Linkwitz–Riley filter

A Linkwitz–Riley (L-R) filter is an infinite impulse response filter used in Linkwitz–Riley audio crossovers, named after its inventors Siegfried Linkwitz and Russ Riley. This filter type was originally described in Active Crossover Networks for Noncoincident Drivers in the Journal of the Audio Engineering Society. It is also known as a Butterworth squared filter. A Linkwitz–Riley "L-R" crossover consists of a parallel combination of a low-pass and a high-pass L-R filter. The filters are usually designed by cascading two Butterworth filters, each of which has −3 dB gain at the cut-off frequency. The resulting Linkwitz–Riley filter has −6 dB gain at the cut-off frequency. This means that, upon summing the low-pass and high-pass outputs, the gain at the crossover frequency will be 0 dB, so the crossover behaves like an all-pass filter, having a flat amplitude response with a smoothly changing phase response. This is the biggest advantage of L-R crossovers compared to even-order Butterworth crossovers, whose summed output has a +3 dB peak around the crossover frequency. Since cascading two n^th-order Butterworth filters will give a (2n)^th-order Linkwitz–Riley filter, theoretically any (2n)^th-order Linkwitz–Riley crossover can be designed. However, crossovers of order higher than 4 may have less usability due to their complexity and the increasing size of the peak in group delay around the crossover frequency.