A Linkwitz–Riley (L-R) filter is an

infinite impulse response Infinite impulse response (IIR) is a property applying to many linear time-invariant systems that are distinguished by having an impulse response h(t) which does not become exactly zero past a certain point, but continues indefinitely. This is in ...

filter used in Linkwitz–Riley

audio crossover Audio crossovers are a type of electronic filter circuitry that splits an audio signal into two or more frequency ranges, so that the signals can be sent to loudspeaker drivers that are designed to operate within different frequency ranges. Th ...

s, named after its inventors

Siegfried Linkwitz Siegfried Linkwitz (November 23, 1935 – September 11, 2018) was a German American engineer who was noted co-inventor of the Linkwitz–Riley filter along with Russ Riley. He submitted several important technical papers to the ''Journal of t ...

and Russ Riley. This filter type was originally described in ''Active Crossover Networks for Noncoincident Drivers'' in the Journal of the Audio Engineering Society. * It is also known as a ''Butterworth squared'' filter. A Linkwitz–Riley "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 Butterworth may refer to: Places * Butterworth (ancient township), a former township centred on Milnrow, in the then Parish of Rochdale, England, United Kingdom * Butterworth, Eastern Cape, now also known as Gcuwa, a town located in South Africa ...

filters, each of which has −3 dB gain at the cut-off frequency. 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 The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...

response with a smoothly changing

phase response In signal processing, phase response is the relationship between the phase of a sinusoidal input and the output signal passing through any device that accepts input and produces an output signal, such as an amplifier or a filter. Amplifiers, f ...

. This is the biggest advantage of L-R crossovers compared to even-order

crossovers, whose summed output has a +3 dB peak around the crossover frequency. Since cascading two ''n''^th-order

filters will give a (2''n'')^th-order Linkwitz–Riley filter, theoretically any (2''n'')^th-order Linkwitz–Riley crossover can be designed. However, crossovers of order higher than 4 may have less usability due to their complexity and the increasing size of the peak in

group delay In signal processing, group delay and phase delay are delay times experienced by a signal's various frequency components when the signal passes through a system that is linear time-invariant (LTI), such as a microphone, coaxial cable, amplifie ...

around the crossover frequency.

Common types

Second-order Linkwitz–Riley crossover (LR2, LR-2)

Second-order Linkwitz–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 Sallen () is a commune in the Calvados department in the Normandy region in northwestern France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions ...

topology with a Q₀ value of 0.5. There is a 180° phase difference between the

low-pass A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...

and

high-pass A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...

output of the filter, which can be corrected by inverting one signal. In

loudspeaker A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or " ...

s this is usually done by reversing the polarity of one driver if the crossover is

passive Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive * Passive language, a language from which an interpreter works * Passivity (behavior), the condition of submitting to the influence of o ...

. For

active Active may refer to: Music * ''Active'' (album), a 1992 album by Casiopea * Active Records, a record label Ships * ''Active'' (ship), several commercial ships by that name * HMS ''Active'', the name of various ships of the British Royal ...

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 2nd-order

filters. Their slope 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 4th-order

filters.

References

Linkwitz Lab: CrossoversGlossary: Linkwitz–Riley
{{DEFAULTSORT:Linkwitz-Riley filter Linear filters Network synthesis filters Audio engineering Filter theory

Common types

Second-order Linkwitz–Riley crossover (LR2, LR-2)

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

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

See also

References