e., it is in phase quadrature with respect to ). Phase-quadrature outputs are often useful in practice, e.g., for generating complex sinusoids.

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