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.

Evaluating the transfer function of the two-pole resonator
(Eq.
(B.1)) at the point
on the unit circle
(the filter's *resonance frequency*
) yields a *gain at resonance* equal to

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:

In the middle frequency between dc and , , Eq. (B.13) with becomes

and, since is real and positive, it coincides with the amplitude response,

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 [90].) 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
- Constant Resonance Gain
- Peak Gain Versus Resonance Gain
- Constant Peak-Gain Resonator
- Four-Pole Tunable Lowpass/Bandpass Filters

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University