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

$$B \isdef - \frac{\ln(R)}{\pi T}$$ (B.4)

$$R = e^{- \pi B T} \protect$$ (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

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

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

Figure B.6: Frequency response of the two-pole filter