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

![$\displaystyle y(n) = x(n) - R\, x(n-2) + [2R\cos(\theta_c)] y(n-1) - R^2 y(n-2).
$](http://www.dsprelated.com/josimages_new/filters/img1519.png)



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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Peak Gain Versus Resonance Gain
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Normalizing Two-Pole Filter Gain at Resonance