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

I V Korytov,S E Golosov

doi:10.1088/1757-899x/142/1/012031

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.

Highlights

Fourier series is applied to expand function that describes a waveform of periodic signal
Trigonometric polynomial as a partial sum of series is a tool for modelling initial waveform [1], [2], [3]
One problem concerning truncated Fourier series consists in increasing amplitude of oscillations at jump discontinuities

Summary

Introduction

Fourier series is applied to expand function that describes a waveform of periodic signal. In any case, when step function is chosen for sampling, an algorithm of calculation of Fourier coefficients appears as a certain quadrature rule. Top-left corner model In this subsection, we introduce model with piecewise constant function, where the points of observed values are jump discontinuities. Midpoint model In this subsection, we consider a model with a piecewise constant function, where the points of observed values are points of continuity. We consider so called weighted quadrature rules which contain given tabulated function as a factor within integrand. As soon as quadrature sum is a linear combination of integrand values, its coefficients are called weights. The following methods are called as below depending on location of the node within interval of integration: if x0 a the rule is called top-left corner method; if x0.

Composite weighted quadrature rules

Conclusion