  A distributed detection problem over fading Gaussian multiple-access channels
is considered. Sensors observe a phenomenon and transmit their observations to
a fusion center using the amplify and forward scheme. The fusion center has
multiple antennas with different channel models considered between the sensors
and the fusion center, and different cases of channel state information are
assumed at the sensors. The performance is evaluated in terms of the error
exponent for each of these cases, where the effect of multiple antennas at the
fusion center is studied. It is shown that for zero-mean channels between the
sensors and the fusion center when there is no channel information at the
sensors, arbitrarily large gains in the error exponent can be obtained with
sufficient increase in the number of antennas at the fusion center. In stark
contrast, when there is channel information at the sensors, the gain in error
exponent due to having multiple antennas at the fusion center is shown to be no
more than a factor of (8/pi) for Rayleigh fading channels between the sensors
and the fusion center, independent of the number of antennas at the fusion
center, or correlation among noise samples across sensors. Scaling laws for
such gains are also provided when both sensors and antennas are increased
simultaneously. Simple practical schemes and a numerical method using
semidefinite relaxation techniques are presented that utilize the limited
possible gains available. Simulations are used to establish the accuracy of the
results.
