In this paper we consider a linear discrete-time system. We are concerned with the existence and design of a measurement feedback compensator such that in the closed loop system any initial state is forced to zero within a prescribed number of time steps. We follow the so-called two-sided approach and we first derive conditions for the existence of a compensator that achieves the required closed loop behaviour. With these conditions we give an expression for the minimal number of time steps in which any initial state can be forced to zero by means of a measurement feedback compensator. Finally, we present a method of computing such a compensator having a reduced order.