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.

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

• Lanczos resampling