systems over Weibull fading channels A comparative simulation study on the performance of LDPC codedcommunication Technology Journal of Applied Researchand

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Availableonlineatwww.sciencedirect.com

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology14(2016)101–107

Original

A comparative simulation study on the performance of LDPC coded communication systems over Weibull fading channels

Ibrahim Develi

^a,∗

, Yasin Kabalci

^b

aDepartmentofElectrical&ElectronicsEngineering,FacultyofEngineering,ErciyesUniversity,38039Kayseri,Turkey

bDepartmentofElectronicsandAutomation,NigdeVocationalCollegeofTechnicalSciences,NigdeUniversity,51200Nigde,Turkey

Received11April2015;accepted8March2016 Availableonline29April2016

Abstract

TheWeibulldistributionisausefulstatisticalmodelthatcanbeusedtodescribethemultipathfadinginnowadayswirelesscommunication environments.Inthispaper,thebiterrorrate(BER)performanceofLow-DensityParity-Check(LDPC)codedcommunicationsystemsusing differentdecodingrulesispresentedoverWeibullfadingchannelsbymeansofcomparativecomputersimulations.Itisshownthat,especially forthecaseoftheBeliefPropagation(BP)decodingrule,significantperformanceimprovementcanbeachievedincomparisonwithuncoded transmissionwhenthechannelisassumedtohaveWeibullfading.

Keywords:Biterrorrateperformancesimulation;Low-DensityParity-Checkcodes;Weibullfadingchannels

1. Introduction

Owingtotheirgoodperformance,Low-DensityParity-Check (LDPC)codesareanimportantfamilyoferror-correctioncodes employedin current datacommunication systems (Bastug &

Sankur,2004).Theyareatypeoflinearblockcodesandhave beenpresentedbyGallagerintheearly1960s(Gallager,1963).

After MacKay and Neal (MacKay, 1999; MacKay & Neal, 1997) haveshowed that LDPC codes could achieve aShan- non limit error performance similar to that of Turbo codes, LDPCcodeshavebeenalmostrediscoveredinthelate1990s.

LDPCcodeshaveseveraladvantagesagainsttheirbiggestrival Turbocodesandtheseadvantagescanbesummarizedas follows:demonstratingbetterblockerrorperformance,errorfloors inmuchlowerBitErrorRate(BER)values,theabilitytoobtain gooderrorperformancewithouttheneed forinterleavers and an iterative-based decoding processinstead of atrellis-based one.Another importantadvantage of LDPCcodes isthat the

PeerreviewundertheresponsibilityofUniversidadNacionalAutónomade México.

∗Correspondingauthor.

E-mailaddress:[email protected](I.Develi).

complexity of decoding increases linearlywith the length of the blocks (Mataracıo˘gluand Aygölü, 2008;Lin &Costello, 2004; Richardson&Urbanke,2001).Currently,LDPC codes havebeenemployedforchannelcodingonsomecommunication standardssuchasDVB-S2,WiMAX(802.16e),Wi-Fi(802.11n) and10GbitEthernet (IEEE802.3an).Also,properly designed LDPCcodescanexhibithighperformanceinthird-generation (3G)mobilecommunicationsystems(Bonello,Chen,&Hanzo 2011;Ohtsuki,2007;3GPP,2005).

TheWeibulldistributionisastatisticalmodelthathasbeen proposedasanappropriatefadingmodeltodescribemultipath fadingchannels for bothindoorandoutdoor environments. It has been used innumerous wireless communication applica- tions (Babich & Lombardi, 2000; Chen, Liou, & Yu, 2010;

