Mathematical Problems in Engineering

Volume 2018, Article ID 9831378, 8 pages

https://doi.org/10.1155/2018/9831378

## Memory Distributed LMS for Wireless Sensor Networks

The School of Communication Engineering, Hangzhou Dianzi University, Hangzhou, Zhejiang, China

Correspondence should be addressed to Fangmin Xu; nc.ude.udh@nimgnafux

Received 5 September 2017; Revised 5 February 2018; Accepted 7 February 2018; Published 4 March 2018

Academic Editor: Raquel Caballero-Águila

Copyright © 2018 Fangmin Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the limited communication resource and power, it is usually infeasible for sensor networks to gather data to a central processing node. Distributed algorithms are an efficient way to resolve this problem. In the algorithms, each sensor node deals with its own input data and transmits the local results to its neighbors. Each node fuses the information from neighbors and its own to get the final results. Different from the existing work, in this paper, we present an approach for distributed parameter estimation in wireless sensor networks based on the use of memory. The proposed approach consists of modifying the cost function by adding extra statistical information. A distributed least-mean squares (-LMS) algorithm, called memory -LMS, is then derived based on the cost function and analyzed. The theoretical performances of mean and mean square are analyzed. Moreover, simulation results show that the proposed algorithm outperforms the traditional -LMS algorithm in terms of convergence rate and mean square error (MSE) performance.

#### 1. Introduction

Wireless Sensor Networks (WSNs) consist of a large number of sensor nodes that can be deployed for monitoring unattainable areas, such as deep oceans, forest fires, and air pollution [1–4]. To estimate some parameter of interest from data which is collected at nodes distributed over a geographic area, it is becoming more and more important to design and analyze estimation algorithms for the networks.

Generally, there are two kinds of estimation algorithms: centralized estimation [5–8] and distributed estimation [9–12]. In the former one, all nodes transmit their measurements to a central fusion center for processing, and the final estimate is sent back to the nodes. This method can obtain the global solution, but it requires a large amount of energy and communication. What is more, it will be nonrobust if the fusion center is out of work. In the latter one, each node only communicates with its immediate neighbor and the signal is processed locally. Compared to the centralized estimation, the distributed estimation can achieve good estimation performance whilst reducing complexity.

Due to the scalability, robustness, and low cost, distributed estimation algorithms have been receiving more and more attention. The paper [10] proposes a diffusion LMS to obtain the distributed estimation of the networks. The paper [11] proposes a distributed incremental RLS solution. The papers [12, 13] propose LMS algorithms for censor data. To the authors’ knowledge, there is no existing work dealing with the past information of the nodes for distributed estimation. Motivated by this fact, in this paper, we state a new cost function where the past information is considered in the cost function. By optimizing the proposed cost function, a diffusion memory-LMS is proposed in this paper.

This paper is organized as follows. Section 2 describes the mathematical problem of the network and memory -LMS. In Section 3, mean convergence is analyzed. Simulation results are provided in Section 4. Finally, we conclude the paper in Section 5.

*Notation. *We use boldface uppercase letters to denote matrices and boldface lowercase letters to denote vectors. We use to denote the expectation operator, for variance, for complex conjugate-transposition, for the trace of a matrix, and for Kronecker product. The notation stands for a vector obtained by stacking the specified vectors. stands for a linear transformation which converts a matrix into a column vector. Similarly, we use to denote the diagonal matrix consisting of the specified vectors or matrices. Other notations will be introduced as necessary.

#### 2. The Mathematical Problem of the Network and Memory -LMS

We consider a sensor network which is composed of sensor nodes distributed over some geographic regions and used to monitor a physical phenomenon, shown in Figure 1. At every time , each node can get a scalar measurement . Here is a 1 × 1 observations vector, and it can be expressed in terms of the following linear combinations:where is an known regression vector which is corresponding to a realization of a stochastic process . is an nonrandom unknown parameter to be estimated. is a random error in observing and is assumed to be spatially and temporally independent. and . To get a global solution of , it requires access to the information across all nodes in the network. Therefore, we explore a distributed algorithm in this section.