Necessary conditions for optimum distributed sensor detectors under the Neyman-Pearson criterion

Distributed signal detection schemes that are optimum under the Neyman-Pearson criterion continue to be of interest. The functional forms of these schemes can be difficult to specify, especially for cases with dependent observations from sensor to sensor. For cases with dependent observations from sensor to sensor, the optimum sensor test statistics are generally not the likelihood ratios of the sensor observations, Equations expressing the forms of the optimum sensor test statistics in terms of the other optimum test statistics and the optimum fusion rule are given, Detailed proofs of these results are given in this correspondence and have not been given previously. In some communication, radar, and sonar system problems the amplitude of the received signal may be unknown, but the signal may be known to be weak. Equations expressing the forms of the optimum sensor test statistics for such cases are given. These expressions have already been shown to be useful for interpreting and finding optimum distributed detection schemes, but detailed proofs of the type given here have not yet been given.