Sigma approximation
In mathematics, σ-approximation adjusts a 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:
![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] ,](../I/m/1e94bc542ff0d6b14d0b25c4a257a590b8005755.svg)
in terms of the normalized sinc function
the term
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.
See also
This article is issued from Wikipedia - version of the 5/13/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.