EPFL | Biomedical Imaging Group

A Note on BIBO Stability

M. Unser

IEEE Transactions on Signal Processing, vol. 68, pp. 5904–5913, September 21, 2020.

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_∞ sense if and only if its impulse response is included in the space of bounded Radon measures, which is a superset of L₁(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.

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