Optimal invariant detection of a monochromatic plane wave with five unknown parameters

This paper analyzes the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival (DOA) in complex white Gaussian noise with unknown variance. The uniformly-most-powerful-invariant (UMPI) test is derived by using statistical invariance principle but doesn´t exist unless the signal-to-noise ratio (SNR) is known. However, it provides performance bound to evaluate any invariant test´s performance when SNR is unknown. Typically, the generalized-likelihood-ratio-test (GLRT) and locally-most-powerful-invariant (LMPI) test are derived in this paper as realizable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Simulation examples corroborate our analysis and indicate that the GLRT is close to the UMPI bound especially in the low-probability-of-false-alarm region of the receiver operating characteristic (ROC) curve while the performance of LMPI test is close to that of UMPI test in low SNR region.