Time-Varying Two-Pole Filters
It is quite common to want to vary the resonance frequency of a resonator in real time. This is a special case of a tunable filter. In the pre-digital days of analog synthesizers, filter modules were tuned by means of control voltages, and were thus called voltage-controlled filters (VCF). In the digital domain, control voltages are replaced by time-varying filter coefficients. In the time-varying case, the choice of filter structure has a profound effect on how the filter characteristics vary with respect to coefficient variations. In this section, we will take a look at the time-varying two-pole resonator.
For simplicity, let in what follows. In the special cases (resonance at dc) and (resonance at ), we have
Since is real, we have already found the gain (amplitude response) at a dc or resonance:
An important fact we can now see is that the gain at resonance depends markedly on the resonance frequency. In particular, the ratio of the two cases just analyzed is
Note that the ratio of the dc resonance gain to the resonance gain is unbounded! The sharper the resonance (the closer is to 1), the greater the disparity in the gain.
Figure B.17 illustrates a number of resonator frequency responses for the case . (Resonators in practice may use values of even closer to 1 than this--even the case is used for making recursive digital sinusoidal oscillators .) For resonator tunings at dc and , we predict the resonance gain to be dB, and this is what we see in the plot. When the resonance is tuned to , the gain drops well below 40 dB. Clearly, we will need to compensate this gain variation when trying to use the two-pole digital resonator as a tunable filter.
Figure B.18 shows the same type of plot for the complex one-pole resonator , for and 10 values of . In this case, we expect the frequency response evaluated at the center frequency to be . Thus, the gain at resonance for the plotted example is db for all tunings. Furthermore, for the complex resonator, the resonance gain is also exactly equal to the peak gain.
Normalizing Two-Pole Filter Gain at Resonance
The question we now pose is how to best compensate the tunable two-pole resonator of §B.1.3 so that its peak gain is the same for all tunings. Looking at Fig.B.17, and remembering the graphical method for determining the amplitude response,B.6 it is intuitively clear that we can help matters by adding two zeros to the filter, one near dc and the other near . A zero exactly at dc is provided by the term in the transfer function numerator. Similarly, a zero at half the sampling rate is provided by the term in the numerator. The series combination of both zeros gives the numerator . The complete second-order transfer function then becomes
Checking the gain for the case , we have
which is better behaved, but now the response falls to zero at dc and rather than being heavily boosted, as we found in Eq.(B.12).
It turns out it is possible to normalize exactly the resonance gain of the second-order resonator tuned by a single coefficient . This is accomplished by placing the two zeros at , where is the radius of the complex-conjugate pole pair . The transfer function numerator becomes , yielding the total transfer function
Thus, the gain at resonance is for all resonance tunings .
Figure B.19 shows a family of amplitude responses for the constant resonance-gain two-pole, for various values of and . We see an excellent improvement in the regularity of the amplitude response as a function of tuning.
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 except for the special cases , , and . 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 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 .
Constant Peak-Gain Resonator
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 §B.6.1:
That is, the two-pole resonator normalized by zeros at has the constant peak-gain property when it has resonant peaks in its response at all. Note, however, that the peak-gain frequency and the pole-resonance frequency (cf. §B.6.3), are generally two different things, as elaborated below. This structure has the added bonus that its difference equation requires only one more addition relative to the unnormalized two-pole resonator, and no new multiply.
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.B.23 (), Fig.B.20 (), and Fig.B.21 (). 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 
When the right-hand side of the above equation exceeds 1 in magnitude, there is no (real) solution for the pole frequency . This happens, for example, when is less than 1 and is too close to 0 or . Conversely, given any pole angle , there always exists a solution for the peak frequency , since when . However, when is small, the peak frequency can be far from the pole resonance frequency, as shown in Fig.B.22.
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.B.22.
As Figures B.23 through B.25 show, 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 B.19 and B.20.
As a practical note, it is worth mentioning that in popular analog synthesizers (both real and virtualB.7), VCFs are typically fourth order rather than second order as we have studied here. Perhaps the best known VCF is the Moog VCF. The four-pole Moog VCF is configured to be a lowpass filter with an optional resonance near the cut-off frequency. When the resonance is strong, it functions more like a resonator than a lowpass filter. Various methods for digitizing the Moog VCF are described in . It turns out to be nontrivial to preserve all desirable properties of the analog filter (such as frequency response, order, and control structure), when translated to digital form by standard means.
Elementary Filter Problems