# How are poles of Bessel filter related to cutoff frequency?

For a 2nd order Bessel low pass, I have a pole-zero map which has a pole couple of the form: $p_{1/2} = a \pm jb$.

Can I deduce the cutoff frequency from the pole location? Does it contain any more information?

EDIT: Since the calculation of the cutoff frequency in the answer depends on the filter type I am wondering, if it is possible to determine the cutoff frequency for any kind of 2nd order low pass solely by the pole location (under the assumption they are complex conjugate)?

Yes - you can. There are filter tables for all the well-known lowpass approximations - including the Thomson-Bessel response. These tables list the relavant pole parameters for the various filter orders.

For second order Bessel response the relevant parameters are

Qp=0.5773 and Wp=wp/wo=1.732.

Explanation: The pole quality factor Qp is a measure for the angle ß between the pole position and the negative-real axis. We have Qp=1/2cos(ß) . This angle can also be expressed using the given values for the real part (a) and the img. part (b) of the pole.

The normalized angular frequency Wp=wp/wo is the ratio of the actual angular pole frequency wp divided by the actual cut-off frequency wo. The actual frequency wp is identical to the magnitude of the line which connects the origin and the pole position. This magnitude can also be expressed using the known values for a and b.

In this context, you should note that the above mentioned value Wp=1.7320 is based on the following definition of cut-off: The cut-off angular frequency wo is defined as the inverse of the group delay, hence: wo=1/tau.

This commonly used definition makes sense because - in most cases - the Bessel response is applied because of its time behaviour (delay filter). This is in contrast to other responses (Butterworth, Chebyshev,..) where the frequency response matters primarily.

If the cut-off frequency wo is defined using the 3dB criterion for the magnitude the following normalized value applies: Wp=wp/wo=1.2723.

• Since there are two definitions for cutoff frequency. Is there a relation which expresses wo for one definition in terms of wo of the other definition, i.e. something like $$w_{o-3dB} = \alpha \cdot w_{o-tau}$$ ? – luis Mar 17 '17 at 15:00