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Quality Factor (Q)

The quality factor (Q) of a two-pole resonator is defined by [20, p. 184]

$\displaystyle Q \isdef \frac{\omega_0}{2\alpha} \protect$ (E.7)

where $ \omega_0$ and $ \alpha$ are parameters of the resonator transfer function

$\displaystyle H(s) = g\frac{s}{s^2 + 2\alpha s + \omega_0^2}. \protect$ (E.8)

Note that Q is defined in the context of continuous-time resonators, so the transfer function $ H(s)$ is the Laplace transform (instead of the z transform) of the continuous (instead of discrete-time) impulse-response $ h(t)$. An introduction to Laplace-transform analysis appears in Appendix D. The parameter $ \alpha$ is called the damping constant (or ``damping factor'') of the second-order transfer function, and $ \omega_0$ is called the resonant frequency [20, p. 179]. The resonant frequency $ \omega_0$ coincides with the physical oscillation frequency of the resonator impulse response when the damping constant $ \alpha$ is zero. For light damping, $ \omega_0$ is approximately the physical frequency of impulse-response oscillation ($ 2\pi$ 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 $ H(s)$ are given by

$\displaystyle p = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \isdef -\alpha \pm \alpha_d . \protect$ (E.9)

Therefore, the poles are complex only when $ Q>1/2$. Since real poles do not resonate, we have $ Q>1/2$ for any resonator. The case $ Q=1/2$ is called critically damped, while $ Q<1/2$ is called overdamped. A resonator ($ Q>1/2$) is said to be underdamped, and the limiting case $ Q=\infty$ is simply undamped.

Relating to the notation of the previous section, in which we defined one of the complex poles as $ p\isdef \sigma_p+j\omega_p$, we have

$\displaystyle \sigma_p$ $\displaystyle =$ $\displaystyle -\alpha$ (E.10)
$\displaystyle \omega_p$ $\displaystyle =$ $\displaystyle \sqrt{\omega_0-\alpha^2}.$ (E.11)

For resonators, $ \omega_p$ coincides with the classically defined quantity [20, p. 624]

$\displaystyle \omega_d \isdef \omega_p = \sqrt{\omega_0^2 -\alpha^2} = \frac{\alpha_d}{j}.

Since the imaginary parts of the complex resonator poles are $ \pm\omega_d$, the zero-crossing rate of the resonator impulse response is $ \omega_d/\pi$ crossings per second. Moreover, $ \omega_d$ is very close to the peak-magnitude frequency in the resonator amplitude response. If we eliminate the negative-frequency pole, $ \omega_d/\pi$ becomes exactly the peak frequency. In other words, as a measure of resonance peak frequency, $ \omega_d$ 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 $ \omega_d/\pi$ gives the impulse-response zero-crossing rate exactly, as is immediately seen from the derivation in the next section.

Previous: Relating Pole Radius to Bandwidth
Next: Decay Time is Q Periods

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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 (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 for details.


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