Cooperative Closed-Loop Coded-MIMO Transmissions for Smart ... · ResearchArticle Cooperative Closed-Loop Coded-MIMO Transmissions for Smart Grid Wireless Applications NdéyeBinetaSarr

Research ArticleCooperative Closed-Loop Coded-MIMO Transmissions for SmartGrid Wireless Applications

Ndeacuteye Bineta Sarr 12 Olufemi J Oyedapo3 Basile L Agba4 Franccedilois Gagnon1

Herveacute Boeglen2 and Rodolphe Vauzelle2

1 Ecole de Technologie Superieure (ETS) Montreal QC Canada H3C 1K32XLIM Institute University of Poitiers 86360 Futuroscope France3McGill University Montreal QC Canada4Institut de Recherche drsquoHydro-Quebec (IREQ) Varennes QC Canada J3X 1S1

Correspondence should be addressed to Ndeye Bineta Sarr ndeye-binetasarr1ensetsmtlca

Received 9 December 2018 Accepted 21 March 2019 Published 7 April 2019

Academic Editor Laurie Cuthbert

Copyright copy 2019 Ndeye Bineta Sarr et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Inherent interfering signals generated by the underlying elements found in power substations have been known to span overconsecutive noise samples resulting in bursty interfering noise samples In the impulsive noise environments we elaborate a space-sensitive technique using multiple-input multiple-output (MIMO) which is particularly well suited in these usually very difficultsituations We assume the availability of channel state information (CSI) at the transmitter to achieve typical MIMO system gainsin ad hoc mode In this paper we show that more than 10 dB gains are obtained with the most efficient system that we propose forachieving smart grid application requirements On the one hand the results obviously illustrate that the max minus 119889119898119894119899 precoderassociated with the rank metric coding scheme is especially adapted to minimize the bit error rate (BER) when a maximumlikelihood (ML) receiver is employed On the other hand it is shown that a novel node selection technique can reduce the requirednodes transmission energies

1 Introduction

Modernization of power grids is underway inmany countriesaround the world Induced by important factors such asnational security economic development the environmentand the integration of renewable energies the provincesstates and countries are prioritizing technological innova-tions to be deployed to make the electricity network smarterSmart Grid (SG) It consists of the integration of commu-nication and information technologies into the networksand makes them communicate considering the actions ofthe players in the electricity system while ensuring a moreefficient economically viable and secure electricity supplyThe aim is always to provide equity between supply anddemand with increased responsiveness and reliability andto optimize network operations In fact such applicationsinvolve regular operations in real time which require mea-surements from several sources Recently wireless sensor

networks (WSN) have been identified as an encouragingtechnology to perform energy-efficient seamless reliableremote monitoring and low-cost control in SG applications[1] Despite these benefits WSN are facing some challengesincluding the dynamic topology the unpredictable commu-nication channel and the limited power sources of nodes Inaddition in high-voltage (HV) substation environments theinherent background additive white Gaussian noise (AWGN)is constantly present but this classical observation is nolonger relevant when the occurrence of an impulse becomesnoticeable In such realistic environments noise signals gen-erated by the underlying elements such as metallic structures(transformers circuit breakers disconnect switches andtransformers) span over several samples [2] giving rise tobursty appearances of impulses To avoid the aforementionedconstraints the use of MIMO cooperative techniques [3ndash7]may be an obvious solution enabling nodes to be grouped intoa set of virtual antennas as depicted in Figure 1

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 4863823 13 pageshttpsdoiorg10115520194863823

2 Wireless Communications and Mobile Computing

MIMO Transmission

DGN

Cluster

Node

Data Gathering Node (DGN)

dc

dlℎ ≫ dc

Figure 1 Cooperative MIMO system model

Closed-loop cooperative transmission ensures that thesource node cooperates with the idle neighbors to providespatial diversity Since the distance between nodes is smallerthan the distance between the cluster and the data gatheringnode (DGN) each cooperating node then precodes thedata before it transmits over the diverse subchannels to thereceiver where data is combined and detected Cooperativetransmissions are well studied for improving the error rateprobability or spectral efficiency performance It has beenshown that single input single output (SISO) and multihopapproaches are less effective than cooperative transmissionin terms of energy over long-haul distance [8] By exploit-ing channel state information at the transmitter (CSI-T) aMIMO precoder can optimize specific criteria to increasesystem performance The max minus 119889119898119894119899 precoder optimizesthe Euclidean distance for improving the performance Thework in [7] highlighted the interest of using max minus 119889119898119894119899by comparing the BERs and the mutual information withwater filling (WF) lattice and mercury water filling (MWF)The obtained results showed that the max minus 119889119898119894119899 achievedthe best performanceMoreover coded-MIMO ensures moreefficiency and reliability in communication systems For thispurpose it has also been largely studied for performanceimprovements In [9 10] the authors proposed a coded-MIMO based on Turbo codes and blockwise concatenatedconvolutional code (BCCC) The obtained results show sig-nificant improvements Despite the encouraging results ofthese techniques the weak points that are among othersinclude the complexity of the decoding the propagationof synchronization errors and important time delays Ina previous study [11] we proposed a coded-orthogonalfrequency division multiplexing (OFDM) system based onrank metric code (RC) and convolutional code (CC) Thesystem approach was simple and robust for mitigating thebursty nature of impulsive noise occurring in the HV sub-stations even in a deterministic ray tracing channel Wehad considered a deterministic channel extracted from a 3Dray tracing software called RapSor [12] It is a 2D3D raytracing open and extensible tool which is associated with theuniform theory of diffraction (UTD) and the geometricaloptic laws (GO) We now confirm that the same order ofcoding gain is maintained even with a closed-loop MIMOtransmission The objective of this paper is to provide areliable and efficient communication system by combiningthe rank metric scheme and MIMO using a max minus 119889119898119894119899

precoder and reduce the energy transmissionwith an efficientnode selection technique in an impulsive noise environmentThemain contributions of this paper consist of the following

(i) The max minus 119889119898119894119899 precoder approximation for binaryphase shift keying (BPSK) modulation

(ii) The proposition of a novel study case which takesinto consideration the joint solution using an outerforward error correction (FEC) based on rank metricapplied to the max minus 119889119898119894119899 MIMO precoder assumingmaximum likelihood (ML) detection at the receiver

(iii) The reduction of the complexity of our node selectiontechnique assuming full channel state information(FCSI)

(iv) The reduction of the overall nodes transmissionenergy in bursty impulsive interferers

The rest of the paper is presented as follows Section 2gives a review of impulsive noise models particularly theAu model [13] while Section 3 considers the fundamentalsof RC codes [14] Section 4 presents the considered MIMOchannel Section 5 deals with the proposed system the nodeselection technique and cooperative MIMO The obtainedresults are highlighted and discussed in Section 6 It firstdetails the BER performance for the Rayleigh fading channelSecondly we assume that the channel is frequency-selectiveusing RapSor Section 7 is about the energy consumptionmodel Conclusion and outlook are provided in Section 8

