Hypothesis testing of complex covariance matrices

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.