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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 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[8-
10]. 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 Fenton-
Wilkinson 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.
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 M-
QAM 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)
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 of
10 , (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 log-
normal PDF with the same mean and variance. The Fenton
approximation (sometimes referred to as the Fenton-

189 NITTTR, Chandigarh EDIT-2015
Wilkinson(FW) method) is simpler to apply for awide
range of log-normal parameters.
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 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-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
diversity branches
L
μ
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.
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
= 3.
1
− 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)
( ) = ( ).
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 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.
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
received SNR
PDFofMRCoutputfordifferentdiversitybranches
L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50

NITTTR, Chandigarh EDIT -2015 190
Fig 3. ABEP of MQAM for MRC Combiner Output M=4
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
= 3.
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
( ) =
∫ ( ).
√
( )
…(12)
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)
( )
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 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.
0 2 4 6 8 10 12 14 16 18 20
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
ABEP of MQAM in Log Normal Using MRC for M=4
SNR Ys(dB)
ABEP L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50
0 2 4 6 8 10 12 14 16 18 20
10
-4
10
-3
10
-2
10
-1
10
0
SNR Ys(dB)
ABEP
L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50
0 2 4 6 8 10 12 14 16 18 20
10
-2
10
-1
10
0
SNR Ys(dB)
ABEP
L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50
0 2 4 6 8 10 12 14 16 18 20
10
-25
10
-20
10
-15
10
-10
10
-5
10
0
ABEP of MPSK in Log Normal using MRC M=4
ABEP
SNR Ys(dB)
L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50
Fig 6. ABEP of MPSK for MRC Combiner Output M=4
( ) = ( ).
σ √2

191 NITTTR, Chandigarh EDIT-2015
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 log-
normal PDF that approximates the sum (MRC) of log-
normal 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.
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approach for non iid lognormal random variables summation in
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[6] Hong Nie, “Lognormal Sum Approximation with
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[8] H. Suzuki, “A Statistical Model for Urban Radio Propagation”, IEEE
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[11] Mohamed-Slim Alouni, Marvin K.Simon, ”Dual Diversity over
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[12] Faissal El Bouanani, Hussain Ben-Azza, MostafaBelkasmi,” New
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[13]Karmeshu, VineetKhandelwal,” On the Applicability of Average
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[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.
0 2 4 6 8 10 12 14 16 18 20
10
-25
10
-20
10
-15
10
-10
10
-5
10
0
SNR Ys(dB)
ABEP L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50
0 2 4 6 8 10 12 14 16 18 20
10
-25
10
-20
10
-15
10
-10
10
-5
10
0
SNR Ys(dB)
ABEP
L=2
L=4
L=6
L=8
L=10
L=15
L=20
L=25
L=30
L=50

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