Formulation and Steady-state Analysis of LMS Adaptive Networks for Distributed Estimation in the Presence of Transmission Errors

This article presents the formulation and steady-state analysis of the distributed estimation algorithms based on the diffusion cooperation scheme in the presence of errors due to the unreliable data transfer among nodes. In particular, we highlight the impact of transmission errors on the least-mean squares (LMS) adaptive networks. We develop the closed-form expressions of the steady-state mean-square deviation (MSD) which is helpful to assess the effects of the imperfect information flow on on the behavior of the diffusion LMS algorithm in terms of the steady-state error. The model is then validated by performing Monte Carlo simulations. It is shown that local and global MSD curves are not necessarily monotonic increasing functions of the error probability. We also assess sufficient conditions that ensure mean and mean-square stability of diffusion LMS strategies in the presence of transmission errors. Moreover, issues such as scalability in the sense of network size and regressor size, spatially correlated observations, as well as the effect of the distribution of the noise variance are studied. While the proposed theoretical framework is general in the sense that it is not confined to a particular source of error during information diffusion, for practical reasons we additionally study a specific scenario where errors occur at the medium access control (MAC) level. We develop a model to quantify the MAC-level transmission errors according to the network topology and system parameters for a set of nodes employing a backoff procedure to access the channel. To overcome the problem of unreliable data exchange, we propose an enhanced combining rule that can be deployed in order to improve the performance of diffusion estimation algorithms by using the knowledge of the properties of the transmission errors. Index Terms Adaptive networks, diffusion LMS algorithm, MAC layer, distributed estimation, distributed signal processing. The authors are with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 0E9, Canada. E-mail: {[email protected]; [email protected]}. A preliminary version of this work considering a two-node network has appeared as a conference paper in the Proceedings of Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, November 2012. This work was supported by Hydro-Quebec, the Natural Sciences and Engineering Research Council of Canada and McGill University in the framework of the NSERC/Hydro-Quebec/McGill Industrial Research Chair in Interactive Information Infrastructure for the Power Grid. ar X iv :1 31 0. 73 68 v1 [ cs .S Y ] 2 8 O ct 2 01 3