2 Review of Impulsive Noise Models

Impulsive noise is not only characteristic to substationsother environments like the industrial domains can introducethis noise and degrade communications as well Severalmodels of impulsive noise exist They can be used depend-ing on assumptions made in terms of the communicationconditions The popularly used models among others areMiddleton Class A [15] and the Symmetric Alpha-Stableprocess [16] However the weak points of these models arethat they do not take into consideration the correlationbetween successive pulsesTherefore in recent years two newmodels have been proposed in the literature The first oneis the partitioned Markov chain model (PMC-6) [17] andthe second is the Au model [13] which will be discussed inthis section The PMC-6 model is a model with one state

Wireless Communications and Mobile Computing 3

representing the background noise considered as Gaussianand the 6 states are the impulse states The transitionsbetween several states are defined as a characterization of theremaining interaction between pulses Nevertheless due tothe computational complexity of the model we do not use itin this paper

21 AuNoise TheldquoAurdquo noisemodel follows the physical con-cern of the mechanism making electromagnetic interference(EMI) in substations mostly generated by partial discharges(PD) Its model is considered as the first model that makes alink between the partial discharge evolution and the inducedfar-field oscillation propagation [13] To characterize thePD they proposed a process whose main components arethe impulse detection composed of a denoising process ashort-time analysis a detection and a statistical analysisLet us define V(120583 119905) as the waveform of impulsive noiseevaluated in volts per meter (Vm) such as 120583 is a total ofrandom elements indicating its occurrence duration andother substantial characteristics Considering V119898(120583 119905) as thewaveform quantified in V one can represent

V (120583 119905) = V119898 (120583 119905) radic 119885041205871198711199031198661199031198911205822 (1)

where 119871119903 represents the load resistance and 119866119903119891 the RFsystem gain while 120582 corresponds to the wideband antennawavelength and 1198850 = 120120587Ω is the free-space impedanceIn practice the final noise received by an antenna can beindicated as follows119909 (120583 119905) = sum

119896

V119896 (120583 119905) + 119861119899 (119905) (2)

where 119861119899(119905) is the background noise generally considered asGaussian During a long observation period the resultantsignal is formed by a superposition of several transientimpulse waveforms For a better location of the impulse adenoising process is used It consists of extracting the pulsesfrom the noise This operation is done using a wavelet trans-formation to which a threshold namely 119862119903 = 1199042radic2 log(119870119894)is exercised 119870119894 is the sample at the moment i and 1199042 isthe variance of the background Gaussian noise The dataobtained from measurements are made up of a sequence ofpulses located arbitrarily in time Partial discharges can beidentified applying a temporal interpretation of the waveformspectrogram 119881(120583 119905119892 119891) given by

119881 (120583 119905119892 119891) = int V (120583 119905) 119892 (119905 minus 119905119892) 119890minus1198952120587119891119905119889119905 (3)

such that 119892(119905) whose length is 119905119892 is a quadratically inte-grable temporal window function The Au model has beencompared to measurements from different levels of voltagesuch as 25 230 315 and 735 kV electrical substations Thesetup of measurement used is well described in [13 17] Tovalidate its model a comparison between experimentationand simulation results was produced in [18] which shows thatthe Au model is the best model to represent impulsive noisein substations

3 Principles of Rank Metric Coding Scheme

Introduced by Delsarte in coding theory [14] and developedby E Gabidulin [19] the RC or Gabidulin codes are widelyemployed in cryptography However recently it has beenintroduced in communication systems to improve the perfor-mance degraded by noise such as impulsive noise representedas a matrix in a row or column In [11 20] the authors usedRC concatenated with a CC in their systems Their resultsshowed that with these codes it is possible to mitigate theimpulsive noise occurring in industrial environments such aspower substations

Considering the significant improvements and the lowcomplexity of these codes compared to the traditional Turbocodes and Reed-Solomon (RS) codes [20] we use this codingscheme in our system For this purpose we start with thedefinition of some meaningful parameters of this codingscheme

Let q be a power of a prime and F119902 designate Galois Fieldwith q elements Let F Vtimes119906

119902 express the V times 119906 matrices overF119902 and set F V

119902 = F Vtimes1119902 Let F119906119902 be an extension of F119902 Every

extension field can be considered as a vector space over thefinite field Let B = 1205730 1205731 sdot sdot sdot 120573119906minus1 be a basis for F119906119902 overF119902 Since F

119906119902 is also a field we may consider a vector isin F119906119902

Whenever isin F V119902119906 we denote by 119909119894 the 119894119905ℎ entry of x that is119909 = [1199090 1199091 119909Vminus1]119879 It is natural to extend the map [∙] to

a bijection from F V119902119906 to F Vtimes119906

119902 such that the 119894119905ℎrow of [119909]B isexpressed by [119909119894]B

RC codes are described as a nonempty subset X sube F Vtimes119906119902

The rank weight of 119909 defined as R119896(119909) is denoted to bethe maximum number of coordinates in 119909 that are linearlyindependent over F119902

The rank distance between two vectors 1199091 and 1199092 is thecolumn rank of their difference R119896(1199091 minus 1199092 | F119902) The rankdistance of a vector rank code X sub F V

119902119906 is expressed as theminimal rank distance119889 (X) = 119889 = min (R119896 (119909119894 minus 119909119895) 119909119894 119909119895 isin X 119894 = 119895) (4)

For 119906 ge V an important class of rank metric codes wasproposed byGabidulin [21] Gabidulin code is a linear (V 119896 119889)block code over F119902119906 defined by the parity-check matrix119875 = [119901119895[119894]] 0 le 119894 le V minus 119896 minus 1 0 le 119895 le V minus 1 where theelements (1199010 1199011 119901Vminus1) isin F119902119906 are linearly independentover F119902 and 119896 = V minus 119889 minus 1 is the dimension of the code Theparity matrix defines a maximum rank distance (MRD) codewith length V le 119906 and 119889 = V minus 119896 + 1 Another method forMRD construction can be obtained using generator matrices[21]

For rank error correction we consider a MRD (V 119896 119889)code X The transmitted signal is 119909 and received signal canbe depicted as y = x + eeff such that eeff is an error Vectorerrors that can be corrected by the codeX are of the form

eeff = e + erow + ecol (5)

where e erow and ecol are a random rank error of rank t avector rank row erasure and a vector rank column erasure

4 Wireless Communications and Mobile Computing

MLreceivery

++

b

bs

b b

feed backchannel matrix

diagonalchannelmatrix

H

n

s

linearprecoder

btimes bFd

Figure 2 Equivalent MIMO system with a linear precoder in virtual channel

respectively Fast correction of rank erasures and randomrank errors was presented in [19] It is called the modifiedBerlekamp-Massey algorithm Formore information readersare referred to [21 22] This is an effective technique fordecoding RC errors and will be used in this paper

