Representations of linear systems

Representations of linear time-invariant discrete- time systems are discussed. A system is defined as a behavior, that is, as a family of trajectories mapping the time axis into the signal space. The following characterizations are equivalent: (i) the system is linear, time-invariant, and complete, (ii) the behavior is linear, shift-invariant, and closed, (iii) the behavior is kernel of a linear difference operator with a polynomial symbol, (iv) the behavior is kernel of a linear difference operator with a rational symbol, (v) the system allows a linear input/output representation in terms of polynomial matrices, (vi) the system allows a linear constant coefficient input/state/output representation. If the system is controllable, then the system also allows (vii) an image representation with a polynomial symbol, and (viii) an image representation with a rational symbol.