Research ArticleEfficient Impulsive Noise Mitigation for OFDM SystemsUsing the Alternating Direction Method of Multipliers

Xin-Rong Lv 12 Youming Li 1 and Yu-Cheng He3

1The Faculty of Information Science and Engineering Ningbo University 315211 China2The School of Intelligent Electronics Zhejiang Business Technology Institute Ningbo 315012 China3The College of Information Science and Engineering National Huaqiao University Xiamen 361021 China

Correspondence should be addressed to Youming Li liyoumingnbueducn

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

Academic Editor Paolo Addesso

Copyright copy 2018 Xin-Rong Lv et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An efficient impulsive noise estimation algorithm based on alternating direction method of multipliers (ADMM) is proposed forOFDM systems using quadrature amplitude modulation (QAM) Firstly we adopt the compressed sensing (CS) method based onthe ℓ1-norm optimization to estimate impulsive noise Instead of the conventional methods that exploit only the received signal innull tones as constraint we add the received signal of data tones and QAMconstellations as constraintsThen a relaxation approachis introduced to convert the discrete constellations to the convex box constraints After that a linear programming is used to solvethe optimization problem Finally a framework of ADMM is developed to solve the problem in order to reduce the computationcomplexity Simulation results for 4-QAM and 16-QAM demonstrate the practical advantages of the proposed algorithm over theother algorithms in bit error rate performance gains

1 Introduction

Power line communications (PLC) have become an attractivecommunication solution for smart grid and other applica-tions due to their advantages of high penetration and lowdeployment costs over other communication technologies [1ndash3] The applications of PLC however are limited by someunfavorable factors among which the impulsive noise (IN)is the major factor that influences the data transmission overpower grid Impulsive noise can be divided roughly into twocategories asynchronous and periodic [4] Asynchronousimpulsive noise is primarily caused by switching transientsof electrical appliances and characterized by short durationand high power impulses with random arrivals Periodicimpulsive noise typically arises from switching mode powersupplies and contains longer bursts of interference spikes thatoccur periodically with half the main cycle of grid

Orthogonal frequency-division multiplexing (OFDM)is less sensitive to impulsive noise than single carrier byspreading the effect of impulsive noise across all subcarriers[5] Thus OFDM has found applications in the physical

layer technology by most modern PLC standards Recentfield measurements however have identified that the powerspectral density (PSD) of impulsive noise can exceed that ofbackground noise by at least 20 dB or even 50 dB in somescenarios [6 7] ConventionalOFDM-based PLC systems candeteriorate sharply in the presence of impulsive noise

In this work we focus on the mitigation of asynchronousimpulsive noise for which the common methods includeclipping or blanking [8ndash11] In [8 9] the optimal clippingand blanking thresholds are derived in closed-form underthe conditions that the occurrence probability of impulsivenoise and the noise power can be perfectly estimated at thereceiver In [10] the method of moments (MoM) is used toestimate the parameters and update adaptively the thresholdson assuming that the statistics of impulsive noise maintainsstationary in a long period In [11] a framework of optimalblanking threshold (OBT) estimation is proposed based onthe peak-to-average power ratio (PAPR) of signals Howeverit is difficult to obtain the PAPR values at the receiver In [12]an MMSE estimator is proposed on the assumption that theimpulsive noise is iid in the time domain which yields an

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4968682 11 pageshttpsdoiorg10115520184968682

2 Mathematical Problems in Engineering

Demapper Detector EqualizerImpulsiveNoise

Mitigation

ChannelEstimation

Modulator(IDFT)

OFDM

OFDMDemodulator

(DFT)

Mapperb

r

i g

b X X Y Y

HH

H

Pilot

PLC

X x

minusCP

+CP

Figure 1 PLC system block diagram

approximate optimal estimation only when the noise stateinformation (NSI) is known

Note that the above methods need to estimate the param-eters of either impulsive noise or OFDM signal Since theimpulsive noise in the PLC environments is time-varyingaccurate estimation of the parameters can be an exhaustingtask To avoid impractical parameter estimation nonpara-metric mitigation methods may be preferred In [13] thecompressed sensing (CS) is applied in the mitigation withobserving that the asynchronous impulsive noise is sparse inthe time domain that is the number of impulses in anOFDMsymbol cannot exceed some threshold In [14] the CS-basedtechnique is extended to the scenarios of bursty impulsivenoise Due to the sparsity property of impulsive noise itcan also be recovered by the Sparse Bayesian Learning(SBL) algorithm [15] Moreover in [15] two enhanced SBLalgorithms for impulsive noise mitigation are proposed SBLusing all tones and SBL using decision feedback These SBLalgorithms can improve the performance and robustness ofmitigation by incorporating some a priori information onthe impulsive noise but at fairly high complexities In [16]the orthogonal clustering (OC) algorithm is developed tofind the dominant support of asynchronous IN and thenthe MMSE estimate of IN is derived With the advantage oflow complexity it needs some correct parameter informationabout IN and the background noise variance at the same time

In this paper we investigate the CS technique for asyn-chronous impulsive noise mitigation in OFDM-based PLCsystems with QAM by exploiting the modulus property ofQAM constellations First we adopt the CS algorithm basedon the ℓ1-normminimization to estimate the impulsive noisewith the constraint on the projection of impulsive noise ontonull tones Then we add the received signal of data tonesand source QAM symbols as constraints As a result weobtain a nonconvex optimization problem based on the ℓ1-norm minimization with constraints on the information ofall tones of OFDM symbols Next we convert the nonconvexproblem into a linear programming (LP) problem withproper relaxation In order to reduce the computationalcomplexity we introduce the ADMM framework to solve

the LP optimization problem Consequently we obtain anIN estimation algorithm based on ADMM which is ableto exploit the information with respect to all tones Withindependence of channel coding the proposed algorithmis suitable for both uncoded and coded PLC systems Incomputer simulations we use a rate-12 convolutional codefor the coded systems and consider twomodels for generatingthe impulsive noise The main novel contributions of thispaper are tripartite (1) both the received signal of all tonesand source QAM symbols property are exploited to improvethe IN estimation performance which has not been reportedin the literature (2) A relaxation approach is introducedto convert the discrete QAM constellations into convex setwhich makes a nonconvex optimization problem become aconvex optimization problem (3) An ADMM framework isdeveloped to solve the resulting optimization problem whichcan obtain the same performance as the LP algorithm but haslower computation complexity than the LP algorithm

The remainder of this paper is organized as follows InSection 2 the system model and the impulsive noise modelsare briefly described In Section 3 we first briefly introducethe CS algorithm with null tones using the system modeland then derive a nonparametric algorithm by extendingthe CS algorithm to include all the tones Furthermore wederive a nonconvex optimization problem and approximateit into a standard LP problem In Section 4 we proposean ADMM framework to solve the convex optimizationproblem In Section 5 simulation results show the advantagesof the proposedmethod in the error performance over a widerange of signal-to-noise ratio (SNR) values Finally Section 6concludes the paper

2 System Model

The complex baseband model for OFDM-based PLC systemsis shown in Figure 1 At the transmitter the frequency-domain OFDM symbols are mapped from the data bits anddenoted in the vector form as X = (1198830 1198831 119883119873)119879where 119873 is the total number of subcarriers The OFDMmodulator as an inverse discrete Fourier transformation

Mathematical Problems in Engineering 3

(IDFT) converts the frequency-domain OFDM symbols tothe time-domain OFDM signals x = (1199090 1199091 119909119873)119879 asfollows

x = FlowastX (1)

where F is an 119873 point DFT matrix and Flowast is the Hermitiantranspose of F Then a cyclic prefix (CP) of length largerthan the channel delay spread is inserted ahead of x forcircumventing the intersymbol interference (ISI) over thePLC channel

Noise in the PLC channel cannot be considered simplyas a stationary AWGN but may be viewed as an additivemixture of the background AWGN and the impulsive noiseIt is known that asynchronous impulsive noise is time-variantwith a short duration random occurrence and a high powerspectral density Thus at the receiver the received time-domainOFDMsignals with the CP dropped can be expressedby

r = Hx + i + g (2)

where H is an 119873 times 119873 circulant matrix whose first columnis formed by the zero-padded channel impulse responsei represents the impulsive noise vector g represents thebackground noise vector of119873 iid AWGN variables and thevectors r x i and g in the time domain are all located in the119873-dimensional complex signal space C119873

Through the OFDM demodulator the received signalvector r in the time domain is converted by DFT into thefrequency domain as

Y = Fr = FHFlowastX + Fi + Fg = ΛX + Fi + G (3)

where Λ = FHFlowast = diag(11986701198671 119867119873minus1) is a diagonalmatrix with diagonal elements 119867119894 being the point DFTcoefficients of channel impulse response G = Fg is the DFTof g which can also be assumed to be AWGN since F is anunitary matrix

Subsequently the impulsive noise i is estimated andremoved through the impulsive noise mitigation module thechannel distortion with respect to Λ can be compensatedusing a simple one-tap frequency-domain equalizer (FEQ)and the detectorworks as if onlyAWGNwere present Finallythe data bits are obtained by demapper

In computer simulations we consider two models forgenerating the impulsive noise components respectivelyGaussian mixture (GM) model and Middleton Class A(MCA) noisemodel which arewidely used inPLC [17] In theGMmodel the probability density function (PDF) of a noisesample 119909 is a weighted summation of 119870 different Gaussianvariables given by

119891 (119909) = 119870minus1sum119896=0

119901119896119892119896 (119909) (4)

where119892119896(119909) is the PDF of the complex Gaussian variable withzero-mean and variance 120574119896 and 119901119896 is the mixing probabilityof the 119896th component such that sum119870minus1119896=0 119901119896 = 1

In the MCA noise model the PDF of the noise sample 119909is given in a different form

119891 (119909) = infinsum119896=0

120587119896119892119896 (119909) (5)

where 120587119896 is the mixing probability subject to the Poissondistribution with parameter 119860 ie 120587119896 = 119890minus119860119860119896119896 for 119896 =0 1 TheMCAmodel can be viewed as a special case of theGMmodel with 119901119896 = 120587119896 and an infinite number of Gaussiancomponents with zero-mean and variance

120574119896 = 119896119860 + Γ1 + Γ (6)

where 119896 = 0 1 119870 the parameter 119860 denotes the impulsiveindex and Γ is the mean power ratio of background-to-impulsive noises [15]

3 The IN Estimation Using LP

In OFDM-based PLC systems 119873 subcarriers of an OFDMsymbol can be divided into two parts null tones and datatones The null tones are occupied by means of setting theall-zero input and the data tones are used to transmit datasymbols or pilot symbols

