### Constant Resonance Gain

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.13). 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.

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Normalizing Two-Pole Filter Gain at Resonance