A class of estimators for optimum adaptive detection

It is shown that in the operation of optimum detection of a signal whose form or descriptive parameters are specified by nonsingular a priori probability distributions, the receiver's structure can be interpreted as an estimator of the unknown signal followed by a detector that treats the estimate as a perfectly known waveform. This result holds, under very broad conditions, for any signal and noise distribution. For some special noise distributions, however, including the Gaussian, the interpretation is found to be obviously not unique. It is shown that the Price-Kailath results on the optimum receiver for the Gaussian channel (identified as an estimator-correlator or adaptive matched filter) correspond to one of the possible interpretations of the detection operation. In addition, it is shown that in this case the resulting estimator is minimum-variance only if the signal also is a realization of a normal process. As an illustrative example of an alternative, equivalent interpretation of the detection operation, a nonminimum-variance estimator is discussed for the case of Rayleigh fading and additive Gaussian noise, and its performance is evaluated