Cvetkovic,Djordjevic,&Stefanovic,2011;Kapucu,Bilim,&

Develi, 2013; Singh, Rai, Mohan, & Singh, 2011; Tzeremes

& Christodoulou, 2002). Babich et al. have emphasizedthat the Weibull distribution provides the optimum fit for Digital EnhancedCordlessTelecommunications(DECT)systemsoper- atingat1.89GHz(Babich&Lombardi,2000).InTzeremesand Christodoulou(2002)theauthorshaveperformedsomeexperi- mentsforaGlobalSystemforMobileCommunications(GSM) network,whichoperates at900MHzand haveproposedthat

http://dx.doi.org/10.1016/j.jart.2016.04.001

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theWeibulldistributioncanalsobeusedasafadingmodelfor outdoor systems. Furthermore, the Weibull fading model has beenrecommendedfortheoreticalstudiesbytheIEEEVehicular TechnologySocietyCommitteeonRadioPropagation(Adawi etal.,1988).WhenstudiesrelatedtotheperformancesofLDPC codesover fadingchannels are examinedit canbeseenthat, sofar,theperformanceofLDPCcodeshasbeenwidelyinves- tigatedoverRayleigh,Rician,Nakagami-m,blockfadingand non-ergodicblockfadingchannels(Boutros,Fabregas,Biglieri,

&Zémor,2010;Djordjevic,Djordjevic,&Ivanis,2009;Tan,Li,

&Teh,2011a,2011b,2012;Yang,An,&Li,2011).However,it isimportanttonotethatnocomprehensivestudyhasbeenper- formedontheperformanceofLDPCcodesoverWeibullfading channelsintheliterature.Therefore,theproposedstudyaimsto makeacontributiontothistopic.

The rest of the paper is organized as follows. Section 2 describestheencodinganddecodingprocessesforLDPCcodes.

Section3describesthestructureoftheanalyzedcommunication systemmodelwiththeWeibullfadingchannel.Thesimulation resultsareexaminedinSection4indetail,andfinally,conclu- sionsaredrawninSection5.

2. EncodinganddecodingprocessesforLDPCcodes

2.1. Encodingprocess

AnLDPCcodeisatypeoflinearblockcodeandisdefinedby aparity-checkmatrixH,whichcontainsmostlyzeros(0s)and asmallnumberofones(1s).Itisrequiredtoset(N−K)×Nas thesizeoftheHmatrixtoobtainan(N,K)LDPCcode,where (N−K)isthenumberofrows,Nisthenumberofcolumnsand Kisthenumberofmessagebits.Thecodingrate(R)isdefined byR=K/N foran (N,K) LDPCcode. Whenthe Hmatrix is considered,thenumberof1bitsinarowiscalledtherowweight (wr)andsimilarly,thenumberof 1bitsinacolumniscalled thecolumnweight(wc).LDPCcodescanbeclassifiedintotwo categoriesaccordingtowrandwcasregularandirregularLDPC codes.Ifthenumberof1sineachofthecolumnsandrowsis constant,itiscalledaregularLDPCcode,otherwiseitiscalled anirregularLDPCcode.Aparity-checkmatrixHforan(8,4) regularLDPCcodewithwc=2andwr=4isshowninEq.(1):

H=

⎡

⎢⎢

⎢⎣

1 0 0 1 1 0 1 0

1 1 1 0 0 1 0 0

0 1 0 1 1 0 0 1

0 0 1 0 0 1 1 1

⎤

⎥⎥

⎥⎦ (1)

AnLDPCcodecanalsobepresentedbyabipartitegraphthat wasproposedbyTannerin1981(Tanner,1981).Tannergraphs contain N numberbit nodes(b1, b2,..., bN), N−K number checknodes(c1,c2,...,cN−K)andseveralconnectionsbetween thesenodesforan(N,K)LDPCcode.Thechecknodescorre- spondtotherowsoftheparity-checkmatrixHandthebitnodes correspond tothe columnsofthe parity-checkmatrixH.The connectionsbetweenthenodesareusedforonly1elementof theparity-checkmatrixH.ATannergraphforEq.(1)isshown inFigure1.

Figure1.Graphicalrepresentationof(8,4)regularLDPCcode.

