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Linkwitz–Riley filter

Linkwitz–Riley filter is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Linkwitz–Riley filter rather than just read about it. In short: 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.

Linkwitz–Riley filter — main illustration
Linkwitz–Riley filter — illustration

Key takeaways

  • Linkwitz–Riley filter belongs to computer science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Linkwitz–Riley filter to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Linkwitz–Riley filter from memory before moving on to harder problems.

Reference excerpt

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. 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 −3 dB gain at the cut-off frequency. The resulting Linkwitz–Riley filter has a −6 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 0 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 +3 dB peak around the crossover frequency. Since cascading two nth-order Butterworth filter filters creates a (2n)th-order Linkwitz–Riley filter, theoretically any (2n)th-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) have a 12 dB/octave (40 dB/decade) slope. They can be realized by cascading two one-pole filters or by using a Sallen Key filter topology with a Q0 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) are currently the most commonly used type of audio crossover. They are constructed by cascading two 2nd-order Butterworth filters. Their slope is 24 dB/octave (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) have a very steep, 48 dB/octave (160 dB/decade) slope. They can be constructed by cascading two 4th-order Butterworth filters.

See also

Partition of unity

References

External links Linkwitz Lab: Crossovers Linkwitz Lab: Active Filters Linkwitz–Riley Crossovers: A Primer Glossary: Linkwitz–Riley

Illustrations

Linkwitz–Riley filter: 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.
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.

Worked examples

Example 1 — a first encounter with Linkwitz–Riley filter

Start with the simplest possible case. Write down what Linkwitz–Riley filter claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linkwitz–Riley filter before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linkwitz–Riley filter ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linkwitz–Riley filter

In research
Linkwitz–Riley filter appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Linkwitz–Riley filter in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Linkwitz–Riley filter is common in secondary-school and first-year university syllabi. It links to neighbouring topics Audio engineering, Filter theory, Linear filters, so understanding it makes those chapters shorter.
In everyday life
Look for Linkwitz–Riley filter outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Linkwitz–Riley filter in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Linkwitz–Riley filter means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Linkwitz–Riley filter out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linkwitz–Riley filter in simple terms?

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.

Why does Linkwitz–Riley filter matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linkwitz–Riley filter?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linkwitz–Riley filter.

Tags

  • Audio engineering
  • Filter theory
  • Linear filters
  • Network synthesis filters

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