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Let be a mean zero complex stationary Gaussian signal process depending on a vector parameter whose components represent parameters of the covariance function R(r) of . These parameters are chosen as phase of , and they are simply related to the parameters of the spectral density of . This paper is concerned with the determination of most powerful (MP) tests that distinguish between random signals having different covariance functions. The tests are based upon correlated pairs of independent observations on . Although the MP test that distinguishes between and the alternative hypothesis has been solved previously , the problem of identifying the random signals is often complicated by the fact that the signal power is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesis and the composite alternative . In addition, the uniformly MP invariant test that distinguishes between the composite hypotheses and has also been found. In all cases, exact probability distributions have been obtained.