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Relating Pole Radius to Bandwidth

Consider the continuous-time complex one-pole resonator with $ s$-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 $ B$, 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 $ -3$ dB attenuation is the same thing as a power scaling by $ 1/2$, the 3-dB bandwidth is also called the half-power bandwidth. It now remains to ``digitize'' the continuous-time resonator and show that relation Eq.$ \,$(8.7) follows. The most natural mapping of the $ s$ plane to the $ z$ plane is

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

where $ T$ 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 $ p$ in the $ s$ 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 $ R_d$ and analog 3-dB bandwidth $ B$ (in Hz) is now shown. Since the mapping $ z=e^{sT}$ becomes exact as $ T\to 0$, we have that $ B$ 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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