Relating Pole Radius to Bandwidth

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

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

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University