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.11)

For simplicity, let

in what follows. In the special cases $\theta_c=0$ (resonance at dc) and $\theta_c=\pi$ (resonance at

), we have

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

(B.12)

Since

is real, we have already found the gain (amplitude response) at a dc or

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

, $\theta_c = \omega_c T = \pi/2$ , Eq. $\,$ (B.11) with

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

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

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

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 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 [89].) For resonator tunings at dc and , 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 , 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 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 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 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 and 10 values of $\theta _c$ uniformly spaced from 0 to $\pi$ . The 5th case is plotted using thicker lines.
$\includegraphics[width=\twidth ]{eps/cresgain}$

Subsections

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Previous: Peaking Equalizers
Next: Normalizing Two-Pole Filter Gain at Resonance

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
Elementary Audio Digital Filters
Time-Varying Two-Pole Filters