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
EDIT -2015
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