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]]

'''σ-approximation''' adjusts a Fourier summation to eliminate the [[Gibbs phenomenon]] which would otherwise occur at discontinuities. A σ-approximated summation can be written as follows,

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>

An ''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}( \frac{k\pi}{m}\right)^{p} \cdot \left[a_{k} \cos \left( \frac{2 \pi k}{T} \theta \right) + b_k \sin \left( \frac{2 \pi k}{T} \theta \right) \right],</math>

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

<math display="block"> \operatorname{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.

The term

<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. This is sampling the right side of the main lobe of the <math>\operatorname{sinc}</math> function to rolloff the higher frequency Fourier Series coefficients.

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).

Here, the term <math>\mathrm{sinc}(\frac{k\pi}{m})</math> is the Lanczos &sigma; factor which is responsible for eliminating the Gibbs ringing phenomenon.

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}}

[[Category:Fourier series]]

[[Category:Numerical analysis]]

{{mathanalysis-stub}}