4 Closed-Loop MIMO

For aMIMO channel with no delay spread comprising F andG which are the precoder and decoder matrices respectivelythe following linear system equation applies

y = GHFs + Gn (6)

such that 119904 is the 119887times1 transmitted symbol vector y is the 119887times1received vector n is an 119899119903 times 1 additive noise vector H is thechannelmatrix of 119899119903times119899119905 here 119899119903 and 119899119905 are the numbers of thereceive and transmit antennas respectively and F is the 119899119905times119887precoder matrix We suppose that 119887 le 119903119886119899119896(H) le min(119899119905 119899119903)and

E sslowast = IbE nnlowast = N0Ib

(7)

The FCSI permits the precoder to diagonalize the channelinto b parallel SISO channels as depicted in Figure 2 If 119864119879is the total available power the following power constraint isapplied to the transmitter

trace [FFlowast] = 119864119879 (8)

The precoding and decoding matrices are separated into twocomponents as F = F

119907F119889 and G = GVG119889 respectively

The unitary matrices GV and FV derived from the singularvalue decomposition (SVD) of H diagonalize the channeland decrease the scope to 2 Hence the received symbol in(6) becomes

y = G119889F119889HVs + G119889nV (9)

such that HV = GVHFV = diag(1205731 1205732 120573119887) is thevirtual channel matrix 120573119894 denote the gains of the subchannelsorted in decreasing structure and nV = GVn is the 119887 times 1channel virtual noise Since the ML detection will be usedin the following sections the decoding matrix G119889 does notinfluence the efficiency and is considered to be Ib

41 MinimumEuclideanDistance Precodingmaxminus119889119898119894119899 Theprecoder max minus 119889119898119894119899 consists of the maximization of theminimum Euclidean distance 119889119898119894119899 between the signal itemsat the receiver as119889119898119894119899 (F119889) = min

(119904119896minus119904119897 )119896 =119897

1003817100381710038171003817HVF119889 (sk minus sl)1003817100381710038171003817 (10)

Let us define e = (skminussl) as the difference between possibletransmitted vectors Thus 119889119898119894119899(F119889) becomes119889119898119894119899 (F119889) = min

119890

1003817100381710038171003817HVF119889e1003817100381710038171003817 (11)

Therefore its optimization problem entails finding the matrix119865119889 which maximizes the criterion

F119889119898119894119899119889

= argmaxF119889

119889119898119894119899 (F119889)= argmax

F119889min119890

1003817100381710038171003817HVF119889e1003817100381710038171003817 (12)

Since the ML detection will be considered this criterion iswell suited because the probability of symbol errors relies onthe minimum Euclidean distance

However determining the solution of F119889 is complicateddue to the large solutions space and the alphabet symbolswhich it processes For this purpose we propose to simplifythe technique and derive a solution for b = 2 virtual channelsHence the channel virtual matrix can be expressed as

HV = (radic1205731 00 radic1205732) = radic2120573 (cos 120572 00 sin 120572) (13)

where 120572 is the channel angle and 120572 isin [0 1205874] and 120573 = (1205731 +1205732)2 This solution does not rely on the SNR but is based onthe channel angle 120572

The SVD applied to the matrix precoder is as follows

F119889 = QsumRlowast (14)

where sum is the diagonal matrix and Q and R are 119887 times 119887 unitarymatrices

Recall that the power constraint at the transmit antennasalways remains sum must fulfill the constraint too and isderived as sum = radic119864119879(cos 120574 00 sin 120574) (15)

with 0 le 120574 le 1205874

Wireless Communications and Mobile Computing 5

Since the matrix Rlowast has no influence on the singularvalues they can be derived fromHVQsum The largest singularvalues are obtained when Q = I2

Proof of Q = I2 Consider the form of the unitary matrix ofQ

Q = ( (cos 120579) 1198901198941205791 (sin 120579) 1198901198941205793minus (sin 120579) 1198901198941205792 (cos 120579) 1198901198941205794) (16)

with the constraints(1205791 + 1205794) = (1205792 + 1205793) mod 2120587 (17)

The angle 120579 isin 0 le 120579 lt 1205872Recall that the single values are null or (positive and real)

and the determinant of a unitary matrix = 1 We define U andVlowast as the single value decomposition of HVQsum and 120590119896 thediagonal components of andThe product of SV is not based onQ In fact we can note that12059011205902 = 10038161003816100381610038161003816det (⋀)10038161003816100381610038161003816 = 1003816100381610038161003816det (U and Vlowast)1003816100381610038161003816 = 10038161003816100381610038161003816det (HVQsum)10038161003816100381610038161003816= 1003816100381610038161003816100381610038161003816radic(12057311205732)119864119879 cos 120574 sin 120574 det (Q)1003816100381610038161003816100381610038161003816= radic(12057311205732)119864119879 cos 120574 sin 120574

(18)

Moreover we have12059012 + 12059022 = trace (and2) = trace (U and VlowastV and Ulowast)= 1003817100381710038171003817U and Vlowast10038171003817100381710038172F = 10038171003817100381710038171003817HVQsum100381710038171003817100381710038172F (19)

Therefore the phases of the constituents of Q do no impacton 12059012 + 12059022 Eventually we deduce that the single values donot rely on the phases of the constituents of Q Thus we justassume real matrices Q whose typical structure is

Q = ( cos 120579 sin 120579minus sin 120579 cos 120579) (20)

where 0 le 120579 lt 1205872We now examine the sum of the square single value of

HVQsum

12059012 + 12059022 = 10038171003817100381710038171003817HVQsum100381710038171003817100381710038172F = trace (HVQsumsumQlowastHV)= 119864119879(1205731sin2120574 + 1205732cos2120574+ (1205731 minus 1205732) cos (2120574) cos2120579(21)

As 1205731 gt 1205732 for every 1205901 the maximum value of 1205902 is acquiredfor 120579 = 0 which is denoted as Q = I2

Hence Rlowast can be simplified as follows

Rlowast = ( cos120603 (sin120603) 119890119894120593minus sin120603 (cos120603) 119890119894120593) (22)

while developing

Rlowast = ( cos120603 sin120603minus sin120603 cos120603) (1 00 119890119894120593) = R120603R120593 (23)

with 0 le 120593 lt 2120587 and 0 le 120603 le 1205872Thus the precoder can be expressed as

F119889 = radic119864119879(cos 120574 00 sin 120574) ( cos120603 sin120603minus sin120603 cos120603) (1 00 119890119894120593) (24)

