A characterisation of discrete time linear periodic systems is presented, based on a general framework using cyclic projection operators on sequences. Two known liftings of periodic systems to a time invariant one, the monodromy and the cyclic representation, are readily derived in this framework. This approach also leads to the definition of the operational transfer function for such systems, and more generally, the operational transfer inclusion. It is shown that the cyclic realisation allows the multiplexing of individual realisations of a periodic system. These descriptions are useful in the realisation problem and the search for canonical forms. Parameterisations for the class of reachable systems are recalled, and their geometric and topological features are illustrated.