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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 $ z=\pm \sqrt{R}$ , where $ R$ is the radius of the complex-conjugate pole pair . The transfer function numerator becomes $ B(z)=(1-\sqrt{R}z^{-1})(1+\sqrt{R}z^{-1}) = (1-Rz^{-2})$ , 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}}

which corresponds to the difference equation

$\displaystyle y(n) = x(n) - R\, x(n-2) + [2R\cos(\theta_c)] y(n-1) - R^2 y(n-2).

We see there is one more multiply-add per sample (the term $ -R x(n-2)$ ) relative to the unnormalized two-pole resonator of Eq.(B.15). The resonance gain is now

H(e^{j\theta_c}) &=& \frac{1 - R e^{-j2\theta_c}}{1-2R\cos(\theta_c)e^{-j\theta_c} + R^2e^{-j2\theta_c}}\\
&=& \frac{1 - R e^{-j2\theta_c}}{1-R\left(1+ e^{-j2\theta_c}\right) + R^2e^{-j2\theta_c}}\\
&=& \frac{1 - R e^{-j2\theta_c}}{(1-R) - (1-R)Re^{-j2\theta_c}}\\
&=& \frac{1}{1 - R}

Thus, the gain at resonance is $ 1/(1-R)$ for all resonance tunings $ \theta _c$ .

Figure B.19 shows a family of amplitude responses for the constant resonance-gain two-pole, for various values of $ \theta _c$ and $ R=0.99$ . We see an excellent improvement in the regularity of the amplitude response as a function of tuning.

Figure B.19: Frequency response overlays for the constant resonance-gain two-pole filter $ H(z)=(1-R)(1-Rz^{-2})/(1-2R\cos (\theta _c)z^{-1}+R^2z^{-2})$ , for $ R=0.99$ and 10 values of $ \theta _c$ uniformly spaced from 0 to $ \pi $ . The 5th case is plotted using thicker lines.
\includegraphics[width=\twidth ]{eps/cgresgain}

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University