42 Solution for BPSK Modulation Considering a BinaryPhase Shift Keying (BPSK) technique where 119887 = 2 the datasymbols are in 1 minus1 and the difference vectors related to e =(sk minus sl) are ( 02 ) ( 0minus2 ) ( 20 ) ( 22 ) ( 2minus2 ) ( minus20 ) ( minus22 ) ( minus2minus2 ) Since some vectors are collinear the solution is reducede119861119875119878119870

= ( 02 ) ( 20 ) ( 22 ) ( minus2minus2 ) A numerical search over 120574120603 and120593whichmaximizes the smallest distance for differencevectors in e

119861119875119878119870demonstrates that whatever the channel ie

whatever the channel angle 120572 the precoder whichmaximizes119889119898119894119899 is obtained for 120574 = 0∘ 120603 = 45∘ and 120593 = 90∘Hence by substituting for the real values we can deduce

the solution for BPSK modulation which is given as follows

F119889 (119861119875119878119870) = F119861119875119878119870 = radic 1198641198792 (1 radicminus10 0 ) (25)

And its 119889119898119894119899 namely 119889119861119875119878119870119898119894119899 is

119889119861119875119878119870119898119894119899 = 1003817100381710038171003817100381710038171003817100381710038171003817HVF119861119875119878119870(20)1003817100381710038171003817100381710038171003817100381710038171003817 = 2radic120573119864119879 cos 120572 (26)

Notice that the second row of (25) is equal to 0 indicatingthat the signal is completely transmitted on the most favoredsubchannelThis solution could be compared to themax SNRthat streams power just on the strongest eigenmode of thechannel [23]

The distance (26) normalized by radic2120573119864119879 is depicted inFigure 3 [24] showing that this distance depends on thechannel angle

5 System Model and Cooperative MIMO

51 Description The system model which is considered inthis paper is depicted in Figure 1 We assume transmissionsfrom a cluster of 119899119888 nodes to the DGN over Rayleigh fadingchannels and a realistic channel model obtained with theRapSor simulator Any node 119894(119894 = 1 2 119899119888) in a cluster 119896is a single-antenna node with the capability to be a slave ora cluster-head A node acting as a cluster head synchronizesits 119899119888 minus 1 neighbors while a slave cooperates with othernodes in cluster 119896 over a relatively short SISO communicationlink The DGN is a multiantenna receiver and equipped withrelatively high processing capabilities and without energyconstraints Assume this scenario where substation elementsand infrastructure are fittedwith several wireless sensors suchas temperature pressure and electrical parameters (voltage

6 Wireless Communications and Mobile Computing

MMSEWF

Nor

mal

ized

dm

in

Channel angle in degrees

15

1

05

00 5 10 15 20 25 30 35 40 45

max(dmin)max(min)

Figure 3 Normalized Euclidean distance for BPSK modulation

current and frequency) Such sensor nodes are required tomeasure and cooperatively transmit measured data wirelesslytoDGNover a distance119889119897ℎ Due to relatively shorter distancesdc between cooperating nodes anAWGNchannel is assumedwith no fading while Rayleigh fading is supposed to be fixedoverall the transmission of the codeword from the cluster tothe DGN over the distance 119889119897ℎ The communication protocoldepicted in Figure 4 can be described as follows

(i) Declaration Phase We assume neighborhood discov-ery had been previously performed Any source nodehaving data to transmit forms a cluster and confirmsitself as the cluster head since the first which declaresis considered as the head of the cluster All the nodeswhich ldquohearrdquo the source node set their ldquostatusrdquo to slaveready to receive from the source In an event that twoor more nodes perform declaration the cluster-headwith the least residual energy Eres wins but nodeswith data can still send to neighboring nodes after thecurrent cluster-head

(ii) Phase 1 The source node multicasts its data to 119899119888 minus1 neighbors over the average distance of dc this is aSISO communication

(iii) Phase 2 Next the 119899119888 minus 1 neighbors as potentialrelays send each training frame 119905119903119886 to the DGNwhich uses this to estimate the multipath coefficientsfor each of its received antennas The DGN alsonotes the identification (ID) of the cluster-head forfuture acknowledgment It then constructs the chan-nel matrix H and selects the best 119899119905 nodes includingthe optimal precoding matrix index for the selectednodes

(iv) Phase 3 The DGN selects 119899119905 nodes that will usethe precoding matrix whose index is found in theprecodingmessage119901119903119890119888 sent by theDGN to 119899119905 nodesThemessage 119901119903119890119888 also includes the ID of the selectednodes

(v) Phase 4 The 119899119905 selected nodes precode with the pre-coding matrix and then transmit the data frames tothe DGN using MIMO transmission over a Rayleighchannel or a channel obtained with RapSor

52 Cooperative MIMO When the FCSI is available FV is aunitary matrix derived from SVD of the channel matrix HIn practical applications the hypothesis of FCSI availability atthe transmitter is unrealistic rather the channel informationmust be made available to the transmitter from the receivervia the rate-limited feedback control channel [25] Thechannel information types that can bemade available includethe channel statistics instantaneous channel and partial orquantized CSI (QCSI) The most practical of these is theQCSI because the feedback amount can be adjusted to theavailable rate of the feedback control channel In the case ofthe limited CSI we implement a finite codebook in which thereceiver selects the optimal matrix F119889 and FV from FV andF119889 dictionaries The optimal dictionary FV containing a set119865V1 119865V2 119865V119873 is implemented according to the algorithmin [26] where 119873 = 21198611 is the dictionary size and 1198611 is thenumber of quantization bits Generally constructing theF119889

dictionary is required for each H realization in conjunctionwith the 119865V dictionary but for the BPSK modulation thecontent of dictionary F119889 will be limited to a single precodermatrix119865119889 since it is independent of the channel angleThe twodictionaries are generated offline combined into a codebookF = FVF119889 = (119865V1 119865V2 119865119873) and are made available toall nodes The codebooks for 2 3 and 4 transmit nodes aregenerated with 3 5 and 7 bits resolution respectively andare used for all our simulations

53 Nodes Selection Node selection is performed by theDGN to select 119899119905 nodes from a cluster of interest by 119889119898119894119899associated with each node as119889119898119894119899 (ℎ(119895)) = min

1198901015840

10038171003817100381710038171003817G(119895)V h(119895)F(119895)e101584010038171003817100381710038171003817 (27)

where G(119895)V [1 times 119899119903] ℎ(119895) is the 119895119905ℎ column of the clusterdestination channel matrix H[119899119903 times 119899119905] 119865(119895) is the associatedprecoding matrix 119895119905ℎ column of H and 1198901015840 is the differencebetween possible transmitted vectors belonging to a setminus1 1 Due to constraint 119887 le min(119899119903 119899119905) F(119895)119889 becomes ascalar The unitary matrix F(119895)V obtained by the method ofdictionary construction explained previously (or by SVD forFCSI) is a scalar ie F(119895)

