The design of optimal fixed-configuration filters for estimating, without bias, a linear transformation of the state of a dynamic system in the presence of coloured measurement noise is studied. The performance index employed is the integral of error variance, and this leads to a matrix variational problem in the filter gains. It is demonstrated that the optimal time-varying filter gains may be computed by solving this variational problem using standard conjugate gradient algorithms. Consideration is also given to both algebraic and differential approximation of the optimal filter gains. The problem is formulated as one of ordinary minimization in parameters defining the approximation, and the required gradient expressions are derived for its solution using standard function minimization algorithms. An illustrative numerical example is presented.