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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Elementary Audio Digital Filters
       Elementary Filter Sections
          Two-Pole
             Resonator Bandwidth in Terms of Pole Radius


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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

\begin{eqnarray*} G(\omega)\! &=& \!\frac{b_0}{\sqrt{[1 + a_1 \cos(\omega T) + a... ... + a_1 \cos(\omega T) + a_2 \cos(2\omega T)}\right]\qquad(b_0>0) \end{eqnarray*}

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.
\begin{figure}\input fig/kfig2p23.pstex_t \end{figure}

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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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