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.
Subsections
Previous: Relating Pole Radius to BandwidthNext: Decay Time is Q Periods
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.