We study the average capacity performance for multiple-input multiple-output (MIMO) free-space optical (FSO) communication systems using multiple partially coherent beams propagating through non-Kolmogorov strong turbulence, assuming equal gain combining diversity configuration and the sum of multiple gamma-gamma random variables for multiple independent partially coherent beams. The closed-form expressions of scintillation and average capacity are derived and then used to analyze the dependence on the number of independent diversity branches, power law α, refractive-index structure parameter, propagation distance and spatial coherence length of source beams. Obtained results show that, the average capacity increases more significantly with the increase in the rank of MIMO channel matrix compared with the diversity order. The effect of the diversity order on the average capacity is independent of the power law, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. The average capacity increases and saturates with the decreasing spatial coherence length, at rates depending on the diversity order, power law and turbulence strength. There exist optimal values of the spatial coherence length and diversity configuration for maximizing the average capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.
©2013 Optical Society of America
Free space optical communication (FSOC)  is a promising technology for high bandwidth wireless communication links over a long distance where a fiber or wire is unfeasible or where RF communication is inadequate .The link performance of FSOC can be severely degraded by atmospheric turbulence induced effects including intensity fluctuations , phase fluctuations, beam wandering and beam jittering . Atmospheric turbulence strength would become stronger with increasing Rytov variance related to the refractive index structure parameter Cn2 and the propagation distance L . Under the strong turbulence regime, scintillation index increases beyond unity, reaches its maximum value in the focusing regime and decreases toward unity in the saturation regimes. Recent experiments have shown significant deviations from the Kolmogorov model in some layers of the atmosphere, which has prompted research on optical wave propagation through non-Kolmogorov atmospheric turbulence [6,7].
The intensity fluctuations or scintillation at the receiver reduces FSOC channel capacity. In order to improve system performance, scintillation can be mitigated by means of reducing the spatial coherence of the transmitted beam [8,9]and the spatial diversity using multiple transmitted beams and multiple receivers [10,11].
Partially coherent beams with reduced spatial coherence show lower scintillation at the cost of larger divergence angle and lower average received power [12,13].Partially coherent beams have a lower scintillation than fully coherent beams. However, a partially coherent beam has a larger beam spreading and forms a large spot in the receiver aperture, which leads to a loss of the transmitted energy being received by the detector. By optimizing the spatial coherence length, the improvement in scintillation reduction can overcome the penalty of power reduction and significant signal-to-noise ratio gains can be obtained in weak atmospheric turbulence [14,15].
Spatial diversity using multiple transmitted beams and multiple receivers can also be employed to reduce scintillation and ultimately improve FSO channel capacity. It has been shown that the scintillation of a beam array can be reduced by carefully adjusting the spatial separation of beamlets . However, scintillation of a beam array will increase significantly if the spatial separation of beamlets is smaller than the correlated length. In addition, the received energy from the beam arrays is low unless the constituent beamlets are inclined to overlap at the receiver aperture, which is difficult to achieve over long propagation distances .The use of multiple transmitters and receivers has also been suggested for use in multiple-input–multiple-output (MIMO) configurations [16,18].
If the transmitted and received signals are assumed to be uncorrelated or independent, the separation space between transmitters and receivers should be larger than the correlation length rc of the irradiance fluctuations, which involves the 1/e2 point of the normalized covariance function of irradiance [1,19,20]. In weak turbulence(Rytov varianceσR2<1), the correlation length is directly related to the size of the first Fresnel zone,where k is the wave number and L is the link range . For example, at wavelength of 780 nm this separation distance corresponds to 1.2 inches at a link range of 1.2 km . For a link distance of L = 1.5 km, a wavelength of 1550 nm, and an aperture diameter of 1 mm, photodetectors separated by as little as 35 mm are practically spatially uncorrelated . The separation distances are the order of centimeters for link distances of the order of kilometers . Whereas in strong fluctuation regimes(σR2?1), the correlation length is defined by the spatial coherence radiusρ0 in atmospheric turbulence . For stronger turbulence conditions, the reduced spatial coherence radius is smaller than the Fresnel Zone. Furthermore, partially coherent beams with smaller initial coherence length can reduce the spatial coherence radius and correlation length in strong atmospheric turbulence. Consequently, the required separation space between independent transmitters and receivers can be reduced by using partially coherent beams propagation through strong atmospheric turbulence.
