Notch Filters

Brief

A second-order biquad notch filter selectively attenuates a narrow band of frequencies around a specified center while allowing others to pass, using a combination of two poles and two zeros to shape its frequency response. It is implemented with a difference equation employing both feedback and feedforward coefficients, providing precise control over gain, center frequency, and bandwidth (Q factor).

Try it

Drag the sliders to see how the magnitude and phase response change. Sample rate is 48 kHz; the dashed line marks \(f_{0}\), and the normalized coefficients update live.

Formulae

In order to construct a biquad notch filter, the sample rate, center frequency, and Q-factor need to be provided. From those, the coefficients for the underlying filter can be calculated.

First, some intermediate values are calculated:

• \(\displaystyle ω_{0} = 2π{f_{0} \over F_{s}}\)
• \(\displaystyle α = {sin(ω_{0}) \over {2Q}}\)

Where \(f_{0}\) is the center frequency, \(F_{s}\) is the sample rate, and \(Q\) is the Q-factor.

Then, the filter coefficients can be calculated:

• \(\displaystyle b_{0} = 1\)
• \(\displaystyle b_{1} = -2cos(ω_{0})\)
• \(\displaystyle b_{2} = 1\)
• \(\displaystyle a_{0} = 1 + α\)
• \(\displaystyle a_{1} = -2cos(ω_{0})\)
• \(\displaystyle a_{2} = 1 - α\)

Implementation

You can find a C++ and Rust implementation of this and other biquad filters on my GitHub page:

alex-parisi/biquad-filters

C++ and Rust implementations of every biquad filter type described here.

Notes

Since Q-factor is a unit-less value and is a little hard to quantify, sometimes it is easier to provide bandwidth instead. Bandwidth can be converted to the Q-factor by using the formula:

\(\displaystyle Q = {1 \over {2sinh(BW ⋅ {log_{10}(2) \over 2})}}\)