Linear Differential 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 equationP(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 Lp functions and DLp 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 =∞. 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 Lp, related to a simple differential equation. We also show a digital simulation of this inverse system.