Peak Gain Versus Resonance Gain

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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 is close to 1, the shifting can be negligible, but in more damped resonators, e.g., when , 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 . 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 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 .

**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 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 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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Elementary Audio Digital Filters
       Time-Varying Two-Pole Filters
          Peak Gain Versus Resonance Gain