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Sigma approximation

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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.

An m-1-term, σ-approximated summation for a series of period T can be written as follows: s ( θ ) = 1 2 a 0 + ∑ k = 1 m − 1 ( sinc ⁡ k m ) p ⋅ [ a k cos ⁡ ( 2 π k T θ ) + b k sin ⁡ ( 2 π k T θ ) ] , {\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}\cdot \left,} in terms of the normalized sinc function: sinc ⁡ x = sin ⁡ π x π x . {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.} a k {\displaystyle a_{k}} and b k {\displaystyle b_{k}} 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 ⁡ k m ) p {\displaystyle \left(\operatorname {sinc} {\frac {k}{m}}\right)^{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 {\displaystyle \operatorname {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

References

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