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In this paper, we use invariant tests for the detection of a complex signal with unknown phase variation and unknown amplitude in additive white Gaussian noise (AWGN). We show that in this problem, the uniformly most powerful invariant (UMPI) detector exists only if the signal-to-noise-ratio (SNR) is known. We derive the UMPI detector in known SNR and use it as the upper performance bound for any invariant test. In addition, we derive the generalized likelihood ratio (GLR) detector and evaluate its performance against the UMPI performance bound. We show that the GLR detector asymptotically approaches the UMPI test in large SNRs. Simulation results illustrate the close performances of the two detectors even at low SNRs, while in contrast of the UMPI test the SNR is unknown in the proposed GLR test. In order to understand, why the knowledge of SNR is not so important in this detection problem, we 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 equivalent with the one derived for unknown SNR, i.e., the knowledge of the SNR is not used in any of GLR tests. This reveals why the knowledge of the SNR is not so useful in this detection problem.