This paper is concerned with the discrete-time $H_\infty$ control problem with strictly proper measurement feedback. We derive necessary and sufficient conditions for the existence of strictly proper compensator which achieves a given $H_\infty$ norm bound. Note that contrary to the continuous time, in discrete time there might exist a suitable proper compensator but no suitable strictly proper compensator. Finally, we given an explicit formula for one controller which achieves this bound.
Keywords: Discrete time systems, Hoo control problem, strictly proper controllers.