where and are constants (generally complex). In this ``parallel one-pole'' form, it can be seen that the peak gain is no longer equal to the resonance gain, since each one-pole frequency response is ``tilted'' near resonance by being summed with the ``skirt'' of the other one-pole resonator, as illustrated in Fig.B.9. This interaction between the positive- and negative-frequency poles is minimized by making the resonance sharper ( ), and by separating the pole frequencies . The greatest separation occurs when the resonance frequency is at one-fourth the sampling rate ( ). However, low-frequency resonances, which are by far the most common in audio work, suffer from significant overlapping of the positive- and negative-frequency poles.
To show Eq.(B.7) is always true, let's solve in general for and given and . Recombining the right-hand side over a common denominator and equating numerators gives
The solution is easily found to be
where we have assumed im, as necessary to have a resonator in the first place.
Note that the inverse z transform of a sum of one-pole transfer functions can be easily written down by inspection. In particular, the impulse response of the PFE of the two-pole resonator (see Eq.(B.7)) is clearly
Resonator Bandwidth in Terms of Pole Radius