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Quality Factor (Q)
The quality factor (Q) of a twopole resonator is defined by
[20, p. 184]

(E.7) 
where
and
are parameters of the resonator transfer
function:

(E.8) 
Note that Q is defined in the context of continuoustime
resonators, so the transfer function
is the Laplace transform
(instead of the z transform) of the continuous (instead of
discretetime) impulseresponse
. An introduction to
Laplacetransform analysis appears in Appendix D. The parameter
is called the damping constant (or ``damping factor'')
of the secondorder transfer function, and
is called the
resonant frequency [20, 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 impulseresponse oscillation
(
times the zerocrossing 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 §B.6 for a more complete discussion of resonators,
in the discretetime case.)
By the quadratic formula, the poles of the transfer function
are given by

(E.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
For resonators,
coincides with the classically defined
quantity [20, p. 624]
Since the imaginary parts of the complex resonator poles are
, the zerocrossing rate of the resonator impulse
response is
crossings per second. Moreover,
is very close to the peakmagnitude frequency in the resonator
amplitude response. If we eliminate the negativefrequency pole,
becomes exactly the peak frequency. In other
words, as a measure of resonance peak frequency,
only
neglects the interaction of the positive and negativefrequency
resonance peaks in the frequency response, which is usually negligible
except for highly damped, lowfrequency resonators. For any amount of
damping
gives the impulseresponse zerocrossing rate
exactly, as is immediately seen from the derivation in the next
section.
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