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It has been shown lately the optimality of uncoded transmission in estimatingGaussian sources over homogeneous/symmetric Gaussian multiple access channels(MAC) using multiple sensors. It remains, however, unclear whether it stillholds 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, wefirst provide a joint source and channel coding approach in estimating Gaussiansources over Gaussian MAC channels, as well as its sufficient and necessarycondition in restoring Gaussian sources with a prescribed distortion value. Aninteresting relationship between our proposed joint approach with a morestraightforward separate source and channel coding scheme is then established.We then formulate constrained power minimization problems and transform them torelaxed convex geometric programming problems, whose numerical results exhibitthat either separate or uncoded scheme becomes dominant over a linear topologynetwork. In addition, we prove that the optimal decoding order to minimize thetotal transmission powers for both source and channel coding parts is solelysubject to the ranking of MAC channel qualities, and has nothing to do with theranking of measurement qualities. Finally, asymptotic results for homogeneousnetworks are obtained which not only confirm the existing optimality of theuncoded approach, but also show that the asymptotic SNR exponents of thesethree approaches are all the same. Moreover, the proposed joint approach sharethe same asymptotic ratio with respect to high SNR and high SMNR as the uncodedscheme.

Energy Efficient Estimation of Gaussian Sources Over Inhomogeneous Gaussian MAC Channels