3. Passive detection of a coherent source

We would like to illustrate here the importance of modeling a problem and show how it affects the performance and robustness of the result. The following four problems all concern the detection in noise of a single radiating source from the signals it generates on the sensors of an antenna array. The same problem is approached in four different ways, depending on the modeling assumptions adopted.

Certain assumptions are common; in particular, the source to be detected is unique and of perfect spatial coherence (existence of antenna variety). The signals are stationary, Gaussian, centered, and we have an observation consisting of the Fourier transforms of the signals received over durations such that we can neglect the measurement bias, and assume that the vectors ${}_{}$