Detection of radar echoes in noise revisited

The analysis of radar detection probability in the presence of unknown parameters, given by Kelly, Reed, and Root for maximum-likelihood detection procedures, is extended in this paper to cases where the detection procedures still belong to the class of maximum-likelihood procedures, but the actual waveform, or noise covariance, or both, may differ from those for which the detection procedure would be maximum likelihood. The analysis covers detection procedures ranging from fully coherent, through partially coherent, to pulse-to-pulse incoherent cases. These cases are distinguished by changing the list of parameters treated as unknown by the detection procedure. The resulting equations for the characteristic function of the detection test variable lead to probability distribution functions which are incomplete Toronto functions or generalizations of them; this is convenient for performance analysis, since these functions have been extensively studied. The analysis is further extended to the case of signals having fluctuating amplitudes and phases, for cases ranging from pulse-to-pulse fluctuations, through partially correlated fluctuations, to the case of complete correlation for the duration of the received waveform on a single scan. In the case of partially correlated fluctuations of amplitude and phase, the situation is considered where such fluctuations impose the limit on the amount of coherent integration which can be accomplished. The analysis is brought within the Kelly, Reed, Root framework by modifying the maximum-likelihood procedure to take account of a priori knowledge of the correlation properties of the signal amplitude and phase fluctuations, so that the resulting detection procedure becomes a maximum a posteriori procedure. A classification of special cases is given, with attention focused on the question of when it is still true (as in the case considered by Kelly, Reed, and Root) that the detection probability is given by an incomplete Toronto function, and when generalizations of these must be used. Also, a classification is given of the appropriate definition of signal-to-noise ratio for various special cases.