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The authors propose invariant tests for the detection of a complex signal with unknown constant amplitude and unknown phase variation in additive white Gaussian noise (AWGN). The authors show that in this problem, the uniformly most powerful invariant (UMPI) detector does exist only if the number of samples N is two. For more than two samples N ges 3, the authors derive the most powerful invariant (MPI) detector in known signal-to-noise ratio (SNR) and use its performance as the upper bound benchmark for any invariant test. In addition, the authors derive the generalised likelihood ratio (GLR) detector and evaluate its performance against the MPI performance bound. This detector is very simple and represents the ratio of the L 1-norm to the L 2-norm of the data. Simulation results illustrate the close performances of the two detectors even at low SNRs, whereas in contrast to the MPI test the SNR is not required in the proposed GLR test. In order to understand why the knowledge of SNR is not so important in this detection problem, the authors also derive the GLR test for the case of known SNR. Interestingly, the resulting GLR detector (derived for the case of known SNR) turns out to be equivalent to the one derived for unknown SNR, i.e. a knowledge of the SNR is not used in any of the GLR tests. This reveals why the knowledge of the SNR is not so useful in this detection problem.