A consistent time-domain and frequency-domain representation for discrete-time linear time-invariant feedback systems

A framework for the consistent time-domain and frequency-domain representation of discrete-time linear time-invariant systems is established. As required, the class of signals constitute a linear space and the class of systems constitute an algebra of linear operators mapping the class of signals into itself. Furthermore, the inverses of the return difference operators exist and they themselves belong to the chosen class of systems. The formalism, amongst other things, supports the consistent analysis of discrete-time linear time-invariant feedback systems when signals are double-sided and the systems are open-loop unstable.