Download - Average Bit Error Probability of MRC Combiner in Log Normal Shadowed Fading

Int. Journal of Electrical & Electronics Engg. Vol. 2, Spl. Issue 1 (2015) e-ISSN: 1694-2310 | p-ISSN: 1694-2426

NITTTR, Chandigarh EDIT -2015 188

Average Bit Error Probability of MRCCombiner in Log Normal Shadowed Fading

Rupender Singh1, S.K. Soni2, P. K. Verma3

1, 2, 3Department of Electronics & Communication EngineeringDelhi Technological University (Formerly Delhi College of Engineering), Delhi

[email protected]

Abstract: In this paper, we provide the average bit errorprobabilities of MQAM and MPSK in the presence of lognormal shadowing using Maximal Ratio Combiningtechnique for L diversity branches. We have derivedprobability of density function (PDF) of received signal tonoise ratio (SNR) for L diversity branches in Log Normalfadingfor Maximal Ratio Combining (MRC). We have usedFenton-Wilkinson method to estimate the parameters for asingle log-normal distribution that approximates the sum oflog-normal random variables (RVs). The results that weprovide in this paper are an important tool for measuring theperformance ofcommunication links in a log-normalshadowing.

Keywords: Log normal random variable, FW Method, MaximalRatio Combining (MRC), Probability of density function(PDF),MQAM, MPSK, Random Variables RVs, ABEP(Average BitError Probability).

1. INTRODUCTIONWireless communication channels are impaired bydetrimental effects such as Multipath Fading andShadowing[1]. Based on various indoor and outdoorempirical measurements, there is general consensus thatshadowing be modeled using Log-normal distribution[8-10]. Fading causesdifficulties in signal recovery. When areceivedsignal experiences fading during transmission, itsenvelope and phase both fluctuate over time.One of the methods used to mitigate thesedegradation arediversity techniques, such as spacediversity [1], [2].Diversity combining has beenconsidered as an efficientway to combat multipathfading and improve the receivedsignal-to-noiseratio (SNR) because the combined SNRcomparedwith the SNR of each diversity branch, isbeingincreased. In this combining, two or more copiesofthe same information-bearing signal are combiningtoincrease the overall SNR. The use of log-normaldistribution [1], [10] tomodel shadowing which is randomvariabledoesn’t lead to a closed formsolution forintegrations involving in sum ofrandom variables at thereceiver. Thisdistribution(PDF) can be approximated byanother log normal random variable using Fenton-Wilkinson method[3].This paper presents Maximal-Ratio Combiningprocedurefor communication system where thediversity combining isapplied over uncorrelated branches (ρ=0), which are givenas channels with log-normal fading.Maximal-Ratio Combining (MRC) is one of themostwidely used diversity combining schemeswhose SNR isthe sum of the SNR’s of eachindividual diversity branch.MRC is the optimalcombining scheme, but its price andcomplexity arehigh, since MRC requires cognition of allfadingparameters of the channel.

The sum of log normal random variables has beenconsidered in [3-6]. Up to now these papers has showndifferent techniques such as MGF, Type IV PearsonDistribution and recursive approximation. In this paper wehave approximated sum (MRC) of log normal randomvariables using FW method. On the basis of FWapproximation, we have given amount of fading (AF),outage probability (Pout) and channel capacity (C) for MRCcombiner.In this paper, a simple accurate closed-form usingHoltzmanin [14] approximation for the expectation of thefunction of a normal variant is also employed. Then,simple analytical approximations for the ABEP of M-QAM modulation schemes for MRC combiner output arederived.

2. SYSTEMS AND CHANNEL MODELSLog-normal DistributionA RV γ is log-normal, i.e. γ ∼ LN(μ, σ2), ifand only if ln(γ)∼ N(μ, σ2). A log-normal RVhas the PDF( ) = √ ( )

……….......(1)For any σ2 > 0. The expected value of γ is( ) = ( . )And the variance of γis( ) = ∗ ( )Where =10/ln 10=4.3429, μ(dB) is the mean of10 , (dB) is standard deviation of 10 .

