###

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

#### Damping and Tuning Parameters

The tuning and damping of the resonator impulse response are governed by the relation#### Eigenvalues in the Undamped Case

When , the eigenvalues reduce towhere 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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