Firstofall,fortheencodingprocess,itisnecessarytocre- ateageneratormatrixGfromtheparity-checkmatrixH.The generator matrix Gcan bederived by applyingthe Gaussian eliminationmethodandsomematrixoperationstotheparity- checkmatrixH.AftertheGaussianeliminationprocess,anew parity-checkmatrixHisobtainedasseeninEq.(2):

H=

−A^T|I_N−K

(2) whereIN−Kisaunitmatrixwith(N−K)×(N−K)sizewhile A^TisthetransposematrixoftheAmatrix.Thegeneratormatrix GcanbedenotedbyarrangingEq.(2)asfollows:

G= [I_k|A] (3)

Thelaststepoftheencodingprocessismultiplyingtheinfor- mationbitsequencesbythegeneratormatrixGasgivenbelow:

z=u·G (4)

whereuistheuncodedinformationsequencewith1×Ksize andzistheencodedinformationsequencewith1×Nsize.

2.2. Decodingprocess

TheLDPCdecodingprocesscanbeimplementedbyusing softorharddecisiondecoderswheretheharddecisiondecoders use the mathematical equations of the Tanner graphs in the decodingprocedures.TheBitFlipping(BF)algorithmisgen- erallyusedinharddecisiondecodersduetoitslowcomplexity.

Several studies have been performed to improve the performance of the BF algorithm andmodified versions of the BF algorithm, such as the Weighted Bit Flipping (WBF) algorithm,ImprovedWeightedBitFlipping(IWBF)algorithmand Implementation-efficientReliabilityRatiobasedWeightedBit Flipping(IRRWBF)algorithm,havebeenderivedbyresearchers (Kou, Lin, & Fossorier, 2001; Lee & Wolf, 2005; Zhang &

Fossorier,2004).BeliefPropagation(BP)decodersusetheBP algorithm for the decodingsteps. TheBP algorithm isa soft decodingmethodthatisanefficientandrobusttechniqueused intheLDPCdecodingprocedure.

Inthefollowing,itisassumedthatBinaryPhaseShiftKey- ing(BPSK)modulationisusedtoexplaintheLDPCdecoding

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procedures. The BPSK mapsa codeword c=(c1, c2,..., cN) into a transmitted sequence x=(x1, x2,..., xN), according to xn=2cn−1equivalencyforn=1,2,...,N.Thereceivedvalue correspondingtoxnafterthedemodulatorisyn=xn+wn,where wnisarandomvariablewithazeromeanandvarianceofN0/2 (Chen&Fossorier,2002;Fossorier,Mihaljevi´c,&Imai,1999;

Kouetal.,2001;Liao,Lin,Chang,&Liu,2007;Ohtsuki,2007).

2.2.1. WeightedBitFlipping(WBF)algorithm

The performance of the BF decoding algorithm can be improvedby including the information on the receivedsym- bolsinthedecodingdecisionprocess(Kouetal.,2001).Firstof all,intheWBFalgorithm,mvaluesarecomputedbythehelp ofEq.(5)intheWBFalgorithmasfollows(Kouetal.,2001;

Zhang&Fossorier,2004):

|y|min−m= min

n:n∈N(n)|yn| (5)

whereN(m)={n:Hmn=1}andm=1,2,...,M.Afterthedeter- minationofmvalues,theWBFdecodingalgorithmisperformed accordingtothestepsgivenbelow(Kouetal.,2001;Zhang&

Fossorier,2004):

Step 1. Syndrome component sm is computed from hard decisionsequencez.

s_m=^N

n=1

z_nH_mn (6)

Step2.EniscomputedusingEq.(7)forn=1,2,...,N.

E_n =

m∈M(n)

(2s_m−1)|y|min−m (7)

Step3.Flipthebitznforn=argmax1≤n≤NEn.

Ifthealgorithmreachesthemaximumnumberofiterations oralltheparitycheckequationsaresatisfiedthenthealgorithm is ended otherwise the algorithm is repeated from Step 1 to Step3.

