  In this paper we establish a general first-order statistical framework for
the detection of a common signal impinging on spatially distributed receivers.
We consider three types of channel models: 1) the propagation channel is
completely known, 2) the propagation is known but channel gains are unknown,
and 3) the propagation channel is unknown. For each problem, we address the
cases of a) known noise variances, b) common but unknown noise variances, and
c) different and unknown noise variances. For all 9 cases, we establish
generalized-likelihood-ratio (GLR) detectors, and show that each one can be
decomposed into two terms. The first term is a weighted combination of the GLR
detectors that arise from considering each channel separately. This result is
then modified by a fusion or cross-validation term, which expresses the level
of confidence that the single-channel detectors have detected a common source.
Of particular note are the constant false-alarm rate (CFAR) detectors that
allow for scale-invariant detection in multiple channels with different noise
powers.
