#### 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.

*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

**Next Section:**

Exercise

**Previous Section:**

Resonator Bandwidth in Terms of Pole Radius