Optimal invariant detection of a sinusoid with unknown parameters

A uniformly most-powerful test does not exist for the problem of detecting a sinusoid of unknown amplitude, phase, and frequency in complex white Gaussian noise of unknown variance. The problem is shown to be invariant to scale and modulation transformations. Using invariance principles, it is found that the uniformly most-powerful-invariant (UMPI) test does not exist. A test is derived that is UMPI if it is given that the signal-to-noise ratio (SNR) is known. For SNR unknown, the derived test can be used as an upper performance bound for any invariant test. Suboptimal tests are typically implemented such as the generalized likelihood ratio and locally most powerful tests, which are also scale and modulation invariant. It is shown analytically that as SNR approaches zero, the locally most-powerful-invariant test performance approaches the UMPI performance bound, and as the SNR becomes large, the generalized likelihood ratio test performance approaches the performance of the UMPI bound. Computer simulation examples indicate that the generalized likelihood ratio test performance is close to the UMPI bound, particularly in the low-probability-of-false-alarm region of the receiver operating characteristic curve.