The Saturation Property of the Signal Restoration Method

Abstract

Fourier series and their modifications occupy an important place in signal processing, in particular, in their restoration and filtering. In foreign literature, scientists often consider the problems related to the study of a signal in the amplitude and frequency plane. In such situations, Fourier series are suggested to use. However, in some works, including foreign ones, this problem is solved using not the series themselves, but their partial sums or other methods based on them. This work shows that partial Fourier sums are not the optimal method for studying signals, including when they are restored to the continuous form. As an alternative to partial Fourier sums, we have proposed to use Fourier series summation methods in this paper. As an example of this use, the Abel-Poisson method has been considered. When comparing the efficiency of the Fourier and Abel-Poisson partial sum methods as signal representation methods, we have shown that the Abel-Poisson method allows getting better signal restoration. We have also revealed that the Abel-Poisson method is saturated and, accordingly, more predictable when being used. This allows building better signal processing algorithms than those based on the partial Fourier sums.