Sigma approximation: Difference between revisions

Revision as of 11:42, 3 July 2006

Revision as of 15:16, 3 July 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 for a series of period T can be written as follows''':'''

:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\left(\frac{\pi k}{m}\right)\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>, &nbsp; in terms of the normalized [[sinc function]] (where<math> T\mathrm{sinc} isx the= period\frac{\sin of\pi thex}{\pi Fourier Seriesx}</math>).

Here, the term

:<math>\mathrm{sinc}\left(\frac{\pi k}{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.