119889= F(119895)V = 1F(j) Sorted in descending

order the 119899119905 indexes of the eigenvalues corresponding to thecolumn vectors of matrix H are the 119899119905 columns of matrix Hof selected nodes Nodes can be selected faster as opposedmaximizing the 119889119898119894119899 of L subcarriers for each H where L =119899119888119899119905(119899119888 minus 119899119905)6 BER Performance Analysis

This section introduces numerical results performed bysimulations under Rayleigh and RapSor channels affectedby Gaussian noise and Au impulsive noise We assume ML

Wireless Communications and Mobile Computing 7

prec

tra

data

ACK

data

Sleep

Sleep

t

t

t

Clusterhead

Slave1 to nc-1

DGN

Twake 2T1+Tdata Ttra Tprec T1 Tdata Tack

2T1 + 2Tack

wake data tra sleep

RxTxWake up

Figure 4 The assumed cooperative protocol

detection at the DGN indeed the average probability oferror limited to the nearest 119889119898119894119899 neighbors [27] can beapproximated as

119875119890 asymp 1198731198992 (radic (119889119898119894119899)2 11986411987941205902 ) (28)

such that 119873119899 is the mean of the nearest neighbors Consider-ing a BPSK modulation the bit error probability is given by

119875119887119894119905 asymp 1198731198992119887 log2 119872 erfc(radic (119889119861119875119878119870119898119894119899 )2 11986411987941205902 ) (29)

where M = 2 is the modulation order and erfc is thecomplementary error function To estimate the performanceof MIMO system with max minus 119889119898119894119899 precoder the MATLABsoftware is utilized The simulation started with uncodedMIMO system and then used concatenated RCCC in thepresence of Gaussian noise and Au impulsive noise Two con-figurations are also considered a transmission without nodeselection and a transmission with node selection MIMOsystem efficiency is investigated for both Rayleigh fading andRapSor channels The reliability of the system is expressedby the correlation between bit error rate (BER) versus thesignal to noise ratio (SNR) Firstly the system described withno channel coding approaches is to demonstrate the impactof employing coding scheme in cooperative MIMO systemby utilizing BPSK modulation over AWGN and impulsivenoise with Rayleigh fading and RapSor channels We alsoinvestigated the performance of concatenated RC and CCThe size of Galois Field for the RC is F119902119906 = 16 while theCC employed has a coding rate 119877 = 12 and generatorpolynomials in octal form 1198751 = 171 and 1198752 = 133 The

decoding of RC is implemented by the modified Berlekamp-Massey while CC decoding is performed by soft decision ofViterbi algorithm

61 AWGN and Impulsive Noise under Rayleigh Channel

611 Transmission without Node Selection Figure 5 depictsBER performance of max minus 119889119898119894119899 MIMO precoding withFCSIwithout node selectionThe results demonstrate that theworst performance of MIMO system is with no channel cod-ing for both AWGN and impulsive noise Uncoded-MIMOindicates a flattening of the BER between -5 and 5 dBThen itis improved by adding coding technique Using concatenatedRCCC with max minus 119889119898119894119899 precoding in MIMO system givesmore improvement to the system Considering the presenceof impulsive noise the coding gain between uncoded andsuggested approach is approximately 8 dB at a target BERof 10minus4 We now compare our results to those obtained in[28] The authors proposed an effective technique to trackthe double-selected multipath channel for MIMO-OFDMsystem A Space Time Block Coding (STBC) is applied andleads to interesting performance However our system ismore robust and presents better performanceWe have a gainof approximately 12 dB compared to the proposed approachdescribed above Furthermore in [29] the authors presenteda MIMO-OFDM system with a concatenated RSCC Thesystem is evaluated in both Rayleigh and Rician channelsThe obtained results are improved compared to an uncodedsystem However our system still has the best performance

612 Transmission with Node Selection The first simulationswemade concerned the transmission without node selectionIn this paragraph we present numerical results when optimaland suboptimal node selection are implemented combinedwith the knowledge of the channel (FCSI orQCSI) Assuming

8 Wireless Communications and Mobile Computing

1050minus5

Uncoded-MIMO AWGNUncoded-MIMO + Imp NoiseCoded-MIMO AWGNCoded-MIMO + Imp Noise

100

10minus2

10minus4

10minus6

10minus8

BER

SNR (dB)

Figure 5 BER performance of max minus 119889119898119894119899 MIMO precoding withFCSI under Rayleigh channel without node selection

0 10987654321

100

10minus2

10minus1

10minus3

10minus4

10minus6

10minus5

BER

SNR (dB)

MIMO + AWGN - FCSIMIMO + Imp Noise - FCSIMIMO + AWGN - QCSIMIMO + Imp Noise - QCSI

Figure 6 Performance comparison between FCSI andQCSI curveswith solid lines represent FCSI while dashed lines represent theQCSI

the full channel knowledge the system model describedin Section 4 is implemented For the QCSI a codebookquantized using 3 5 and 7 bits for 2 3 and 4 selected nodesis considered respectively The performances are shown inFigures 6 and 7 Results are only shown for 4 transmit nodesIn Figure 5 the results of uncoded systems are presentedand the performances between FCSI andQCSI are comparedAs can be seen FCSI outperforms QCSI for both AWGNand impulsive noise Since FCSI yields better performanceresults than QCSI we represent only results in FCSI with the

minus5 3210minus1minus2minus3minus4

100

10minus2

10minus4

10minus6

10minus10

10minus8

BER

SNR (dB)

MIMO + AWGNMIMO + Imp Noise

Figure 7 Coded-BER performance of max minus 119889119898119894119899 precoding underRayleigh fading channel with FCSI and node selection

node selection in Figure 7 which shows simulation resultswith a coded system As for the case without selection aperformance improvement can be noticed Considering achannel impaired by impulsive noise and a concatenatedRCCC a target BER of 10minus4 is achieved at an SNR ofapproximately 1 dB It leads to a coding gain of 47 dB betweenuncoded and coded MIMO systems

62 AWGN and Impulsive Noise under a RapSor ChannelIn the preceding section we studied the impact of coded-MIMO communications under a Rayleigh fading channelaffected by impulsive noise and AWGNThe results obtainedshowed that good performances are achieved Howeverit was the perfect case The reality of power substationsconsiders multipath components due to the presence ofmetallic structures equipment and devices In order to takeinto account the aforementioned aspects we now considera deterministic channel extracted from the RapSor software[12] Our objective is to acquire the channel impulse response(CIR) of the simulated channel matrix [119899119903 times 119899119905] coefficientsFor this purpose we select a HV substation located inQuebec (Canada) operated by the energy company Hydro-Quebec Our WSN application consists of a 6times4 virtualMIMO system made up with the DGN node as the receiverplaced on a tower of 60 m and the sensors forming a 10-nodecluster mounted on transformers serving as the emitters Theclustering distance is approximately 14m while the long-hauldistance is 1029 m

