The paper is concerned with H-2 optimal reduced-order filtering for a linear discrete time : signal model, The signal. model is allowed to be unstable. The objective is : to obtain a reduced-order filter that not only gives rise to a : stable filtering error transfer function but also minimises the HT norm. The authors first derive a parameterisation of a set of filters of fixed order which lead to stable filtering error transfer functions. Such a parameterisation is given in terms of an arbitrary orthogonal projection matrix. As a consequence, the problem of minimising the Hz norm of the filtering error transfer function over the set : of; reduced-order filters is formulated as an equivalent unconstrained parametric optimisation, problem over a compact manifold. Two gradient-based : algorithms are then, proposed to compute an. optimal reduced-order filter. The good properties of the algorithms, including : the convergence property, are established theoretically as well as demonstrated numerically with an example..