A study on interpolation and weighting function for numerical Fourier transform

Abstract

In order to mitigate the Gibbs oscillation, a very simple and effective linear mid-point interpolation method is proposed. The relationship between proposed linear mid-point interpolation in time domain and window function in numerical inverse Fourier transform is also investigated in this paper. It is proved that the linear mid-point interpolation in time domain is equivalent to the cosine window function defined as Gcos(ω)=cos(π2ωωmax), where ωmax is the maximum angular frequency used in the transform. Furthermore, the cosine window function and sinc window function (also known as sigma-factor) show the similar characteristic. A weighting order n, which is originally defined as the power to which the window function is raised, can also be applied to the interpolation method when n is an integer. The nth-time interpolation is equivalent to applying the window function [Gcos(ω)]n in frequency domain.