Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
e-ISSN: 1694-2310 | p-ISSN: 1694-2426
Average Bit Error Probability of MRC Combiner in Log Normal Shadowed Fading Rupender Singh1, S.K. Soni2, P. K. Verma3 1, 2, 3
Department of Electronics & Communication Engineering Delhi Technological University (Formerly Delhi College of Engineering), Delhi
rupendersingh04cs39@gmail.com Abstract: In this paper, we provide the average bit error probabilities of MQAM and MPSK in the presence of log normal shadowing using Maximal Ratio Combining technique for L diversity branches. We have derived probability of density function (PDF) of received signal to noise ratio (SNR) for L diversity branches in Log Normal fadingfor Maximal Ratio Combining (MRC). We have used Fenton-Wilkinson method to estimate the parameters for a single log-normal distribution that approximates the sum of log-normal random variables (RVs). The results that we provide in this paper are an important tool for measuring the performance ofcommunication links in a log-normal shadowing. Keywords: Log normal random variable, FW Method, Maximal Ratio Combining (MRC), Probability of density function(PDF), MQAM, MPSK, Random Variables RVs, ABEP(Average Bit Error Probability).
1. INTRODUCTION Wireless communication channels are impaired by detrimental effects such as Multipath Fading and Shadowing[1]. Based on various indoor and outdoor empirical measurements, there is general consensus that shadowing be modeled using Log-normal distribution[810]. Fading causesdifficulties in signal recovery. When a receivedsignal experiences fading during transmission, its envelope and phase both fluctuate over time. One of the methods used to mitigate thesedegradation are diversity techniques, such as spacediversity [1], [2]. Diversity combining has beenconsidered as an efficient way to combat multipathfading and improve the received signal-to-noiseratio (SNR) because the combined SNR comparedwith the SNR of each diversity branch, is beingincreased. In this combining, two or more copies ofthe same information-bearing signal are combiningto increase the overall SNR. The use of log-normal distribution [1], [10] tomodel shadowing which is random variabledoesn’t lead to a closed formsolution for integrations involving in sum ofrandom variables at the receiver. Thisdistribution(PDF) can be approximated by another log normal random variable using FentonWilkinson method[3]. This paper presents Maximal-Ratio Combiningprocedure for communication system where thediversity combining is applied over uncorrelated branches (ρ=0), which are given as channels with log-normal fading. Maximal-Ratio Combining (MRC) is one of themost widely used diversity combining schemeswhose SNR is the sum of the SNR’s of eachindividual diversity branch. MRC is the optimalcombining scheme, but its price and complexity arehigh, since MRC requires cognition of all fadingparameters of the channel. NITTTR, Chandigarh
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The sum of log normal random variables has been considered in [3-6]. Up to now these papers has shown different techniques such as MGF, Type IV Pearson Distribution and recursive approximation. In this paper we have approximated sum (MRC) of log normal random variables using FW method. On the basis of FW approximation, we have given amount of fading (AF), outage probability (Pout) and channel capacity (C) for MRC combiner. In this paper, a simple accurate closed-form using Holtzmanin [14] approximation for the expectation of the function of a normal variant is also employed. Then, simple analytical approximations for the ABEP of MQAM modulation schemes for MRC combiner output are derived. 2. SYSTEMS AND CHANNEL MODELS Log-normal Distribution A 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)
2
For any σ > 0. The expected value of γ is ( )= ( . ) And the variance of γis ( )= ∗ ( Where 10
)
=10/ln 10=4.3429, μ(dB) is the mean of , (dB) is standard deviation of 10 .
3. MAXIMAL RATIO COMBINING The total SNR at the output of the MRC combiner is given by: 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 not permit integration in closed-form. So above convolution can never be possible to present. Fenton (1960) estimate the PDF for a sum of log-normal RVs using another lognormal PDF with the same mean and variance. The Fenton approximation (sometimes referred to as the Fenton188
Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
Wilkinson(FW) method) is simpler to apply for awide range of log-normal parameters.
1.5
PDF of MRC output for different diversity branches
3.1Consider the sum of L uncorrelated log-normal RVs, , as specified in (1) where each ∼LN(μ, σ2) with the expected value and varianceμ and σ respectively. The expected value and variance of γ MRC are ( )= ( ) And ( )= ( )
The FW approximationis a log-normal PDF with parametersμMand σ2M such that (
And
.
