The problem of the strong stabilization with infinite gain margin, with the additional requirement of a prescribed rate of convergence of the free responses, is addressed for linear time-invariant discrete-time multivariable plants in the case when unknown different scalar gains act either on the inputs or on the outputs. Necessary and sufficient conditions for the solvability of the problem by means of a stable linear periodic discrete-time output feedback dynamic controller are derived. Algorithmic procedures are given for designing the proposed periodic controllers.