Distributed bias-compensated normalized least-mean squares algorithms with noisy input

In this paper, we study the problem of distributed normalized least-mean squares (NLMS) estimation over multi-agent networks, where all nodes collaborate to estimate a common parameter of interest. We consider the situations that all nodes in the network are corrupted by both input and output noise. This yields into biased estimates by the distributed NLMS algorithms. In our analysis, we take all the noise into consideration and prove that the bias is dependent on the input noise variance. Therefore, we propose a bias compensation method to remove the noise-induced bias from the estimated results. In our development, we first assume that the variances of the input noise are known a priori and develop a series of distributed-based bias-compensated NLMS (BCNLMS) methods. Under various practical scenarios, the input noise variance is usually unknown a priori, therefore it is necessary to first estimate for its value before bias removal. Thus, we develop a real-time estimation method for the input noise variance, which overcomes the unknown property of this noise. Moreover, we perform some main analysis results of the proposed distributed BCNLMS algorithms. Furthermore, we illustrate the performance of the proposed distributed bias compensation method via graphical simulation results.