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

We present a new solution for the fixed interval linear least-squares smoothing of a stationary random signal in additive white noise. By using the generalized Sobolev identity for the Fredholm resolvent of a covariance kernel, the smoothed estimate is expressed entirely in terms of time-invariant causal and anticausal filtering operations. These operations are interpreted from a stochastic point of view as giving some constrained (time-invariant) filtered estimates of the signal. From a computational pointof view, the implementation presented here is particularly convenient, not only because time-invariant filters can be used to find the smoothed estimate, but also because a fast algorithm based on Levinson recursions can be used to compute the time-invariant filters themselves.