{{Short description|Type of electronic filter used in audio}} {{Linear analog electronic filter|filter2=hide|filter3=hide}}
[[File:Linkwitz vs Butterworth.svg|thumb|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.|300px|right]]
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''.<ref name=Linkwitz1976/><ref name=LinkwitzLab/> 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 {{val|-3|ul=decibel{{!}}dB}} gain at the cut-off frequency. The resulting Linkwitz–Riley filter has a {{val|-6|u=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 {{val|0|u=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 {{val|+3|u=dB}} peak around the crossover frequency.
Since cascading two ''n''<sup>th</sup>-order Butterworth filter filters creates a (2''n'')<sup>th</sup>-order Linkwitz–Riley filter, theoretically any (2''n'')<sup>th</sup>-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.
==Common types==
===Second-order Linkwitz–Riley crossover===
Second-order Linkwitz–Riley crossovers (LR2, LR12)<ref name=Linkwitz1976/><ref name=Bohn1989a/><ref name=Bohn1989b/><ref name=Bohn2005/> have a {{nowrap|12 dB/octave}} ({{nowrap|40 dB/decade}}) slope. They can be realized by cascading two one-pole filters or by using a Sallen Key filter topology with a Q<sub>0</sub> value of 0.5. There is a 180° phase difference between the low-pass and high-pass outputs, 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 typically achieved using a unity gain inverting op-amp.
===Fourth-order Linkwitz–Riley crossover===
Fourth-order Linkwitz–Riley crossovers (LR4, LR24)<ref name=Linkwitz1976/><ref name=Bohn1989a/><ref name=Bohn1989b/><ref name=Bohn2005/> are currently the most commonly used type of audio crossover.{{Citation needed|date=October 2025}} They are constructed by cascading two 2nd-order Butterworth filters. Their slope is {{nowrap|24 dB/octave}} ({{nowrap|80 dB/decade}}). The phase difference is 360°, meaning the two drivers appear in phase, although the low-pass section has a full period time delay.
===Eighth-order Linkwitz–Riley crossover===
Eighth-order Linkwitz–Riley crossovers (LR8, LR48)<ref name=Bohn1989a/><ref name=Bohn1989b/><ref name=Bohn2005/> have a very steep, {{nowrap|48 dB/octave}} ({{nowrap|160 dB/decade}}) slope. They can be constructed by cascading two 4th-order Butterworth filters.
==See also== {{commons category|Linkwitz–Riley filters}} *Partition of unity
==References==
{{Reflist|refs=
<ref name=Linkwitz1976>{{cite journal |last=Linkwitz |first=Siegfried H. |title=Active Crossover Networks for Noncoincident Drivers |url=http://www.aes.org/e-lib/browse.cfm?elib=2649 |journal=Journal of the Audio Engineering Society |volume=24 |issue=1 |pages=2–8 |date= February 1976 |access-date=2024-05-05 |language=en }}</ref>
<ref name=LinkwitzLab>{{cite web |last=Linkwitz |first=Siegfried H. |date=1976 |title=Active Crossover Networks for Noncoincident Drivers |website=Linkwitz Lab |url=http://www.linkwitzlab.com/JAES/jaes_papers76.htm |access-date=2024-05-05 }}</ref>
<ref name=Bohn2005>{{cite web |last=Bohn |first=Dennis |title=Linkwitz-Riley Crossovers: A Primer (RaneNote 160) |publisher=Rane Corporation |date=2005 |url=https://www.ranecommercial.com/legacy/pdf/ranenotes/Linkwitz_Riley_Crossovers_Primer.pdf |access-date=2024-05-05 }}</ref>
<ref name=Bohn1989a>{{cite web |last=Bohn |first=Dennis |title=Linkwitz-Riley Active Crossovers up to 8th-Order: An Overview (RaneNote 119) |publisher=Rane Corporation |date=1989 |url=https://schematicsforfree.com/files/Audio/Circuits/Equalizers%2C%20Filters%2C%20Loudness%20%26%20Tone%20Controls/Filters/Filters%20-%20Analog/Other/Linkwitz-Riley%20Active%20Crossovers.pdf |access-date=2025-05-25 }}</ref>
<ref name=Bohn1989b>{{cite journal |last=Bohn |first=Dennis |title=An Overview of Linkwitz-Riley Active Crossovers |journal=Sound & Video Contractor |date=1989 |issue=20 September |pages=42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64 }}</ref>
}}
==External links== *[http://www.linkwitzlab.com/crossovers.htm Linkwitz Lab: Crossovers] *[http://www.linkwitzlab.com/filters.htm Linkwitz Lab: Active Filters] *[https://web.archive.org/web/20140227064203/http://www.rane.com/note160.html Linkwitz–Riley Crossovers: A Primer] *[https://web.archive.org/web/20070310185350/http://www.sweetwater.com/expert-center/glossary/t--LinkwitzRiley Glossary: Linkwitz–Riley]
{{DEFAULTSORT:Linkwitz-Riley filter}} Category:Linear filters Category:Network synthesis filters Category:Audio engineering Category:Filter theory