This paper investigates robust filtering design problems in H-2 and H-infinity spaces for discrete-time systems subjected to parameter uncertainty which is assumed to belong to a convex bounded polyhedral domain. It is shown that, by a suitable change of variables, both design problems can be converted into convex programming problems written in terms of linear matrix inequalities (LMI). The results generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured. Then, assuming the order of the uncertain system is known, the optimal guaranteed performance H-2 and H-infinity filters are proven to be of the same order as the order of the system. Comparisons with robust filters for systems subjected to norm-bounded uncertainty are provided in both theoretical and practical settings. In particular, it is shown that under the same assumptions the results here are generally better as far as the minimization of a guaranteed cost expressed in terms of H-2 or H-infinity norms is considered. Some numerical examples illustrate the theoretical results.