Linear coherent decentralized estimation

We consider the distributed estimation of an unknown vector signal in a bandwidth constrained sensor network with a fusion center (FC). Due to power and bandwidth limitations, each sensor compresses its data in order to minimize the amount of information that needs to be communicated to the FC. In this context, we design a linear decentralized estimation scheme (DES), where each sensor linearly encodes its observations before the transmission to the FC, which performs a minimum mean squared error (MMSE) estimation for the unknown vector signal based on the received messages. When the channels between sensors and the FC are orthogonal, it has been shown previously that the complexity of designing the optimal encoding matrices is NP-hard in general. In this paper, we study the optimal design of linear DES for the case of non-orthogonal multiple access channel (MAC) under both bandwidth and power constraints. We show that when the MAC between sensors and the FC is noiseless, the resulting problem has a closed-form solution, while in the noisy MAC case, the problem can be efficiently solved by semi-definite programming (SDP).