2.2.2. ImprovedWeightedBitFlipping(IWBF)algorithm The IWBF algorithm is a modified type of the bit nodes informationwhichtakesplaceintheWBFalgorithm(Zhang&

Fossorier,2004).TheIWBFalgorithmdeterminesthereliability ofeachbitnodebyusingchecknodeandbitnodeinformation, incontrasttotheWBFalgorithmthatonlytakesadvantageofthe informationfromchecknodes.IntheIWBFalgorithmaweight- ingfactor(␣)isused forthebitmessageascanbeseenfrom Eq.(9).Theweightingfactorisarealnumberandisgreaterthan zero.Ifα=0theIWBFalgorithmturnsintothestandardWBFin thiscase.ThestepsoftheIWBFalgorithmcanbesummarized asfollows(Zhang&Fossorier,2004):

s_m=^N

n=1

z_nH_mn (8)

E_n=

m∈M(n)

(2s_m−1)|y|min−m−α· |y_n| (9)

ThealgorithmrepeatsfromStep1toStep3untilthealgo- rithmreachesthemaximumnumberofiterationsoralltheparity checkequationsaresatisfied.

2.2.3. ImprovedWeightedBitFlipping(IWBF)algorithm Even though the reliability ratio based weighted bit flip- pingalgorithmisanefficientharddecisiondecodingalgorithm, it requires a long time for the decoding process. The Implementation-EfficientReliabilityRatiobasedWeightedBit Flipping (IRRWBF) algorithmis proposed tosolve thistime problemindecoding.TheIRRWBF algorithmcanbedivided into four steps: initialization, check node, variable node and decisionsteps. Thisalgorithmcanbe summarizedas follows (Lee&Wolf,2005):

Initialization: Tm=

n:n∈N(m)

|yn| (10)

s_m=^N

n=1

z_nH_mn (11)

En= 1

|yn|

m∈M(n)

(2sm−1)Tm (12)

Intheeventthatthealgorithmreachesthemaximumnumber ofiterationsoralltheparitycheckequationsaresatisfied,the algorithmisended.

2.2.4. BeliefPropagation(BP)algorithm

TheBeliefPropagation(BP)algorithmrealizesthedecoding processasthetransferofinformationfromonenodetoanother nodeinthe Tannergraph.The transferredinformationmeans thepossibilityoftherebeing1or0bitsofinformation.TheBP algorithmisalsoknownastheMessagePassing(MP)algorithm intheliterature.Itisnecessarytodefinesomenotationsasso- ciatedwithagiveniterationasfollows(Fossorieretal.,1999;

Kouetal.,2001;Liaoetal.,2007;Ohtsuki,2007):

Fn:Thelog-likelihoodratio(LLR)ofbitnwhichisderived fromthereceivedvalueyn.

Lⁱ_mn:TheLLRofthebitnsentfromthechecknodemtothe bitnodenintheithiteration.

zⁱ_mn:TheLLR of thebitn sentfromthe bitnoden tothe checknodemintheithiteration.

zⁱ_n:TheaposterioriLLRofthebitcomputedateachiteration.

TheBPalgorithmcanbemathematicallydescribedasfollows byreferringtotheparametersgivenabove(Fossorieretal.,1999;

Kouetal.,2001;Liaoetal.,2007;Ohtsuki,2007):

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Initialization:Theaprioriprobability(L(ci))iscalculated bythedecoderinthisstepusingthefollowingequation:

L(ci)=logP(y_n|x_n=0)

P(y_n|x_n=1) (13)

Step 1. In the first step, Lⁱ_mn isupdated for each m,n as follows:

Lⁱ_mn=ln

1+T_mnⁱ 1−T_mnⁱ

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where T_mnⁱ =

n∈N(m)\n

tanh

zⁱ⁻¹_mn

2

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Step2.Inthesecondstep,zⁱ_mnandzⁱ_nareupdatedforeach m,nasfollows:

zⁱ_mn=Fn+

m∈M(n)\m

Lⁱ_mn (16)

zⁱ_n=F_n+

m∈M(n)

Lⁱ_mn (17)

Decision:Forstoppingcriteria,itisfirstofallnecessaryto createavectorsuchasxˆⁱ =

xˆⁱ_n

,wherexˆⁱ_n=1ifzⁱ_n>0and xˆⁱ_n =0ifzⁱ_n<0.

a. IfH ˆxⁱ^T =0,thedecodingalgorithmstopsand ˆxⁱisconsid- eredas avaliddecodingresult,otherwise thealgorithm is repeatedfromStep1.

b. Ifthealgorithmreachesthemaximumnumberofiterations, thealgorithmisended.