31 The IN Estimation with Null Tones Let 119863 denote theindex set of null tones with a total number 119872 Let (sdot) denotethe operation of forming a submatrix (or subvector) byselecting the rows (or elements) indexed by 119863 Accordingto the arrangement of tone clusters the received OFDMsymbols at the null tones are obtained from (3) as

Y119863 = (ΛX)119863 + F119863i + G119863 (7)

where G119863 sim CN(0 1205902I119872) is a complex AWGN vector withzero-mean and variance matrix 1205902I119872

With known zero symbols 119883119894 119894 isin 119863 at the null tones(7) becomes

Y119863 = F119863i + G119863 (8)

which is a group of linear observations of the impulsivenoise Notice that the matrix F119863 is underdetermined since119872 ≪ 119873 Thus (8) can also be viewed as underdeterminedequations with measurement noise It is generally acceptedthat the asynchronous impulsive noise is sparse in the timedomain [13ndash15] Thus the CS technique can be employed toestimate the impulsive noise i according to (8) [13] Using theminimum ℓ1-norm reconstruction algorithm for CS [18] wecan formulate the impulsive noise estimation as the followingconvex optimization problem

min i1st 1003817100381710038171003817Y119863 minus F119863i

100381710038171003817100381722 le 1205761 (9)

where 1205761 gt 0 is the maximum tolerable root-mean-squareerror (RMSE) due to the background noise and i119901 denotes

4 Mathematical Problems in Engineering

the ℓ119901-norm for a real number 119901 ge 1 that is given by i119901 =(sum119899119896=1 |119894119896|119901)1119901Let i be a solution to the problem in (9) The impacts of

the impulsive noise can be removed by subtraction as follows

Y ≜ Y minus Fi = ΛX + F (i minus i) + G (10)

Assuming the impulsive noise residual (iminusi) can be negligiblethe receiver can then proceed as if only Gaussian noise ispresent and apply the conventional zero-forcing equalizationalgorithm

32 The IN Estimation with All Tones It is known thatincreasing the number of null-data tones is equivalent toincreasing the rank of matrix F119863 in (9) which can promotethe accuracy of impulsive noise estimation However theimprovement will be obtained at a cost of losing systemthroughput To avoid this loss we should not increase thenumber of non-data tones but resort to exploiting theinformation of data tones for the purpose of improving theimpulsive noise estimation

Somemodern PLC standards such as Ghnem [19] adoptQAM modulation which requires a coherent receiver Manysimulation results have shown that the coherent receiver cansignificantly increase system performance even in scenar-ios with strong impulsive noise and time-varying channelbehavior [19] Let Ω be the set of QAM constellation pointsExploiting the information of data tones we can formulate anew impulsive noise estimation optimization problem

min i1st 1003817100381710038171003817Y119863 minus F119863i

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minus (ΛX)119863 minus F119863i

100381710038171003817100381722 le 1205762119883119896 isin Ω forall119896 isin 119863

(11)

where 119863 denotes the index set of data tones with a totalnumber 119873 minus 119872 Compared with (9) (11) adds some newconstraints corresponding to the information of data tones

Defining a new vector t = [X119879 i119879]119879 isin C2119873 the sourcesymbol vector X and the impulsive noise vector i can beexpressed as

X = Ut

i = Vt (12)

where the matrix U ≜ [I119873 0119873] isin R119873times2119873 V ≜ [0119873 I119873] isinR119873times2119873 We use 0119873 to denote the119873times119873 zeromatrix and I119873 todenote the119873times119873 identity matrix Then (11) can be rewrittenas

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762Ut (119896) isin Ω forall119896 isin 119863

(13)

where Φ ≜ [Λ119863 F119863] Clearly the optimization problem in(13) belongs to the class of NP-hard problems because there isa discrete constraint [20] To make the optimization problemfeasible we convert the nonconvex constraint into a convexconstraint in the sequel

We use ΩRe = Re(Ω) and ΩIm = Im(Ω) to representrespectively the sets of real and imaginary coordinates ofthe set Ω According to the origin-symmetric property of Ωthe constellation covers a rectangular region with boundariesgiven by

|Re (119878)| le max 119886 isin ΩRe ≜ 119906119877 forall119878 isin Ω|Im (119878)| le max 119886 isin ΩIm ≜ 119906119868 forall119878 isin Ω (14)

where 119906119877 = 119906119868 holds for square QAM constellationsSubstituting (14) into (13) the nonconvex constraint is

relaxed to the convex box constraint (15)

minus119906119877 le Re (Ut (119896)) le 119906119877minus119906119868 le Im (Ut (119896)) le 119906119868 (15)

Consequently we obtain the following convex problem

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(16)

1119873 denotes the 119873 times 1 column vector with all elements beingone

In problems (9) and (16) it is not easy to choose theconstants 1205761 and 1205762 that are dependent on the noise powerestimation Recalling (2) for the received OFDM signals wedenote the sum of impulsive noise and background noise ase = i+g In reality the amplitudes of asynchronous impulsivenoise samples are much larger than those of backgroundnoise samples So it is reasonable to consider e1 as anapproximation to i1 with g omitted such that the constants1205761 and 1205762 diminish in (9) and (16) Upon replacing i1 by e1we can rewrite the problem (16) as

min Vt1st Y = At

minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(17)

where

t = [X119879 e119879]119879Y = [Y119879

119863Y119879119863]119879

A = [F119863VΦ

] (18)

Mathematical Problems in Engineering 5

It can verify that problem (17) is a typical linear programming(LP) problems which can be solved by using the on-the-shelfsoftware packages such asCVX [21]We refer to the algorithmas ldquoLP-Allrdquo

4 The IN Estimation Using ADMM

Although in general the above LP-based approaches canprovide satisfying results they become computationallydemanding as the signal dimension increases because ofmatrix inversion [22] In this section we propose the ADMMframework to solve the IN estimation optimization problems

41 Framework of ADMM Let us consider the optimizationproblem in the following form [23]

min 119891 (119909) + 119892 (119911)st 119860119909 + 119861119911 = 119888 (19)

with variable 119909 isin R119899 and 119911 isin R119898 where 119860 isin R119901times119899119861 isin R119901times119898 and 119888 isin R119901 We also assume that 119891 and 119892 areconvex functions The augmented Lagrangian function of theproblem (19) is given by

L (119909 120582 119911) = 119891 (119909) + 119892 (119911) + 120582119879 (119860119909 + 119861119911 minus 119888)+ 1205882 119860119909 + 119861119911 minus 11988822 (20)

where 120582 isin R119901 is the Lagrangian multiplier and 120588 gt 0 isa penalty parameter Given (119911119896 120582119896) ADMM consists of theiterations as follows

119909119896+1 larr997888 argmin119909

L (119909 119911119896 120582119896)119911119896+1 larr997888 argmin

119911L (119909119896+1 119911 120582119896)

120582119896+1 larr997888 120582119896 + 120588 (119860119909119896+1 + 119861119911119896+1 minus 119888) (21)

Defining the scaled dual variable 119906 = (1120588)120582 we can expressADMM iterations as

119909119896+1 larr997888 argmin119909

(119891 (119909) + 1205882 10038171003817100381710038171003817119860119909 + 119861119911119896 minus 119888 + 1199061198961003817100381710038171003817100381722)119911119896+1 larr997888 argmin

119911(119892 (119911) + 1205882 10038171003817100381710038171003817119860119909119896+1 + 119861119911 minus 119888 + 1199061198961003817100381710038171003817100381722)

119906119896+1 larr997888 119906119896 + 119860119909119896+1 + 119861119911119896+1 minus 119888(22)

ADMM is proved to converge for all positive values of 120588under quite mild conditions which makes the selection of 120588rather flexible [24] Hence we can simply use a fixed positiveconstant for 120588 = 1 in our work

42The IN Estimation with All Tones Using ADMM (ADMM-All) In ADMM form the IN estimation algorithm (17) canbe written as

min I (t) +G (z1) + 1003817100381710038171003817z210038171003817100381710038171st Ut minus z1 = 0

Vt minus z2 = 0(23)

where I(t) is the indicator function of equality constrainsand G(z1) is the indicator function of the box constraint of(17)

We introduce an auxiliary vector z = [z1198791 z1198792 ]119879 isin C2119873

with z1 = Uz and z2 = VzThen the two constraints Utminusz1 =0 andVtminus z2 = 0 can be combined into one constraint tminus z =0 So we can express the augmented Lagrangian function asfollows

L (t z 120582) = I (t) +G (Uz) + Vz1 + 120582119879 (t minus z)+ 1205882 t minus z22 (24)

The corresponding ADMM iterations consisted of

t119896+1 larr997888 prod(z119896 minus u119896) (25)

z119896+11 larr997888 P (U (t119896+1 + u119896)) (26)

z119896+12 larr997888 S1120588 (V (t119896+1 + u119896)) (27)

u119896+1 larr997888 u119896 + t119896+1 minus z119896+1 (28)

where prod is projection onto t | At = Y and P is Euclidianprojection onto the box constraint (15)

The t-update (25) involves solving a linearly constrainedminimum Euclidean norm problem which can be expressedas

mint

10038171003817100381710038171003817t minus (z119896 minus u119896)1003817100381710038171003817100381722st At = Y (29)

The KKT (Karush-Kuhn-Tucker) conditions [22] for thisproblem are

[ I A119879

A 0][tlowast

]lowast] = [z119896 minus u119896

Y] (30)

where ]lowast is the optimal dual variable The solution tlowast of theabove equations is

tlowast = (I minus A119879 (AA119879)minus1 A) (z119896 minus u119896)+ A119879 (AA119879)minus1 Y

(31)

In z1-update (26) theP is elementwise Euclidian projec-tion onto the box constraint (15) which is defined as

P (119886) =

119906119877 119886 gt 119906119877119886 |119886| le 119906119877minus119906119877 119886 lt minus119906119877

(32)

In z2-update (27) S1120588(x119896+1 + u119896) is the well-knownelementwise soft thresholding operator [23] which is definedas

S120581 (119886) =

119886 minus 120581 119886 gt 1205810 |119886| le 120581119886 + 120581 119886 lt minus120581

(33)

6 Mathematical Problems in Engineering

43 The IN Estimation with Null Tones Using ADMM(ADMM-Null) We first recast problem (9) as LP form

min e1st Y119863 = F119863e (34)

And then (34) named ldquoLP-Nullrdquo can be written as

min I (e) + z1st e minus z = 0 (35)

whereI(e) is the indicator function of e isin C119873 | F119863e = Y119863The augmented Lagrangian function of problem (35) is

