Mathematical Problems in Engineering

Volume 2018, Article ID 4968682, 11 pages

https://doi.org/10.1155/2018/4968682

## Efficient Impulsive Noise Mitigation for OFDM Systems Using the Alternating Direction Method of Multipliers

^{1}The Faculty of Information Science and Engineering, Ningbo University 315211, China^{2}The School of Intelligent Electronics, Zhejiang Business Technology Institute, Ningbo 315012, China^{3}The College of Information Science and Engineering, National Huaqiao University, Xiamen 361021, China

Correspondence should be addressed to Youming Li; nc.ude.ubn@gnimuoyil

Received 24 January 2018; Accepted 2 May 2018; Published 7 June 2018

Academic Editor: Paolo Addesso

Copyright © 2018 Xin-Rong Lv 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

An efficient impulsive noise estimation algorithm based on alternating direction method of multipliers (ADMM) is proposed for OFDM systems using quadrature amplitude modulation (QAM). Firstly, we adopt the compressed sensing (CS) method based on the -norm optimization to estimate impulsive noise. Instead of the conventional methods that exploit only the received signal in null tones as constraint, we add the received signal of data tones and QAM constellations as constraints. Then a relaxation approach is introduced to convert the discrete constellations to the convex box constraints. After that a linear programming is used to solve the optimization problem. Finally, a framework of ADMM is developed to solve the problem in order to reduce the computation complexity. Simulation results for -QAM and -QAM demonstrate the practical advantages of the proposed algorithm over the other algorithms in bit error rate performance gains.

#### 1. Introduction

Power line communications (PLC) have become an attractive communication solution for smart grid and other applications due to their advantages of high penetration and low deployment costs over other communication technologies [1–3]. The applications of PLC, however, are limited by some unfavorable factors, among which the impulsive noise (IN) is the major factor that influences the data transmission over power grid. Impulsive noise can be divided roughly into two categories: asynchronous and periodic [4]. Asynchronous impulsive noise is primarily caused by switching transients of electrical appliances and characterized by short duration and high power impulses with random arrivals. Periodic impulsive noise typically arises from switching mode power supplies and contains longer bursts of interference spikes that occur periodically with half the main cycle of grid.

Orthogonal frequency-division multiplexing (OFDM) is less sensitive to impulsive noise than single carrier by spreading the effect of impulsive noise across all subcarriers [5]. Thus, OFDM has found applications in the physical layer technology by most modern PLC standards. Recent field measurements, however, have identified that the power spectral density (PSD) of impulsive noise can exceed that of background noise by at least 20 dB or even 50 dB in some scenarios [6, 7]. Conventional OFDM-based PLC systems can deteriorate sharply in the presence of impulsive noise.

In this work, we focus on the mitigation of asynchronous impulsive noise, for which the common methods include clipping or blanking [8–11]. In [8, 9], the optimal clipping and blanking thresholds are derived in closed-form under the conditions that the occurrence probability of impulsive noise and the noise power can be perfectly estimated at the receiver. In [10], the method of moments (MoM) is used to estimate the parameters and update adaptively the thresholds on assuming that the statistics of impulsive noise maintains stationary in a long period. In [11], a framework of optimal blanking threshold (OBT) estimation is proposed based on the peak-to-average power ratio (PAPR) of signals. However it is difficult to obtain the PAPR values at the receiver. In [12], an MMSE estimator is proposed on the assumption that the impulsive noise is i.i.d. in the time domain, which yields an approximate optimal estimation only when the noise state information (NSI) is known.

Note that the above methods need to estimate the parameters of either impulsive noise or OFDM signal. Since the impulsive noise in the PLC environments is time-varying, accurate estimation of the parameters can be an exhausting task. To avoid impractical parameter estimation, nonparametric mitigation methods may be preferred. In [13], the compressed sensing (CS) is applied in the mitigation with observing that the asynchronous impulsive noise is sparse in the time domain; that is, the number of impulses in an OFDM symbol cannot exceed some threshold. In [14], the CS-based technique is extended to the scenarios of bursty impulsive noise. Due to the sparsity property of impulsive noise, it can also be recovered by the Sparse Bayesian Learning (SBL) algorithm [15]. Moreover, in [15], two enhanced SBL algorithms for impulsive noise mitigation are proposed: SBL using all tones and SBL using decision feedback. These SBL algorithms can improve the performance and robustness of mitigation by incorporating some a priori information on the impulsive noise, but at fairly high complexities. In [16], the orthogonal clustering (OC) algorithm is developed to find the dominant support of asynchronous IN, and then the MMSE estimate of IN is derived. With the advantage of low complexity, it needs some correct parameter information about IN and the background noise variance at the same time.

In this paper, we investigate the CS technique for asynchronous impulsive noise mitigation in OFDM-based PLC systems with QAM by exploiting the modulus property of QAM constellations. First, we adopt the CS algorithm based on the -norm minimization to estimate the impulsive noise with the constraint on the projection of impulsive noise onto null tones. Then, we add the received signal of data tones and source QAM symbols as constraints. As a result, we obtain a nonconvex optimization problem based on the -norm minimization with constraints on the information of all tones of OFDM symbols. Next, we convert the nonconvex problem into a linear programming (LP) problem with proper relaxation. In order to reduce the computational complexity, we introduce the ADMM framework to solve the LP optimization problem. Consequently, we obtain an IN estimation algorithm based on ADMM which is able to exploit the information with respect to all tones. With independence of channel coding, the proposed algorithm is suitable for both uncoded and coded PLC systems. In computer simulations, we use a rate- convolutional code for the coded systems and consider two models for generating the impulsive noise. The main novel contributions of this paper are tripartite: (1) both the received signal of all tones and source QAM symbols property are exploited to improve the IN estimation performance, which has not been reported in the literature. (2) A relaxation approach is introduced to convert the discrete QAM constellations into convex set, which makes a nonconvex optimization problem become a convex optimization problem. (3) An ADMM framework is developed to solve the resulting optimization problem, which can obtain the same performance as the LP algorithm but has lower computation complexity than the LP algorithm.

The remainder of this paper is organized as follows. In Section 2, the system model and the impulsive noise models are briefly described. In Section 3, we first briefly introduce the CS algorithm with null tones using the system model and then derive a nonparametric algorithm by extending the CS algorithm to include all the tones. Furthermore, we derive a nonconvex optimization problem and approximate it into a standard LP problem. In Section 4, we propose an ADMM framework to solve the convex optimization problem. In Section 5, simulation results show the advantages of the proposed method in the error performance over a wide range of signal-to-noise ratio (SNR) values. Finally, Section 6 concludes the paper.

#### 2. System Model

The complex baseband model for OFDM-based PLC systems is shown in Figure 1. At the transmitter, the frequency-domain OFDM symbols are mapped from the data bits and denoted in the vector form as , where is the total number of subcarriers. The OFDM modulator, as an inverse discrete Fourier transformation (IDFT), converts the frequency-domain OFDM symbols to the time-domain OFDM signals as follows:where is an point DFT matrix and is the Hermitian transpose of . Then, a cyclic prefix (CP) of length larger than the channel delay spread is inserted ahead of for circumventing the intersymbol interference (ISI) over the PLC channel.