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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 $ e^{j\theta_c}$ on the unit circle (the filter's resonance frequency $ \omega_c=\theta_c/T$ ) yields a gain at resonance equal to

$\displaystyle H(e^{j\theta_c})$ $\displaystyle =$ $\displaystyle \frac{b_0}{(1-Re^{j\theta_c}e^{-j\theta_c})(1-Re^{-j\theta_c}e^{-j\theta_c})}$  
  $\displaystyle =$ $\displaystyle \frac{b_0}{1-R}\cdot \frac{1}{1-Re^{-j2\theta_c}}
\protect$ (B.13)

For simplicity, let $ b_0 = 1$ in what follows. In the special cases $ \theta_c=0$ (resonance at dc) and $ \theta_c=\pi$ (resonance at $ f=f_s/2$ ), we have

$\displaystyle H(\pm1) = \frac{1}{(1-R)^2} \protect$ (B.14)

Since $ R$ is real, we have already found the gain (amplitude response) at a dc or $ f_s/2$ resonance:

$\displaystyle G(0) = G(\pi/T) \isdef \left\vert H(\pm1)\right\vert = \left\vert\frac{1}{(1-R)^2}\right\vert = \frac{1}{(1-R)^2}

In the middle frequency between dc and $ f_s/2$ , $ \theta_c = \omega_c T
= \pi/2$ , Eq.(B.13) with $ b_0 = 1$ becomes

$\displaystyle H(j) = \frac{1}{(1-Re^{j2\frac{\pi}{2}})(1-R)} = \frac{1}{(1+R)(1-R)} = \frac{1}{1-R^2}

and, since $ H(j)$ is real and positive, it coincides with the amplitude response, i.e., $ H(j)=G(\pi/2)=1/(1-R^2)$ .

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

$\displaystyle \left\vert\frac{H(1)}{H(j)}\right\vert = \frac{1-R^2}{(1-R)^2} = \frac{1+R}{1-R} =
\frac{\hbox{maximum resonance gain}}{\hbox{minimum resonance gain}}

We did not show that resonance gain is maximized at $ e^{j\theta_c}=\pm
1$ and minimized at $ e^{j\theta_c}=\pm j$ , 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 $ f_s/4$ resonance gain is unbounded! The sharper the resonance (the closer $ R$ is to 1), the greater the disparity in the gain.

Figure B.17 illustrates a number of resonator frequency responses for the case $ R=0.99$ . (Resonators in practice may use values of $ R$ even closer to 1 than this--even the case $ R=1$ is used for making recursive digital sinusoidal oscillators [90].) For resonator tunings at dc and $ f_s/2$ , we predict the resonance gain to be $ 20\log_{10}[1/(1-R)^2] = -20\log_{10}[(1-0.99)^2] =
-40\log_{10}(0.01) = 80$ dB, and this is what we see in the plot. When the resonance is tuned to $ f_s/4$ , 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.17: Frequency response overlays for the two-pole resonator $ H(z)=1/(1-2R\cos (\theta _c)z^{-1}+ R^2z^{-2})$ , for $ R=0.99$ and 10 values of $ \theta _c$ uniformly spaced from 0 to $ \pi $ . The 5th case is plotted using thicker lines.
\includegraphics[width=\twidth ]{eps/resgain}

Figure B.18 shows the same type of plot for the complex one-pole resonator $ H(z)=1/(1-Re^{j\theta _c}z^{-1})$ , for $ R=0.99$ and 10 values of $ \theta _c$ . In this case, we expect the frequency response evaluated at the center frequency to be $ H(e^{j\omega_c T})
=1/(1-Re^{j\theta_c}e^{-j\theta_c})=1/(1-R)$ . Thus, the gain at resonance for the plotted example is $ 1/(1-0.99)=100=40$ db for all tunings. Furthermore, for the complex resonator, the resonance gain is also exactly equal to the peak gain.

Figure B.18: Frequency response overlays for the one-pole complex resonator $ H(z)=1/(1-Re^{j\theta _c}z^{-1})$ , for $ R=0.99$ and 10 values of $ \theta _c$ uniformly spaced from 0 to $ \pi $ . The 5th case is plotted using thicker lines.
\includegraphics[width=\twidth ]{eps/cresgain}

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University