Sigma approximation

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: $${\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}\cdot \left[a_{k}\cos \left({\frac {2\pi k}{T}}\theta \right)+b_{k}\sin \left({\frac {2\pi k}{T}}\theta \right)\right],}$${\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}\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],} in terms of the normalized sinc function: $${\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.}$${\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.} ${\displaystyle a_{k}}${\displaystyle a_{k}} and ${\displaystyle b_{k}}${\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 $${\displaystyle \left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}}$${\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 ${\displaystyle \operatorname {sinc} }${\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 (equivalent to large amounts 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).

• Lanczos resampling

• Fourier series
• Numerical analysis
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