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 H(z) = \frac{B(z)}{A(z)} = \frac{1 - Rz^{-2}}{1-2R\cos(\theta_c)z^{-1}+ R^2z^{-2}}
$](http://www.dsprelated.com/josimages_new/filters/img1518.png)
![$\displaystyle y(n) = x(n) - R\, x(n-2) + [2R\cos(\theta_c)] y(n-1) - R^2 y(n-2).
$](http://www.dsprelated.com/josimages_new/filters/img1519.png)
![$ -R x(n-2)$](http://www.dsprelated.com/josimages_new/filters/img1520.png)
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
![\begin{eqnarray*}
H(e^{j\theta_c}) &=& \frac{1 - R e^{-j2\theta_c}}{1-2R\cos(\th...
...\theta_c}}{(1-R) - (1-R)Re^{-j2\theta_c}}\\
&=& \frac{1}{1 - R}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img1521.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