Linkwitz–Riley filter

A Linkwitz–Riley (L-R) filter is an infinite impulse response filter used in Linkwitz–Riley audio crossovers. It is named after its inventors Siegfried Linkwitz and Russ Riley and was originally described in Active Crossover Networks for Noncoincident Drivers. It is also known as a Butterworth squared filter.

A Linkwitz–Riley crossover consists of a parallel combination of a low-pass and a high-pass L-R filter. These filters are typically designed by cascading two Butterworth filters, each providing a −3 dB gain at the cut-off frequency. The resulting Linkwitz–Riley filter has a −6 dB gain at the cut-off frequency. This means that when summing the low-pass and high-pass outputs, the gain at the crossover frequency is 0 dB. As a result, the crossover network behaves like an all-pass, exhibiting a flat amplitude response with a smoothly changing phase response. This is a primary advantage of L-R crossovers compared to even-order Butterworth filter crossovers, whose summed output has a +3 dB peak around the crossover frequency.

Since cascading two nth-order Butterworth filter filters creates 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 be less practical due to their complexity and an increasing peak in group delay around the crossover frequency.