![Sigma-approximation of a Square Wave](https://upload.wikimedia.org/wikipedia/commons/b/bd/Sigma-approximation_of_a_Square_Wave_.gif)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, σ-approximation adjusts a
Fourier summation to greatly reduce the
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...
, which would otherwise occur at
discontinuities.
A σ-approximated summation for a series of period ''T'' can be written as follows:
in terms of the normalized
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
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
*
Lanczos resampling
filtering and Lanczos resampling are two applications of a mathematical formula. It can be used as a low-pass filter or used to smoothly interpolate the value of a digital signal between its samples. In the latter case it maps each sample of t ...
References
Fourier series
Numerical analysis
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