High-accuracy formula for discrete calculation of fourier transforms

Abstract

The conventional formula for discrete calculation of Fourier transform is to use the rectangular rules to numerically compute the oscillatory integral appearing in the transform over a finite interval. Due to the notable impact of oscillatory kern in integral, the large error by the present formula must occur, even it will lead to wrong results, especially for analysis of the higher order spectra. In order to avoid this fatal fault and to get the better results for the original Fourier transform, this paper tries to derive the modified formula with high-accuracy to the conventional formula. The related procedure includes: (1) to approximately discretize the original signal function at the sampling points, and (2) to use piecewise linear function to represent it within each interval, then (3) to exactly calculate the oscillatory integral of Fourier transform over each interval for each spectrum. The resulted expression (called as modified formula) is particularly simple and useful since only two multipliers are needed to act on the conventional formula. The related formulae and examples are presented in detail.