All Pass Filters

Digital Biquad Filtering

---

Brief:

A second-order biquad all-pass filter preserves the amplitude of all frequencies while altering their phase, using a combination of two poles and two zeros arranged symmetrically. It is implemented with a difference equation employing both feedback and feedforward coefficients, allowing precise control over the phase shift characteristics around a specified center frequency and bandwidth (Q factor). All-pass filters are often used in phase correction, group delay adjustment, and filter design applications.

---

Formulae:

In order to construct a biquad all-pass filter, the sample rate, cutoff 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:

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

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:

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

---

Implementation

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

---

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:

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