Simultaneous signal detection and estimation under multiple hypotheses

Simultaneous detection and estimation under multiple hypotheses when data from only one observation interval are available, are treated on the basis of statistical decision theory. Estimation is carried out under the assumption that the signal of interest is not present with probability 1, which is necessary if detection is to be a meaningful operation. Also, we consider the case where the operations of detection and estimation are coupled. Specific detector and estimator structures are determined for the case of strong coupling when the cost of estimation error is given by a quadratic function. The detector structures are in general complex nonlinear functions of the received data. However, a detailed analysis of the Gaussian case resulted in a type of correlation detector, which correlates the received data with the least square estimators of the possible signals in the absence of uncertainty. The associated optimum estimator structure is found to be a weighted sum of least square estimators in the absence of uncertainty. Also, joint detection and estimation under multiple hypotheses is discussed for the case of a simple cost function. The estimators that result can be interpreted as generalized maximum-likelihood estimators. Finally, optimum prediction and filtering are briefly considered.