Signal detection in the presence of weakly dependent noise--II: Robust detection

The problem of designing robust systems for detecting constant signals in the presence of weakly dependent noise with uncertain statistics is considered. As in Part I of this study, which treated optimum detection in weakly dependent noise, a moving-average representation is used to model the dependence structure of the noise process, with the degree of dependence being parameterized by the averaging weights. Weak dependence is then modeled as the situation in which quantities depending to second or higher order on the averaging weights can be considered to be negligible. Uncertainty in the noise statistics is introduced within this framework by allowing a general type of uncertainty in the univariate statistics of the independent sequence that drives the moving average. To find robust detectors for signals in this type of weakly dependent noise environment, related results concerning robust location estimation in an analogous dependent situation are applied to modify a robust detection system for the corresponding independent-noise case. It is argued here that a robust detector for this dependent noise model is characterized by a least favorable noise distribution which coincides with the distribution that is least favorable for the corresponding independent-noise case. However, the resulting detector design for dependent noise differs from that for independent noise; in particular, the robust detector for dependent noise is based on a linearly corrected version of the influence curve that defines the independent-noise robust detector. The worst-case performance of the proposed robust detector relative to that of the independent-noise robust detector is also analyzed, with the conclusion that the performance of the proposed technique is better, to first order in the averaging weights, in this respect. A modification of this robust detector is also proposed which eliminates some practical disadvantages of this system while retaining equivalent performance to first order. The specific situation of contaminated Gaussian noise is treated in order to illustrate the analysis.