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 "Passive Crossover Networks for Noncoincident Drivers" in [http://www.aes.org/e-lib/browse.cfm?elib=3287 JAES Volume 26 Number 3 pp. 149-150; March 1978] .It is also known as a "Butterworth squared" filter. An 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^th order Linkwitz-Riley crossover can be designed. However, crossovers of higher order than 4^th may have less usability due to their increasing peak in group delay around crossover frequency and complexity.

Common types

econd order Linkwitz-Riley crossover (LR2, LR-2)

Second order Linwkwitz-Riley crossovers (LR2) have a 12 dB/octave (40 dB/decade) slope. They can be realized by cascading two one-pole filters, or using a Sallen Key filter topology with a Q₀ value of 0.5. There's 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 op-amp.

Fourth order Linkwitz-Riley crossover (LR4, LR-4)

Fourth order Linkwitz-Riley crossovers (LR4) are probably today's most commonly used type of audio crossover. They are constructed by cascading two second order 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 low-pass section.

Eighth order Linkwitz-Riley crossover (LR8, LR-8)

Eighth order Linkwitz-Riley crossovers (LR8) have a very steep, 48 dB/octave (160 dB/decade) slope. They can be constructed by cascading two 4^th order Butterworth filters.

References

* [http://www.linkwitzlab.com/crossovers.htm Linkwitz Lab: Crossovers]
* [http://www.linkwitzlab.com/filters.htm Linkwitz Lab: Active Filters]
* [http://www.rane.com/note160.html Linkwitz-Riley Crossovers: A Primer]
* [http://www.sweetwater.com/expert-center/glossary/t--LinkwitzRiley Glossary: Linkwitz-Riley]

ee also

*Audio crossover
*Butterworth filter
*Siegfried Linkwitz