A robust diffusion algorithm using logarithmic hyperbolic cosine cost function for channel estimation in wireless sensor network under impulsive noise environment

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Abstract

In a wireless sensor network (WSN), the performance of the error-squared based adaptive channel estimation algorithm degrades in the presence of impulsive noise. Robust methods are used to minimize the impulsive noise impact at the cost of a slow convergence rate. We propose a robust diffusion adaptive channel estimation algorithm using the logarithmic hyperbolic cosine cost function assuming that the received signal at each node is corrupted by impulsive noise. The sensor nodes experiencing the common channel with the base station are grouped using their initial channel estimates. After grouping, the robust diffusion algorithm is applied in each group to estimate the corresponding channel. The steady-state mean square deviation (MSD) and excess mean square error (EMSE) of the robust algorithm are derived. The algorithm's steady-state and convergence rate performance are simulated for Rayleigh fading channel corrupted by Bernoulli-Gaussian distributed impulsive noise. The simulation results demonstrate that the proposed algorithm gives faster convergence without compromising steady-state performance compared to other robust algorithms.

Introduction

In a wireless sensor network (WSN), each sensor node communicates among themselves and the base station to perform different tasks like environmental monitoring, precision agriculture, source localization, and tracking [1]. In the WSN, the channel between the sensor node and the base station suffers multipath fading and is corrupted with noise. The receiver needs to have precise channel information to combat the effects of multipath and noise effect on the channel [2], [3]. The gradient-based adaptive algorithms like least mean square (LMS) and its variants are popularly used to estimate the channel in an additive white Gaussian noise environment [4]. However, the WSN channel is also affected by impulsive noise due to environmental and other mechanical effects [5]. The presence of impulsive noise in the wireless channel degrades the steady-state performance of conventional error squared-based adaptive algorithms [6]. It is because the error term in the weight update equation becomes significant in the presence of impulsive noise that causes the estimation algorithm to diverge. The divergence issue is mitigated using l1 or mixed norm of the error as a cost function [7], [8].

The adaptive channel estimation algorithm uses different robust cost functions to mitigate the harsh effect of impulsive noise. Among the robust cost functions, Huber's cost function is popularly used, which switches the cost function between l1 and l2 norm of the error, based on a defined threshold [9]. Its efficiency depends on the choice of the threshold value [10]. The disadvantage of l1 and the mixed norm-based cost function is its slow convergence rate and sensitive to the threshold, respectively. The convergence rate of the robust algorithms can be enhanced using logarithmic-based cost functions like least mean logarithmic square (LMLS) and logarithmic cost least mean absolute (LCLMA) algorithms [11], [12]. Subsequently, other robust methods such as maximum correntropy criteria (MCC), minimum kernel risk-sensitive loss (MKRSL), sigmoidal LMS (SLMS) cost functions are proposed to mitigate the effect of impulsive noise [13], [14], [15]. The MCC cost function has the advantage of a faster convergence rate with a lower steady-state error than the LCLMA cost function. The MCC uses an exponential term that causes more computational complexity and steady-state misalignment, which is minimized with the maximum Versoria criterion (MVC) algorithm. It has lower computational complexity compared to MCC algorithm [16]. The aforementioned cost function-based robust adaptive algorithms give satisfactory steady-state error compared to a non-robust cost function. However, these algorithms provide a slower convergence rate with an increase in the percentage of impulsive noise. The robust algorithm using logarithmic hyperbolic cosine cost function is insensitive to the percentage of the impulsive noise in the channel [17], [18]. It combines mean square error (MSE) and mean absolute error (MAE), to enhance the steady-state performance in an impulsive noise channel. Hence, the proposed algorithm considers the use of logarithmic hyperbolic cosine cost function for channel estimation in the presence of impulsive noise.

In a resource-constrained WSN, the channel estimation algorithm needs to be adaptive to the environmental changes, energy-efficient, and robust. The energy efficiency is enhanced using different distributed strategies like an incremental, consensus, and diffusion by further improving the convergence rate [19], [20]. The incremental cooperation is based on a Hamiltonian cycle; hence, a single link failure could result in complete network failure. Although the consensus strategy is more robust than the incremental strategy, it is unstable at some instances, even though the individual nodes of the network are stable [21]. The instability issue can be resolved using a fully decentralized diffusion strategy with added advantages such as robust to link failure, adaptive to the change in the environment, low steady-state error with faster convergence. Although there has been extensive study on the application of distributed algorithms for parameter estimation, its use in WSN channel estimation is very limited. Incremental sensor node cooperation is used for estimating the Rayleigh fading channel [22]. Similarly, diffusion cooperation is being used in distributed channel estimation [23]. In both the distributed algorithms, the noise is considered as additive white Gaussian. Recently, Lorentzian adaptive filter (LAF) and Geman cost function based algorithms are proposed for distributed channel estimation in a non-Gaussian noise environment [5], [24], [25]. The current work proposes a diffusion-based distributed algorithm with a robust logarithmic hyperbolic cosine cost function for channel estimation to combat the effect of impulsive noise in the channel.

