Low Shelf Filters

Digital Biquad Filtering

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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.

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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:

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

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:

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

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Implementation

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

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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:

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