Butterworth Filter Design

Butterworth Filter Design is a collection of C++ classes and an accompanying suite of unit tests for designing high order Butterworth IIR and EQ filters using the bilinear transform.

The generated filter coefficients are split out into cascaded biquad sections, for easy use in any biquad or second-order section (SOS) implementation.

Features

• Lowpass, Highpass, Bandpass and Bandstop IIR & EQ filter design
• Low and High Shelving filter design
• High order Parametric boost/cut EQ filter design *Biquad and Biquad Chain implementations (for filtering audio buffers with cascaded biquad sections)
• Compact, readable and well-commented codebase
• Convert from SOS to Transfer Function coefficients
• Convoluttion function

Unit tests

As with any good audio signal processing toolkit, there are unit tests that provide basic proof of correctness. There are currently 6 primary test cases that check 113 different conditions.

Unit tests live in main.cpp and are written using the compact Catch test framework for C++.

Prerequisites

• cmake
• libsndfile: brew install libsndfile

Usage

The unit tests are a good place to start for a survey of usage.

For example, to design an 8-tap Butterworth lowpass filter with a cutoff @ 500Hz (which will generate coefficients for 4 biquad filters), running @ 44100Hz, with unity gain (1.0) execute the following:

vector <Biquad> coeffs;  // array of biquad filters (for this case, array size = 4 )
Butterworth butterworth;
bool designedCorrectly = butterworth.loPass(44100,  // fs
					    500,    // freq1
					    0,      // freq2. N/A for lowpass
					    8, 	    // filter order,
					    coeffs, // coefficient array being filled
					    1.0);   // overall gain

To generate the same set of coefficients in MATLAB (R14) as a comparison, to double-check our work, execute the following MATLAB commands:

[z, p, k] = butter(8, 500, 's');		% designs a 8-tap lowpass s-domain filter
[Zd, Pd, Kd] = bilinear(z, p, k, 44100);	% analog-to-digital filter conversion
[sos, g] = zpk2sos(Zd, Pd, Kd)			% zero-pole-gain form to second-order sections (SOS)

To convert from SOS to Transfer Function Form

The Butterworth class get the Second Order Sections form coefficients of the Butter-Worth filter bank, but you can use sos2tf() to get the Transfer Function coefficients.

vector <Biquad> coeffs;  // array of biquad filters (for this case, array size = 4 )
Butterworth butterworth;
bool designedCorrectly = butterworth.loPass(44100,  // fs
					    500,    // freq1
					    0,      // freq2. N/A for lowpass
					    8, 	    // filter order,
					    coeffs, // coefficient array being filled
					    1.0);   // overall gain
// get Transfer Function coefficients
vectord b, a;
sos2tf(coeffs, gain, b, a);
// you can use b and a now.

Other filter design repos on GitHub

• Vinnie Falco DSP Filters
• Mofr Butterworth

Terms and Conditions

For my Master's thesis, and in the course of work writing audio plugins and music apps, I've implemented a few different IIR and EQ filter design classes (e.g. RBJ's EQ Cookbook, Cheby and Butter IIRs). These classes are a fresh rewrite and should be considered unoptimized reference code, with an emphasis on clarity (and for pedagogical reasons, being able to refer to the textbook, see the math and understand the code). For this reason, in the hope they can be useful, these classes are provided under GPL v3.

Butterworth Filter Design - Copyright © 2013   iroro orife

Source code is provided under GPL v3