A procedure is presented for linear discrete system state estimation in which the observation does not contain additive white noise. The order of the system for estimation is reduced according to the number of noiseless outputs and noiseless output differences (here noiseless implies the absence of white noise). The advantage of this approach is its general application. Unlike earlier methods, it is not necessary for the noise states to be separable or even distinguishable from other system states. A systematic approach to the formulation of the required estimation model from a general system model is presented. The procedure is straightforward but somewhat lengthy to describe for the general problem. However, once the basic scheme is understood, it can be applied without undue difficulty to a large number of problems. For the modified estimation model, required filtering algorithms are derived which are of essentially the same complexity as the usual white noise algorithms. Direct estimation of the original system states, using the reduced order system for calculation of gains and the required error variances, is described. Modelling errors often occur in representing physical systems, and error analysis algorithms are derived. The use of the estimation and error analysis algorithms is demonstrated by examples and the results compared with the usual discrete (augmented states) Kalman filter approach.