)
=
= ln
And
μ = ln(
+ 1 … … … ………..(5)
) + 0.5(
−
)…… (6)
So PDF of the sum of L diversity branches using F-W method is given as (γ
)=
√
(
)
.......................(7)
The different μ and σ have been calculated for different numbers of branches L using above F-W approximationfrom (5) and (6) and shown in table 1.For calculations we have considered ∼LN(0.69,1.072). Table 1 μ and for different number of diversity branches Number of μ diversity branches L 2 2.04 0.85 4 2.70 0.65 6 3.51 0.55 8 4.05 0.48 10 4.51 0.44 15 5.51 0.36 20 6.35 0.31 25 7.09 0.28 30 7.76 0.26 50 10.01 0.20 In Fig (1) PDF of received SNR using MRC diversity techniques has presented. As we can see from the Fig that as the number of branches increases, PDF of received SNR tends towards Gaussian distribution shape. So we can conclude that FW approximation method also satisfies central limit theorem.
189
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
1
0.5
0
( )
( ) ∗ = ( ) Solving above equations for μMand σ2M gives
e-ISSN: 1694-2310 | p-ISSN: 1694-2426
0
1
2
3
4
5 received SNR
6
7
8
9
10
Fig 1. PDF of received SNR of MRC combiner output
3. ABEP of MQAM for MRC Combiner Output The instantaneous BEP obtained by using maximum likelihood coherent detection for different modulation types employing Gray encoding at high SNR can be written in generic form[15], [16] as ( )= . ( )…….(8) Where 4 1 = = 3. −1 isreceived SNR in additive white-gaussiannoise, and is a non-negative random variable depends on the fading type. ABEP of MQAM for MRC Combiner Output can be written as ( )= ∫
( ).
√
(
)
…(9)
It is difficult to calculate the results directly, in this work, weadopt the efficient tool proposed by Holtzmanin[9] to simplifyEg. (5). Taking Eg. (5-7) in [14], we have Using 10 = in (9) ( )=
( ).
(μ) +
μ + √3
Then finally we have ABEP ( )≈
Where
σ √2
+
(
)
μ − √3
….(10)
( ) = . ( exp ( ) ) In Fig (3), (4) and (5)ABEP of MQAM has been shown for different numbers of diversity branches from L=2 to 50. We can conclude that as the number of L increases ABEP decreases. We can see that MRC not only improves SNR but also improves performance in sense of ABEP. Also we have concluded that with increasing M=4, 16, 64 ABEP also increases.
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Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-2
10
-4
10
ABEP
isreceived SNR in additive white-gaussiannoise, and is a non-negative random variable depends on the fading type. ABEP of MPSK for MRC Combiner Output can be written as ( )=
ABEP of MQAM in Log Normal Using MRC for M=4
0
10
-6
10
-8
-10
-12
2
4
6
8
10 12 SNR Ys(dB)
14
16
18
20
Fig 3. ABEP of MQAM for MRC Combiner Output M=4 L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-1
ABEP
10
-2
10
-3
10
( )≈
Where
0
2
4
6
8
10 12 SNR Ys(dB)
14
16
( )=
( ).
(μ) +
μ + √3
( )=
18
…(12)
σ √2
. (
+
(
)
μ − √3
….(19)
exp ( ) )
In Fig (6), (7) and (8)ABEP of MPSK has been shown for different numbers of diversity branches from L=2 to 50. We can conclude that as the number of L increases ABEP decreases. We can see that MRC not only improves SNR but also improves performance in sense of ABEP. Also we have concluded that with increasing M=4, 16, 64 ABEP also increases.
-4
10
√
)
Then finally we have ABEP
ABEP of MQAM in Log Normal Using MRC for M=16
0
10
(
It is difficult to calculate the results directly, in this work, weadopt the efficient tool proposed by Holtzmanin[9] to simplifyEg. (5). Taking Eg. (5-7) in [14], we have Using 10 = in (12)
10
0
( ).
