On the detection of known binary signals in Gaussian noise of exponential covariance

The optimum test statistic for the detection of binary sure signals in stationary Gaussian noise takes a particularly simple form, that of a correlation integral, when the solution, denoted by
q(t)
, of a given integral equation is well behaved
(L_{2})
. For the case of a rational noise spectrum, a solution of the integral equation can always be obtained if delta functions are admitted. However, it cannot be argued that the test statistic obtained by formally correlating the receiver input with a
q(t)
which is not
L_{2}
is optimum. In this paper, a rigorous derivation of the optimum test statistic for the case of exponentially correlated Gaussian noise
R(\tau) = \sigma^{2} e^{-\alpha|\tau|}
is obtained. It is proved that for the correlation integral solution to yield the optimum test statistic when
q(t)
is not
L_{2}
, it is sufficient that the binary signals have continuous third derivatives. Consideration is then given to the case where a, the bandwidth parameter of the exponentially correlated noise, is described statistically. The test statistic which is optimum in the Neyman-Pearson sense is formulated. Except for the fact that the receiver employs
\alpha_{\infty}
(which in general depends on the observed sample function) in place of
\alpha
, the operations of the optimum detector are unchanged by the uncertainty in
\alpha
. It is then shown that almost all sample functions can be used to yield a perfect estimate of
\alpha
. Using this estimate of
\alpha
, a test statistic equivalent to the Neyman-Pearson statistic is obtained.