L (e z 120582) = I (e) + z1 + 120582119879 (e minus z) + 1205882 e minus z22 (36)

According the ADMM iterations (22) we can obtain theupdate primal variables e z and a dual variable u as follows

e119896+1 larr997888 prod(z119896 minus u119896) (37)

z119896+1 larr997888 S1120588 (x119896+1 + u119896) (38)

u119896+1 larr997888 u119896 + x119896+1 minus z119896+1 (39)

whereprod is projection onto e isin C119873 | F119863e = Y119863 and S1120588 isthe elementwise soft thresholding operator

The solution of e-update (37) can be derived using thesimilar method in (29)sim(31) as follows

elowast = (I minus F119879119863(F119863F119879119863)minus1 F119863) (z119896 minus u119896)

+ F119879119863(F119863F119879119863)minus1 Y119863

(40)

44 Complexity Analysis Theworst-case complexity of solv-ing LP-Null and LP-All is O(1198733) by using the interior pointmethod [22] For the proposedADMM-Null and ADMM-Allalgorithms the leading computation cost is the matrix inver-sions andmatrixmultiplications in (40) and (31) However wenotice that these inverses only have to be computed once Wecan caching the matrix inversion and use this cached valuein subsequent steps The resulting complexity is O(1198732) +1198723for ADMM-Null and O(1198732) + 1198733 for ADMM-All So thecomputation complexity of the IN estimation algorithmusingADMM is significantly lower than that using LP

5 Simulation Results

In this section we present computer simulation results for theproposed impulsive noise mitigation method in the complexbaseband OFDM-based PLC systems over an ideal (unit-gain nonfading) channel and a 15-path multipath power linechannel model [25] respectively In our simulations theimpulsive noise samples are generated using the GM modelin (4) and the MCA model in (5) respectively In orderto represent the typical noise scenarios given in [15] themean power ratio of impulsive-to-background noises is set

Table 1 Parameters of simulations

OFDM Carriers FFT length119873 = 128 null tones119872 = 56 datatones119873 minus119872 = 72

Modulation 4-QAM 16-QAMFEC Code A rate-12 convolutional codeGMModel 119870 = 3 119901119896 = 09 007 003 120574119896 = 1 100 1000MCAmodel 119860 = 01 Γ = 001 and the first 10 terms are

truncated

to be 30 dB in the GM model and 20 dB in the MCA modelrespectively The signal-to-noise ratio (SNR) is defined asSNR = 119875s119875e where 119875s is the power of the transmitted signaland 119875e is the total noise power including the impulsive noiseand background noise The detailed simulation parametersare listed in Table 1

In the same simulation environments we compare thebit error rate (BER) performances between our proposedmethod and several available mitigation methods in theliterature In Figures 2ndash7 we identified the three SBL-basedalgorithms in [15] respectively as their original notationsldquoSBL with null tonesrdquo ldquoSBL with all tonesrdquo and ldquoSBL withDFrdquo and the conventional OFDM system without impul-sive noise mitigation as ldquoNo mitigationrdquo For comparisonsbetween nonparametric and parametric methods we alsopresent the results for the MMSE detectors with noise stateinformation (NSI) in [12] which is identified as ldquoMMSE withNSIrdquo Note that the MMSE detector with NSI is optimalamong the parametric methods [12]

51 Ideal Channel Figures 2ndash5 compare the BER perfor-mances of all the above mitigation techniques for uncodedand coded systems using 4-QAM and 16-QAM respectivelyNo result of the ldquoSBL with DFrdquo method is shown in Figures 2and 4 since it is not applicable for uncoded systems

We first analyze the results for the uncoded 4-QAMsystem in Figure 2 Obviously our proposed ldquoLP-Allrdquo andldquoADMM-Allrdquo outperforms all the other methods except forthe MMSE detector with NSI which appears the best in alower SNR region However as SNR increases our methodbecomes the most advantageous in moderate to high SNRregion When only the nondata tones are considered forimpulsive noise estimation the algorithm ldquoLP-Nullrdquo andldquoADMM-Nullrdquo achieve less SNR gains than the SBL algo-rithm with null tones under both impulsive noise modelsConversely with all the tones being taken into account theldquoLP-Allrdquo and ldquoADMM-Allrdquo exploiting the modulus of con-stellation points present advantages over the SBL algorithmwith all tones achieving an SNR gain more than 2 dB in theGM model or between 4 dB and 6 dB in the MCA modelrespectively

Then we analyze the results presented in Figure 3 for thecoded systems using 4-QAM Again our proposed methoddemonstrates a favorable capability of outperforming most ofother algorithms in a similar tendency to that for the uncodedsystems in Figure 2 Compared with the SBL with DF algo-rithm our method presents a marginal performance loss inlower SNR region with GM impulsive noise Conversely in

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

Mathematical Problems in Engineering 3

(IDFT) converts the frequency-domain OFDM symbols tothe time-domain OFDM signals x = (1199090 1199091 119909119873)119879 asfollows

x = FlowastX (1)

where F is an 119873 point DFT matrix and Flowast is the Hermitiantranspose of F Then a cyclic prefix (CP) of length largerthan the channel delay spread is inserted ahead of x forcircumventing the intersymbol interference (ISI) over thePLC channel

Noise in the PLC channel cannot be considered simplyas a stationary AWGN but may be viewed as an additivemixture of the background AWGN and the impulsive noiseIt is known that asynchronous impulsive noise is time-variantwith a short duration random occurrence and a high powerspectral density Thus at the receiver the received time-domainOFDMsignals with the CP dropped can be expressedby

r = Hx + i + g (2)

where H is an 119873 times 119873 circulant matrix whose first columnis formed by the zero-padded channel impulse responsei represents the impulsive noise vector g represents thebackground noise vector of119873 iid AWGN variables and thevectors r x i and g in the time domain are all located in the119873-dimensional complex signal space C119873

Through the OFDM demodulator the received signalvector r in the time domain is converted by DFT into thefrequency domain as

Y = Fr = FHFlowastX + Fi + Fg = ΛX + Fi + G (3)

where Λ = FHFlowast = diag(11986701198671 119867119873minus1) is a diagonalmatrix with diagonal elements 119867119894 being the point DFTcoefficients of channel impulse response G = Fg is the DFTof g which can also be assumed to be AWGN since F is anunitary matrix

Subsequently the impulsive noise i is estimated andremoved through the impulsive noise mitigation module thechannel distortion with respect to Λ can be compensatedusing a simple one-tap frequency-domain equalizer (FEQ)and the detectorworks as if onlyAWGNwere present Finallythe data bits are obtained by demapper

In computer simulations we consider two models forgenerating the impulsive noise components respectivelyGaussian mixture (GM) model and Middleton Class A(MCA) noisemodel which arewidely used inPLC [17] In theGMmodel the probability density function (PDF) of a noisesample 119909 is a weighted summation of 119870 different Gaussianvariables given by

119891 (119909) = 119870minus1sum119896=0

119901119896119892119896 (119909) (4)

where119892119896(119909) is the PDF of the complex Gaussian variable withzero-mean and variance 120574119896 and 119901119896 is the mixing probabilityof the 119896th component such that sum119870minus1119896=0 119901119896 = 1

In the MCA noise model the PDF of the noise sample 119909is given in a different form

119891 (119909) = infinsum119896=0

120587119896119892119896 (119909) (5)

where 120587119896 is the mixing probability subject to the Poissondistribution with parameter 119860 ie 120587119896 = 119890minus119860119860119896119896 for 119896 =0 1 TheMCAmodel can be viewed as a special case of theGMmodel with 119901119896 = 120587119896 and an infinite number of Gaussiancomponents with zero-mean and variance

120574119896 = 119896119860 + Γ1 + Γ (6)

where 119896 = 0 1 119870 the parameter 119860 denotes the impulsiveindex and Γ is the mean power ratio of background-to-impulsive noises [15]

3 The IN Estimation Using LP

In OFDM-based PLC systems 119873 subcarriers of an OFDMsymbol can be divided into two parts null tones and datatones The null tones are occupied by means of setting theall-zero input and the data tones are used to transmit datasymbols or pilot symbols

31 The IN Estimation with Null Tones Let 119863 denote theindex set of null tones with a total number 119872 Let (sdot) denotethe operation of forming a submatrix (or subvector) byselecting the rows (or elements) indexed by 119863 Accordingto the arrangement of tone clusters the received OFDMsymbols at the null tones are obtained from (3) as

Y119863 = (ΛX)119863 + F119863i + G119863 (7)

where G119863 sim CN(0 1205902I119872) is a complex AWGN vector withzero-mean and variance matrix 1205902I119872

With known zero symbols 119883119894 119894 isin 119863 at the null tones(7) becomes

Y119863 = F119863i + G119863 (8)

which is a group of linear observations of the impulsivenoise Notice that the matrix F119863 is underdetermined since119872 ≪ 119873 Thus (8) can also be viewed as underdeterminedequations with measurement noise It is generally acceptedthat the asynchronous impulsive noise is sparse in the timedomain [13ndash15] Thus the CS technique can be employed toestimate the impulsive noise i according to (8) [13] Using theminimum ℓ1-norm reconstruction algorithm for CS [18] wecan formulate the impulsive noise estimation as the followingconvex optimization problem

min i1st 1003817100381710038171003817Y119863 minus F119863i

100381710038171003817100381722 le 1205761 (9)

where 1205761 gt 0 is the maximum tolerable root-mean-squareerror (RMSE) due to the background noise and i119901 denotes

4 Mathematical Problems in Engineering

the ℓ119901-norm for a real number 119901 ge 1 that is given by i119901 =(sum119899119896=1 |119894119896|119901)1119901Let i be a solution to the problem in (9) The impacts of

the impulsive noise can be removed by subtraction as follows

Y ≜ Y minus Fi = ΛX + F (i minus i) + G (10)

Assuming the impulsive noise residual (iminusi) can be negligiblethe receiver can then proceed as if only Gaussian noise ispresent and apply the conventional zero-forcing equalizationalgorithm

32 The IN Estimation with All Tones It is known thatincreasing the number of null-data tones is equivalent toincreasing the rank of matrix F119863 in (9) which can promotethe accuracy of impulsive noise estimation However theimprovement will be obtained at a cost of losing systemthroughput To avoid this loss we should not increase thenumber of non-data tones but resort to exploiting theinformation of data tones for the purpose of improving theimpulsive noise estimation

Somemodern PLC standards such as Ghnem [19] adoptQAM modulation which requires a coherent receiver Manysimulation results have shown that the coherent receiver cansignificantly increase system performance even in scenar-ios with strong impulsive noise and time-varying channelbehavior [19] Let Ω be the set of QAM constellation pointsExploiting the information of data tones we can formulate anew impulsive noise estimation optimization problem

min i1st 1003817100381710038171003817Y119863 minus F119863i

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minus (ΛX)119863 minus F119863i

