Linkwitz–Riley filter
Linear analog electronic filters 


Simple filters 
A Linkwitz–Riley (LR) 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. 28; February 1976. It is also known as a Butterworth squared filter. An LR crossover consists of a parallel combination of a lowpass and a highpass LR filter. The filters are usually designed by cascading two Butterworth filters, each of which has −3 dB gain at the cutoff frequency. The resulting Linkwitz–Riley filter has a −6 dB gain at the cutoff frequency. This means that summing the lowpass and highpass outputs, the gain at the crossover frequency will be 0 dB, so the crossover behaves like an allpass filter, having a flat amplitude response with a smoothly changing phase response. This is the biggest advantage of LR 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 2nd order Linkwitz–Riley filter, theoretically any 2nd order Linkwitz–Riley crossover can be designed. However, crossovers of higher order than 4th may have less usability due to their increasing peak in group delay around crossover frequency and complexity.
Common types
Second order Linkwitz–Riley crossover (LR2, LR2)
Secondorder Linkwitz–Riley crossovers (LR2) have a 12 dB/octave (40 dB/decade) slope. They can be realized by cascading two onepole filters, or using a Sallen Key filter topology with a Q_{0} value of 0.5. There is a 180° phase difference between the lowpass and highpass output of the filter, which can be corrected by inverting one signal. In loudspeakers this is usually done by reversing the polarity of one driver if the crossover is passive. For active crossovers inversion is usually done using a unity gain inverting opamp.
Fourth order Linkwitz–Riley crossover (LR4, LR4)
Fourthorder Linkwitz–Riley crossovers (LR4) are probably today's most commonly used type of audio crossover. They are constructed by cascading two 2ndorder Butterworth filters. Their steepness is 24 dB/octave (80 dB/decade). The phase difference amounts to 360°, i.e. the two drives appear in phase, albeit with a full period time delay for the lowpass section.
Eighth order Linkwitz–Riley crossover (LR8, LR8)
Eighthorder Linkwitz–Riley crossovers (LR8) have a very steep, 48 dB/octave (160 dB/decade) slope. They can be constructed by cascading two 4thorder Butterworth filters.
See also
Wikimedia Commons has media related to Linkwitz–Riley filters. 
References
 Linkwitz Lab: Crossovers
 Linkwitz Lab: Active Filters
 Linkwitz–Riley Crossovers: A Primer
 Glossary: Linkwitz–Riley