Linkwitz–Riley filter

This filter type was originally described in Active Crossover Networks for Noncoincident Drivers in the Journal of the Audio Engineering Society.

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.

There is a 180° phase difference between the low-pass and high-pass output of the filter, which can be corrected by inverting one signal.

For active crossovers, inversion is usually done using a unity gain inverting op-amp.

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.