Low Shelf Filters

Brief

A second-order biquad low-shelf filter boosts or attenuates frequencies below a specified cutoff point while leaving higher frequencies relatively unaffected. It uses two poles and two zeros to shape the transition between the affected and unaffected frequency ranges, creating a shelving response. Implemented using a difference equation with feedback and feedforward coefficients, it provides precise control over the gain, cutoff frequency, and slope of the shelf, making it useful for tonal adjustments in audio processing.

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 low shelf filter, the sample rate, cutoff frequency, Q-factor, and gain (dB) need to be provided. From those, the coefficients for the underlying filter can be calculated.

First, some intermediate values are calculated:

• \(\displaystyle ω_{0} = 2π{f_{0} \over F_{s}}\)
• \(\displaystyle α = {sin(ω_{0}) \over {2Q}}\)
• \(\displaystyle A = 10 ^ {{gain_{dB}} \over 40}\)

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

Then, the filter coefficients can be calculated:

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

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:

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