Relating Pole Radius to Bandwidth

Consider the continuous-time complex one-pole resonator with -plane transfer function

$\displaystyle H(s) = \frac{-\sigma_p}{s-p}.$

where $s=\sigma + j\omega$ is the Laplace-transform variable, and $p\isdef \sigma_p+j\omega_p$ is the single complex pole. The numerator scaling has been set to $-\sigma_p$ so that the frequency response is normalized to unity gain at resonance:

$\displaystyle H(j\omega_p) = \frac{-\sigma_p}{j\omega_p-\sigma_p-j\omega_p} = \frac{-\sigma_p}{-\sigma_p} = 1.$

The amplitude response at all frequencies is given by

$\displaystyle G(\omega) \isdef \left\vert H(j\omega)\right\vert = \frac{\left\v... ...\frac{\left\vert\sigma_p\right\vert}{\sqrt{(\omega-\omega_p)^2 + \sigma_p^2}}.$

Without loss of generality, we may set $\omega_p=0$ , since changing $\omega_p$ merely translates the amplitude response with respect to $\omega$ . (We could alternatively define the translated frequency variable $\nu\isdef \omega-\omega_p$ to get the same simplification.) The squared amplitude response is now

$\displaystyle G^2(\omega) = \frac{\sigma_p^2}{\omega^2+\sigma_p^2}.$

Note that

$\begin{eqnarray*} G^2(0) &=& 1 = 0 \hbox{ dB},\\ G^2(\pm\sigma_p) &=& \frac{1}{2} = - 3 \hbox{ dB}. \end{eqnarray*}$

This shows that the 3-dB bandwidth of the resonator in radians per second is $2\left\vert\sigma_p\right\vert$ , or twice the absolute value of the real part of the pole. Denoting the 3-dB bandwidth in Hz by , we have derived the relation $2\pi B = 2\left\vert\sigma_p\right\vert$ , or

$\displaystyle \zbox {B=\frac{\left\vert\sigma_p\right\vert}{\pi}=\frac{\left\vert\mbox{re}\left\{p\right\}\right\vert}{\pi}.}$

Since a

dB attenuation is the same thing as a power scaling by

, the 3-dB bandwidth is also called the half-power bandwidth.

It now remains to ``digitize'' the continuous-time resonator and show that relation Eq. $\,$ (8.7) follows. The most natural mapping of the plane to the plane is

$\displaystyle z = e^{sT},$

where

is the sampling period. This mapping follows directly from sampling the Laplace transform to obtain the z transform. It is also called the impulse invariant transformation [68, pp. 216-219], and for digital poles it is the same as the matched z transformation [68, pp. 224-226]. Applying the matched z transformation to the pole

in the

plane gives the digital pole

$\displaystyle p_d = R_d e^{j\theta_d} \isdef e^{p T} = e^{(\sigma_p+j\omega_p)T} = e^{\sigma_p T} e^{j\omega_p T}$

from which we identify

$\displaystyle R_d = e^{\sigma_p T} = e^{-\pi B T}$

and the relation between pole radius

and analog 3-dB bandwidth

(in Hz) is now shown. Since the mapping $z=e^{sT}$ becomes exact as $T\to 0$ , we have that

is also the 3-dB bandwidth of the digital resonator in the limit as the sampling rate approaches infinity. In practice, it is a good approximate relation whenever the digital pole is close to the unit circle ( $R_d \approx 1$ ).

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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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Introduction to Digital Filters
Analog Filters
Relating Pole Radius to Bandwidth