∫
10
10
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ABEP of MPSK in Log Normal using MRC M=4
0
20
10
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-5
10
Fig 4. ABEP of MQAM for MRC Combiner Output M=16 ABEP of MQAM in Log Normal Using MRC for M=64
ABEP
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-1
10
-10
10 ABEP
0
10
-15
10
-20
10
-25
10
0
2
4
6
8
10 12 SNR Ys(dB)
14
16
18
20
Fig 6. ABEP of MPSK for MRC Combiner Output M=4 -2
10
0
2
4
6
8
10 12 SNR Ys(dB)
14
16
18
20
Fig 5. ABEP of MQAM for MRC Combiner Output M=64
4.
ABEP OF MPSK FOR MRC COMBINER OUTPUT The instantaneous BEP obtained by using maximum likelihood coherent detection for different modulation types employing Gray encoding at high SNR can be written in generic form[15], [16] as ( )= . ( )…….(11) Where =
2
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Int. Journal of Electrical & Electronics Engg.
Vol. 2, Spl. Issue 1 (2015)
ABEP of MPSK in Log Normal using MRC M=16
0
10
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-5
10
-10
ABEP
10
-15
10
-20
10
-25
10
0
2
4
6
8
10 12 SNR Ys(dB)
14
16
18
20
Fig 7. ABEP of MPSK for MRC Combiner Output M=16 ABEP of MPSK in Log Normal using MRC M=64
0
10
L=2 L=4 L=6 L=8 L=10 L=15 L=20 L=25 L=30 L=50
-5
10
-10
ABEP
10
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Type IV Pearson Distribution”, IEEE Communication letters, Vol. 11, No. 10, Oct 2007. [8] H. Suzuki, “A Statistical Model for Urban Radio Propagation”, IEEE Trans. Comm., Vol. 25, pp. 673–680,1977. [9] H. Hashemi, “Impulse response modeling of indoor radio propagation channels,” IEEE J. Select. Areas Commun., vol. SAC-11, September 1993, pp. 967–978. [10] F. Hansen and F.I. Mano, “Mobile Fading-Rayleigh and Lognormal Superimposed”, IEEE Trans. Vehic.Tech., Vol. 26, pp. 332–335, 1977. [11] Mohamed-Slim Alouni, Marvin K.Simon, ”Dual Diversity over correlated Log-normal Fading Channels”, IEEE Trans. Commun., vol. 50, pp.1946-1959, Dec 2002. [12] Faissal El Bouanani, Hussain Ben-Azza, MostafaBelkasmi,” New results for Shannon capacity over generalized multipath fading channels with MRC diversity”, El Bouananiet al. EURASIP Journal onWireless Communications andNetworking 2012, 2012:336. [13]Karmeshu, VineetKhandelwal,” On the Applicability of Average Channel Capacityin Log-Normal Fading Environment”, Wireless PersCommun (2013) 68:1393–1402 DOI 10.1007/s11277-0120529-2. [14] J. M.Holtzman, "A simple, accurate method to calculate spread multipleaccesserror probabilities," IEEE Trans.Commu., vol. 40, no. 3, pp. 461- 464, Mar. 1992. [15] Y. Khandelwal, Karmeshu, "A new approximation for average symbol error probability over Log-normal channels," IEEE Wireless Commun.Lett., vol.3, pp. 58-61.2014.
-15
10
-20
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-25
10
0
2
4
6
8
10 12 SNR Ys(dB)
14
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Fig 8. ABEP of MPSK for MRC Combiner Output M=64
5. CONCLUSION This paper has established a process for estimating the distribution of MRC combiner output for lognormal distributed 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 lognormal PDF that approximates the sum (MRC) of lognormal RVs. Fenton Wilkinson (FW) approximation was shown to be general enough to cover the cases of sum of uncorrelated log normal RVs. We have tabulated μ and σ . ABEP for MQAM and MPSK for MRC combiner output in Log Normal fading channel also plotted from Fig (2) to (8) for different diversity branches L. We can conclude that MRC improves performance as well as ABEP 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 wireless communication, vol. 6, No. 7, July 2007. [4] Rashid Abaspour, MehriMehrjoo, “A recursive approximation approach for non iid lognormal random variables summation in cellular system”, IJCIT-2012-Vol.1-No.2 Dec. 2012. [5] Norman C. Beaulieu, “An Optimal Lognormal Approximation to Lognormal Sum Distributions”, IEEE Trans on vehicular technology, vol. 53, No. 2, March 2004. [6] Hong Nie, “Lognormal Sum Approximation with
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