100381710038171003817100381722 le 1205762119883119896 isin Ω forall119896 isin 119863

(11)

where 119863 denotes the index set of data tones with a totalnumber 119873 minus 119872 Compared with (9) (11) adds some newconstraints corresponding to the information of data tones

Defining a new vector t = [X119879 i119879]119879 isin C2119873 the sourcesymbol vector X and the impulsive noise vector i can beexpressed as

X = Ut

i = Vt (12)

where the matrix U ≜ [I119873 0119873] isin R119873times2119873 V ≜ [0119873 I119873] isinR119873times2119873 We use 0119873 to denote the119873times119873 zeromatrix and I119873 todenote the119873times119873 identity matrix Then (11) can be rewrittenas

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762Ut (119896) isin Ω forall119896 isin 119863

(13)

where Φ ≜ [Λ119863 F119863] Clearly the optimization problem in(13) belongs to the class of NP-hard problems because there isa discrete constraint [20] To make the optimization problemfeasible we convert the nonconvex constraint into a convexconstraint in the sequel

We use ΩRe = Re(Ω) and ΩIm = Im(Ω) to representrespectively the sets of real and imaginary coordinates ofthe set Ω According to the origin-symmetric property of Ωthe constellation covers a rectangular region with boundariesgiven by

|Re (119878)| le max 119886 isin ΩRe ≜ 119906119877 forall119878 isin Ω|Im (119878)| le max 119886 isin ΩIm ≜ 119906119868 forall119878 isin Ω (14)

where 119906119877 = 119906119868 holds for square QAM constellationsSubstituting (14) into (13) the nonconvex constraint is

relaxed to the convex box constraint (15)

minus119906119877 le Re (Ut (119896)) le 119906119877minus119906119868 le Im (Ut (119896)) le 119906119868 (15)

Consequently we obtain the following convex problem

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(16)

1119873 denotes the 119873 times 1 column vector with all elements beingone

In problems (9) and (16) it is not easy to choose theconstants 1205761 and 1205762 that are dependent on the noise powerestimation Recalling (2) for the received OFDM signals wedenote the sum of impulsive noise and background noise ase = i+g In reality the amplitudes of asynchronous impulsivenoise samples are much larger than those of backgroundnoise samples So it is reasonable to consider e1 as anapproximation to i1 with g omitted such that the constants1205761 and 1205762 diminish in (9) and (16) Upon replacing i1 by e1we can rewrite the problem (16) as

min Vt1st Y = At

minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(17)

where

t = [X119879 e119879]119879Y = [Y119879

119863Y119879119863]119879

A = [F119863VΦ

] (18)

Mathematical Problems in Engineering 5

It can verify that problem (17) is a typical linear programming(LP) problems which can be solved by using the on-the-shelfsoftware packages such asCVX [21]We refer to the algorithmas ldquoLP-Allrdquo

4 The IN Estimation Using ADMM

Although in general the above LP-based approaches canprovide satisfying results they become computationallydemanding as the signal dimension increases because ofmatrix inversion [22] In this section we propose the ADMMframework to solve the IN estimation optimization problems

41 Framework of ADMM Let us consider the optimizationproblem in the following form [23]

min 119891 (119909) + 119892 (119911)st 119860119909 + 119861119911 = 119888 (19)

with variable 119909 isin R119899 and 119911 isin R119898 where 119860 isin R119901times119899119861 isin R119901times119898 and 119888 isin R119901 We also assume that 119891 and 119892 areconvex functions The augmented Lagrangian function of theproblem (19) is given by

L (119909 120582 119911) = 119891 (119909) + 119892 (119911) + 120582119879 (119860119909 + 119861119911 minus 119888)+ 1205882 119860119909 + 119861119911 minus 11988822 (20)

where 120582 isin R119901 is the Lagrangian multiplier and 120588 gt 0 isa penalty parameter Given (119911119896 120582119896) ADMM consists of theiterations as follows

119909119896+1 larr997888 argmin119909

L (119909 119911119896 120582119896)119911119896+1 larr997888 argmin

119911L (119909119896+1 119911 120582119896)

120582119896+1 larr997888 120582119896 + 120588 (119860119909119896+1 + 119861119911119896+1 minus 119888) (21)

Defining the scaled dual variable 119906 = (1120588)120582 we can expressADMM iterations as

119909119896+1 larr997888 argmin119909

(119891 (119909) + 1205882 10038171003817100381710038171003817119860119909 + 119861119911119896 minus 119888 + 1199061198961003817100381710038171003817100381722)119911119896+1 larr997888 argmin

119911(119892 (119911) + 1205882 10038171003817100381710038171003817119860119909119896+1 + 119861119911 minus 119888 + 1199061198961003817100381710038171003817100381722)

119906119896+1 larr997888 119906119896 + 119860119909119896+1 + 119861119911119896+1 minus 119888(22)

ADMM is proved to converge for all positive values of 120588under quite mild conditions which makes the selection of 120588rather flexible [24] Hence we can simply use a fixed positiveconstant for 120588 = 1 in our work

42The IN Estimation with All Tones Using ADMM (ADMM-All) In ADMM form the IN estimation algorithm (17) canbe written as

min I (t) +G (z1) + 1003817100381710038171003817z210038171003817100381710038171st Ut minus z1 = 0

Vt minus z2 = 0(23)

where I(t) is the indicator function of equality constrainsand G(z1) is the indicator function of the box constraint of(17)

We introduce an auxiliary vector z = [z1198791 z1198792 ]119879 isin C2119873

with z1 = Uz and z2 = VzThen the two constraints Utminusz1 =0 andVtminus z2 = 0 can be combined into one constraint tminus z =0 So we can express the augmented Lagrangian function asfollows

L (t z 120582) = I (t) +G (Uz) + Vz1 + 120582119879 (t minus z)+ 1205882 t minus z22 (24)

The corresponding ADMM iterations consisted of

t119896+1 larr997888 prod(z119896 minus u119896) (25)

z119896+11 larr997888 P (U (t119896+1 + u119896)) (26)

z119896+12 larr997888 S1120588 (V (t119896+1 + u119896)) (27)

u119896+1 larr997888 u119896 + t119896+1 minus z119896+1 (28)

where prod is projection onto t | At = Y and P is Euclidianprojection onto the box constraint (15)

The t-update (25) involves solving a linearly constrainedminimum Euclidean norm problem which can be expressedas

mint

10038171003817100381710038171003817t minus (z119896 minus u119896)1003817100381710038171003817100381722st At = Y (29)

The KKT (Karush-Kuhn-Tucker) conditions [22] for thisproblem are

[ I A119879

A 0][tlowast

]lowast] = [z119896 minus u119896

Y] (30)

where ]lowast is the optimal dual variable The solution tlowast of theabove equations is

tlowast = (I minus A119879 (AA119879)minus1 A) (z119896 minus u119896)+ A119879 (AA119879)minus1 Y

(31)

In z1-update (26) theP is elementwise Euclidian projec-tion onto the box constraint (15) which is defined as

P (119886) =

119906119877 119886 gt 119906119877119886 |119886| le 119906119877minus119906119877 119886 lt minus119906119877

(32)

In z2-update (27) S1120588(x119896+1 + u119896) is the well-knownelementwise soft thresholding operator [23] which is definedas

S120581 (119886) =

119886 minus 120581 119886 gt 1205810 |119886| le 120581119886 + 120581 119886 lt minus120581

(33)

6 Mathematical Problems in Engineering

43 The IN Estimation with Null Tones Using ADMM(ADMM-Null) We first recast problem (9) as LP form

min e1st Y119863 = F119863e (34)

And then (34) named ldquoLP-Nullrdquo can be written as

min I (e) + z1st e minus z = 0 (35)

whereI(e) is the indicator function of e isin C119873 | F119863e = Y119863The augmented Lagrangian function of problem (35) is

L (e z 120582) = I (e) + z1 + 120582119879 (e minus z) + 1205882 e minus z22 (36)

According the ADMM iterations (22) we can obtain theupdate primal variables e z and a dual variable u as follows

e119896+1 larr997888 prod(z119896 minus u119896) (37)

z119896+1 larr997888 S1120588 (x119896+1 + u119896) (38)

u119896+1 larr997888 u119896 + x119896+1 minus z119896+1 (39)

whereprod is projection onto e isin C119873 | F119863e = Y119863 and S1120588 isthe elementwise soft thresholding operator

The solution of e-update (37) can be derived using thesimilar method in (29)sim(31) as follows

elowast = (I minus F119879119863(F119863F119879119863)minus1 F119863) (z119896 minus u119896)

+ F119879119863(F119863F119879119863)minus1 Y119863

(40)

44 Complexity Analysis Theworst-case complexity of solv-ing LP-Null and LP-All is O(1198733) by using the interior pointmethod [22] For the proposedADMM-Null and ADMM-Allalgorithms the leading computation cost is the matrix inver-sions andmatrixmultiplications in (40) and (31) However wenotice that these inverses only have to be computed once Wecan caching the matrix inversion and use this cached valuein subsequent steps The resulting complexity is O(1198732) +1198723for ADMM-Null and O(1198732) + 1198733 for ADMM-All So thecomputation complexity of the IN estimation algorithmusingADMM is significantly lower than that using LP

5 Simulation Results

In this section we present computer simulation results for theproposed impulsive noise mitigation method in the complexbaseband OFDM-based PLC systems over an ideal (unit-gain nonfading) channel and a 15-path multipath power linechannel model [25] respectively In our simulations theimpulsive noise samples are generated using the GM modelin (4) and the MCA model in (5) respectively In orderto represent the typical noise scenarios given in [15] themean power ratio of impulsive-to-background noises is set

Table 1 Parameters of simulations

OFDM Carriers FFT length119873 = 128 null tones119872 = 56 datatones119873 minus119872 = 72

Modulation 4-QAM 16-QAMFEC Code A rate-12 convolutional codeGMModel 119870 = 3 119901119896 = 09 007 003 120574119896 = 1 100 1000MCAmodel 119860 = 01 Γ = 001 and the first 10 terms are

truncated

to be 30 dB in the GM model and 20 dB in the MCA modelrespectively The signal-to-noise ratio (SNR) is defined asSNR = 119875s119875e where 119875s is the power of the transmitted signaland 119875e is the total noise power including the impulsive noiseand background noise The detailed simulation parametersare listed in Table 1

