Sigma approximation: Difference between revisions

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Revision as of 00:16, 8 June 2006

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

A σ-approximated summation can be written as follows,''':'''

:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\left(\frac{k\pi}{m}\right)\cdot \left[a_{k} \cos \left( k\theta \right) +b_k\sin\left(k \theta \right) \right].</math>, &nbsp; in terms of the normalized [[sinc function]].

Here, the term

:<math>\mathrm{sinc}\left(\frac{k\pi}{m}\right)</math>

is the '''Lanczos &sigma; factor''', which is responsible for eliminating most of the Gibbs ringing 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.