  This paper is concerned with decentralized estimation of a Gaussian source
using multiple sensors. We consider a diversity scheme where only the sensor
with the best channel sends their measurements over a fading channel to a
fusion center, using the analog amplify and forwarding technique. The fusion
centre reconstructs an MMSE estimate of the source based on the received
measurements. A distributed version of the diversity scheme where sensors
decide whether to transmit based only on their local channel information is
also considered. We derive asymptotic expressions for the expected distortion
(of the MMSE estimate at the fusion centre) of these schemes as the number of
sensors becomes large. For comparison, asymptotic expressions for the expected
distortion for a coherent multi-access scheme and an orthogonal access scheme
are derived. We also study for the diversity schemes, the optimal power
allocation for minimizing the expected distortion subject to average total
power constraints. The effect of optimizing the probability of transmission on
the expected distortion in the distributed scenario is also studied. It is seen
that as opposed to the coherent multi-access scheme and the orthogonal scheme
(where the expected distortion decays as 1/M, M being the number of sensors),
the expected distortion decays only as 1/ln(M) for the diversity schemes. This
reduction of the decay rate can be seen as a tradeoff between the simplicity of
the diversity schemes and the strict synchronization and large bandwidth
requirements for the coherent multi-access and the orthogonal schemes,
respectively.
