IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 8, AUGUST 2008 3727 Decentralized Quantized Kalman Filtering With Scalable Communication Cost Eric J. Msechu, Student Member, IEEE, Stergios I. Roumeliotis, Member, IEEE, Alejandro Ribeiro, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE Abstract—Estimation and tracking of generally nonstationary Markov processes is of paramount importance for applications such as localization and navigation. In this context, ad hoc wireless sensor networks (WSNs) offer decentralized Kalman filtering (KF) based algorithms with documented merits over centralized alternatives. Adhering to the limited power and bandwidth re- sources WSNs must operate with, this paper introduces two novel decentralized KF estimators based on quantized measurement innovations. In the first quantization approach, the region of an observation is partitioned into contiguous, nonoverlapping intervals where each partition is binary encoded using a block of bits. Analysis and Monte Carlo simulations reveal that with minimal communication overhead, the mean-square error (MSE) of a novel decentralized KF tracker based on 2-3 bits comes stunningly close to that of the clairvoyant KF. In the second quantization approach, if intersensor communications can afford bits at time , then the th bit is iteratively formed using the sign of the difference between the th observation and its estimate based on past observations (up to time ) along with previous bits (up to ) of the current observation. Analysis and simu- lations show that KF-like tracking based on bits of iteratively quantized innovations communicated among sensors exhibits MSE performance identical to a KF based on analog-amplitude observations applied to an observation model with noise variance increased by a factor of . Index Terms—Decentralized state estimation, Kalman filtering, limited-rate communication, quantized observations, target tracking, wireless sensor networks. I. INTRODUCTION C ONSIDER an ad-hoc wireless sensor network (WSN) deployed to track a Markov stochastic process. Each sensor node acquires observations which are noisy linear transformations of a common state. The sensors then transmit Manuscript received August 12, 2007; revised April 14, 2008. The associate editor coordinating the review of this manuscript and approving it for publi- cation was Dr. Hongbin Li. Prepared through collaborative participation in the Communication and Networks Consortium sponsored by the U. S. Army Re- search Lab under the Collaborative Technology Alliance Program, Coopera- tive Agreement DAAD19-01-2-0011. The U.S. Government is authorized to re- produce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. Parts of this work were presented at the Asilomar Conference of Signals, Systems, and Computers, Pacific Grove, CA, November 2007, and at the IEEE Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV,April 2008. E. J. Msechu, A. Ribeiro, and G. B. Giannakis are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: emsechu@ece.umn.edu; aribeiro@ece.umn.edu; geor- gios@ece.umn.edu). S. I. Roumeliotis is with the Department of Computer Science and Engi- neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: ster- gioscs.umn.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2008.925931 observations to each other in order to form a state estimate. If observations were available at a common location, minimum mean-square error (MMSE) estimates could be obtained using a Kalman filter (KF). However, since observations are distributed in space and there is limited communication bandwidth, the ob- servations have to be quantized before transmission. Thus, the original estimation problem is transformed into decentralized state estimation based on quantized observations. The problem is further complicated by the harsh environment typical of WSNs; see e.g., [3] and [4]. Quantizing observations to estimate a parameter of interest, is not the same as quantizing a signal for later reconstruction [7]. Instead of a reconstruction algorithm, the objective is finding, e.g., MMSE optimal, estimators using quantized observations [19], [20]. Furthermore, optimal quantizers for reconstruction are, generally, different from optimal quantizers for estimation. State estimation using quantized observations is a nonlinear estimation problem that can be solved using e.g., unscented (U)KFs [11] or particle filters [5]. Surprisingly, for the case where quantized observations are defined as the sign of the in- novation (SoI) sequence, it is possible to derive a filter with complexity and performance very close to the clairvoyant KF based on the analog-amplitude observations [21]. Even though promising, the approach of [21] is limited to a particular 1-bit per observation quantizer. This paper builds on and considerably broadens the scope of [21] by addressing the middle ground between estimators based on severely quantized (1-bit) data and those based on un-quan- tized data. The end result is quantizer-estimator structures that offer desirable trade-offs between bandwidth requirements (dic- tating the number of quantization bits used for intersensor com- munications) and overall tracking performance (assessed by the mean-square state estimation error). The rest of the paper is organized as follows. Problem state- ment including the modeling assumptions are in Section II. Section III presents a quantizer-estimator based on multi-level batch quantization, whereas Section IV describes a second quantizer-estimator that relies on iterative multi-bit quantiza- tion. (For a high-level description of the batch and iterative approaches, see also Fig. 1.) Performance analysis of the iterative quantizer-estimator is also detailed in Section IV. Simulations in Section V are used to corroborate the analytical discourse and compare the two quantization approaches. Notation: Vectors (resp. matrices) are denoted using lower (upper) case bold face letters. The probability density function (pdf) of conditioned on is represented by , where denotes the random variable as well as the value it takes. The Gaussian pdf with mean and covariance ma- trix is represented as and . The probability mass function for a discrete random variable is denoted as . Estimators 1053-587X/$25.00 © 2008 IEEE