## 1 Introduction
While, as mentioned, generalisations of the Kalman filter for the case of correlated noise terms are available, generalisations of the RTS smoother for this case are rarely discussed in the literature. In the present paper, these generalisations will be derived using the weighted least-squares approach, both for contemporaneous-form and future-form state space models. It will also be pointed out that by switching between the contemporaneous form and future form, lag-zero cross-correlations between the two noise terms become lag-one cross-correlations, and vice versa; this will explain the fact that the corresponding generalised Kalman filter algorithms are not equivalent. The next step will then be to define a further generalised state estimation problem that contains both lag-zero and lag-one cross-correlations; it will be demonstrated that for this case, through the weighted least-squares approach, no optimal recursive filtering and smoothing algorithms can be derived. Instead, an approximate recursive algorithm will be presented, which can be obtained by merging the available Kalman filter algorithms for contemporaneous-form and future-form state space models. Related work has been presented recently by [PERSON] and coauthors (2007), although these authors arrive at a different recursive algorithm. In practical applications, it is often assumed that, from prior knowledge, it would be known that the noise terms were uncorrelated. In many cases, this assumption may actually be justified, while in other cases, it has to be regarded as an unwarranted constraint. Sometimes, the same source of randomness affects both the dynamical process itself and the observation process; as an example, [PERSON] and [PERSON] (2007) mention aircraft inertial navigation. There are also applications where the state space model is used mainly as a device for a quantitative description of the temporal correlation structure present in the original data and not as a model of an underlying physical reality, and, in such cases, it would be difficult to justify the constraint of uncorrelated noise terms. If it is decided to set the cross-correlation to zero, this should be the result of a corresponding model comparison step, i.e., by proving that a model with vanishing cross-correlation displays the best performance in modelling the data. The structure of the present paper is as follows. In Section 2, linear state space models are discussed. In Section 3, Kalman filters are discussed for the case of correlated noise terms, including the corresponding information filters. In Section 4, a recursive approach to solving a certain class of linear equations is reviewed; later, this approach is employed to solve the weighted least-squares problem. Section 5 contains the main results of the paper, including the discussion and generalisation of the weighted least-squares approach to state estimation. In Section 6, the case of both lag-zero and lag-one cross-correlations being present simultaneously is discussed. In Section 7, an example of the application of the proposed algorithms to simulated data is presented, while in Section 8 an example of their application to real-world data follows. The paper closes in Section 9 with a final discussion and conclusions. The Appendix A contains a somewhat lengthy derivation, which establishes the equivalence of the two equations arising in Section 5.