Butterworth Filter

The Butterworth filter was described by Stephen Butterworth in 1930 and remains one of the most commonly used filter topologies in audio engineering. Its defining characteristic is the maximally flat magnitude response, meaning the passband has no amplitude ripple whatsoever. The tradeoff is a slower roll-off rate compared to Chebyshev or elliptic filters of the same order. A first-order Butterworth filter rolls off at 6 dB per octave (20 dB per decade). A second-order at 12 dB per octave. A fourth-order at 24 dB per octave. The Linkwitz-Riley crossover, which is the standard in professional audio, is actually two cascaded Butterworth filters of the same order, producing an in-phase alignment at the crossover point. In speaker system design, Butterworth alignments also describe sealed enclosure behavior. A Butterworth (B2) alignment with Qtc of 0.707 gives the flattest possible response from a sealed box before it rolls off. Higher Qtc values produce a resonant peak; lower values roll off earlier but more gently. For measurement, SonaVyx's transfer function display shows filter behavior in real time. You can measure the actual crossover slopes of a loudspeaker system and compare them against the theoretical Butterworth response. Deviations indicate driver interaction issues, incorrect crossover settings, or room boundary effects. The filter's phase response is also important. A second-order Butterworth introduces 90 degrees of phase shift at the cutoff frequency and 180 degrees in the stopband. This phase behavior must be accounted for when aligning multiple drivers in a speaker system.