# Notch Filter - Bandwidth vs pole / zero location vs Q factor

I am reading up about notch filters. I understand the closer the pole is to the zero, the narrower the width gets and thus the bandwidth gets smaller. Is there an equation that links the bandwidth to the location of a pole? I have tried searching for so long and I couldn't get any useful answers.

I want to be able to determine the position of the pole given the required bandwidth and sampling rate and the frequency of the noise. I do not want to do it via trial and error

Thanks

• do you want bandwidth in Hz (relative to Nyquist) or in octaves? there is a formula that relates BW in octaves to the analog filter Q and that can be mapped to digital with bilinear transform. Commented Apr 8, 2016 at 23:08

In this paper by Nehora, "A Minimal Parameter Adaptive Notch Filter With Constrained Poles and Zeros" IEEE Trans ASSP, vol 33, No 4, 1985 link

The Bandwidth is given as $$BW=\pi(1-\rho)$$ where $\rho$ is the radius of the pole. The zero is assumed to be on the unit circle.

Excerpt from the relevant section of that paper:

And the references 1 through 3:

Another paper gives the same bandwidth and it is listed as the 3dB bandwidth - see (Yong, Kechu, Xiaofan, "A gradient algorithm for complex adaptive notch filters" - Intl Conf on Signal Processing, 2004) link.

Yet another paper with bandwidth details is by Regalia,"A Complex Adaptive Notch Filter" IEEE Sig Proc Letters, Vol 17, No 11, 2010, Link. Note that Regalia's notch filter is based on All-pass filters and so it has a different equation for bandwidth.

The relevant section from the Regalia paper: