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.
Evaluating the transfer function of the two-pole resonator
(B.1)) at the point
on the unit circle
(the filter's resonance frequency
) yields a gain at resonance equal to
In the middle frequency between dc and , , Eq. (B.13) with becomes
and, since is real and positive, it coincides with the amplitude response, i.e., .
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
We did not show that resonance gain is maximized at and minimized at , but this is straightforward to show, and strongly suggested by Fig.B.17 (and Fig.B.9).
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.