3. MAXIMAL RATIO COMBININGThe total SNR at the output of the MRC combiner is givenby: γ MRC=∑ γ i ………………………….………..(2)γMRC= 1 + 2 + 3 + 4 … … … … . + L...........(3)

whereL is number of branches.Since the L lognormal RVs are independentlydistributed,the PDF of the lognormal sum[3]

p( MRC)=p( 1)⊗p( 2)⊗p( 3)⊗…………⊗p( L)……………………………………………………….……(4)

where⊗denotes the convolution operation.The functional form of the log-normal PDF does notpermit integration in closed-form. So above convolutioncan never be possible to present. Fenton (1960) estimatethe PDF for a sum of log-normal RVs using another log-normal PDF with the same mean and variance. The Fentonapproximation (sometimes referred to as the Fenton-


189 NITTTR, Chandigarh EDIT-2015

Wilkinson(FW) method) is simpler to apply for awiderange of log-normal parameters.

3.1Consider the sum of L uncorrelated log-normalRVs, , as specified in (1) where each ∼LN(μ, σ2) withthe expected value and varianceμ and σ respectively. Theexpected value and variance of γ MRC are( ) = ( )And ( ) = ( )The FW approximationis a log-normal PDF withparametersμMand σ2

M such that

( . ) = ( )And ( ) ∗ = ( )Solving above equations for μMand σ2

M gives= ln + 1 … … … ………..(5)And μ = ln( ) + 0.5( − )…… (6)

So PDF of the sum of L diversity branches using F-Wmethod is given as(γ ) =

√ ( ).......................(7)

The different μ and σ have been calculated fordifferent numbers of branches L using above F-Wapproximationfrom (5) and (6) and shown in table 1.Forcalculations we have considered ∼LN(0.69,1.072).

Table 1 μ and for different number of diversitybranches

Number ofdiversity branches

L

μ2 2.04 0.854 2.70 0.656 3.51 0.558 4.05 0.48

10 4.51 0.4415 5.51 0.3620 6.35 0.3125 7.09 0.2830 7.76 0.2650 10.01 0.20

In Fig (1) PDF of received SNR using MRC diversitytechniques has presented. As we can see from the Fig thatas the number of branches increases, PDF of received SNRtends towards Gaussian distribution shape. So we canconclude that FW approximation method also satisfiescentral limit theorem.

Fig 1. PDF of received SNR of MRC combiner output

3. ABEP of MQAM for MRC Combiner OutputThe instantaneous BEP obtained by using maximumlikelihood coherent detection for different modulationtypes employing Gray encoding at high SNR can bewritten in generic form[15], [16] as( ) = . ( )…….(8)Where = 4 = 3. 1− 1

isreceived SNR in additive white-gaussiannoise, and isa non-negative random variable depends on the fadingtype.ABEP of MQAM for MRC Combiner Output can bewritten as( ) =∫ ( ). √ ( )

…(9)It is difficult to calculate the results directly, in this work,weadopt the efficient tool proposed by Holtzmanin[9] tosimplifyEg. (5). Taking Eg. (5-7) in [14], we haveUsing 10 = in (9)( ) = ( ). σ √2 ( )Then finally we have ABEP( ) ≈ (μ) + μ + √3 + μ − √3 ….(10)

Where ( ) = . ( exp ( ) )In Fig (3), (4) and (5)ABEP of MQAM has been shown fordifferent numbers of diversity branches from L=2 to 50.We can conclude that as the number of L increases ABEPdecreases. We can see that MRC not only improves SNRbut also improves performance in sense of ABEP. Also wehave concluded that with increasing M=4, 16, 64 ABEPalso increases.

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Fig 3. ABEP of MQAM for MRC Combiner Output M=4



4. ABEP OF MPSK FOR MRC COMBINEROUTPUT

The instantaneous BEP obtained by using maximumlikelihood coherent detection for different modulationtypes employing Gray encoding at high SNR can bewritten in generic form[15], [16] as( ) = . ( )…….(11)Where

= 2 = 3.

isreceived SNR in additive white-gaussiannoise, and isa non-negative random variable depends on the fadingtype.ABEP of MPSK for MRC Combiner Output can be writtenas( ) =∫ ( ). √ ( )

…(12)

It is difficult to calculate the results directly, in this work,weadopt the efficient tool proposed by Holtzmanin[9] tosimplifyEg. (5). Taking Eg. (5-7) in [14], we haveUsing 10 = in (12) ( )Then finally we have ABEP( ) ≈ (μ) + μ + √3 + μ − √3 ….(19)

Where ( ) = . ( exp ( ) )In Fig (6), (7) and (8)ABEP of MPSK has been shown fordifferent numbers of diversity branches from L=2 to 50.We can conclude that as the number of L increases ABEPdecreases. We can see that MRC not only improves SNRbut also improves performance in sense of ABEP. Also wehave concluded that with increasing M=4, 16, 64 ABEPalso increases.

