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).

**Next Section:** Eigenvalues in the Undamped Case**Previous Section:** Miscellaneous Properties