Linear Differential Equations and Related Continuous LTI Systems

Abstract

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 $$\mathscr {L}$$ such that for every input signal f yields an output $$\mathscr {L}(f)$$ which verifies $$P(D)\mathscr {L}(f ) = Q(D)f$$ . 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 $${\mathscr {D'}_{L^{p}}}$$ 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 $$p = \infty $$ . 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.