Sigma approximation

Animation of the additive synthesis of a square wave with an increasing number of harmonics by way of the σ-approximation with p=1

In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.[1][2]

An m-1-term, σ-approximated summation for a series of period T can be written as follows: s(θ)=12a0+∑k=1m−1(sinc⁡km)p⋅[akcos⁡(2πkTθ)+bksin⁡(2πkTθ)], in terms of the normalized sinc function: sinc⁡x=sin⁡πxπx. ak and bk are the typical Fourier Series coefficients, and p, a non negative parameter, determines the amount of smoothening applied, where higher values of p further reduce the Gibbs phenomenon but can overly smoothen the representation of the function.

The term (sinc⁡km)p is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the sinc function to rolloff the higher frequency Fourier Series coefficients.

As is known by the Uncertainty principle, having a sharp cutoff in the frequency domain (cutting off the Fourier Series abruptly without adjusting coefficients) causes a wide spread of information in the time domain (lots of ringing).

This can also be understood as applying a Window function to the Fourier series coefficients to balance maintaining a fast rise time (analogous to a narrow transition band) and small amounts of ringing (analogous to stopband attenuation).

## See also

* Lanczos resampling

## References

1. ↑ Chhoa, Jannatul Ferdous (2020-08-01). "An Adaptive Approach to Gibbs' Phenomenon". Master's Theses. https://aquila.usm.edu/masters_theses/762.
2. ↑ Recktenwald, Steffen M.; Wagner, Christian; John, Thomas (2021-06-29). "Optimizing pressure-driven pulsatile flows in microfluidic devices" (in en). Lab on a Chip 21 (13): 2605–2613. doi:10.1039/D0LC01297A. ISSN 1473-0189. PMID 34008605.

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