In this technical brief, the problem of the estimation of a remote signal generated by a discrete-time dynamical system with periodic coefficients subject to multiplicative and additive white noise perturbations is investigated. To measure the quality of the estimation achieved by an admissible filter, we introduced a performance criterion described by the Cesaro limit of the mean square of the deviation between the estimated signal z(F)(t)and the remote signal z(t). The dimension of the state space of the admissible filters is not prefixed. The state-space representation of the optimal filter is constructed based on the unique periodic solution of a discrete-time linear equation together with the stabilizing solution of a suitable discrete-time Riccati equation with periodic coefficients.