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

$$H(z) = \frac{B(z)}{A(z)} = \frac{1 - Rz^{-2}}{1-2R\cos(\theta_c)z^{-1}+ R^2z^{-2}}$$

which corresponds to the difference equation

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

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

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

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.