In the same simulation environments we compare thebit error rate (BER) performances between our proposedmethod and several available mitigation methods in theliterature In Figures 2ndash7 we identified the three SBL-basedalgorithms in [15] respectively as their original notationsldquoSBL with null tonesrdquo ldquoSBL with all tonesrdquo and ldquoSBL withDFrdquo and the conventional OFDM system without impul-sive noise mitigation as ldquoNo mitigationrdquo For comparisonsbetween nonparametric and parametric methods we alsopresent the results for the MMSE detectors with noise stateinformation (NSI) in [12] which is identified as ldquoMMSE withNSIrdquo Note that the MMSE detector with NSI is optimalamong the parametric methods [12]

51 Ideal Channel Figures 2ndash5 compare the BER perfor-mances of all the above mitigation techniques for uncodedand coded systems using 4-QAM and 16-QAM respectivelyNo result of the ldquoSBL with DFrdquo method is shown in Figures 2and 4 since it is not applicable for uncoded systems

We first analyze the results for the uncoded 4-QAMsystem in Figure 2 Obviously our proposed ldquoLP-Allrdquo andldquoADMM-Allrdquo outperforms all the other methods except forthe MMSE detector with NSI which appears the best in alower SNR region However as SNR increases our methodbecomes the most advantageous in moderate to high SNRregion When only the nondata tones are considered forimpulsive noise estimation the algorithm ldquoLP-Nullrdquo andldquoADMM-Nullrdquo achieve less SNR gains than the SBL algo-rithm with null tones under both impulsive noise modelsConversely with all the tones being taken into account theldquoLP-Allrdquo and ldquoADMM-Allrdquo exploiting the modulus of con-stellation points present advantages over the SBL algorithmwith all tones achieving an SNR gain more than 2 dB in theGM model or between 4 dB and 6 dB in the MCA modelrespectively

Then we analyze the results presented in Figure 3 for thecoded systems using 4-QAM Again our proposed methoddemonstrates a favorable capability of outperforming most ofother algorithms in a similar tendency to that for the uncodedsystems in Figure 2 Compared with the SBL with DF algo-rithm our method presents a marginal performance loss inlower SNR region with GM impulsive noise Conversely in

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

4 Mathematical Problems in Engineering

the ℓ119901-norm for a real number 119901 ge 1 that is given by i119901 =(sum119899119896=1 |119894119896|119901)1119901Let i be a solution to the problem in (9) The impacts of

the impulsive noise can be removed by subtraction as follows

Y ≜ Y minus Fi = ΛX + F (i minus i) + G (10)

Assuming the impulsive noise residual (iminusi) can be negligiblethe receiver can then proceed as if only Gaussian noise ispresent and apply the conventional zero-forcing equalizationalgorithm

32 The IN Estimation with All Tones It is known thatincreasing the number of null-data tones is equivalent toincreasing the rank of matrix F119863 in (9) which can promotethe accuracy of impulsive noise estimation However theimprovement will be obtained at a cost of losing systemthroughput To avoid this loss we should not increase thenumber of non-data tones but resort to exploiting theinformation of data tones for the purpose of improving theimpulsive noise estimation

Somemodern PLC standards such as Ghnem [19] adoptQAM modulation which requires a coherent receiver Manysimulation results have shown that the coherent receiver cansignificantly increase system performance even in scenar-ios with strong impulsive noise and time-varying channelbehavior [19] Let Ω be the set of QAM constellation pointsExploiting the information of data tones we can formulate anew impulsive noise estimation optimization problem

min i1st 1003817100381710038171003817Y119863 minus F119863i

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minus (ΛX)119863 minus F119863i

100381710038171003817100381722 le 1205762119883119896 isin Ω forall119896 isin 119863

(11)

where 119863 denotes the index set of data tones with a totalnumber 119873 minus 119872 Compared with (9) (11) adds some newconstraints corresponding to the information of data tones

Defining a new vector t = [X119879 i119879]119879 isin C2119873 the sourcesymbol vector X and the impulsive noise vector i can beexpressed as

X = Ut

i = Vt (12)

where the matrix U ≜ [I119873 0119873] isin R119873times2119873 V ≜ [0119873 I119873] isinR119873times2119873 We use 0119873 to denote the119873times119873 zeromatrix and I119873 todenote the119873times119873 identity matrix Then (11) can be rewrittenas

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762Ut (119896) isin Ω forall119896 isin 119863

(13)

where Φ ≜ [Λ119863 F119863] Clearly the optimization problem in(13) belongs to the class of NP-hard problems because there isa discrete constraint [20] To make the optimization problemfeasible we convert the nonconvex constraint into a convexconstraint in the sequel

We use ΩRe = Re(Ω) and ΩIm = Im(Ω) to representrespectively the sets of real and imaginary coordinates ofthe set Ω According to the origin-symmetric property of Ωthe constellation covers a rectangular region with boundariesgiven by

|Re (119878)| le max 119886 isin ΩRe ≜ 119906119877 forall119878 isin Ω|Im (119878)| le max 119886 isin ΩIm ≜ 119906119868 forall119878 isin Ω (14)

where 119906119877 = 119906119868 holds for square QAM constellationsSubstituting (14) into (13) the nonconvex constraint is

relaxed to the convex box constraint (15)

minus119906119877 le Re (Ut (119896)) le 119906119877minus119906119868 le Im (Ut (119896)) le 119906119868 (15)

Consequently we obtain the following convex problem

min Vt1st 1003817100381710038171003817Y119863 minus F119863Vt

100381710038171003817100381722 le 12057611003817100381710038171003817Y119863 minusΦt100381710038171003817100381722 le 1205762minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(16)

1119873 denotes the 119873 times 1 column vector with all elements beingone

In problems (9) and (16) it is not easy to choose theconstants 1205761 and 1205762 that are dependent on the noise powerestimation Recalling (2) for the received OFDM signals wedenote the sum of impulsive noise and background noise ase = i+g In reality the amplitudes of asynchronous impulsivenoise samples are much larger than those of backgroundnoise samples So it is reasonable to consider e1 as anapproximation to i1 with g omitted such that the constants1205761 and 1205762 diminish in (9) and (16) Upon replacing i1 by e1we can rewrite the problem (16) as

min Vt1st Y = At

minus 119906119877 sdot 1119873 ⪯ Re (Ut) ⪯ 119906119877 sdot 1119873minus 119906119868 sdot 1119873 ⪯ Im (Ut) ⪯ 119906119868 sdot 1119873

(17)

where

t = [X119879 e119879]119879Y = [Y119879

119863Y119879119863]119879

A = [F119863VΦ

] (18)

Mathematical Problems in Engineering 5

It can verify that problem (17) is a typical linear programming(LP) problems which can be solved by using the on-the-shelfsoftware packages such asCVX [21]We refer to the algorithmas ldquoLP-Allrdquo

4 The IN Estimation Using ADMM

Although in general the above LP-based approaches canprovide satisfying results they become computationallydemanding as the signal dimension increases because ofmatrix inversion [22] In this section we propose the ADMMframework to solve the IN estimation optimization problems

41 Framework of ADMM Let us consider the optimizationproblem in the following form [23]

min 119891 (119909) + 119892 (119911)st 119860119909 + 119861119911 = 119888 (19)

with variable 119909 isin R119899 and 119911 isin R119898 where 119860 isin R119901times119899119861 isin R119901times119898 and 119888 isin R119901 We also assume that 119891 and 119892 areconvex functions The augmented Lagrangian function of theproblem (19) is given by

L (119909 120582 119911) = 119891 (119909) + 119892 (119911) + 120582119879 (119860119909 + 119861119911 minus 119888)+ 1205882 119860119909 + 119861119911 minus 11988822 (20)

where 120582 isin R119901 is the Lagrangian multiplier and 120588 gt 0 isa penalty parameter Given (119911119896 120582119896) ADMM consists of theiterations as follows

119909119896+1 larr997888 argmin119909

L (119909 119911119896 120582119896)119911119896+1 larr997888 argmin

119911L (119909119896+1 119911 120582119896)

120582119896+1 larr997888 120582119896 + 120588 (119860119909119896+1 + 119861119911119896+1 minus 119888) (21)

Defining the scaled dual variable 119906 = (1120588)120582 we can expressADMM iterations as

119909119896+1 larr997888 argmin119909

(119891 (119909) + 1205882 10038171003817100381710038171003817119860119909 + 119861119911119896 minus 119888 + 1199061198961003817100381710038171003817100381722)119911119896+1 larr997888 argmin

119911(119892 (119911) + 1205882 10038171003817100381710038171003817119860119909119896+1 + 119861119911 minus 119888 + 1199061198961003817100381710038171003817100381722)

119906119896+1 larr997888 119906119896 + 119860119909119896+1 + 119861119911119896+1 minus 119888(22)

ADMM is proved to converge for all positive values of 120588under quite mild conditions which makes the selection of 120588rather flexible [24] Hence we can simply use a fixed positiveconstant for 120588 = 1 in our work

42The IN Estimation with All Tones Using ADMM (ADMM-All) In ADMM form the IN estimation algorithm (17) canbe written as

min I (t) +G (z1) + 1003817100381710038171003817z210038171003817100381710038171st Ut minus z1 = 0

Vt minus z2 = 0(23)

where I(t) is the indicator function of equality constrainsand G(z1) is the indicator function of the box constraint of(17)

We introduce an auxiliary vector z = [z1198791 z1198792 ]119879 isin C2119873

with z1 = Uz and z2 = VzThen the two constraints Utminusz1 =0 andVtminus z2 = 0 can be combined into one constraint tminus z =0 So we can express the augmented Lagrangian function asfollows

L (t z 120582) = I (t) +G (Uz) + Vz1 + 120582119879 (t minus z)+ 1205882 t minus z22 (24)

The corresponding ADMM iterations consisted of

t119896+1 larr997888 prod(z119896 minus u119896) (25)

z119896+11 larr997888 P (U (t119896+1 + u119896)) (26)

z119896+12 larr997888 S1120588 (V (t119896+1 + u119896)) (27)

u119896+1 larr997888 u119896 + t119896+1 minus z119896+1 (28)

where prod is projection onto t | At = Y and P is Euclidianprojection onto the box constraint (15)

The t-update (25) involves solving a linearly constrainedminimum Euclidean norm problem which can be expressedas

mint

10038171003817100381710038171003817t minus (z119896 minus u119896)1003817100381710038171003817100381722st At = Y (29)

The KKT (Karush-Kuhn-Tucker) conditions [22] for thisproblem are

[ I A119879

A 0][tlowast

]lowast] = [z119896 minus u119896

Y] (30)

where ]lowast is the optimal dual variable The solution tlowast of theabove equations is

tlowast = (I minus A119879 (AA119879)minus1 A) (z119896 minus u119896)+ A119879 (AA119879)minus1 Y

