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## Quality Factor (Q)

The quality factor (Q) of a two-pole 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 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 D. The parameter is called the damping constant (or damping factor'') of the second-order 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 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 §B.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 [20, p. 624]

 (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.

Since the imaginary parts of the complex resonator poles are , 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 radians per second. 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.

Subsections
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