In the paper the problem of designing a full-order state observer for a gen
eral discrete-time linear system with unknown inputs is analyzed. Necessary
and sufficient conditions for the existence either of an asymptotic or of
a deadbeat observer are provided, and a constructive design procedure, toge
ther with some examples, are discussed. Finally, the equivalence of these c
onditions for problem solvability with those derived in previous contributi
ons by means of different approaches is explicitly proved.