Two-Pole Partial Fraction Expansion

Note that every real two-pole resonator can be broken up into a sum of two complex one-pole resonators:

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

which implies

The solution is easily found to be

where we have assumed im , as necessary to have a resonator in the first place.

Breaking up the two-pole real resonator into a parallel sum of two
complex one-pole resonators is a simple example of a *partial
fraction expansion* (PFE) (discussed more fully in §6.8).

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

Since is real, we must have , as we found above without assuming it. If , then is a real sinusoid created by the sum of two complex sinusoids spinning in opposite directions on the unit circle.

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