Optimal Linear Decentralized Estimation in a Bandwidth Constrained Sensor Network Zhi-Quan Luo Dept. Elec. & Comp. Engineering University of Minnesota Minneapolis, MN 55455 Email: luozq@ece.umn.edu Georgios B. Giannakis Dept. Elec. & Comp. Engineering University of Minnesota Minneapolis, MN 55455 Email: georgios@ece.umn.edu Shuzhong Zhang Dept. Systems Eng. & Eng. Management Chinese University of Hong Kong Shatin, New Territory, Hong Kong Email: zhang@se.cuhk.edu.hk Abstract— Consider a bandwidth constrained sensor network in which a set of distributed sensors and a fusion center (FC) collaborate to estimate an unknown vector. Due to power and cost limitations, each sensor must compress its data in order to minimize the amount of information that need to be communicated to the FC. In this paper, we consider the design of a linear decentralized estimation scheme (DES) whereby each sensor transmits over a noisy channel to the FC a fixed number of real-valued messages which are linear functions of its observations, while the FC linearly combines the received messages to estimate the unknown parameter vector. Assuming each sensor collects data according to a local linear model, we propose to design optimal linear message functions and linear fusion function according to the minimum mean squared error (MMSE) criterion. We show that the resulting design problem is nonconvex and NP-hard in general, and identify two special cases for which the optimal linear DES design problem can be efficiently solved either in closed form or by Semi-definite programming (SDP). I. I NTRODUCTION Consider a distributed sensor network where data are col- lected at different sensor sites and are transmitted, after a possible compression, to a fusion center (FC) through a set of independent (non-interfering) channels. The FC (e.g., an unmanned aerial vehicle) needs these data for a specific signal processing task, such as signal detection or parameter estimation. Local data compression is achieved when each sensor sends to the FC only a summary of its data (in the form of a local message function). Upon receiving the sensor messages, the FC combines them according to a fusion rule to generate the final result. Within this framework, the tradi- tional centralized solution corresponds to the case whereby all available data are transmitted to the fusion center without data compression or channel distortion. However, if communication is costly, as is the case in wireless sensor networks, there can be a significant power-saving advantage if less information is transmitted. We may thus pose the following problem: given a fixed bandwidth budget between each sensor and the FC, how should we choose the local message functions and the final fusion rule optimally so as to maximize the overall signal processing performance? There are at least two ways to model the finite bandwidth constraint. First, we can measure the bandwidth in terms of the number of binary bits that need to be transmitted from each sensor to the FC. This bandwidth measure is natural from the digital communication point of view and was adopted widely in various prior studies on communication complexity [8], [16], [21], [25], distributed optimization [21], distributed detection [4], [18]–[20], [22], [24] and estimation [11], [12], [26]. Another measure of communication bandwidth is in terms of the number of real-valued messages that need to be sent from each sensor to the FC. Such a bandwidth measure is adopted in this paper, and we will consider analog transmission of the real-valued sensor messages, both in the presence and in the absence of channel distortions. Although distortion-less analog communication cannot be implemented in practice, it is nonetheless a useful idealization for certain types of problems. For example, most (if not all) of the parallel and distributed numerical optimization algorithms are usually described and analyzed as if real numbers can be computed and transmitted exactly [2]. In addition, there is a fair body of literature in which data are communicated and combined for the purpose of obtaining a centralized optimal estimate [17], [23]. This literature invariably assumes that real-valued messages are transmitted and received without distortion. A main motivation for using an analog communication model (with or without channel distortion) is that it opens up the possibility of applying tools from analysis and convex optimization to the design of optimal decentralized signal processing schemes. At this point, we should make an important (and natural) assumption on the local message functions: they must be continuously differentiable. This assumption is introduced in order to eliminate some uninteresting communication strate- gies when sensor channels are distortion-less. For example, if no smoothness condition is imposed, then each sensor can simply interleave the bits in the binary expansions of each component of its data vector and send the resulting real number to the FC, thus sending a single real-valued message. Upon receiving this message, the FC can easily decode it and determine all entries of the transmitted sensor data vector. Thus, with exactly one real-valued message per sensor node, the FC can recover the observations of the entire sensor network, thus trivially reducing the problem to a centralized one. Such a communication model is not interesting since it basically amounts to sending all the information collected by the sensors to the fusion center. We are interested instead