Consider the
continuous-time complex one-pole
resonator with

-plane
transfer function

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:
The
amplitude response at all frequencies is given by
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 [
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
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 (

).
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