(31)

In z1-update (26) theP is elementwise Euclidian projec-tion onto the box constraint (15) which is defined as

P (119886) =

119906119877 119886 gt 119906119877119886 |119886| le 119906119877minus119906119877 119886 lt minus119906119877

(32)

In z2-update (27) S1120588(x119896+1 + u119896) is the well-knownelementwise soft thresholding operator [23] which is definedas

S120581 (119886) =

119886 minus 120581 119886 gt 1205810 |119886| le 120581119886 + 120581 119886 lt minus120581

(33)

6 Mathematical Problems in Engineering

43 The IN Estimation with Null Tones Using ADMM(ADMM-Null) We first recast problem (9) as LP form

min e1st Y119863 = F119863e (34)

And then (34) named ldquoLP-Nullrdquo can be written as

min I (e) + z1st e minus z = 0 (35)

whereI(e) is the indicator function of e isin C119873 | F119863e = Y119863The augmented Lagrangian function of problem (35) is

L (e z 120582) = I (e) + z1 + 120582119879 (e minus z) + 1205882 e minus z22 (36)

According the ADMM iterations (22) we can obtain theupdate primal variables e z and a dual variable u as follows

e119896+1 larr997888 prod(z119896 minus u119896) (37)

z119896+1 larr997888 S1120588 (x119896+1 + u119896) (38)

u119896+1 larr997888 u119896 + x119896+1 minus z119896+1 (39)

whereprod is projection onto e isin C119873 | F119863e = Y119863 and S1120588 isthe elementwise soft thresholding operator

The solution of e-update (37) can be derived using thesimilar method in (29)sim(31) as follows

elowast = (I minus F119879119863(F119863F119879119863)minus1 F119863) (z119896 minus u119896)

+ F119879119863(F119863F119879119863)minus1 Y119863

(40)

44 Complexity Analysis Theworst-case complexity of solv-ing LP-Null and LP-All is O(1198733) by using the interior pointmethod [22] For the proposedADMM-Null and ADMM-Allalgorithms the leading computation cost is the matrix inver-sions andmatrixmultiplications in (40) and (31) However wenotice that these inverses only have to be computed once Wecan caching the matrix inversion and use this cached valuein subsequent steps The resulting complexity is O(1198732) +1198723for ADMM-Null and O(1198732) + 1198733 for ADMM-All So thecomputation complexity of the IN estimation algorithmusingADMM is significantly lower than that using LP

5 Simulation Results

In this section we present computer simulation results for theproposed impulsive noise mitigation method in the complexbaseband OFDM-based PLC systems over an ideal (unit-gain nonfading) channel and a 15-path multipath power linechannel model [25] respectively In our simulations theimpulsive noise samples are generated using the GM modelin (4) and the MCA model in (5) respectively In orderto represent the typical noise scenarios given in [15] themean power ratio of impulsive-to-background noises is set

Table 1 Parameters of simulations

OFDM Carriers FFT length119873 = 128 null tones119872 = 56 datatones119873 minus119872 = 72

Modulation 4-QAM 16-QAMFEC Code A rate-12 convolutional codeGMModel 119870 = 3 119901119896 = 09 007 003 120574119896 = 1 100 1000MCAmodel 119860 = 01 Γ = 001 and the first 10 terms are

truncated

to be 30 dB in the GM model and 20 dB in the MCA modelrespectively The signal-to-noise ratio (SNR) is defined asSNR = 119875s119875e where 119875s is the power of the transmitted signaland 119875e is the total noise power including the impulsive noiseand background noise The detailed simulation parametersare listed in Table 1

In the same simulation environments we compare thebit error rate (BER) performances between our proposedmethod and several available mitigation methods in theliterature In Figures 2ndash7 we identified the three SBL-basedalgorithms in [15] respectively as their original notationsldquoSBL with null tonesrdquo ldquoSBL with all tonesrdquo and ldquoSBL withDFrdquo and the conventional OFDM system without impul-sive noise mitigation as ldquoNo mitigationrdquo For comparisonsbetween nonparametric and parametric methods we alsopresent the results for the MMSE detectors with noise stateinformation (NSI) in [12] which is identified as ldquoMMSE withNSIrdquo Note that the MMSE detector with NSI is optimalamong the parametric methods [12]

51 Ideal Channel Figures 2ndash5 compare the BER perfor-mances of all the above mitigation techniques for uncodedand coded systems using 4-QAM and 16-QAM respectivelyNo result of the ldquoSBL with DFrdquo method is shown in Figures 2and 4 since it is not applicable for uncoded systems

We first analyze the results for the uncoded 4-QAMsystem in Figure 2 Obviously our proposed ldquoLP-Allrdquo andldquoADMM-Allrdquo outperforms all the other methods except forthe MMSE detector with NSI which appears the best in alower SNR region However as SNR increases our methodbecomes the most advantageous in moderate to high SNRregion When only the nondata tones are considered forimpulsive noise estimation the algorithm ldquoLP-Nullrdquo andldquoADMM-Nullrdquo achieve less SNR gains than the SBL algo-rithm with null tones under both impulsive noise modelsConversely with all the tones being taken into account theldquoLP-Allrdquo and ldquoADMM-Allrdquo exploiting the modulus of con-stellation points present advantages over the SBL algorithmwith all tones achieving an SNR gain more than 2 dB in theGM model or between 4 dB and 6 dB in the MCA modelrespectively

Then we analyze the results presented in Figure 3 for thecoded systems using 4-QAM Again our proposed methoddemonstrates a favorable capability of outperforming most ofother algorithms in a similar tendency to that for the uncodedsystems in Figure 2 Compared with the SBL with DF algo-rithm our method presents a marginal performance loss inlower SNR region with GM impulsive noise Conversely in

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

Mathematical Problems in Engineering 5

It can verify that problem (17) is a typical linear programming(LP) problems which can be solved by using the on-the-shelfsoftware packages such asCVX [21]We refer to the algorithmas ldquoLP-Allrdquo

4 The IN Estimation Using ADMM

Although in general the above LP-based approaches canprovide satisfying results they become computationallydemanding as the signal dimension increases because ofmatrix inversion [22] In this section we propose the ADMMframework to solve the IN estimation optimization problems

41 Framework of ADMM Let us consider the optimizationproblem in the following form [23]

min 119891 (119909) + 119892 (119911)st 119860119909 + 119861119911 = 119888 (19)

with variable 119909 isin R119899 and 119911 isin R119898 where 119860 isin R119901times119899119861 isin R119901times119898 and 119888 isin R119901 We also assume that 119891 and 119892 areconvex functions The augmented Lagrangian function of theproblem (19) is given by

L (119909 120582 119911) = 119891 (119909) + 119892 (119911) + 120582119879 (119860119909 + 119861119911 minus 119888)+ 1205882 119860119909 + 119861119911 minus 11988822 (20)

where 120582 isin R119901 is the Lagrangian multiplier and 120588 gt 0 isa penalty parameter Given (119911119896 120582119896) ADMM consists of theiterations as follows

119909119896+1 larr997888 argmin119909

L (119909 119911119896 120582119896)119911119896+1 larr997888 argmin

119911L (119909119896+1 119911 120582119896)

120582119896+1 larr997888 120582119896 + 120588 (119860119909119896+1 + 119861119911119896+1 minus 119888) (21)

Defining the scaled dual variable 119906 = (1120588)120582 we can expressADMM iterations as

119909119896+1 larr997888 argmin119909

(119891 (119909) + 1205882 10038171003817100381710038171003817119860119909 + 119861119911119896 minus 119888 + 1199061198961003817100381710038171003817100381722)119911119896+1 larr997888 argmin

119911(119892 (119911) + 1205882 10038171003817100381710038171003817119860119909119896+1 + 119861119911 minus 119888 + 1199061198961003817100381710038171003817100381722)

119906119896+1 larr997888 119906119896 + 119860119909119896+1 + 119861119911119896+1 minus 119888(22)

ADMM is proved to converge for all positive values of 120588under quite mild conditions which makes the selection of 120588rather flexible [24] Hence we can simply use a fixed positiveconstant for 120588 = 1 in our work

42The IN Estimation with All Tones Using ADMM (ADMM-All) In ADMM form the IN estimation algorithm (17) canbe written as

min I (t) +G (z1) + 1003817100381710038171003817z210038171003817100381710038171st Ut minus z1 = 0

Vt minus z2 = 0(23)

where I(t) is the indicator function of equality constrainsand G(z1) is the indicator function of the box constraint of(17)

We introduce an auxiliary vector z = [z1198791 z1198792 ]119879 isin C2119873

with z1 = Uz and z2 = VzThen the two constraints Utminusz1 =0 andVtminus z2 = 0 can be combined into one constraint tminus z =0 So we can express the augmented Lagrangian function asfollows

L (t z 120582) = I (t) +G (Uz) + Vz1 + 120582119879 (t minus z)+ 1205882 t minus z22 (24)

The corresponding ADMM iterations consisted of

t119896+1 larr997888 prod(z119896 minus u119896) (25)

z119896+11 larr997888 P (U (t119896+1 + u119896)) (26)

z119896+12 larr997888 S1120588 (V (t119896+1 + u119896)) (27)

u119896+1 larr997888 u119896 + t119896+1 minus z119896+1 (28)

where prod is projection onto t | At = Y and P is Euclidianprojection onto the box constraint (15)

The t-update (25) involves solving a linearly constrainedminimum Euclidean norm problem which can be expressedas

mint

10038171003817100381710038171003817t minus (z119896 minus u119896)1003817100381710038171003817100381722st At = Y (29)

The KKT (Karush-Kuhn-Tucker) conditions [22] for thisproblem are

[ I A119879

A 0][tlowast

]lowast] = [z119896 minus u119896

Y] (30)

where ]lowast is the optimal dual variable The solution tlowast of theabove equations is

tlowast = (I minus A119879 (AA119879)minus1 A) (z119896 minus u119896)+ A119879 (AA119879)minus1 Y

(31)

In z1-update (26) theP is elementwise Euclidian projec-tion onto the box constraint (15) which is defined as

P (119886) =

119906119877 119886 gt 119906119877119886 |119886| le 119906119877minus119906119877 119886 lt minus119906119877

(32)

In z2-update (27) S1120588(x119896+1 + u119896) is the well-knownelementwise soft thresholding operator [23] which is definedas

S120581 (119886) =

119886 minus 120581 119886 gt 1205810 |119886| le 120581119886 + 120581 119886 lt minus120581

