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Peak Gain Versus Resonance Gain

While the constant resonance-gain filter is very well behaved, it is not ideal, because, while the resonance gain is perfectly normalized, the peak gain is not. The amplitude-response peak does not occur exactly at the resonance frequencies $ \omega
T=\pm\theta_c$ except for the special cases $ \theta_c=0$ , $ \pm\pi/2$ , and $ \pi $ . At other resonance frequencies, the peak due to one pole is shifted by the presence of the other pole. When $ R$ is close to 1, the shifting can be negligible, but in more damped resonators, e.g., when $ R<0.9$ , there can be a significant difference between the gain at resonance and the true peak gain.

Figure B.20 shows a family of amplitude responses for the constant resonance-gain two-pole, for various values of $ \theta _c$ and $ R=0.9$ . We see that while the gain at resonance is exactly the same in all cases, the actual peak gain varies somewhat, especially near dc and $ f_s/2$ when the two poles come closest together. A more pronounced variation in peak gain can be seen in Fig.B.21, for which the pole radii have been reduced to $ R=0.5$ .

Figure: 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=0.9$ and 10 values of $ \theta _c$ uniformly spaced from 0 to $ \pi $ . The 5th case is plotted using thicker lines.
\includegraphics[width=\twidth ]{eps/cgresgaindamped}

Figure: 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=0.5$ and 10 values of $ \theta _c$ uniformly spaced from 0 to $ \pi $ . The 5th case is plotted using thicker lines. Note the more pronounced variation in peak gain (the resonance gain does not vary).
\includegraphics[width=\twidth ]{eps/cgresgaindampedp5}


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-01-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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