Recursive least squares filter

Key takeaways

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

Reference excerpt

Recursive least squares (RLS) is an adaptive filter algorithm that recursively finds the coefficients that minimize a weighted linear least squares cost function relating to the input signals. This approach is in contrast to other algorithms such as the least mean squares (LMS) that aim to reduce the mean square error. In the derivation of the RLS, the input signals are considered deterministic, while for the LMS and similar algorithms they are considered stochastic. Compared to most of its competitors, the RLS exhibits extremely fast convergence. However, this benefit comes at the cost of high computational complexity.

Motivation RLS was discovered by Gauss but lay unused or ignored until 1950 when Plackett rediscovered the original work of Gauss from 1821. In general, the RLS can be used to solve any problem that can be solved by adaptive filters. For example, suppose that a signal d ( n ) {\displaystyle d(n)} is transmitted over an echoey, noisy channel that causes it to be received as

x ( n ) = ∑ k = 0 q b n ( k ) d ( n − k ) + v ( n ) {\displaystyle x(n)=\sum _{k=0}^{q}b_{n}(k)d(n-k)+v(n)}

where v ( n ) {\displaystyle v(n)} represents additive noise. The intent of the RLS filter is to recover the desired signal d ( n ) {\displaystyle d(n)} by use of a p + 1 {\displaystyle p+1} -tap FIR filter, w {\displaystyle \mathbf {w} } :

d ( n ) ≈ ∑ k = 0 p w ( k ) x ( n − k ) = w T x n {\displaystyle d(n)\approx \sum _{k=0}^{p}w(k)x(n-k)=\mathbf {w} ^{\mathit {T}}\mathbf {x} _{n}}

where x n = [ x ( n ) x ( n − 1 ) … x ( n − p ) ] T {\displaystyle \mathbf {x} _{n}=[x(n)\quad x(n-1)\quad \ldots \quad x(n-p)]^{T}} is the column vector containing the p + 1 {\displaystyle p+1} most recent samples of x ( n ) {\displaystyle x(n)} . The estimate of the recovered desired signal is

d ^ ( n ) = ∑ k = 0 p w n ( k ) x ( n − k ) = w n T x n {\displaystyle {\hat {d}}(n)=\sum _{k=0}^{p}w_{n}(k)x(n-k)=\mathbf {w} _{n}^{\mathit {T}}\mathbf {x} _{n}}

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Worked examples

Example 1 — a first encounter with Recursive least squares filter

Start with the simplest possible case. Write down what Recursive least squares filter claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Recursive least squares 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 Recursive least squares 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 Recursive least squares filter

In research: Recursive least squares filter appears in mathematics 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 Recursive least squares 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: Recursive least squares filter is common in secondary-school and first-year university syllabi. It links to neighbouring topics Digital signal processing, Filter theory, Statistical signal processing, so understanding it makes those chapters shorter.
In everyday life: Look for Recursive least squares 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.

How to study Recursive least squares filter in 20 minutes

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

Frequently asked questions

What is Recursive least squares filter in simple terms?

Recursive least squares (RLS) is an adaptive filter algorithm that recursively finds the coefficients that minimize a weighted linear least squares cost function relating to the input signals. This approach is in contrast to other algorithms such as the least mean squares (LMS) that aim to reduce t…

Why does Recursive least squares filter matter?

Because it connects several mathematics 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 Recursive least squares 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 Recursive least squares filter.