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Fast time-invariant implementations for linear least-squares smoothing filters

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4 Author(s)
Levy, B.C. ; Stanford University, Stanford, CA, USA ; Kailath, T. ; Ljung, L. ; Morf, M.

We present a new solution for the fixed interval linear least-squares smoothing of a random signal, finite dimensional or not, inadditive white noise. By using the so-called Sobolev identity of radiative transfer theory, the smoothed estimate for stationary processes is expressed entirely in terms of time-invariant causal and anticausal filtering operations; these are interpreted from a stochastic point of view as giving certain constrained (time-invariant) filtered estimates of the signal. Then by using a recently introduced notion of processes close to stationary, these results are extended in a natural way to general nonstationary processes. From a computational point of view, the representations presented here are particularly convenient, not only because time-invariant filters can be used to find the smoothed estimate, but also because a fast algorithm based on the so-called generalized Krein-Levinson recursions can be used to compute the time-invariant filters themselves.

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Automatic Control, IEEE Transactions on  (Volume:24 ,  Issue: 5 )