On the relationship between the GLRT and UMPI tests for the detection of signals with unknown parameters

The generalized likelihood ratio test (GLRT) is widely used in signal processing applications such as image processing, wireless communications, medical imaging, classification, and signal detection. However, the GLRT does not have many known properties, other than that it is invariant, uniformly most powerful invariant (UMPI) for problems that fit the linear model, and asymptotically (N→∞) UMPI in general. Since it is invariant, it belongs to the class of tests for which the UMPI test is optimal. In this paper, we consider a general class of detection problems in which unknown signal parameters imply a problem invariance that can be described analytically by orthogonal subgroups. This invariance is natural for problems with unknown signal parameters and, for example, include those of the matched subspace detectors of Scharf and Friedlander. We derive the GLRT and UMPI detectors for this general signal class for the case of Gaussian noise. An expression is found that relates the two test statistics showing the UMPI statistic to be the sum of two terms, one of which is the GLRT. Using this, we find that the GLRT and UMPI tests are asymptotically equivalent as signal-to-noise ratio (SNR) approaches infinity (or as probability of false alarm approaches zero). These results are illustrated by extending an example given by Nicolls and de Jager to show the analytic relationship between the GLRT and UMPI tests. The results indicate that the performance between the tests becomes close at signal-to-noise ratios (SNRs) associated with operating points of the receiver operating curve that are typically of interest in signal detection applications.