This paper is concerned with the H-2-optimal estimation of a linear combination of the state and of the input of a discrete-time linear time-invariant dynamic system. We reformulate such problem in terms of a set of three linear matrix inequalities (LMI) and we provide explicit formulas to Compute a family of solutions for such LMIs. This family is explicitly parameterized by a real parameter epsilon. The H-2 performance of the corresponding filter may be rendered arbitrarily close to the optimum by choosing a sufficiently small epsilon. This procedure is shown to be computationally very efficient.