where
is the Laplace-transform variable, and
is the single complex pole. The numerator scaling
has been set to so that the frequency response is normalized to
unity gain at resonance:

Without loss of generality, we may set
, since changing
merely translates the amplitude response with respect to .
(We could alternatively define the translated frequency variable
to get the same simplification.) The squared amplitude response is now

Note that

This shows that the 3-dB bandwidth of the resonator in radians
per second is
, 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
, or

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

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 [65, pp.
216-219], and for digital poles it is the same as the
matched z transformation [65, pp. 224-226].
Applying the matched z transformation to the pole in the
plane gives the digital pole

from which we identify

and the relation between pole radius and analog 3-dB bandwidth
(in Hz) is now shown. Since the mapping
becomes
exact as , 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 (
).