Does analog transmission achieve OPTA in an asymmetric Gaussian sensor network ?

We consider the estimation of a Gaussian source by a Gaussian sensor network where L distributed sensors transmit noisy observations of the source through a Gaussian multiple access channel to a fusion center. In a recent work, Gastpar showed that for a symmetric sensor network with no fading, analog (uncoded) transmission achieves the optimal performance theoretically attainable (OPTA). In this work, by comparing lower and upper bounds on the OPTA, we provide optimality conditions for analog transmission in an asymmetric Gaussian sensor network with deterministic fading. We also obtain an optimal power allocation scheme to minimize the mean-squared error distortion given a linear combination of powers (LCP) constraint. We determine optimality conditions for analog transmission under an LCP constraint, which includes the sum-power constraint as a special case.