On singular and nonsingular optimum (Bayes) tests for the detection of normal stochastic signals in normal noise

The necessary and sufficient (
n
. and
s
.) conditions for the nonsingularity, i.e., regularity, and for the singularity of optimum tests for the presence of one Gaussian process vs another on a finite sample are established, for both nonstationary and stationary processes, including those with nonrational spectra. In the stationary cases, the condition may be expressed alternatively in terms of an integral of suitable spectral ratios when the random processes possess rational spectra and for certain classes of nonrational spectra as well. Equivalently, for rational spectra the
n
. and
s
. condition for nonsingularity is that the spectral ratio approach unity as frequency becomes infinite and that the spectral ratio be finite and nonzero for all frequencies, while for singularity the
n
. and
s
. condition requires that this ratio differ from unity in the limit or if unity in the limit, that this ratio vanish or be unbounded at some one (or more) finite frequencies. Some of the implications of these results in applications to signal detection are considered, and a method of solution of an associated class of integral equations, of the type \begin{equation} \int_{0-}^{T+} L(\tau, u)K(\bar u - t \bar) du = G(t, \tau), O - < t, \tau < T+ \end{equation} where
K
is a rational kernel and
G
is suitably specified, is briefly outlined. Specific results in the case of RC and LRC noise kernels, with
G
correspondingly the difference of two (different) RC or LRC covariance functions, are also given.