(33)

6 Mathematical Problems in Engineering

43 The IN Estimation with Null Tones Using ADMM(ADMM-Null) We first recast problem (9) as LP form

min e1st Y119863 = F119863e (34)

And then (34) named ldquoLP-Nullrdquo can be written as

min I (e) + z1st e minus z = 0 (35)

whereI(e) is the indicator function of e isin C119873 | F119863e = Y119863The augmented Lagrangian function of problem (35) is

L (e z 120582) = I (e) + z1 + 120582119879 (e minus z) + 1205882 e minus z22 (36)

According the ADMM iterations (22) we can obtain theupdate primal variables e z and a dual variable u as follows

e119896+1 larr997888 prod(z119896 minus u119896) (37)

z119896+1 larr997888 S1120588 (x119896+1 + u119896) (38)

u119896+1 larr997888 u119896 + x119896+1 minus z119896+1 (39)

whereprod is projection onto e isin C119873 | F119863e = Y119863 and S1120588 isthe elementwise soft thresholding operator

The solution of e-update (37) can be derived using thesimilar method in (29)sim(31) as follows

elowast = (I minus F119879119863(F119863F119879119863)minus1 F119863) (z119896 minus u119896)

+ F119879119863(F119863F119879119863)minus1 Y119863

(40)

44 Complexity Analysis Theworst-case complexity of solv-ing LP-Null and LP-All is O(1198733) by using the interior pointmethod [22] For the proposedADMM-Null and ADMM-Allalgorithms the leading computation cost is the matrix inver-sions andmatrixmultiplications in (40) and (31) However wenotice that these inverses only have to be computed once Wecan caching the matrix inversion and use this cached valuein subsequent steps The resulting complexity is O(1198732) +1198723for ADMM-Null and O(1198732) + 1198733 for ADMM-All So thecomputation complexity of the IN estimation algorithmusingADMM is significantly lower than that using LP

5 Simulation Results

In this section we present computer simulation results for theproposed impulsive noise mitigation method in the complexbaseband OFDM-based PLC systems over an ideal (unit-gain nonfading) channel and a 15-path multipath power linechannel model [25] respectively In our simulations theimpulsive noise samples are generated using the GM modelin (4) and the MCA model in (5) respectively In orderto represent the typical noise scenarios given in [15] themean power ratio of impulsive-to-background noises is set

Table 1 Parameters of simulations

OFDM Carriers FFT length119873 = 128 null tones119872 = 56 datatones119873 minus119872 = 72

Modulation 4-QAM 16-QAMFEC Code A rate-12 convolutional codeGMModel 119870 = 3 119901119896 = 09 007 003 120574119896 = 1 100 1000MCAmodel 119860 = 01 Γ = 001 and the first 10 terms are

truncated

to be 30 dB in the GM model and 20 dB in the MCA modelrespectively The signal-to-noise ratio (SNR) is defined asSNR = 119875s119875e where 119875s is the power of the transmitted signaland 119875e is the total noise power including the impulsive noiseand background noise The detailed simulation parametersare listed in Table 1

In the same simulation environments we compare thebit error rate (BER) performances between our proposedmethod and several available mitigation methods in theliterature In Figures 2ndash7 we identified the three SBL-basedalgorithms in [15] respectively as their original notationsldquoSBL with null tonesrdquo ldquoSBL with all tonesrdquo and ldquoSBL withDFrdquo and the conventional OFDM system without impul-sive noise mitigation as ldquoNo mitigationrdquo For comparisonsbetween nonparametric and parametric methods we alsopresent the results for the MMSE detectors with noise stateinformation (NSI) in [12] which is identified as ldquoMMSE withNSIrdquo Note that the MMSE detector with NSI is optimalamong the parametric methods [12]

51 Ideal Channel Figures 2ndash5 compare the BER perfor-mances of all the above mitigation techniques for uncodedand coded systems using 4-QAM and 16-QAM respectivelyNo result of the ldquoSBL with DFrdquo method is shown in Figures 2and 4 since it is not applicable for uncoded systems

We first analyze the results for the uncoded 4-QAMsystem in Figure 2 Obviously our proposed ldquoLP-Allrdquo andldquoADMM-Allrdquo outperforms all the other methods except forthe MMSE detector with NSI which appears the best in alower SNR region However as SNR increases our methodbecomes the most advantageous in moderate to high SNRregion When only the nondata tones are considered forimpulsive noise estimation the algorithm ldquoLP-Nullrdquo andldquoADMM-Nullrdquo achieve less SNR gains than the SBL algo-rithm with null tones under both impulsive noise modelsConversely with all the tones being taken into account theldquoLP-Allrdquo and ldquoADMM-Allrdquo exploiting the modulus of con-stellation points present advantages over the SBL algorithmwith all tones achieving an SNR gain more than 2 dB in theGM model or between 4 dB and 6 dB in the MCA modelrespectively

Then we analyze the results presented in Figure 3 for thecoded systems using 4-QAM Again our proposed methoddemonstrates a favorable capability of outperforming most ofother algorithms in a similar tendency to that for the uncodedsystems in Figure 2 Compared with the SBL with DF algo-rithm our method presents a marginal performance loss inlower SNR region with GM impulsive noise Conversely in

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

6 Mathematical Problems in Engineering

43 The IN Estimation with Null Tones Using ADMM(ADMM-Null) We first recast problem (9) as LP form

min e1st Y119863 = F119863e (34)

And then (34) named ldquoLP-Nullrdquo can be written as

min I (e) + z1st e minus z = 0 (35)

whereI(e) is the indicator function of e isin C119873 | F119863e = Y119863The augmented Lagrangian function of problem (35) is

L (e z 120582) = I (e) + z1 + 120582119879 (e minus z) + 1205882 e minus z22 (36)

According the ADMM iterations (22) we can obtain theupdate primal variables e z and a dual variable u as follows

e119896+1 larr997888 prod(z119896 minus u119896) (37)

z119896+1 larr997888 S1120588 (x119896+1 + u119896) (38)

u119896+1 larr997888 u119896 + x119896+1 minus z119896+1 (39)

whereprod is projection onto e isin C119873 | F119863e = Y119863 and S1120588 isthe elementwise soft thresholding operator

The solution of e-update (37) can be derived using thesimilar method in (29)sim(31) as follows

elowast = (I minus F119879119863(F119863F119879119863)minus1 F119863) (z119896 minus u119896)

+ F119879119863(F119863F119879119863)minus1 Y119863

(40)

44 Complexity Analysis Theworst-case complexity of solv-ing LP-Null and LP-All is O(1198733) by using the interior pointmethod [22] For the proposedADMM-Null and ADMM-Allalgorithms the leading computation cost is the matrix inver-sions andmatrixmultiplications in (40) and (31) However wenotice that these inverses only have to be computed once Wecan caching the matrix inversion and use this cached valuein subsequent steps The resulting complexity is O(1198732) +1198723for ADMM-Null and O(1198732) + 1198733 for ADMM-All So thecomputation complexity of the IN estimation algorithmusingADMM is significantly lower than that using LP

5 Simulation Results

In this section we present computer simulation results for theproposed impulsive noise mitigation method in the complexbaseband OFDM-based PLC systems over an ideal (unit-gain nonfading) channel and a 15-path multipath power linechannel model [25] respectively In our simulations theimpulsive noise samples are generated using the GM modelin (4) and the MCA model in (5) respectively In orderto represent the typical noise scenarios given in [15] themean power ratio of impulsive-to-background noises is set

Table 1 Parameters of simulations

OFDM Carriers FFT length119873 = 128 null tones119872 = 56 datatones119873 minus119872 = 72

Modulation 4-QAM 16-QAMFEC Code A rate-12 convolutional codeGMModel 119870 = 3 119901119896 = 09 007 003 120574119896 = 1 100 1000MCAmodel 119860 = 01 Γ = 001 and the first 10 terms are

truncated

to be 30 dB in the GM model and 20 dB in the MCA modelrespectively The signal-to-noise ratio (SNR) is defined asSNR = 119875s119875e where 119875s is the power of the transmitted signaland 119875e is the total noise power including the impulsive noiseand background noise The detailed simulation parametersare listed in Table 1

In the same simulation environments we compare thebit error rate (BER) performances between our proposedmethod and several available mitigation methods in theliterature In Figures 2ndash7 we identified the three SBL-basedalgorithms in [15] respectively as their original notationsldquoSBL with null tonesrdquo ldquoSBL with all tonesrdquo and ldquoSBL withDFrdquo and the conventional OFDM system without impul-sive noise mitigation as ldquoNo mitigationrdquo For comparisonsbetween nonparametric and parametric methods we alsopresent the results for the MMSE detectors with noise stateinformation (NSI) in [12] which is identified as ldquoMMSE withNSIrdquo Note that the MMSE detector with NSI is optimalamong the parametric methods [12]

51 Ideal Channel Figures 2ndash5 compare the BER perfor-mances of all the above mitigation techniques for uncodedand coded systems using 4-QAM and 16-QAM respectivelyNo result of the ldquoSBL with DFrdquo method is shown in Figures 2and 4 since it is not applicable for uncoded systems

We first analyze the results for the uncoded 4-QAMsystem in Figure 2 Obviously our proposed ldquoLP-Allrdquo andldquoADMM-Allrdquo outperforms all the other methods except forthe MMSE detector with NSI which appears the best in alower SNR region However as SNR increases our methodbecomes the most advantageous in moderate to high SNRregion When only the nondata tones are considered forimpulsive noise estimation the algorithm ldquoLP-Nullrdquo andldquoADMM-Nullrdquo achieve less SNR gains than the SBL algo-rithm with null tones under both impulsive noise modelsConversely with all the tones being taken into account theldquoLP-Allrdquo and ldquoADMM-Allrdquo exploiting the modulus of con-stellation points present advantages over the SBL algorithmwith all tones achieving an SNR gain more than 2 dB in theGM model or between 4 dB and 6 dB in the MCA modelrespectively

Then we analyze the results presented in Figure 3 for thecoded systems using 4-QAM Again our proposed methoddemonstrates a favorable capability of outperforming most ofother algorithms in a similar tendency to that for the uncodedsystems in Figure 2 Compared with the SBL with DF algo-rithm our method presents a marginal performance loss inlower SNR region with GM impulsive noise Conversely in

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

Mathematical Problems in Engineering 7

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

640 2 8 10minus4 minus2minus6SNR(dB)

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 2 BER performance comparison for various mitigation methods in uncoded 4-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 3 BER performance comparison for various mitigation methods in coded 4-QAM system

the scenario of MCA our method achieves a relatively largerSNR gain where the SBL with DF algorithm no longer worksas well as in GM since it needs to impose a priori informationto the covariance matrix of impulsive noise [15]