The major contributions of the paper are as follows:

  • A robust diffusion logarithmic hyperbolic cosine adaptive filtering (DLHCAF) algorithm is proposed for estimating the common channel experienced by individual nodes of each group in an impulsive noise environment.

  • The analytical formulation for steady-state mean square deviation (MSD) and excess mean square error (EMSE) of the proposed algorithm is carried out and validated with the simulation results.

  • The steady-state and convergence performance of the proposed algorithm is compared with other robust algorithms such as MVC [16], LAF [5], LCLMA [12] in the distributed environment. Further, it is also compared with the non-robust diffusion LMS (DLMS) [20] and non-cooperative (NC) version of DLHCAF (LHCAF-NC).

The remaining paper is framed as follows. Section 2 presents distributed channel estimation model with reference to the proposed method. The analytical formulation for the steady-state performance is carried out in Section 3. Section 4 deals with the results and analysis, followed by conclusions in Section 5.

Section snippets

Distributed channel estimation

The current research focuses on channel estimation with a distributed framework in a multipath fading environment with impulsive noise. The type of multipath fading can be deduced from the distribution of channel gain. Its nature is determined by the behavior of the channel gain from an observation over a long enough time window to obtain the statistics. A non-line-of-sight transmission is inferred if the channel coefficients follow a Rayleigh distribution [2]. The channel estimate for Rayleigh

Theoretical analysis of the proposed DLHCAF algorithm

This section presents, an analytical study on the mean and mean-square stability of the proposed algorithm. The update equation defined in (16) can be rewritten concisely ashˆk(n)=φk(n1)+μkρk(n)ek(n)xk(n), where ρk(n) is the robust weighted error parameter defined asρk(n)=tanh(λek(n))ek(n). In order to analyze the performance of the algorithm analytically for each group we need to extend the analysis to a global environment, where the global data model is defined asd(n)=X(n)h+ϑ(n). The global

Results and analysis

This section analyzes the proposed robust distributed algorithm's (DLHCAF) steady-state and transient performance in an impulsive noise environment. The derived analytical expressions for steady-state MSD and EMSE in (50) and (51) are also verified. A simulation setup is created in Matlab with the following specifications. A connected network with fifteen nodes is considered, with three clusters of five nodes each, where the channel information for each cluster are independent with each other.

Conclusions

In this paper, we proposed a robust and distributed adaptive algorithm to estimate the channel coefficients of a wireless sensor network in the presence of impulsive noise. The robustness was achieved using the logarithmic hyperbolic cosine cost function. Diffusion-based cooperation among sensor nodes was incorporated to estimate the channel coefficients in a distributed environment. The analytical formulation for steady-state mean squared deviation (MSD) and excess mean squared error (EMSE) of

CRediT authorship contribution statement

Bishnu Prasad Mishra: Methodology, Software, Writing – original draft. Trilochan Panigrahi: Conceptualization, Writing – review & editing. Annet Mary Wilson: Software, Writing – review & editing. Samrat L. Sabat: Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Bishnu Prasad Mishra received his M.Tech. in Electronics and Communication Engineering from Biju Patnaik University of Technology Rourkela, India, in 2015. He is currently pursuing the Ph.D. degree in Electronics and Communication Engineering at National Institute of Technology, Goa, India. His research interest includes adaptive signal processing for sensor networks.

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      On the other hand, in diffusion cooperation, each node accumulates the information from the neighbours, then estimates the desired parameters locally and subsequently shares it among the neighbours. Therefore, the diffusion approach is adaptive in real time to the changes in the network [25]. In the presence of impulsive noise, low complex, robust diffusion affine projection adaptive algorithms are reported for distributed parameter estimation [34,35].

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    Bishnu Prasad Mishra received his M.Tech. in Electronics and Communication Engineering from Biju Patnaik University of Technology Rourkela, India, in 2015. He is currently pursuing the Ph.D. degree in Electronics and Communication Engineering at National Institute of Technology, Goa, India. His research interest includes adaptive signal processing for sensor networks.

    Trilochan Panigrahi received the M.Tech. degree in electronics and communication engineering from the Biju Patnaik University of Technology, Rourkela, India, in 2005, and the Ph.D. degree in electronics and communication engineering from the National Institute of Technology Rourkela, India in 2012. He is currently an Associate Professor with the Department of Electronics and Communication Engineering, National Institute of Technology Goa, India. His research interests include signal processing for wireless communication, wireless sensor network, application of evolutionary algorithms in signal processing, and source localization.

    Annet Mary Wilson received her B. Tech and M.Tech degree in Electronics and Communication Engineering from Calicut University, India, in 2012 and 2015 respectively. She is currently pursuing the Ph.D. degree in Electronics and Communication Engineering at National Institute of Technology, Goa, India. Her research interest includes adaptive signal processing and signal processing for sensor networks.

    Samrat L. Sabat received the M.Sc. and Ph.D. degrees in Electronics Science from Berhampur University in 1997 and 2004, respectively. He is currently a Professor with the Centre for Advanced Studies in Electronics Science and Technology, University of Hyderabad, India. His current research includes signal processing algorithm for wireless communication, spectrum sensing, and VLSI architecture development for signal processing algorithms.

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