Robust Wiener- Kolmogorov theory

A minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete-time wide-sense stationary signals when knowledge of the signal and/or noise spectra is inexact. The solution is given (under mild regularity conditions) in terms of a least favorable pair of spectra, thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily. It is noted that this design method leads, in particular, to robust <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -step predictors, robust causal filters, and robust <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -lag smoothers. The method of design is illustrated by a thorough development of the special case of one-step noiseless prediction. Further, solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise, and numerical examples are given for this case which illustrate the effectiveness of this design.