Minimax linear smoothing for capacities: The case of correlated signals and noise

Results of an earlier paper giving minimax linear smoothers for the problem of estimating a homogeneous signal field in an additive orthogonal noise field when both have uncertain spectral properties, are extended to the case in which the signal and noise fields are arbitrarily correlated. As before, spectral uncertainty is modeled by assuming that the spectral measures of the signal and noise fields lie in classes of measures generated by two-alternating Choquet capacities. It is demonstrated that this problem admits a general solution in terms of the Huber-Strassen derivative between the capacities that generate the uncertainty sets, and that the least favorable spectra for smoothing in orthogonal noise are also the least favorable marginal spectra for smoothing in correlated noise. The resulting filter is seen to be a zonal filter that also arises as the solution to an analogous problem in (nonparametric) minimax hypothesis testing. These new results extend the applicability of minimax robust smoothing techniques to application involving signal-dependent noise phenomena, such as multipath and clutter, which are usually difficult to model precisely.