Journal ofScience & Technology 101 (2014) 134-139 Channel Capacity of Free-Space Optical MIMO Systems Over Atmospheric Turbulence Channels Ha Duyen Trung Hanoi University ofScience and Technology, No 1, Dai Co Viet Str., Hai Ba Trung, Ha Noi, Viet Nam Abstract in this paper we theoretically analyze the pertormance of multiple-input multiple-output (MIMO) free-space optical (FSO) systems The liAIMO/FSO average channel capacity (ACC), which is expressed in (e/ms of average spectral efficiency (ASE) is denved taking into account the atmospheric turbulence effects on the MIMO/FSO channel They are modeled by log-normal and the gamma-gamma distnbutions for the cases of weak-to-strong turbulence conditions We quantitatively discuss the influence of turbulence strength, link distance, and different MIMO configurations on the system ASE Numencal and computer simulation results are presented in order to verify the validity of the mathematical analysis Keywords: Free-space optical communications, MIMO, channel capacity Introduction' Free-space optical (FSO) communications, a cost-effective, license-free, high security and high bandwidth access techmque, has received considerable attention recently for a variety of applications [1] One of major degradations to the performance of FSO commumcations is the mfluence of atmos-phenc turbulence caused by vanations in the refractive index due to uihomogeneities in tempera-ture, pressure fluctuations in the air along the propa-gation path of the laser beam [2], Recent studies have revealed that, similar to wireless communications, the effect of turbulence on FSO links can be significantly reduced by employ-ing a multiple-in put multiple-output (MIMQ) with multiple lasers at the transmitter and multiple photodetectors at the receiver The first use of advantage of spatial diversity in ESQ systems has been proposed in [3] In [4] and [5] Lee el al have derived the outage probability of FSO MIMO systems over log-normal turbulence channels assu-ming Gaussian noise statistics In [6] and [7] Wilson et al have formulated and analyzed symbol-enor probability (SEP) and bitenor probability (BEP) of ESQ MiMO transmissions assuming pulse-position-modulation (PPM) and ^-ary PPM in both log-normal and Rayleigh fading channels In all of these studies however, FSO systems using either OOK or PPM signaling are considered thanks to Its simpli-city and low cost Whereas, because of the presence of atmospheric turbulence, OOK modulation requires to select adaptive thresholds appropriately to achieve its optimal performance and PPM modu-lation has poor bandwidth efficiency In order to overcome ' Corresponding author Tel.: (-F844) 3868,0974 Email trung haduyen@hust edu.vn the limitations of OOK and PPM, FSO systems using sub-carrier (SC) intensity modulation schemes, such as sub-carrier phase shift keying (SC-PSK) and subcamer quadrature amplitude modu-lation (SC-QAM), have been proposed The use of the SC intensity modulation scheme also allows the combmation of several radio frequency SC streams into an intensity modulated laser signal, which results in the higher sysiem throughput and flexibility in signal multiplexing The performance of FSO systems using SC-PSK has been extensively investigated [8]-[ll], Regarding the SC-QAM systems, the average SEP of the FSO SISQ systems using SC-QAM over atmospheric turbulence chan-nel can be found m [12], Recently, in [13] Hassan e( a/ presented the ASER for sub-carrier mtensity modulated wireless optical commumcations using SC-QAM and series expansion of the modified Bessel function from [14], in [15] Tning presented die ASER of MIMO/FSO system usmg SC-QAM signals In addition to ASER performance, average capacity performance for MIMO/FSO systems is recently reported in [16], [17], In [16] Deng el al presented analytic expressions and statistical models of the scintillation index and average capacity for multiple partiall coherent beams propagatmg through non-kolmogorov sttong turbulence MIMO FSO links In [17] Shaima etal evaluated the capacity of MIMOOFDM FSO with mtersymbol interference (ISI) under strong atmosphenc turbulence conditions However, to the best of our knowledge, the performance on average channel capacity of MIMO/FSO systems for both weak and strong atmospheric turbulence chatmels has not been clarified In this woilt, we therefore present a comprehen-sive study on the performance of FSO Journal ofScience & Technology 101 (2014) 134-139 employing MIMO configurations m order to unprove systems' capacity performance over atmospheric turbulence channels In particular, we theoretically derive and discuss the MIMO/FSO average channel capacity (ACC), which is expressed in terms of average spectral efficiency (ASE), under the impact of vari-ous channel conditions, system parameters and con-figurations The numerical results that generated by analytical formulas accurately approximate computer simulation results The remainder of the paper is organized as follows In Section the system descriptions are described in detail In Section atmosphenc turbulence models are presented In Section channel capacity of MIMO/FSO links is derived for weak and sttong turbulence channels In the Section the numerical and simulation results are