Sandberg's Representation Theorem and Classification of Linear Systems

It is shown that the theorem on representation of linear time-invariant system input-output maps recently published by Sandberg can be used to formulate an interesting classification of the known linear system maps. A certain property of a system map defined in this theorem and closely related to the notion of approximately-finite or vanishing memory can be also used as the system classifier. Restricting in it to consideration of only causal systems, we can distinguish between three cases: systems with no memory, with vanishing memory, and with infinitely long memory. Each of these cases is specific with regard to possessing or not possessing its own representation.