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Savitzky–Golay filter

Savitzky–Golay filter is a 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 Savitzky–Golay filter rather than just read about it. In short: A Savitzky–Golay filter is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. This is achieved, in a process known as convolution, by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares.

Savitzky–Golay filter — main illustration
Savitzky–Golay filter — illustration

Key takeaways

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

Reference excerpt

A Savitzky–Golay filter is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. This is achieved, in a process known as convolution, by fitting successive sub-sets of adjacent data points with a low-degree polynomial by the method of linear least squares. When the data points are equally spaced, an analytical solution to the least-squares equations can be found, in the form of a single set of "convolution coefficients" that can be applied to all data sub-sets, to give estimates of the smoothed signal, (or derivatives of the smoothed signal) at the central point of each sub-set. The method, based on established mathematical procedures, was popularized by Abraham Savitzky and Marcel J. E. Golay, who published tables of convolution coefficients for various polynomials and sub-set sizes in 1964. Some errors in the tables have been corrected. The method has been extended for the treatment of 2- and 3-dimensional data. Savitzky and Golay's paper is one of the most widely cited papers in the journal Analytical Chemistry and is classed by that journal as one of its "10 seminal papers" saying "it can be argued that the dawn of the computer-controlled analytical instrument can be traced to this article".

Applications The data consists of a set of points { ( x j , y j ) } j = 1 n {\displaystyle \left\{(x_{j},y_{j})\right\}_{j=1}^{n}} , where x j {\displaystyle x_{j}} is an independent variable and y j {\displaystyle y_{j}} is an observed value. They are treated with a set of m {\displaystyle m} convolution coefficients, C i {\displaystyle C_{i}} , according to the expression

Y j = ∑ i = − s s C i y j + i , s + 1 ≤ j ≤ n − s , s = m − 1 2 {\displaystyle Y_{j}=\sum _{i=-s}^{s}C_{i}\,y_{j+i},\qquad s+1\leq j\leq n-s,\;\;s={\frac {m-1}{2}}}

Selected convolution coefficients are shown in the tables, below. For example, for smoothing by a 5-point quadratic polynomial, m = 5 , i = − 2 , − 1 , 0 , 1 , 2 {\displaystyle m=5,i=-2,-1,0,1,2} and the j {\displaystyle j} -th smoothed data point, Y j {\displaystyle Y_{j}} , is given by

Y j = 1 35 ( − 3 y j − 2 + 12 y j − 1 + 17 y j + 12 y j + 1 − 3 y j + 2 ) , {\displaystyle Y_{j}={\frac {1}{35}}\left(-3y_{j-2}+12y_{j-1}+17y_{j}+12y_{j+1}-3y_{j+2}\right),}

… excerpt ends here. Continue reading the full article.

Illustrations

Savitzky–Golay filter: The Savitzky-Golay smoothing being applied, passing through the data from left to right in a moving window fashion. The red line represents the local polynomial being used to fit a subset of the data. From this, the smoothed values - shown as a purple line - are derived by evaluating the polynomial at the central point of the moving window.
The Savitzky-Golay smoothing being applied, passing through the data from left to right in a moving window fashion. The red line represents the local polynomial being used to fit a subset of the data. From this, the smoothed values - shown as a purple line - are derived by evaluating the polynomial at the central point of the moving window.
Savitzky–Golay filter: Application of the smoothing to a noisy signal with Gaussian peaks via a discrete convolution with a subsequent polynomial fit at the boundaries. Compared to the sequential direct polynomial fitting shown above, this can be achieved more efficiently. The sudden drops to 0 at the edges of the filter cause discontinuities which limit the noise suppression capabilities of the Savitzky-Golay filter in practice.
Application of the smoothing to a noisy signal with Gaussian peaks via a discrete convolution with a subsequent polynomial fit at the boundaries. Compared to the sequential direct polynomial fitting shown above, this can be achieved more efficiently. The sudden drops to 0 at the edges of the filter cause discontinuities which limit the noise suppression capabilities of the Savitzky-Golay filter in practice.
Savitzky–Golay filter illustration
Savitzky–Golay filter illustration
Savitzky–Golay filter illustration

Worked examples

Example 1 — a first encounter with Savitzky–Golay filter

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

In research
Savitzky–Golay filter appears in 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 Savitzky–Golay 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
Savitzky–Golay filter is common in secondary-school and first-year university syllabi. It links to neighbouring topics Filter theory, Signal estimation, so understanding it makes those chapters shorter.
In everyday life
Look for Savitzky–Golay 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 Savitzky–Golay filter in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Savitzky–Golay 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 Savitzky–Golay filter out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Savitzky–Golay filter in simple terms?

A Savitzky–Golay filter is a digital filter that can be applied to a set of digital data points for the purpose of smoothing the data, that is, to increase the precision of the data without distorting the signal tendency. This is achieved, in a process known as convolution, by fitting successive su…

Why does Savitzky–Golay filter matter?

Because it connects several 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 Savitzky–Golay 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 Savitzky–Golay filter.

Tags

  • Filter theory
  • Signal estimation

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