Proceedings PaperBackward consistency concept and a new decomposition of the error propagation dynamics in RLS algorithms
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We present here some preliminary results on a new approach for the analysis of the propagation of round-off errors in recursive algorithms. This approach is based on the concept of backward consistency. In general, this concept leads to a decomposition of the state space of the algorithm, and, in fact, to a manifold. This manifold is the set of state values that are backward consistent. Perturbations within the manifold can be interpreted as perturbations on the input data. Hence, the error propagation on the manifold corresponds exactly (without averaging or even linearization) to the propagation of the effect of a perturbation of the input data at some point in time on the state of the algorithm at future times. In this paper, we apply these ideas to the Kalman filter and its various derivatives. In particular, we consider the conventional Kalman filter, some minor variations of it, and its square-root forms. Next we consider the Chandrasekhar equations, which apply to time-invariant filtering problems. Recursive least-squares parameter (RLS) estimation is a special case of Kalman filtering and, hence, the previous results also apply to the RLS algorithms. We shall furthermore consider in detail two groups of fast RLS algorithms: the fast transversal filter (FTF) algorithms and the fast lattice fast QR (FLA/FQR) RLS algorithms.