3. LDPCcodedcommunicationsystemwithWeibull fadingchannel

TheWeibulldistributionisastatisticalmodelandhasbeen proposedasanappropriatefadingmodeltodescribemultipath fadingforbothindoorandoutdoorenvironments.TheWeibull distributionisusedinmanywirelesscommunicationandimage processingapplications(Babich&Lombardi,2000;Chenetal., 2010;Cvetkovicetal.,2011;Kapucuetal.,2013;Singh,Rai, Mohan, & Singh, 2011; Tzeremes & Christodoulou, 2002).

WhenaWeibullfadingchannelisconsidered,thereceivedsignal s(t)canbedefinedasfollows(Cvetkovicetal.,2011):

s(t)=r(t)exp

j(2πfct+ψ(t)+γ(t))

+n(t) (18)

wherefcisthecarrierfrequency,ψ(t)istheinformationbearing phase,γ(t)istherandomphaseuniformlydistributedin[0,2␲), r(t)istheWeibulldistributedrandomvariablewiththeproba- bilitydensityfunction(PDF)andn(t)istheAWGNnoisethat isarandomvariablewithazeromeanandvarianceN0/2.The PDFoftheWeibulldistributioncanbedefinedas

pr(r)= β

r^β−1 exp

−r^β

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where=E r^β

(E [·] representstheexpectedvalueoper- ator), isalso knownas the scalingparameter, andβ is the Weibull fadingparameterwithvalues 0<β<∞.Asa special case, if β=1, theWeibulldistributionturns intoan exponen- tialdistributionandifβ=2theWeibulldistributionturnsintoa Rayleighdistributioninthiscase(Chenetal.,2010;Cvetkovic etal.,2011).

The LDPC codedcommunication systemmodel isshown inFigure2.Thesimulationprocessesareperformedforareg- ular LDPC code, whichhas (1008,504) block length and½ code rate. The messagegenerator block isshown inthe first part of Figure 2 and generates random binary data as mes- sageinformation.Theoutputsequenceofthemessagegenerator block (u) consists of0 and1 bitsof the messagewith1×K length.TheLDPCencoderblockperformsthechannelcoding for messagebitssothatresulting bitsgainrobustnessandare less changedbyvariouseffectssuchasnoiseandinterference originatingfromdisruptivefactorswhiletheypassthroughthe channel.

TheLDPCencoderblockencodestheinputmessageinfor- mation(u)with1×Klengthusingtheproceduresdescribedin Section2.1andproducesa1×NlengthzsequenceintheLDPC encoderblockoutput.Themodulatorblockcarriesoutthemodu- lationprocessforthezsequenceaccordingtoBPSKmodulation type.ThemodulatedandLDPCcodedmessagebitsarestoredin theoutputofthemodulatorblocknamedx.Afterthemodulation andchannelcodingprocesses,messagebitsareappliedtothe communicationchannelwhichiscomposedofaWeibullfading channelandAWGNnoise.Themessageinformationfromthe outputofthechannelisshownbytheysequenceandisappliedas inputtothedemodulatorblock,asseeninFigure2.Thedemod- ulator blockperforms thedemodulation processfor theinput ysequenceandproducesazˆ sequencewith1×Nlength.The LDPCdecoderblockperformsthedecodingprocessaccording toWBF,IWBF, IRRWBFor BP decoder.Auˆ sequencewith 1×KlengthisobtainedfromtheoutputoftheLDPCdecoder blockafterthedecodingprocess.TheBERperformanceofthe analyzedsystemisobtainedbycomparinginputsequenceuwith outputsequenceu.ˆ

