  New problems arise when the standard theory of joint detection and estimation
is applied to a set of signals drawn from a continuous family; decision
thresholds must be determined as a function of the continuous parameter x
characterizing the signals, and false alarms occur, not with a discrete
probability, but with a density in x. A Bayes decision structure over the
domain of signal parameters yields a state estimate of the signal parameter x
as an integral part of a signal declaration. The decision criterion is
converted to a form in which detection and false alarm densities appear and
from which is derived a relation between them for all x. The limiting case of
additive Gaussian noise and a high detection threshold allows a simplified
decision criterion and a state estimate of signal location in x that approaches
the Cramer-Rao bound. Also in this limit, an analytic form for the false alarm
probability density over x, a quantity not readily obtained in general, is
evaluated here through its relation to the detection probability. The false
alarm density expression and state accuracy prediction are tested through Monte
Carlo simulations, and the comparison demonstrates excellent agreement.
