  We consider a two sensor distributed detection system transmitting a binary
non-uniform source over a Gaussian multiple access channel (MAC). We model the
network via binary sensors whose outputs are generated by binary symmetric
channels of different noise levels. We prove an optimal one dimensional
constellation design under individual sensor power constraints which minimizes
the error probability of detecting the source. Three distinct cases arise for
this optimization based on the parameters in the problem setup. In the most
notable case (Case III), the optimal signaling design is to not necessarily use
all of the power allocated to the more noisy sensor (with less correlation to
the source). We compare the error performance of the optimal one dimensional
constellation to orthogonal signaling. The results show that the optimal one
dimensional constellation achieves lower error probability than using
orthogonal channels.
