Sigma approximation: Difference between revisions

Revision as of 01:15, 7 February 2017

Revision as of 03:49, 11 February 2017

Line 1:

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

A σ-approximated summation for a series of period ''T'' can be written as follows:

:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrmoperatorname{sinc}\Bigl( \frac{k}{m}\Bigr) \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> \mathrmoperatorname{sinc}\, x = \frac{\sin \pi x}{\pi x}.</math>

theThe term

:<math>\mathrmoperatorname{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 Phenomenonphenomenon in the most extreme cases.

== See also ==