Linkwitz-Riley filter
Encyclopedia
A Linkwitz–Riley filter is an infinite impulse response
filter used in Linkwitz–Riley audio crossover
s, named after its inventors Siegfried Linkwitz
and Russ Riley, which was originally described in Passive Crossover Networks for Noncoincident Drivers in 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 nth order Butterworth
filters will give a 2nth order Linkwitz–Riley filter, theoretically any 2nth 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.
topology with a Q0 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 loudspeaker
s 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.
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.
filters.
Infinite impulse response
Infinite impulse response is a property of signal processing systems. Systems with this property are known as IIR systems or, when dealing with filter systems, as IIR filters. IIR systems have an impulse response function that is non-zero over an infinite length of time...
filter used in Linkwitz–Riley audio crossover
Audio crossover
Audio crossovers are a class of electronic filter used in audio applications. Most individual loudspeaker drivers are incapable of covering the entire audio spectrum from low frequencies to high frequencies with acceptable relative volume and lack of distortion so most hi-fi speaker systems use a...
s, named after its inventors Siegfried Linkwitz
Siegfried Linkwitz
Siegfried Linkwitz is well known as the co-inventor of the Linkwitz-Riley filter along with Russ Riley. He has submitted several important technical papers to the and other related publications, which have become foundational to modern loudspeaker theory. Examples of his recent work include...
and Russ Riley, which was originally described in Passive Crossover Networks for Noncoincident Drivers in 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
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
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
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
response with a smoothly changing phase response
Phase response
In signal processing and electrical engineering, 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....
. This is the biggest advantage of L-R crossovers compared to Butterworth
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
crossovers, whose summed output has a +3 dB peak around the crossover frequency. Since cascading two nth order Butterworth
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
filters will give a 2nth order Linkwitz–Riley filter, theoretically any 2nth 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
Group delay
Group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component...
around crossover frequency and complexity.
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 filterSallen Key filter
The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source filter topology...
topology with a Q0 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 loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
s 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 2nd-order ButterworthButterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
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 4th-order ButterworthButterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
filters.
See also
- Audio crossoverAudio crossoverAudio crossovers are a class of electronic filter used in audio applications. Most individual loudspeaker drivers are incapable of covering the entire audio spectrum from low frequencies to high frequencies with acceptable relative volume and lack of distortion so most hi-fi speaker systems use a...
- Butterworth filterButterworth filterThe Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
- Siegfried LinkwitzSiegfried LinkwitzSiegfried Linkwitz is well known as the co-inventor of the Linkwitz-Riley filter along with Russ Riley. He has submitted several important technical papers to the and other related publications, which have become foundational to modern loudspeaker theory. Examples of his recent work include...