It turns out it is possible to normalize exactly the resonance gain of the second-order resonator tuned by a single coefficient [89]. This is accomplished by placing the two zeros at , where is the radius of the complex-conjugate pole pair . The transfer function numerator becomes , yielding the total transfer function
which corresponds to the difference equation
We see there is one more multiply-add per sample (the term ) relative to the unnormalized two-pole resonator of Eq. (B.15). The resonance gain is now
Thus, the gain at resonance is for all resonance tunings .
Figure B.19 shows a family of amplitude responses for the constant resonance-gain two-pole, for various values of and . We see an excellent improvement in the regularity of the amplitude response as a function of tuning.