Optimality in detecting targets with unknown location

An optimal test does not exist for the problem of detecting a known target with unknown location in additive Gaussian noise. A common solution uses a generalised likelihood ratio testing (GLRT) formalism, where a maximum likelihood estimate of the unknown location parameter is used in a likelihood ratio test. The performance of this test is commonly assessed by comparing it to the ideal matched filter, which assumes the target location known in advance. This comparison is of limited utility, however, since the fact that the location is unknown has a significant effect on the detectability of the target. We demonstrate that a uniformly most powerful invariant (UMPI) optimal test exists for a specific class of unknown target location problems, where observations are discrete and shifts are defined circularly. Since this approach explicitly models the location as unknown, an assessment of the suboptimality of competing tests becomes meaningful. It is shown that for certain examples in this class the GLRT performance is negligibly different from that of the optimal test.