مرکز منطقه ای اطلاع رساني علوم و فناوري - Hypothesis testing of complex covariance matrices

Abstract :

Let $cal y$ be a mean zero complex stationary Gaussian signal process depending on a vector parameter $\\theta \\prime = { \\theta_{1}, \\theta_{2}, \\theta_{3} }$ whose components represent parameters of the covariance function R(r) of $cal y$ . These parameters are chosen as $\\theta_{1} = R(0), \\theta_{2} = |R( \\tau )| /R(0), \\theta_{3} =$ phase of $R( \\tau )$ , and they are simply related to the parameters of the spectral density of $cal y$ . 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 upon $N$ correlated pairs of independent observations on $cal y$ . Although the MP test that distinguishes between $\\theta = \\theta_{o}$ and the alternative hypothesis $\\theta = \\theta_{1}$ has been solved previously [11], the problem of identifying the random signals is often complicated by the fact that the signal power $\\theta_{1} = R(0)$ is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesis $\\lambda \\equiv R( \\tau )/R(0) = \\lambda _{0}$ and the composite alternative $\\lambda = \\lambda _{1}$ . In addition, the uniformly MP invariant test that distinguishes between the composite hypotheses $\\theta_{2} < _{=} | \\lambda _{o} |$ and $\\theta_{2} > | \\lambda _{0} |$ has also been found. In all cases, exact probability distributions have been obtained.