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, which was originally described in Active Crossover Networks for Noncoincident Drivers in JAES Volume 24 Issue 1 pp. 2–8; February 1976. 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 a −6 dB gain at the cutoff frequency. This means that 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 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-order Linkwitz–Riley crossover can be designed. However, crossovers of higher order than 4th may have less usability due to their complexity and increasing peak in group delay around crossover frequency.

Comparison of the magnitude response of the summed Butterworth and Linkwitz–Riley low-pass and high-pass 2nd-order filters. The Butterworth filters have a +3dB peak at the crossover frequency, whereas the L-R filters have a flat summed output.