4. Comparativeperformanceanalysis

TheBERperformancesofLDPCcodedcommunicationsys- temsareevaluatedbyMonte-Carlosimulations,andthenumber of Monte-Carlotrialswassetto1000.The BERperformance resultsofthesystemwith(1008,504)LDPCcodeareshownin Figures3–5wherethesimulationsareperformedfor25itera- tionsineachLDPCdecoderemployed.Inallfigures,theeffect of the Weibullchannel isincorporatedintothe simulationby varying the valueof β whilethe is setto1ineach condi- tion. Figure 3 illustratesthe BER performanceof the system that runs inaWeibull fadingchannel. In thissimulation, the fading parameter(β)is setto1. Itis evidentfromthe figure that theBERperformancesof WBF,IWBFandIRRWBF are observedclosetoeachother,whiletheBPdecoderoutperforms theotherharddecisiondecoders,andtheBERperformanceof

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Figure2.BlockdiagramofLDPCcodedcommunicationsystemoverWeibullfadingchannel.

Figure3.BERperformanceofLDPCcodeintheWeibullfadingchannelwith β=1.

thisdecodingruleisupto6.5dBbetterthanthatoftheuncoded caseataBERlevelof10⁻².

Figure 4 depicts the performance of the system when the Weibullfading parameter (β)is increased to2. Even though theharddecisiondecodersgivebetterresultsaccordingtothe previousanalysis,thesedecodersstill cannotprovideasgood performanceasthatobtainedintheBPdecoder.Ascanbeseen, theBERperformanceof theBPdecoderisalwaysbetterthan

Figure4.BERperformanceofLDPCcodeintheWeibullfadingchannelwith β=2.

that of the other decoders.Also notethat theIWBF decoder presentstheworstBERvalueseventhoughitsperformanceis almostthesameasthat oftheuncodedcase.Whencompared withthe uncodedcase, the improvement achievedby theBP decoderisnearly2.5dBforBERof10⁻¹.

Thefinalanalysisisperformedbysettingβto2.5forthesame system.TheperformancecurvesareseeninFigure5.Although theBPdecodercannotgiveasatisfactoryperformanceaswell

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Figure5.BERperformanceofLDPCcodeintheWeibullfadingchannelwith β=2.5.

astheharddecisiondecodersuntil4.5dB,thisdetectoroutper- formstheotherthreetypesdecoders forEb/N0valuesgreater than4.5dB.

Comparisonbetweentheperformancefortheuncodedcase andtheBPdecodingperformancerevealsthattheimprovement achievedbytheBP decoder isnearly7dB ataBERlevel of 10⁻².

5. Conclusions

TheWeibulldistributionhasbeenemployedasanappropriate fadingmodelintheliteraturefortheoreticalchanneldefinition.

Inthisstudy,theBERperformanceofLDPCcodesinWeibull fadingchannels wasanalyzed for differentdecodingrules by meansofcomparativecomputersimulations.AnLDPCcoded BPSKcommunicationsystemwasdesignedasacommunication infrastructuretoperformtheanalysis.Fourdifferentdecoding rulesandaregularLDPCcodewith(1008,504)blocklength and½coderatewereusedinsimulations.Fromallthesimula- tionstudiesrealizedinaWeibullfadingchannel,wecanseethat significantperformanceimprovementcanbeobtainedincom- parisonwiththeuncodedtransmissionespeciallywhentheBP decodingruleisemployedinthereceiver.

Conﬂictofinterest

Theauthorshavenoconflictsofinteresttodeclare.

Acknowledgements

ThisworkwassupportedbytheResearchFundof Erciyes University,grantnumber:FBD-10-3092.Theauthorsthankto Proofreading&EditingOfficeofErciyesUniversityforcareful readingandcorrectionsonthemanuscript.

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