presented, analyzing the influence of turbulence strength, link distance and various MIMO configurations on the systems' ASE The paper concludes with a summary given in Section 6, System Descriptions We consider a general MxN FSO MIMO system using SC-QAM signals with M transmitting lasers pointing toward an /^'-aperture receiver as depicted in Fig, I, Data transmission employing the same SC-QAM signal is transmitted with perfect synch-ronization by each of the A/transmit telescopes through an turbulence channel toward N photodetectors (PDs) with the light beamwidths of each telescope assume to be wide enough to illuminate the entire receiver anay The transmitter's telescope anay is assumed to produce the same total optical power irrespective of M to enforce a fair compari-son with the single ttansmitter case The distance between the individual transmit telescope and receive telescope is assumed to be sufficient so that spatial conelation is negligible r^"'it) = ^P,Ke{t)Y^X^il) + o"{t),n = l2 ,N, (1) where e(ij represents the QAM signal, X^it) denotes the stationary random process for the mrbulence channel between the mth laser to the «th PD The X,^ 's are unconelated with independent and identically distnbuted (i.i.d) random vanables (RVs) o'it) is the AWGN with zero mean and variance A'J"' Assuming the equal gain combl-mng (EGC) detector is employed at the receiver to estimate the transmitted signal Such that Eq (1) is valid with uii) = y' Li'"'(f) , and the combined electncal intensity at the receiver output can be expressed as yA')-Y.'-'°'(''> = ^PM'')Xf v„„-K;(r) (2) Under the assumption of perfect channel estima-tion at the receiver side, the conditional input SNR at the PD can be written as a finite sum of sub channel SNRs as where y^, are RVs defined as the instantaneous electrical SNR at the output of the nth PD caused by signal from the mth laser ;'^„ are given in the following {MN N^ = r„,„xl^ (4) in which, we denote y^,^ a the average SNR contributed by the sub-channel between the m* laser and thcM^PD, y.^ IS given by Atmospheric Turbilence Models Fig I Block diagram of MxN FSO MIMO system using SC-QAM signals over atmospheric turbulence channel The MIMO channel is modeled and can be denoted by an A/x/^ matrix of the turbulence channel X =[^"^(0)1 '^^, • The electrical signal at the output of the PD corresponding to the nth receive aperture output can be expressed as follows There are several statistical models that can be used to describe inadiance fluctuation For weak atmospheric turbulence condition, the turbulence included fading is assumed to be a random process that follows that log-normal distribution [12], whereas for sttong turbulence conditions, a gammagamma distribution is used [18] Journal of Science & Technology 101 (2014) 134-139 3.1 The log-normal turbulence model Channel Capacity In log-normal fadmg channel, the probability density function (pdf) for an normalized inadiance with log-normal RV, X^„ > , is described as [12] In this section, we analytically derive the average channel capacity (ACC) for the M^N MIMO/FSO Imk in the presence of atmospheric turbulences This is a crucial mettic for evaluating the optical link performance The ACC can also be expressed in terms of average spectral efficiency (ASE) in bits/s/Hz if the frequency response of the channel is known We assume that the optical channel is memoryless, stationary, ergodic with i i.d nirbulence statistics and perfect charmel state information (CSI) IS available at both the ttansmitting lasers and the aperture receivers, die system ASE can be defined (InW + where CT/=exp((w,+(W2)-1 with (i'l and v'z ^ ^ respecnvely given by (1 + 18^^-I-0.56CTJ'''*) = l^\og,{l-\-r)xfAV)dr, (bit/s/Hz) (12) 0.510-;^ [\ + 9d^+0.62cr^^^'^f^ InEq.ilXd^yJkD-/4L, where k^2!t/X is the optical wave number, L is the link distance in meters, A is the optical wavelength, and D is the receiver aperture diameter of the PD The parameter !T, is the Rytov variance and in this case, is expressed by [18] crj=1.23C„'A:"U"" (8) where C„ stands for die sttength of the atmospheric turbulence, which is the altitude-dependent and varying fi-om lO"" to 10'%-^'^ according to the turbulence conditions 3.2 where B is the channel' bandwidth and is the total channel SNR and r = {}'^,n = l, ,N,m = \, ,M} is the mafrix of the MIMO atmospheric turbulence channels The joint p.d.f /j.(r)can be reduced to a product of the firt-order p.d.f of each element y^ The pdfs of F are respectively described in lognormal and gamma-gamma distributions as follows nKj= (^i^)^a])^ 2y^^tj^ -ihr Ar.J = The gamma-gamma turbulence model The pdf of a normalized gamma-gamma RV ^''„„ > arises from the product of two independent gamma distnbuted RVs and given as [ 18] 2iam ' / r „ W - r{a)r(^)'' "V„_^(2V^), (9) 4.1 Capacity channels of log-normal Using (12) and (13), the ASE of a log-nomial MIMO/FSO charmel capacity can be e h(n-r-) 2c^,^/2Iln(2) where r ( a ) is the Gamma function and K^_^i) is the modified Bessel fiinction of the second kind of order (a-p), while the parameters a and fi are directly related to atmospheric conditions through the foilowmg expressions: a = [exp((c,)-l]"' (10) MIMO/FSO dy„,„- Using the equality hi(l-i-j:) = ^ ( - l ) * " ' V / f t , (0