The transfer-function form of the stationary algebraic Riccati equation is investigated. A generalized spectral factorization technique that copes with unstable systems is introduced. This factorization is used to provide an efficient way of solving the Riccati equation and to establish the exact equivalence between the time domain and the transfer-function approaches to the linear stationary filtering and the deterministic optimal control problems. The proposed method is easily extended to cope, in the filtering problem, with coloured signals and the superiority of its computational method for systems having a small number of inputs and outputs is demonstrated. Finally, an application of the transfer-function approach in determining the class of all systems that share the same optimal solution is introduced.