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Eigenstructure

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

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 (C.141) gives us two equations in two unknowns:

Substituting the first into the second to eliminate , we get

As approaches (no damping), we obtain

They are linearly independent provided . In the undamped case (), this holds whenever . The eigenvectors are finite when . Thus, the nominal range for is the interval .

We can now use Eq.(C.142) to find the eigenvalues:

#### Damping and Tuning Parameters

The tuning and damping of the resonator impulse response are governed by the relation

To obtain a specific decay time-constant , we must have

Therefore, given a desired decay time-constant (and the sampling interval ), we may compute the damping parameter for the digital waveguide resonator as

To obtain a desired frequency of oscillation, we must solve

for , which yields

#### Eigenvalues in the Undamped Case

When , the eigenvalues reduce to

where denotes the angular advance per sample of the oscillator. Since corresponds to the range , we see that in this range can produce oscillation at any digital frequency.

For , the eigenvalues are real, corresponding to exponential growth and/or decay. (The values were excluded above in deriving Eq.(C.144).)

In summary, the coefficient in the digital waveguide oscillator () and the frequency of sinusoidal oscillation is simply

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