Peaking EQ Filters

Digital Biquad Filtering

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Brief:

A second-order biquad peaking EQ filter boosts or attenuates a narrow band of frequencies around a specified center frequency while leaving others largely unaffected. It uses a combination of two poles and two zeros to sculpt the frequency response, creating a peak or dip at the target frequency. Implemented with a difference equation employing both feedback and feedforward coefficients, it provides precise control over gain, center frequency, and bandwidth (Q factor), making it ideal for tone shaping and targeted frequency adjustments.

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Formulae:

In order to construct a biquad notch 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:

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

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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:

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