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Resonator Bandwidth in Terms of Pole Radius

The magnitude $ R$ of a complex pole determines the damping or bandwidth of the resonator. (Damping may be defined as the reciprocal of the bandwidth.)

As derived in §8.5, when $ R$ is close to 1, a reasonable definition of 3dB-bandwidth $ B$ is provided by

$\displaystyle B$ $\displaystyle \isdef$ $\displaystyle - \frac{\ln(R)}{\pi T}$ (B.4)
$\displaystyle R$ $\displaystyle =$ $\displaystyle e^{- \pi B T}
\protect$ (B.5)

where $ R$ is the pole radius, $ B$ is the bandwidth in Hertz (cycles per second), and $ T$ is the sampling interval in seconds.

Figure B.6 shows a family of frequency responses for the two-pole resonator obtained by setting $ b_0 = 1$ and varying $ R$ . The value of $ \theta _c$ in all cases is $ \pi /4$ , corresponding to $ f_c =
f_s/8$ . The analytic expressions for amplitude and phase response are

G(\omega)\! &=&
\!\frac{b_0}{\sqrt{[1 + a_1 \cos(\omega T) + a_2 \cos(2\omega T)]^2
+ [-a_1 \sin(\omega T) - a_2 \sin(2\omega T)]^2}}\\ [10pt]
\Theta(\omega)\! &=&
\!-\tan^{-1}\left[ \frac{-a_1 \sin(\omega T) - a_2 \sin(2\omega T)}{1 + a_1 \cos(\omega T) + a_2 \cos(2\omega T)}\right]\qquad(b_0>0)

where $ a_1 = - 2R \cos(\theta_c)$ and $ a_2 = R^2$ .

Figure B.6: Frequency response of the two-pole filter
$ y(n) = x(n) + 2R \cos (\theta _c) y(n - 1) - R^2 y(n - 2)$
with $ \theta _c$ fixed at $ \pi /4$ and for various values of $ R$ . (a) Amplitude response. (b) Phase response.

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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