On Accuracy Order of Fourier Coefficients Computation for Periodic Signal Processing Models

Abstract

The article is devoted to construction piecewise constant functions for modelling periodic signal. The aim of the paper is to suggest a way to avoid discontinuity at points where waveform values are obtained. One solution is to introduce shifted step function whose middle points within its partial intervals coincide with points of observation. This means that large oscillations of Fourier partial sums move to new jump discontinuities where waveform values are not obtained. Furthermore, any step function chosen to model periodic continuous waveform determines a way to calculate Fourier coefficients. In this case, the technique is certainly a weighted rectangular quadrature rule. Here, the weight is either unit or trigonometric. Another effect of the solution consists in following. The shifted function leads to application midpoint quadrature rules for computing Fourier coefficients. As a result the formula for zero coefficient transforms into trapezoid rule. In the same time, the formulas for other coefficients remain of rectangular type.