Linear dierential equations and related continuous LTI systems

In this paper, we consider the problem, usual in analog signal processing, to find a continuous linear time-invariant system related to a linear differential equation P(D) x = Q(D) f, i.e. a system L such that for every input signal f yields an output L(f) which verifies P(D) L(f) = Q(D) f. We give a systematic theoretical analysis of the existence and uniqueness of such systems (both causal and non-causal ones) defined on L-p functions and D'(Lp) distributions (input spaces which include signals with not necessarily left-bounded support), for every p. More precisely, by finding all their possible impulse responses, we characterise all these systems apart two pathologies arising when p = infinity. Finally, we give necessary and sufficient conditions on P, Q for causality and stability of the systems. As an application, we consider the problem of finding the inverse of a causal continuous linear time-invariant system, defined on L-p, related to a simple differential equation. We also show a digital simulation of this inverse system.