621 Transmission without Node Selection We consider thesame situation as for the Rayleigh fading channel Howeveronly results for 4times4 MIMO are depicted since they achievethe best performance The results obtained are plotted inFigure 8 For the uncoded system we notice a performancedegradation when the channel is affected by impulsive noiseAs for Rayleigh channel a flattening of the BER curve

Wireless Communications and Mobile Computing 9

1050minus5

Uncoded-MIMO AWGNUncoded-MIMO + Imp NoiseCoded-MIMO AWGNCoded-MIMO + Imp Noise

SNR (dB)

100

10minus2

10minus4

10minus6

10minus8

BER

Figure 8 BER performance for max minus 119889119898119894119899 MIMO precoding withFCSI under RapSor channel without node selection

between -3 and 2 dB can be noticed in the presence of impul-sive noise However by combining the concatenated codesand maxminus 119889119898119894119899 precoder with MIMO we have an increase ofsystem performance Target BER of 10minus4 is achieved at SNRof 0 and 87 dB for coded and uncoded MIMO respectivelywhen the channel is affected by impulsive noise It yields toa coding gain of 87 dB between uncoded and coded MIMOsystems

622 Transmission with Node Selection In this section opti-mal node selection is implemented to select 2 and 4 transmitnodes from the cluster of 10 Assuming the full channelknowledge we explore the BER results for both coded anduncodedMIMO systemsThe results are depicted in Figure 9For the uncoded case we can note the degradation of theperformance This is improved when the concatenation ofcodes is added Target BER of 10minus4 is achieved at SNR = -1 dBfor coded MIMO while it is 8 dB for uncoded system whenthe channel is affected by impulsive noise

7 Energy Consumption

71 Energy Model The max minus 119889119898119894119899 protocol employs coop-erative MIMO with the distributed nodes serving as multipleantennas Hence we are concerned with the total energy con-sumption 119864119888119900119900119901 of the nodes for a complete communicationAccording to the protocol description the total energy of thecooperating nodes can now be expressed as119864119888119900119900119901 = 119864119897119900119888 + 119864119894119899119894119905 + 119864119891119887119896 + 119864119872119868119872119874 (30)

where 119864119897119900119888 is the local transmission energy ie the SISOcommunication between the nodes 119864119894119899119894119905 is the initializationphase 119864119891119887119896 is the feedback control channel energy and

1050

100

10minus2

10minus4

10minus6

10minus8

BER

minus5

Uncoded-MIMO AWGNUncoded-MIMO + Imp NoiseCoded-MIMO AWGNCoded-MIMO + Imp Noise

SNR (dB)

Figure 9 BER performance for max minus 119889119898119894119899 MIMO precoding withFCSI under RapSor channel with node selection

119864119872119868119872119874 is the energy of the data packet for MIMO transmis-sion

The average energy consumption of a radio frequency(RF) system can broadly be separated into 119875119860119898119901 and 119875119888119888119905which are the power consumption of power amplifiers andother circuits blocks respectively The model of typical RFblocks [30] representing the emitter is depicted in Figure 10while the receiver can be seen in Figure 11

The 119875119860119898119901 is expressed as

119875119860119898119901 = 120589120576119875119900119906119905 = 120589120576 1198641198871198730 (2120587)2 11988911987101198711198981198631199031198601198921199051198601198921199031205822 119877119887 (31)

120589 is the peak-to-average ratio (PAR) 120576 corresponds to thepower amplifier efficiency 1198641198871198730 is the ratio energy per bitto the noise 119860119892119905 and 119860119892119903 are the emitter and the receiverantenna gains respectively 119871119898 is the margin componentwhich compensates for the variations of the hardware processand other noises 120582 is the wavelength 119863119903 is the power densityat the receiver 119889 is the long-haul distance 1198710 is the path-losscomponent and 119877119887 is the bit rate The total power dissipatedin circuit 119875119888119888119905 for 119899119905 transmitters and 119899119903 receivers can beapproximately expressed as

119875119888119888119905 = (119875119863119860119862 + 119875119891119894119897119905 + 119875119898119894119909 + 119875119904119910119899119905ℎ)+ (119875119891119894119897119903 + 119875119871119873119860 + 119875119898119894119909 + 119875119868119865119860 + 119875119860119863119862)= 119899119905119875119879119909119888 + 119899119903119875119877119909119888

(32)

where 119875119863119860119862 and 119875119860119863119862 are consumed energy for the digital-to-analogue converter (DAC) and the analogue-to-digitalconverter (ADC) respectively 119875119891119894119897119905t is the power consumedfor the active filters at the transmitter whereas 119875119898119894119909 and119875119891119894119897119903 are the energy consumed for the mixer and the active

10 Wireless Communications and Mobile Computing

FilterFilterFilter

LNA

Mixer

LO

IFA

ADC

Figure 10 Transmitter circuit block

Filter Filter

Mixer

LO

DAC PA

Figure 11 Receiver circuit block

filters at the receiver respectively 119875119871119873119860 119875119904119910119899119905ℎ and 119875119868119865119860 arethe power consumption for the Low-Noise Amplifier (LNA)the frequency synthesizer and the Intermediate FrequencyAmplifier respectively Parameters119875119879119909119888 represent power dissi-pated in the circuit for a single node during data transmissionand119875119877119909119888 for reception Total energy consumed per bit119864119887119894119905 fora fixed-rate system is evaluated in the following equation

119864119887119894119905 = 119875119860119898119901 + 119875119888119888119905119877119887 (33)

Assuming a packet size of D symbols is to be transmittedand training symbols size of 119901119899119905 is inserted (119901 symbols aretransmitted by each node) the effective bit rate 119877119890119891119891119887 is

119877119890119891119891119887

= (119863 minus 119901119899119905119863 ) 119877119887 (34)

Note that replacing 119877119887 in equation (31) by 119877119890119891119891119887

we obtainedthe energy consumption model which accounts for thesupplementary energy due to the 119901 training symbols For themax minus 119889119898119894119899 MIMO precoding transmission the bit rate 119877119887can thus be calculated as follows119877119887 = 119877119898119861 (35)

where 119877 is the MIMO transmission rate expressed as aratio of the number of symbols transmitted 119873119878 over thenumber of periods 119873119875 (ie 119877 = 119873119878119873119875) 119898 = log2(119872)where 119872 is the constellation size and 119861 is the modulationbandwidth Parameter 119864119897119900119888 is the total local transmissionenergy expended within a cluster 119896 that consists of 119899119888 nodesseparated by an average distance of 119889119888 Each source node cantransmit to 119899119903 = (119899119888 minus 1) receivers thus 119864119897119900119888 is expressed as

