High Shelf Filters

Digital Biquad Filtering

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Brief:

A second-order biquad high-shelf filter boosts or attenuates frequencies above a specified cutoff point while leaving lower frequencies relatively unaffected. It employs two poles and two zeros to create a shelving response that shapes the high end of the frequency spectrum. Implemented via a difference equation with both feedback and feedforward coefficients, it allows precise control over the gain, cutoff frequency, and slope of the shelf, making it ideal for enhancing or reducing brightness in audio signals.

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Formulae:

In order to construct a biquad high 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:

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