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