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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