Sigma approximation: Difference between revisions

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[[File:Sigma-approximation of a Square Wave .gif|thumb|Animation of the additive synthesis of a square wave with an increasing number of harmonics by way of the σ-approximation with p=1]]

In [[mathematics]], '''σ-approximation''' adjusts a [[Fourier series|Fourier summation]] to eliminate the [[Gibbs phenomenon]] which would otherwise occur at [[Discontinuity (mathematics)|discontinuities]].

In [[mathematics]], '''σ-approximation''' adjusts a [[Fourier series|Fourier summation]] to greatly reduce the [[Gibbs phenomenon]], which would otherwise occur at [[Discontinuity (mathematics)|discontinuities]].<ref>{{Cite journal |last=Chhoa |first=Jannatul Ferdous |date=2020-08-01 |title=An Adaptive Approach to Gibbs' Phenomenon |url=https://aquila.usm.edu/masters_theses/762 |journal=Master's Theses}}</ref><ref>{{Cite journal |last1=Recktenwald |first1=Steffen M. |last2=Wagner |first2=Christian |last3=John |first3=Thomas |date=2021-06-29 |title=Optimizing pressure-driven pulsatile flows in microfluidic devices |journal=Lab on a Chip |language=en |volume=21 |issue=13 |pages=2605–2613 |doi=10.1039/D0LC01297A |pmid=34008605 |issn=1473-0189|doi-access=free }}</ref>

AAn ''m-1''-term, σ-approximated summation for a series of period ''T'' can be written as follows:

:<math display="block">s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrmleft(\operatorname{sinc}\Bigl( \frac{k}{m}\Bigrright)^{p} \cdot \left[a_{k} \cos \Biglleft( \frac{2 \pi k}{T} \theta \Bigrright) + b_k \sin \Biglleft( \frac{2 \pi k}{T} \theta \Bigrright) \right] ,</math>

in terms of the normalized [[sinc function]] :

:<math display="block"> \mathrmoperatorname{sinc}\, x = \frac{\sin \pi x}{\pi x}.</math>

<math> a_{k} </math> and <math> b_{k} </math> are the typical Fourier Series coefficients, and ''p'', a non negative parameter, determines the amount of smoothening applied, where higher values of ''p'' further reduce the Gibbs phenomenon but can overly smoothen the representation of the function.

theThe term

:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\Bigl(\frac{k}{m}\Bigr)\cdot \left[a_{k} \cos \Bigl( \frac{2 \pi k}{T} \theta \Bigr) +b_k\sin\Bigl( \frac{2 \pi k}{T} \theta \Bigr) \right] ,</math>

<math display="block">\left(\operatorname{sinc} \frac{k}{m}\right)^{p}</math>

is the '''Lanczos σ factor''', which is responsible for eliminating most of the Gibbs phenomenon. ItThis doesis notsampling dothe soright entirely,side however,of butthe onemain canlobe squareof orthe even<math>\operatorname{sinc}</math> cube the expressionfunction to seriallyrolloff attenuatethe Gibbshigher Phenomenonfrequency inFourier theSeries most extreme casescoefficients.

As is known by the [[Uncertainty principle]], having a sharp cutoff in the frequency ___domain (cutting off the Fourier Series abruptly without adjusting coefficients) causes a wide spread of information in the time ___domain (lots of ringing).

in terms of the normalized [[sinc function]]

:<math> \mathrm{sinc}\, x = \frac{\sin \pi x}{\pi x}.</math>

the term

:<math>\mathrm{sinc}\Bigl(\frac{k}{m}\Bigr)</math>

is the '''Lanczos σ factor''', which is responsible for eliminating most of the Gibbs phenomenon. It does not do so entirely, however, but one can square or even cube the expression to serially attenuate Gibbs Phenomenon in the most extreme cases.

This can also be understood as applying a [[Window function]] to the Fourier series coefficients to balance maintaining a fast rise time (analogous to a narrow transition band) and small amounts of ringing (analogous to stopband attenuation).

== See also ==

* [[Lanczos resampling]]

==References==

{{Reflist}}

{{Unreferenced|date=January 2007}}

[[Category:Fourier series]]

[[Category:Numerical analysis]]