119864119897119900119888 = 119873119901119896119905(119875119860119898119901 + 119875119888119888119905119877119890119891119891119887

)119908119894119905ℎ 119875119888119888119905 = 119875119879119909119896119888 + (119899119888 minus 1) 119875119877119909119896119888 (36)

where 119873119901119896119905 = 119899119888119871 is the total number of bits in all sentpackets and for the random nodes cooperative transmissionscenario 119899119888 = 119899119905 In (37) the training phase energy 119864119894119899119894119905 isgiven where 119873119879119878 is the amount of training bits

119864119894119899119894119905 = 119873119879119878 (119875119860119898119901 + 119875119888119888119905119877119874119878119879119861119862119887

) 119908119894119905ℎ 119875119888119888119905 = (119899119888) 119875119879119909119896119888 (37)

Only Alamoutirsquos code yields a rate 119877 = 1 for complexmodulations The OSTBC solution for any value of 119899119905 butwith 119877 = 12 is presented in [31] Solutions for 119899119905 = 3 and4 but with 119877 = 34 are similarly presented To implementour training phase for 10 (119899119888) cluster nodes we consider 4times 4 OSTBC transmission Then we average rate to obtain119877119874119878119879119861119862119887 = 23 and the 1198641198871198730 at the target BER On thefeedback channel the energy 119864119891119887119896 consumed is

119864119891119887119896 = (119873119891119887119896 119899119905119875119877119909119896119888119877119887 ) (38)

where 119899119905 sensor nodes act as receivers in this case 119873119891119887119896 isthe number of bits sent on the feedback channel The valuesof 3 5 and 7 bits are considered for 119873119891119887119896 when 2 3 and 4nodes are selected respectively Note that max minus 119889119898119894119899 basedselection requires lceillog2119871rceil bits which have been included in119873119891119887119896 where lceil∙rceil denotes the nearest higher integer Energy

Wireless Communications and Mobile Computing 11

Energy Consumption under Rayleigh fading channel

200 400 600 800 1000 1200 14000Distance (m)

0

005

01

015

02

025

03

Ener

gy (J

bits

)

Rand AWGNRand Imp Noise

Sel AWGNSel Imp Noise

Figure 12 Energy consumption for maxminus119889119898119894119899 precoding with FCSIunder Rayleigh fading channel

Table 1 Nodes and PAR parameters for energy computation

Parameters ValuesGains 25 dBiFrequency carrier 25 GHzBandwidth 20 MHzPower Amp efficiency 035BER 10minus4required for the transmission of the data packets by MIMOtechnique using the max minus 119889119898119894119899 precoder is

119864119872119868119872119874 = 119873119901119896119905(119875119860119898119901 + 119875119888119888119905119877119890119891119891minus119901119903119890119888119887

) (39)

119877(∙)119887is the efficiency of theMIMO technique used in transmit-

ting the symbols over b subchannels Hence 119877119890119891119891minus119901119903119890119888119887 = 272 Energy Consumption Evaluation This section analyzessimulation results for energy consumption according to equa-tion (30)Theparameters used to compute the simulations areprovided in Table 1

We compute the total consumed energy for 4 times 4 MIMOsystems Figure 12 compares the consumed energy with andwithout selecting nodes for transmission The AWGN andimpulsive noise are considered in a Rayleigh fading channel

As can be seen in this figure the node selection techniquereduces the total consumed energy The energy is reducedfrom 027 to 016 Jbits corresponding to about 40 whenthe channel is corrupted by impulsive noise and the nodeselection is appliedHowever we notice that the nodes require

Energy Consumption under RapSor channel

200 400 600 800 1000 1200 14000Distance (m)

0

005

01

015

02

025

03

Ener

gy (J

bits

)Rand AWGNRand Imp Noise

Sel AWGNSel Imp Noise

Figure 13 Energy consumption for maxminus119889119898119894119899 precoding with FCSIunder RapSor channel

more energy to transmit to the DGN at the target BER whenthe channel is affected by impulsive noise compared to theclassical Gaussian noise

Finally Figure 13 shows the energy consumption per bitin a RapSor channel affected by AWGN and impulsive noiseAs for the BER only the results for 4 cooperative nodes arepresented

Similarly for a Rayleigh channel the energy is alsoreduced by 18 that is from 029 to 024 Jbits

8 Conclusion

Linear precoders optimize a particular criterion using chan-nel knowledgeThey are based on the SVD to diagonalize thechannel Among these precoders the max minus 119889119898119894119899 precodermaximizes the minimum distance of the constellation inreception It presents a maximal order of diversity 119899119905 times 119899119903Based on the benefits of these techniques we have proposeda reliable and efficient communication system by combininga concatenated RC with CC scheme and MIMO using max minus119889119898119894119899 precoder We have also reduced the energy transmissionwith an efficient node selection technique in impulsive noiseenvironment With BPSK modulation over Rayleigh fadingchannel and deterministic ray tracing RapSor channel wehave explored the performance of the suggested proposalThe concatenation of max minus 119889119898119894119899 and RCCC leads to alarge performance improvement The SNR at the target BERreduces as the spatial diversity of MIMO system increasesNotice that the use of rank metric code also improves thereliability of the system compared to the uncoded case FCSIis more useful than QCSI as can be observed in the resultsGlobally the obtained results clearly demonstrate that the

12 Wireless Communications and Mobile Computing

max minus 119889119898119894119899 with FCSI and RC code are suitable for theimpulsive noise mitigation

We also investigate the energy efficiency for the nodeselection algorithm To evaluate the robustness of the nodeselection technique we useMIMO transmissions in commu-nication channel impaired by noisesThe disrupted noise wasimplemented using Au model validated by measurementsResults show that the node selection technique can achievea maximum average amount of 40 in energy saving for4 selected nodes in Rayleigh fading channels However theenergy saving is about 18 in RapSor channel In conclusionas for the BER this technique minimizes the energy con-sumption in both Rayleigh and realistic RapSor channels

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest regarding thepublication of this paper

Acknowledgments

The authors would like to thank Dr Minh Au at Hydro-Quebec Research Institute (IREQ) Canada for the impulsivenoise measurements and Abdul Karim Yazbeck PhD stu-dent at XLIM Institute Limoges France for his work on thechannel coding The work in this paper is supported partiallyby grants from the Richard J Marceau Research chair onWireless IP technology forDevelopingCountries partially byInstitut de Recherche drsquoHydro-Quebec (IREQ) and partiallyby Labex sum-Lim

References

[1] P P Parikh M G Kanabar and T S Sidhu ldquoOpportunities andchallenges of wireless communication technologies for smartgrid applicationsrdquo in Proceedings of the IEEE Power and EnergySociety General Meeting 2010

[2] F Sacuto F Labeau J Beland et al ldquoImpulsive noise measure-ment in power substations for channel modeling in ISM bandrdquoin Proceedings of the CIGRE Canada Conference September2012

