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.141),
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.137), 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).