Tao Liu*, Hailin Zhang Jiaoying Wang, Huihua Fu Ping Wang Jing Li State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, Xi’an, 7007 China Xi’an Institute of Applied Optics, Xi’an 7007 China

I. INTRODUCTION

Recently, free-space optical (FSO) communication as a cost-effective, license-free, high security and high bandwidth access technology has attracted enormous attention [1-4].For a general FSO system, the signal is sent by transmit telescope toward photo detector(PD) through atmosphere media. When an optical wave propagates in such channel, both the amplitude and the phase of the electrical field suffer from random fluctuations which are caused by the variation of the refractive index due to temperature and pressure changes, resulting in the performance degradation[1, 5]. Viable solutions have been proposed to overcome this problem, for example, partially coherent beam [6], aperture averaging [7],adaptive optics [8] etc. In addition, similar to RF communications, a spatial diversity technique employing multiple laser transmitters/photodetectors can be used to mitigate the turbulence-induced fading [1, 9-14]. Up to now, three main diversity combing methods including equal gain combining (EGC) [12],maximal ratio combining (MRC), and selection combining (SC) have been investigated.Among them, MRC has received considerable attention because of its advantage of maximizing the received signal-to-noise ratio (SNR)and giving the optimum performance. S. M.Navidpour et al. investigated the bit error rate(BER) performance of FSO links with multiple photodetectors employing MRC over lognormal fading channels [13]. T. A. Tsiftsiset al. studied the error rate performance of FSO systems with MRC over K-distributed atmospheric turbulence channels [14]. However,lognormal distribution and K distribution are only applicable to weak and strong turbulence conditions, respectively. Recently, it is found that the so called Gamma-Gamma distribution has excellent fit with experiment data over a very wide range of turbulence conditions(weak-to-moderate and moderate-to-strong)[15]. E. Bayaki et al. have investigated the performances of SIMO and MIMO systems based on MRC over independent and identical Gamma-Gamma fading channels [16]. Nevertheless, subchannels are not always identically distributed for a realistic spatial diversity system. Thus, it is necessary to study diversity system with MRC under Gamma-Gamma independent and non-identically distributed fading channels for practical application. In this work, the new analytical BER expressions of dual- and triple-branch FSO systems using MRC diversity technique over Gamma-Gamma independent and non-identically distributed fading channels are presented based on the moment generating function (MGF) approach.The correctness of the analytical results is veried by Monte Carlo (MC) simulations.

II. SYSTEM MODEL AND BER ANALYSIS

Here, we consider a binary phase shift keying(BPSK) subcarrier intensity modulation (SIM)system with dual or triple independent channels. Without loss of generality, it is assumed that the noise consisting of background radiation and electronics thermal noise is modeled by an additive white Gaussian noise (AWGN).Hence, the received electrical signal from thei-th path is given by

whereμis the photodetector responsivity,I(i) is the irradiance which is assumed to follow a Gamma-Gamma distribution,S∈{?1,1}represents the source signal level,?is the modulation index,m(t) represents the subcarrier with a signal symbol durationT.N(i) is the AWGN with zero mean and variance ofThus, the received signal without direct current (DC) bias isThe instantaneous electrical SNR isAssumingso the average SNR is

2.1 The probability density function(PDF) of received signal

Considering the system model, the average BER of MRC at the receiver in dual-branch system is calculated by [17, Eq. (12)] as follows

The fading induced by turbulence follows the Gamma-Gamma distribution [15, 19], and the corresponding PDF ati-th branch has the form:

whereαiis a channel parameter related to the effective number of large-scale cells of the scattering andβirepresents the effective number of small-scale cells.Ku(·) is the modied Bessel function of the second kind of orderu[18, Eq.(8.407/1)]. Γ(·) represents the Gamma function [18, Eq. (8.310/1)]. The PDF of random variableis given by

According to the definition of MGF [20,Eq.(5.113)], the MGF ofYican be expressed as

Applying [21, Eq. (07.34.21.0088.01)] to the above equation, the integral in (5) can be solved. Moreover, by assuming that the channels are independent for each branch of the spatial diversity system, through the product of the MGF of every single channel, the MGFcan be given as

With the help of [21, Eq.(07.34.02.0001.01)],the MGF ofYcan be written in the following form

whereσ1andσ2are used to describe upper and lower bounds of integral, Imandjrepresents the imaginary unity.

Applying the inverse Laplace transform to(8) and interchanging the order of integration,we can obtain whereσis a constant, and used for upper and lower bounds of inverse Laplace transform.

The inner integral can be solved with the help of [18, Eq.(8.315/1)], and the above expression can be rewritten as

Finally, with the help of equationthe PDF of variableIcan be achieved.

2.2 Average BER of dual-branch system

The above mathematical expression can be further simplified in the form of two-fold integrals by using [22, Eq.(2.8.2/1)]. Finally,the BER of dual-branch system under Gamma-Gamma fading channels can be written as (12) in terms of Fox’s H-function [23, Eq.(A.1)], which provided an efcient algorithm for the average BER of diversity system with MRC to obtain immediate design insights for FSO systems [12], [24] and avoided using bounds or generalized innite power series to obtain the exact results [18], [25]-[26].

where, the parameterAi=((2?αi)/2,1),((1?αi)/2, 1), ((2?βi)/2, 1), ((1?βi)/2, 1).

