Note that Q is defined in the context of continuous-time
resonators, so the transfer function is the Laplace transform
(instead of the z transform) of the continuous (instead of
discrete-time) impulse-response . An introduction to
Laplace-transform analysis appears in Appendix B. The parameter
is called the damping constant (or ``damping factor'')
of the second-order transfer function, and is called the
resonant frequency [21, p. 179].
The resonant frequency coincides with the physical
oscillation frequency of the resonator impulse response when the
damping constant is zero. For light damping, is
approximately the physical frequency of impulse-response oscillation
( times the zero-crossing rate of sinusoidal oscillation under
an exponential decay). For larger damping constants, it is better to
use the imaginary part of the pole location as a definition of
resonance frequency (which is exact in the case of a single complex
pole). (See §10.6 for a more complete discussion of resonators,
in the discrete-time case.)

By the quadratic formula, the poles of the transfer function
are given by

(C.9)

Therefore, the poles are complex only when . Since real poles
do not resonate, we have for any resonator. The case
is called critically damped, while is called
overdamped. A resonator () is said to be
underdamped, and the limiting case is simply
undamped.

Relating to the notation of the previous section, in which we defined
one of the complex poles as
, we have

(C.10)

(C.11)

For resonators, coincides with the classically defined
quantity [21, p. 624]

Since the imaginary parts of the complex resonator poles
, the zero-crossing rate of the resonator impulse
response is
crossings per second. Moreover,
is very close to the peak-magnitude frequency in the resonator
amplitude response. If we eliminate the negative-frequency pole,
becomes exactly the peak frequency. In other
words, as a measure of resonance peak frequency, only
neglects the interaction of the positive- and negative-frequency
resonance peaks in the frequency response, which is usually negligible
except for highly damped, low-frequency resonators. For any amount of
damping
gives the impulse-response zero-crossing rate
exactly, as is immediately seen from the derivation in the next
section.