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ABEP of MQAM in Log Normal Using MRC for M=4

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Fig 6. ABEP of MPSK for MRC Combiner Output M=4

( ) = ( ). σ √2


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5. CONCLUSIONThis paper has established a process for estimating thedistribution of MRC combiner output for lognormaldistributed SNR (a single log-normal RV is a special case).The procedure uses the Fenton- Wilkinson approximation(Fenton, 1960) to estimate the parameters for a single log-normal PDF that approximates the sum (MRC) of log-normal RVs. Fenton Wilkinson (FW) approximation wasshown to be general enough to cover the cases of sum ofuncorrelated log normal RVs. We have tabulated μ andσ . ABEP for MQAM and MPSK for MRC combineroutput in Log Normal fading channel also plotted from Fig(2) to (8) for different diversity branches L. We canconclude that MRC improves performance as well asABEP of communication systems in fading environment.

REFERENCES[1] Marvin K. Simon, Mohamad Slim Alouni“Digital Communication

over Fading Channels”, Wiley InterScience Publication.[2] A. Goldsmith, Wireless Communications, Cambridge University

Press, 2005.[3] Neelesh B. Mehta, Andreas F. Molisch,” Approximating a Sum of

Random Variables with a Lognormal”, IEEE Trans on wirelesscommunication, vol. 6, No. 7, July 2007.

[4] Rashid Abaspour, MehriMehrjoo, “A recursive approximationapproach for non iid lognormal random variables summation incellular system”, IJCIT-2012-Vol.1-No.2 Dec. 2012.

[5] Norman C. Beaulieu, “An Optimal Lognormal Approximation toLognormal Sum Distributions”, IEEE Trans on vehiculartechnology, vol. 53, No. 2, March 2004.

[6] Hong Nie, “Lognormal Sum Approximation with

Type IV Pearson Distribution”, IEEE Communication letters, Vol. 11,No. 10, Oct 2007.

[8] H. Suzuki, “A Statistical Model for Urban Radio Propagation”, IEEETrans. Comm., Vol. 25, pp. 673–680,1977.

[9] H. Hashemi, “Impulse response modeling of indoor radio propagationchannels,” IEEE J. Select. Areas Commun., vol. SAC-11,September 1993, pp. 967–978.

[10] F. Hansen and F.I. Mano, “Mobile Fading-Rayleigh and LognormalSuperimposed”, IEEE Trans. Vehic.Tech., Vol. 26, pp. 332–335,1977.

[11] Mohamed-Slim Alouni, Marvin K.Simon, ”Dual Diversity overcorrelated Log-normal Fading Channels”, IEEE Trans. Commun.,vol. 50, pp.1946-1959, Dec 2002.

[12] Faissal El Bouanani, Hussain Ben-Azza, MostafaBelkasmi,” Newresults for Shannon capacity over generalized multipath fadingchannels with MRC diversity”, El Bouananiet al. EURASIP JournalonWireless Communications andNetworking 2012, 2012:336.

[13]Karmeshu, VineetKhandelwal,” On the Applicability of AverageChannel Capacityin Log-Normal Fading Environment”, WirelessPersCommun (2013) 68:1393–1402 DOI 10.1007/s11277-012-0529-2.

[14] J. M.Holtzman, "A simple, accurate method to calculate spreadmultipleaccesserror probabilities," IEEE Trans.Commu., vol. 40, no.3, pp. 461- 464, Mar. 1992.

[15] Y. Khandelwal, Karmeshu, "A new approximation for averagesymbol error probability over Log-normal channels," IEEE WirelessCommun.Lett., vol.3, pp. 58-61.2014.

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Download - Average Bit Error Probability of MRC Combiner in Log Normal Shadowed Fading

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