2.3 Average BER of triple-branch system

The methodology discussed above can be extended to the case of three receivers. Thenal form for the BER of triple-branch system can be given as

and (13) in terms of H-fox function can be expressed as

whereB=(0,1).

III. RESULTS AND DISCUSSION

The analytical results for dual- and triple-branch systems are given in this part. The acceptance/rejection method is adopted in MC simulation to generate random numbers of Gamma-Gamma distribution. With anite integration pathLto replace the innite path,it is found that a value ofLequal to 20 can achieve accurate results. In addition, the parametersσ1andσ2should be selected carefully to separate the poles of Γ(·). We choosevia test cases [12, 24], and our MC numerical simulations conrm that it satises the requirement.

Figure 1 illustrates the average BER results of non-identical channels for the studied system with two sets of parameters (α,)βfor different turbulence conditions. The parameters are directly linked with the Rytov varianceby the [25, Eq.(3)]. The atmospheric turbulence can be classified in a continuum of regimes, from weak to strong turbulence conditions, depending on the value ofThe weak fluctuations regime occurs whenand the stronguctuations regime is associated withwhile forthe regime is said to be moderate. The solid curve represents analytical results, while the circles,squares, and asterisks represent the results of MC simulation for weak, moderate, and strong turbulence conditions, respectively. It can be seen that the results of proposed model have excellent agreements with the MC simulations for all the turbulence conditions, showing the validity of the presented BER expression.As can be found, the FSO system with Rytov variance (0.1, 0.5), (1.2, 1.6) and (3.0,3.5) shows better performances than that of Rytov variance (0.2, 0.5), (2.0, 1.6) and (4.1,3.5), respectively, under weak, moderate, and strong turbulence conditions. It is also found that the performance of the system depends on the channel condition of the branches and the BER for the system decreases with the increase of the value of Rytov variance.

A comparison of the average BER between the dual- and triple-branch for all turbulence conditions with the corresponding Rytov variance values [12] of 0.5, 1.2 and 3.5 has been shown ingure 2. The theoretical results are conrmed by MC simulations. It is found that the required SNR is 48 dB for no diversity to get a BER of 10?8in a weak turbulence regime while the required SNRs for dual and triple receiver apertures to get the same BER are33 dB and 27 dB, respectively. In the case of moderate turbulence condition, to get a BER of 10?8, the required SNRs are 87dB, 51dB and 39dB respectively, for direct link, dualand triple- branch FSO systems with MRC. As expected, the value of the SNR and the estimated error decreases rapidly when increasing the number of the branches.

Based on the numerical results fromgure 2, the relative diversity orders (RDOs) for dual- and triple- branch can be obtained by[26, Eq. (50)]. Assuming the direct link as the benchmark, figure 3 demonstrates the RDOs of FSO systems with two or three receiver apertures over Gamma-Gamma turbulence channels. It is clearly seen that RDOs for dual- and triple- branch FSO systems approximate 2 and 3 respectively when SNR tends to innite.

To further verify the conclusion of our works, the comparison of the average BER between MRC and EGC [12] for dual-branch system has been also carried out. It is seen from figure 4 that the BER performance of MRC system is better than that of EGC over all the turbulence regimes, demonstrating the advantages of MRC.

IV. CONCLUSION

In this work, the performances of the dualand triple- branch FSO communication systems with MRC over Gamma-Gamma fading channel shave been analyzed deeply. Using a MGF approach, the analytical expressions of BER based on MRC reception under identically or non-identically distributed channels have been achieved. The generalized BER expressions are more efficient in computation. The accuracy of the presented analytical results is veried by MC simulation. The theoretical results show that the dual- and triple- branch systems can improve the BER performance compared with the direct link system, and the space diversity reception is able to mitigate the impairments caused by the atmospheric turbulence. Moreover, the study provides an efcient calculation algorithm for the average BER of diversity system with MRC, which can be also used for the immediate design insights for FSO systems.

Fig. 1. The average BER of dual-branch MRC over Gamma-Gamma fading channels with two sets of parameters.

Fig. 2. The average BER of direct link, dual- and triple-branch FSO systems over Gamma-Gamma fading channels under weak to strong turbulence conditions.

ACKNOWLEDGEMENTS

This work has been supported by the National Natural Science Foundation of China (Grant No. 61671347), the Fundamental Research Funds for the Central Universities (Grant No.20106151859 & 20106161859), and this work is also partly supported by 111 Project of China (B08038).

Fig. 3. The RDOs for dual- and triple- branch FSO systems over Gamma-Gamma fading channels in (a) weak turbulence regime, (b) moderate turbulence regime and (c) strong turbulence regime

Fig. 4. The comparison between the average BER of MRC and EGC dual-branch FSO systems over Gamma-Gamma fading channels in (a) weak turbulence regime,(b) moderate turbulence regime and (c) strong turbulence regime

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