Signal detection in Gaussian noise of unknown level: An invariance application

The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to be
K(t,u) = \sigma^2 \pho(t,u)
where
\rho(t,u)
is the known form of the covariance function and
\sigma^2
is the unknown level. Classical approaches to signal detection depend on the assumption that
K(t,u)
is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test of
H_o
(signal absent) versus
H_1
(signal present) can be derived. When
\sigma^2
is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test of
H_o
versus
H_1
that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise level
\sigma^2
, since the derived test (receiver) contains an error-free estimate of
\sigma^2
. This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has a
t
-distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.