A Note on BIBO Stability

TL;DR

This paper clarifies the conditions for BIBO stability of continuous-time convolution systems, correcting misconceptions in textbooks by precisely characterizing stability in terms of impulse responses within bounded Radon measures.

Contribution

It provides a rigorous mathematical characterization of BIBO stability, including the case of the identity operator, using the space of bounded Radon measures.

Findings

BIBO stability in the $L_ abla$-sense is equivalent to the impulse response being in the space of bounded Radon measures.

Classical stability results are recovered when impulse responses are measurable functions.

The note corrects and clarifies common misconceptions in engineering literature about BIBO stability.

Abstract

The statements on the BIBO stability of continuous-time convolution systems found in engineering textbooks are often either too vague (because of lack of hypotheses) or mathematically incorrect. What is more troubling is that they usually exclude the identity operator. The purpose of this note is to clarify the issue while presenting some fixes. In particular, we show that a linear shift-invariant system is BIBO-stable in the $L_\infty$-sense if and only if its impulse response is included in the space of bounded Radon measures, which is a superset of $L_1(\mathbb{R})$ (Lebesgue's space of absolutely integrable functions). As we restrict the scope of this characterization to the convolution operators whose impulse response is a measurable function, we recover the classical statement.