  It has been shown lately the optimality of uncoded transmission in estimating
Gaussian sources over homogeneous/symmetric Gaussian multiple access channels
(MAC) using multiple sensors. It remains, however, unclear whether it still
holds for any arbitrary networks and/or with high channel signal-to-noise ratio
(SNR) and high signal-to-measurement-noise ratio (SMNR). In this paper, we
first provide a joint source and channel coding approach in estimating Gaussian
sources over Gaussian MAC channels, as well as its sufficient and necessary
condition in restoring Gaussian sources with a prescribed distortion value. An
interesting relationship between our proposed joint approach with a more
straightforward separate source and channel coding scheme is then established.
We then formulate constrained power minimization problems and transform them to
relaxed convex geometric programming problems, whose numerical results exhibit
that either separate or uncoded scheme becomes dominant over a linear topology
network. In addition, we prove that the optimal decoding order to minimize the
total transmission powers for both source and channel coding parts is solely
subject to the ranking of MAC channel qualities, and has nothing to do with the
ranking of measurement qualities. Finally, asymptotic results for homogeneous
networks are obtained which not only confirm the existing optimality of the
uncoded approach, but also show that the asymptotic SNR exponents of these
three approaches are all the same. Moreover, the proposed joint approach share
the same asymptotic ratio with respect to high SNR and high SMNR as the uncoded
scheme.
