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

• Published in: IEEE access 2022, Vol.10, p.29451-29460
• Main Authors: Juric, Zeljko, Siljak, Harun
• Format: Article
• Language: English
• Subjects: Analog signal theory; Convolution; Convolution integrals; Engineering education; Fourier transform; Fourier transforms; Functional analysis; generalized integral; Linear systems; Schwartz distribution; Tensors; Topology
• ISSN: 2169-3536; 2169-3536
• DOI: 10.1109/ACCESS.2022.3158929

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.