Weak limits and their calculation in analog signal theory

Abstract

Purpose

This paper aims to improve the mathematical justification of certain analog signal theory concepts and offer a rigorous framework for it.

Design/methodology/approach

The framework relies on functional analysis, namely theory of distributions and the concept of weak limit. Its notation is adjusted to resemble the notation usually used in engineering signal theory. It can be used to prove in a rigorous manner already established results in signal theory, but also to establish new ones.

Findings

Examples have shown the lack of rigour caused by using ordinary calculus in proving fundamental signal theoretic results. On that basis, concepts of limit, Fourier transform and derivative are revisited in the spirit of functional analysis. A new useful formula for weak limit computation is proved.

Originality/value

Functional analysis is efficiently used in signal theory in a manner approachable by engineers. An original and efficient formula for weak limit computation is presented and proved.

Citation

Jurić, Ž. and Šiljak, H. (2013), "Weak limits and their calculation in analog signal theory", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 32 No. 3, pp. 1009-1029. https://doi.org/10.1108/03321641311306015