Resonator Bandwidth in Terms of Pole Radius
The magnitude 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 is close to 1, a reasonable
definition of 3dB-bandwidth
is provided by
where



Figure B.6 shows a family of frequency responses for the
two-pole resonator obtained by setting and varying
. The
value of
in all cases is
, corresponding to
. 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*}](http://www.dsprelated.com/josimages_new/filters/img1385.png)
where
and
.
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Two-Pole Partial Fraction Expansion
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Trigonometric Identities, Continued