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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