[3] M Vu and A Paulraj ldquoSome asymptotic capacity results forMIMO wireless with and without channel knowledge at thetransmitterrdquo in Proceedings of the Conference Record of theThirty-Seventh Asilomar Conference on Signals Systems andComputers pp 258ndash262 USA November 2003

[4] M Vu and A Paulraj ldquoOptimal linear precoders for MIMOwireless correlated channels with nonzero mean in space-timecoded systemsrdquo IEEE Transactions on Signal Processing vol 54no 6 I pp 2318ndash2332 2006

[5] C Tepedelenlioglu and P Gao ldquoOn diversity reception overfading channels with impulsive noiserdquo IEEE Transactions onVehicular Technology vol 54 no 6 pp 2037ndash2047 2005

[6] S Al-Dharrab and M Uysal ldquoCooperative diversity in thepresence of impulsive noiserdquo IEEE Transactions on WirelessCommunications vol 8 no 9 pp 4730ndash4739 2009

[7] O J Oyedapo B Vrigneau R Vauzelle and H BoeglenldquoPerformance analysis of closed-loop MIMO precoder basedon the probability of minimum distancerdquo IEEE Transactions onWireless Communications vol 14 no 4 pp 1849ndash1857 2015

[8] T Nguyen O Berder and O Sentieys ldquoCooperative MIMOschemes optimal selection for wireless sensor networksrdquo in Pro-ceedings of the 2007 IEEE 65th Vehicular Technology Conferencepp 85ndash89 Dublin Ireland April 2007

[9] N Mysore Turbo-Coded MIMO Systems Receiver Design andPerformance Analysis [PhD thesis] McGill University 2006

[10] Y Li and M Salehi ldquoCoded MIMO systems with modulationdiversity for block-fading channelsrdquo in Proceedings of the 201246th Annual Conference on Information Sciences and Systems(CISS) September 2012

[11] N B Sarr A K Yazbek H Boeglen J Cances R Vauzelleand F Gagnon ldquoAn impulsive noise resistant physical layer forsmart grid communicationsrdquo in Proceedings of the ICC 2017 -2017 IEEE International Conference on Communications pp 1ndash7 Paris France May 2017

[12] C Liebe P Combeau A Gaugue et al ldquoUltra-wideband indoorchannel modelling using ray-tracing software for through-the-wall imaging radarrdquo International Journal of Antennas andPropagation vol 2010 Article ID 934602 14 pages 2010

[13] M Au F Gagnon and B L Agba ldquoAn experimental character-ization of substation impulsive noise for a RF channel modelrdquoin Proceedings of the Progress in Electromagnetics Research Sym-posium PIERS 2013 Stockholm vol 1 pp 1371ndash1376 SwedenAugust 2013

[14] P Delsarte ldquoBilinear forms over a finite field with applicationsto coding theoryrdquo Journal of Combinatorial Theory Series A vol25 no 4 pp 226ndash241 1978

[15] A Spaulding and D Middleton ldquoOptimum reception in animpulsive interference environment-part 1 coherent detectionrdquoIEEE Transactions on Communications vol 25 no 9 pp 910ndash923 1977

[16] GA Tsihrintzis andC LNikias ldquoFast estimation of the param-eters of alpha-stable impulsive interferencerdquo IEEE Transactionson Signal Processing vol 44 no 6 pp 1492ndash1503 1996

[17] F Sacuto F Labeau and B L Agba ldquoWide band time-correlatedmodel for wireless communications under impulsivenoise within power substationrdquo IEEE Transactions on WirelessCommunications vol 13 no 3 pp 1449ndash1461 2014

[18] N B Sarr H Boeglen B L Agba F Gagnon and R VauzelleldquoPartial discharge impulsive noise in 735 kV electricity sub-stations and its impacts on 24 GHz ZigBee communicationsrdquoin Proceedings of the 2016 International Conference on SelectedTopics in Mobile and Wireless Networking MoWNeT 2016Egypt April 2016

[19] E E M Gabidulin ldquoTheory of codes with maximum rankdistancerdquo Problemy Peredach Informatsii vol 21 no 1 pp 3ndash161985

[20] A W Kabore V Meghdadi J Cances P Gaborit and ORuatta ldquoPerformance of Gabidulin codes for narrowband PLCsmart grid networksrdquo in Proceedings of the 2015 InternationalSymposium on Power Line Communications and its Applications(ISPLC) pp 262ndash267 Austin TX USA March 2015

[21] E M Gabidulin ldquoRank-metric codes and applicationsrdquo httpiitpruuploadcontent839Gabidulinpdf

Wireless Communications and Mobile Computing 13

[22] S Plass G Richter and A J Han Vinck Coding Schemesfor Crisscross Error Patterns Springer Science+Business MediaLLC 2007

[23] P Stoica and G A Ganesan ldquoMaximum-SNR spatial-temporalformatting designs for MIMO channelsrdquo IEEE Transactions onSignal Processing vol 50 no 12 pp 3036ndash3042 2002

[24] L Collin O Berder P Rostaing and G Burel ldquoOptimal min-imum distance-based precoder for MIMO spatial multiplexingsystemsrdquo IEEE Transactions on Signal Processing vol 52 no 3pp 617ndash627 2004

[25] D J Love R W Heath Jr V K N Lau D Gesbert B D Raoand M Andrews ldquoAn overview of limited feedback in wirelesscommunication systemsrdquo IEEE Journal on Selected Areas inCommunications vol 26 no 8 pp 1341ndash1365 2008

[26] D J Love and J Heath ldquoLimited feedback unitary precodingfor spatial multiplexing systemsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol51 no 8 pp 2967ndash2976 2005

[27] A GoldsmithWireless Communications Cambridge UniversityPress 2005

[28] A Charrada ldquoNonlinear complex M-SVR for LTE MIMO-OFDM channel with impulsive noiserdquo in Proceedings of the2016 7th International Conference on Sciences of ElectronicsTechnologies of Information and Telecommunications (SETIT)pp 10ndash13 Hammamet Tunisia December 2016

[29] G A Hussain M B Mokhtar and R S A B Raja ldquoCon-catenated RS-convolutional codes for MIMO-OFDM systemrdquoAsian Journal of Applied Sciences vol 4 no 7 pp 720ndash727 2011

[30] S Cui A J Goldsmith and A Bahai ldquoEnergy-efficiency ofMIMO and cooperativeMIMO techniques in sensor networksrdquoIEEE Journal on Selected Areas in Communications vol 22 no6 pp 1089ndash1098 2004

[31] V Tarokh H Jafarkhani and A R Calderbank ldquoSpace-timeblock codes from orthogonal designsrdquo IEEE Transactions onInformation Theory vol 45 no 5 pp 1456ndash1467 1999

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