It is surprisingly easy to normalize exactly the peak gain in a second-order resonator tuned by a single coefficient . The filter structure that accomplishes this is the one we already considered in §10.6.1:
The peak gain is , so multiplying the transfer function by normalizes the peak gain to one for all tunings. It can also be shown  that the peak gain coincides with the variance gain when the resonator is driven by white noise. That is, if the variance of the driving noise is , the variance of the noise at the resonator output is . Therefore, scaling the resonator input by will normalize the resonator such that the output signal power equals the input signal power when the input signal is white noise.
Frequency response overlays for the constant-peak-gain resonator are shown in Fig.10.24 (), Fig.10.21 (), and Fig.10.22 (). While the peak frequency may be far from the resonance tuning in the more heavily damped examples, the peak gain is always normalized to unity. The normalized radian frequency at which the peak gain occurs is related to the pole angle by 
Thus, must be close to 1 to obtain a resonant peak near dc (a case commonly needed in audio work) or half the sampling rate (rarely needed in practice). When is much less than 1, the peak frequency cannot leave a small interval near one-fourth the sampling rate, as can be seen at the far left in Fig.10.23.
Figure 10.23 predicts that for , the lowest peak-gain frequency should be around radian per sample. Figure 10.22 agrees with this prediction.
As Figures 10.24 through 10.26 shows, the peak gain remains constant even at very low and very high frequencies, to the extent they are reachable for a given . The zeros at dc and preclude the possibility of peaks at exactly those frequencies, but for near 1, we can get very close to having a peak at dc or , as shown in Figures 10.20 and 10.21.