From the simulation results for 16-QAM in Figures 4and 5 it is observed that the BER performances of allthe mitigation methods are degraded in both uncoded andcoded cases This can be explained in that the constellation

points for high-order modulation become much tighter fortransmit power given and hence the detection of OFDMdata symbols are more susceptible to the impulsive noiseresidual As a consequence over a wide range of SNR valuesthe MMSE with NSI becomes even worse than the SBL withnull tones in the uncoded system and than the SBL with DFin the coded system respectively Moreover the SBL withnull tones becomes more effective than the SBL with all

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

8 Mathematical Problems in Engineering

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 4 BER performance comparison for various mitigation methods in uncoded 16-QAM system

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

640 2 8 10minus4 minus2minus6SNR(dB)

10minus7

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 5 BER performance comparison for various mitigation methods in coded 16-QAM system

tones It is promising to observe that in the 16-QAM systemsour method can obtain substantial performance gains overall other methods for moderate to high SNR values and itbecomes the best method for achieving the BERs below 10minus2for practical purposes

52 PLC Channel In this section the 15-path multipathmodel is adopted as the PLC channel parameters of whichare the same as those listed in Table IV in [25]The frequency

range is 35sim91kHz which is adopted by the NBPLC standard[19 26] Using the model we can obtain the subcarrier gainwhich are used as the diagonal elements of matrix in (11) Atthe receiver the zero-forcing equalizer is adopted to equalizethe received symbols As the PLC channel is deep fadingchannel we set the SNR range is 0sim20 dB Figures 6 and 7compare the BER performances of all the above mitigationtechniques for uncoded and coded systems using 4-QAMrespectively

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

Mathematical Problems in Engineering 9

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

(b) MCA model

Figure 6 BER performance comparison for various mitigation methods in uncoded 4-QAM system The PLC channel frequency responseis known at the receiver

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(a) GM model

No mitigationMMSE with NSISBL with null tonesSBL with all tones

LP-NullADMM-NullLP-AllADMM-All

2 4 6 8 10 12 14 16 18 200SNR(dB)

10minus6

10minus5

10minus4

10minus3

10minus2

10minus1

100

Bit e

rror

rate

SBL with DF

(b) MCA model

Figure 7 BER performance comparison for various mitigation methods in coded 4-QAM system The PLC channel frequency response isknown at the receiver

We first analyze the result for the uncoded 4-QAMsystem in Figure 6 Obviously our method outperforms allthe other methods except for the MMSE detector with NSIwhich appears the best in the lower to moderate SNR regionHowever as SNR increases the ldquoLP-Allrdquo and ldquoADMM-Allrdquo becomes the most advantageous in high SNR region

Moreover the SBL with null tones and the CS algorithm ldquoLP-Nullrdquo and ldquoADMM-Nullrdquo are more effective than the SBLwith all tones

Then we analyze the results presented in Figure 7 forthe coded systems using 4-QAM Again our proposed ldquoLP-Allrdquo and ldquoADMM-Allrdquo demonstrate a favorable capability of

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

10 Mathematical Problems in Engineering

outperformingmost of other algorithms in a similar tendencyto that for the uncoded systems in Figure 6 Compared withthe SBL with DF algorithm our method presents a marginalperformance loss in moderate SNR region However as SNRincreases our method achieves a relatively larger SNR gain

6 Conclusion

In this paper we have explored a nonparametric CS methodfor mitigating the asynchronous impulsive noise in OFDM-based PLC systems The proposed method extends the con-ventional CS technique to exploit information with respectto all tones of OFDM symbol Upon imposing detectionconstraints on the equalized symbols the impulsive noiseestimation has been formulated as a nonconvex optimiza-tion problem and then changed to be an LP problemwith polynomial complexities The ADMM framework isadopted to reduce the computation complexity Simulationresults have shown that the proposed method can achieveBER performance improvements at reduced computationalcomplexities compared with the conventional methods Inparticular it provides a practical technique of asynchronousimpulsive noise mitigation in PLC systems which employhigh-order QAM for increasing the system throughput Itis worth mentioning that the proposed method can alsobe applied in the periodic impulsive noise mitigation whenincorporating the time domain interleaving (TDI-OFDM)technique of [27]

Data Availability

TheMatlab Source Code data used to support the findings ofthis study are available from the corresponding author uponrequest

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant no 61571250the Natural Science Foundation of Zhejiang Province underGrant no LY18F010010 the Zhejiang Open Foundation ofthe Most Important Subjects under Grant no XKXL1415the Science Foundation of Zhejiang Business TechnologyInstitute under Grant no 2017Z01 the K C Wong MagnaFund in Ningbo University and Key Laboratory of MobileInternet Application Technology of Zhejiang Province

References

[1] S Galli A Scaglione and Z Wang ldquoFor the grid and throughthe grid the role of power line communications in the smartgridrdquo Proceedings of the IEEE vol 99 no 6 pp 998ndash1027 2011

[2] Y Yan YQianH Sharif andD Tipper ldquoA survey on smart gridcommunication infrastructuresMotivations requirements and

challengesrdquo IEEE Communications Surveys amp Tutorials vol 15no 1 pp 5ndash20 2013

[3] G Bumiller L Lampe and H Hrasnica ldquoPower line com-munication networks for large-scale control and automationsystemsrdquo IEEE Communications Magazine vol 48 no 4 pp106ndash113 2010

[4] M Nassar J Lin Y Mortazavi A Dabak I H Kim andB L Evans ldquoLocal utility power line communications in the3ndash500 kHz band channel impairments noise and standardsrdquoIEEE Signal Processing Magazine vol 29 no 5 pp 116ndash127 2012

[5] M Ghosh ldquoAnalysis of the effect of impulse noise on multi-carrier and single carrier QAM systemsrdquo IEEE Transactions onCommunications vol 44 no 2 pp 145ndash147 1996

[6] M Zimmermann and K Dostert ldquoAnalysis and modeling ofimpulsive noise in broad-band powerline communicationsrdquoIEEETransactions on Electromagnetic Compatibility vol 44 no1 pp 249ndash258 2002

[7] H Meng Y L Guan and S Chen ldquoModeling and analysis ofnoise effects on broadband power-line communicationsrdquo IEEETransactions on PowerDelivery vol 20 no 2 pp 630ndash637 2005

[8] S V Zhidkov ldquoAnalysis and comparison of several simpleimpulsive noise mitigation schemes for OFDM receiversrdquo IEEETransactions on Communications vol 56 no 1 pp 5ndash9 2008

[9] F H Juwono Q Guo D Huang and K P Wong ldquoDeepclipping for impulsive noisemitigation in OFDM-based power-line communicationsrdquo IEEE Transactions on Power Deliveryvol 29 no 3 pp 1335ndash1343 2014

[10] G Ndo P Siohan and M-H Hamon ldquoAdaptive noise mit-igation in impulsive environment Application to power-linecommunicationsrdquo IEEE Transactions on Power Delivery vol 25no 2 pp 647ndash656 2010

[11] E Alsusa and K M Rabie ldquoDynamic peak-based thresholdestimationmethod formitigating impulsive noise in power-linecommunication systemsrdquo IEEE Transactions on Power Deliveryvol 28 no 4 pp 2201ndash2208 2013

[12] M Nassar Graphical models and message passing receivers forinterference limited communication systems [PhD dissertation]University of Texas Austin TX USA 2013

[13] G Caire T Y Al-Naffouri andAK Narayanan ldquoImpulse noisecancellation in OFDM an application of compressed sensingrdquoin Proceedings of the 2008 IEEE International Symposium onInformationTheory - ISIT pp 1293ndash1297 Toronto ON CanadaJuly 2008

[14] L Lampe ldquoBursty impulse noise detection by compressed sens-ingrdquo in Proceedings of the 2011 IEEE International Symposium onPower Line Communications and Its Applications (ISPLC) pp29ndash34 Udine Italy April 2011

[15] J Lin M Nassar and B L Evans ldquoImpulsive noise mitigationin powerline communications using sparse bayesian learningrdquoIEEE Journal on Selected Areas in Communications vol 31 no7 pp 1172ndash1183 2013

[16] T Y Al-Naffouri A A Quadeer and G Caire ldquoImpulse noiseestimation and removal for OFDM systemsrdquo IEEE Transactionson Communications vol 62 no 3 pp 976ndash989 2014

[17] M Nassar K Gulati Y Mortazavi and B L Evans ldquoStatisti-cal Modeling of Asynchronous Impulsive Noise in PowerlineCommunication Networksrdquo in Proceedings of the 2011 IEEEGlobal Communications Conference (GLOBECOM 2011) pp 1ndash6 Houston TX USA December 2011

[18] R G Baraniuk ldquoCompressive sensingrdquo IEEE Signal ProcessingMagazine vol 24 no 4 pp 118ndash121 2007

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

Mathematical Problems in Engineering 11

[19] V Oksman and J Zhang ldquoGHNEM the new ITU-T standardon narrowband PLC technologyrdquo IEEE Communications Maga-zine vol 49 no 12 pp 36ndash44 2011

[20] N D Sidiropoulos and Z-Q Luo ldquoA semidefinite relaxationapproach to MIMO detection for high-order QAM constella-tionsrdquo IEEE Signal Processing Letters vol 13 no 9 pp 525ndash5282006

[21] CVX Matlab software for disciplined convex programminghttpcvxrcomcvx

[22] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[23] S Boyd N Parikh E Chu B Peleato and J Eckstein ldquoDis-tributed optimization and statistical learning via the alternatingdirection method of multipliersrdquo Foundations and Trends inMachine Learning vol 3 no 1 pp 1ndash122 2010

[24] E Ghadimi A Teixeira I Shames andM Johansson ldquoOptimalparameter selection for the alternating direction method ofmultipliers (ADMM) quadratic problemsrdquo Institute of Electricaland Electronics Engineers Transactions on Automatic Controlvol 60 no 3 pp 644ndash658 2015

[25] M Zimmermann and K Dostert ldquoA multipath model for thepowerline channelrdquo IEEE Transactions on Communications vol50 no 4 pp 553ndash559 2002

[26] M Hoch ldquoComparison of PLC G3 and PRIMErdquo in Proceedingsof the 2011 IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC) pp 165ndash169 UdineItaly April 2011

[27] M Mirahmadi A Al-Dweik and A Shami ldquoBER reductionof OFDM based broadband communication systems over mul-tipath channels with impulsive noiserdquo IEEE Transactions onCommunications vol 61 no 11 pp 4602ndash4615 2013

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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