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
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 time axis in 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 ( ).