Robust Signal Detection and Estimation: A Geometric Approach

In this paper, we consider the problems of f2 estimation and signal detection in the presence of uncertainty regarding the statistical structure of the problem. These problems, which fall within the more general context of robust signal processing, have been studied in recent years by several investigators. In this paper, we formulate and analyze the robust estimation and detection problems in the context of reproducing kernel Hilbert space (RKHS) theory. The relationship between classical detection/estimation and RKHS theory is well-known, but as far as we know, the theory has not been applied previously to the study of robust estimation and detection. By using an RKHS approach, we are able to generalize the notion of a linear filter and to give necessary and sufficient conditions for such a filter to be robust in the minimax sense for the general L2-estimation problem in which there is uncertainty in both the covariance structure of the observed process X = X(t) and the cross-covariance structure of X and Z, the variable to be estimated. We have shown that, under mild regularity conditions, the robust filter can be found by solving a related minimization problem. We also give conditions sufficient to insure that the robust filter exists. In particular, we show that, if the Covariance structure of the observed process is assumed to be known, so that the only uncertainty is in the crosscovariance structure of X and Z, then a robust filter will always exist and can be found by solving a straightforward minimization problem. A somewhat surprising consequence of this analysis is the striking similarity between these results for the general robust f2-estimation problem and results given previously by the second author relating to robust matched filtering. Reformulating the robust matched filtering problem in an RKHS context allows us to generalize these previous results and clearly reveals the underlying similarity between the robust estimation and matched filtering problems. In fact, the structures of minimax solutions to the two problems are seen to be virtually identical. As a final application of the RKHS approach to robustness, we consider the problem of robust quadratic detection of a Gaussian signal in the presence of Gaussian noise, in which the deflection ration is used as a performance criterion. We show that this problem also can be formulated in an RKHS context, and, when the structure of the noise covariance is assumed to be known, is exactly analogous to the robust matched filtering problem. If the covariance structure of the noise is also unknown, the robust quadratic detection problem can be embedded in a larger problem, which is again analogous to the robust matched filtering problem. A robust filter for this larger problem will then possess desirable robustness properties when applied to the quadratic detection problem. The approach presented in this paper, in addition to providing a unified view of the problems discussed above, provides a formulation which may be applied to investigate robustness properties in other problems to which RKHS theory applies.