The idea of duality in control theory results from the striking symmetry between the properties of controllability and observability of linear systems. It is well known that the duality between the optimum regulator problem (ORP) and the optimum estimator problem (OEP) can be used to obtain properties of the OEP from the ORP and vice versa; moreover, computer programs designed for ORPscan be used for solving OEPsand vice versa. Emphasis is laid on the duality between a controller design technique, based on frequency–shaped performance indices, the extended optimum regulator problem, and a technique for solving the optimum state estimation problem for an LT1 system with additive coloured noise, the extended optimum estimator problem.