### Eigenstructure

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

we get, using Eq.(C.137),

 (C.140)

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:

 (C.141) (C.142)

Substituting the first into the second to eliminate , we get

As approaches (no damping), we obtain

Thus, we have found both eigenvectors:

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

where denotes the sampling interval, is the time constant of decay, and is the frequency of oscillation in radians per second. The eigenvalues are presumed to be complex, which requires, from Eq.(C.144),

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

Note that this conclusion follows directly from the determinant analysis of Eq.(C.140), provided it is known that the poles form a complex-conjugate pair.

To obtain a desired frequency of oscillation, we must solve

for , which yields

Note that this reduces to when (undamped case).

#### Eigenvalues in the Undamped Case

When , the eigenvalues reduce to

Assuming , the eigenvalues can be expressed as

 (C.143)

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

Next Section:
Summary
Previous Section:
State-Space Analysis