The quality factor (Q) of a real two-pole resonator may be defined in terms of its complex one-pole components, where the one-pole components have poles at and ; as discussed above, each of these complex resonators has quality factor . However, this is only accurate for their sum when the poles can be treated as fully separated in frequency. In other words, we must assume that the center-frequency and bandwidth of the complex resonators are not significantly affected by summing their frequency responses. In the high-Q case, , this assumption is accurate. For greater accuracy, we should calculate where the peak-gain frequency has moved, and what the modified bandwidths may be.
Let's look at what happens when we add a one-pole complex resonator to its complex-conjugate counterpart to form a real second-order resonator:
We see that a zero at has been induced symmetrically between the poles. This zero, due to phase cancellation from adding the resonators in parallel, symmetrically affects the bandwidth and peak-gain frequency of the two one-pole resonators. For simplicity below, we will let the zero be simply at , which is a typical choice for a real second-order resonator in which no dc gain is desired, and there is little difference when the damping is light ( ). This choice also simplifies calculation of the peak-gain frequency , leading to the incredibly simple result .^{E.1} Moreover, the peak gain at is similarly incredibly simple:
Since we always have for stability, it is convenient to define the positive number
which is called the damping constant (or damping factor) for real, second-order resonators [20, p. 179]. The damping constant is simply minus the real part of the poles. (We are assuming the poles form a complex-conjugate pair--excluding the case of two different real poles.)
We also have
That is, is both the pole modulus and the peak-gain frequency for a sum of complex-conjugate one-pole resonators with a zero at separating them. When damping is light, we have . However, for any nonzero damping we have that the peak gain occurs precisely at frequency , and the gain there is , again when the induced zero has been moved to .
The transfer function of the real second-order resonator with one zero at dc and the other at infinity, with peak gain normalized to , can be written in terms of our new notation as
In view of the above, it is now reasonable to define the quality factor (Q) of the real second-order resonator Eq.(E.7) by [20, p. 184]
The peak-gain frequency (pole modulus) coincides with the physical oscillation frequency of the resonator impulse response (pole imaginary part) when the damping constant is zero. The physical oscillation frequency of the impulse response remains (twice) the zero-crossing rate of the sinusoidal oscillation underneath the exponential decay of the impulse response. See also §B.6 for a discussion of discrete-time resonators.