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Hypothesis testing of complex covariance matrices

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2 Author(s)

Letcal ybe a mean zero complex stationary Gaussian signal process depending on a vector parametertheta prime = { theta_{1}, theta_{2}, theta_{3} }whose components represent parameters of the covariance function R(r) ofcal y. These parameters are chosen astheta_{1} = R(0), theta_{2} = |R( tau )| /R(0), theta_{3} =phase ofR( tau), and they are simply related to the parameters of the spectral density ofcal y. 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 uponNcorrelated pairs of independent observations oncal y. Although the MP test that distinguishes betweentheta = theta_{o}and the alternative hypothesistheta = theta_{1}has been solved previously [11], the problem of identifying the random signals is often complicated by the fact that the signal powertheta_{1} = R(0)is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesislambda equiv R( tau)/R(0) = lambda_{0}and the composite alternativelambda = lambda_{1}. In addition, the uniformly MP invariant test that distinguishes between the composite hypothesestheta_{2} <_{=} | lambda_{o} |andtheta_{2} > | lambda_{0} |has also been found. In all cases, exact probability distributions have been obtained.

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Information Theory, IEEE Transactions on  (Volume:22 ,  Issue: 1 )