This paper describes a family of minimally parameterized state-space models for completely n-step observable multi-input multi-output time-varying systems and their algebraical links with equivalent input-output models. These results are then used to develop an algorithm for the realization of generic input-output sequences generated by a periodic system. A complete numerical example is also developed to illustrate all steps of the algorithm described in the paper. (C) 2003 Elsevier Ltd. All rights reserved.