Rigorous Interpretation of Engineering Formulae for the Convolution and the Fourier Transform Based on the Generalized Integral

Zeljko Juric,Harun Siljak

doi:10.1109/access.2022.3158929

Abstract

This paper aims at providing a framework suitable for justification of classical convolution integral and Fourier transform in many cases not covered by the usual definition of integral used for signal theory applications. Generalized functions approach from functional analysis is used, simplifying it to be approachable for engineers while retaining the rigor. The generalized functions approach results in an elegant and applicable definition of integral known before in the mathematical literature which is readily applicable in signal theory, justifying formulae usually seen as dubious and criticised for lack of rigor. The study offers a rigorous, simple and understandable definition of integral for use in analog signal theory, helping the formalization of engineering education by means of rigor. Main advantage of this approach is retaining the classical notation used in signal theory as well as its straightforward justification of key formulae in signal theory resulting from convolution and/or Fourier transform.

Highlights

T HE convolution integral and the Fourier transform are the most frequently used tools in the linear system analysis in time and frequency domain respectively [1].in some cases that often arise in theoretical considerations, their usual definitions cannot be applied without certain special precautions
The derivation based on the Fourier transform and the Convolution theorem as usually presented in engineering literature is not rigorous, because the Fourier transform of sin ωt involves singular objects like δ-function that cannot be treated rigorously in the framework of the ordinary calculus
In this paper, some problems which arise with the usual interpretation of the convolution and the Fourier transform are presented

Summary

INTRODUCTION

T HE convolution integral and the Fourier transform are the most frequently used tools in the linear system analysis in time and frequency domain respectively [1]. The derivation based on the Fourier transform and the Convolution theorem as usually presented in engineering literature is not rigorous, because the Fourier transform of sin ωt involves singular objects like δ-function that cannot be treated rigorously in the framework of the ordinary calculus Another important case when (1) cannot be applied directly is one when either f (t) or g(t), or both, contain nonintegrable singularities. That is why various different methods were introduced to justify steps involving divergent integrals in the process of Fourier transform determination, but they again rely on complex mathematical apparatus [5] In this introduction, some basic problems that arise from the usual interpretation of the definitions of the convolution and the Fourier transform are presented. Before presenting the key points of this paper, some basic facts from a highly advanced branch of mathematics known as functional analysis related to these problems will be recalled

IMPORTANT FACTS FROM THE THEORY OF DISTRIBUTIONS

CONCLUSION