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 10.21 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.10.22, 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.

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).