where and are parameters of the resonator transfer function
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
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 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 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.
Decay Time is Q Periods
Another well known rule of thumb is that the of a resonator is the number of ``periods'' under the exponential decay of its impulse response. More precisely, we will show that, for , the impulse response decays by the factor in cycles, which is about 96 percent decay, or -27 dB.
The impulse response corresponding to Eq.(E.8) is found by inverting the Laplace transform of the transfer function . Since it is only second order, the solution can be found in many tables of Laplace transforms. Alternatively, we can break it up into a sum of first-order terms which are invertible by inspection (possibly after rederiving the Laplace transform of an exponential decay, which is very simple). Thus we perform the partial fraction expansion of Eq.(E.8) to obtain
as the respective residues of the poles .
The impulse response is thus
Assuming a resonator, , we have , where (using notation of the preceding section), and the impulse response reduces to
Yet another meaning for is as follows [20, p. 326]
Proof. The total stored energy at time is equal to the total energy of the remaining response. After an impulse at time 0, the stored energy in a second-order resonator is
Assuming as before, so that
Analog Allpass Filters
Relating Pole Radius to Bandwidth