However, MIMO FSO links studied so far employ the fully coherent beams whose spatial correlated length is large and the induced scintillation is strong. The mutually independent MIMO branches require a relatively larger separation between the transmitter and receiver apertures than the correlated length . Partially coherent beams with shorter spatial coherence lengths can help reduce the scintillation and correlation length in strong turbulence, and eventually reduce the required separation space between independent apertures. Thus, a question naturally arises: can we develop MIMO FSO links whose spatially separated beamlets can have lower spatial coherence lengths that lead to reduced scintillation and smaller separation between transmitter and receiver apertures? Therefore, we incorporate partially coherent beams into the MIMO FSO links to reduce scintillations and correlation length and to improve the system performance in strong turbulence, performance of MIMO FSO links using multiple partially coherent beams propagation through non-Kolmogorov moderate to strong turbulence should be considered, and the effect of spatial coherence length of partially coherent beams on capacity of MIMO FSO links should be analyzed.
In this paper, the average capacity performance for MIMO FSO communication systems using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence are analyzed, assuming equal gain combining diversity configuration and the sum of multiple gamma-gamma random variables for multiple, independent partially coherent beams in atmospheric turbulence. The analytic expressions are obtained and then used to analyze the effects of the number of independent diversity branches, power law α, refractive-index structure parameter, propagation distance and spatial coherence length of source beams on scintillations and average capacity under various turbulence conditions. In non-Kolmogorov strong turbulence conditions, obtained results show that the average capacity increases more significantly with the increase in the rank of MIMO channel matrix Nm rather than the diversity order M × N. The diversity order is independent of the effects of power law, propagation distance, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. Furthermore, as the spatial coherence length decreases, the average capacity increases significantly and then reaches the maximum value at certain coherence length, at last saturates slightly as it approaches incoherent beams. The results would provide a useful approach for the optimization of diversity configuration and spatial coherence length to maximize channel capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.
2. System and channel model
Consider a MIMO FSO communication system [1,11,23] where the information signal is transmitted via Mapertures and received by N apertures over Non-Kolmogorov strong atmospheric turbulence. It is assumed that the aperture diameter of each receiver in the array is less than the spatial correlation width of the irradiance fluctuations so that each receiver acts like a point detector . Moreover, the array elements are spatially separated by a sufficient distance so that each acts independently of the others. Furthermore, a large field of view is considered for each receiver  indicating that multiple transmitters  are simultaneously observed by each receiver. This actually leads to the collection of larger amount of background radiation which justifies the use of the AWGN model as a good approximation of the Poisson photon counting detection model. Assuming on-off keying (OOK), the detected signal at the receive aperture is given by 1] is given by Imn = I0exp(2Xmn), where I0is the signal light intensity without turbulence and Xmn are identically distributed normal random variables with mean μx and varianceσx2.
Various statistical models have been proposed over the years for modeling atmospheric turbulence. For the weak turbulence regime, the fluctuations are generally considered to be lognormal distributed, and for very strong turbulence, it is assumed negative exponential distributed . However, lognormal distribution is mainly restricted to weak turbulence conditions as it can underestimate the behavior in the tails as compared with measured data. The negative exponential distribution is applicable only in the very strong fluctuation regimes, which happens only far into the saturation regime. Recently the Gamma–Gamma distribution has received considerable attention because of its excellent fit with measurement data for a wide range of turbulence conditions (from weak to strong turbulence) .The Gamma-Gamma parameters can be directly related to atmospheric conditions and be consistent with large-scale and small-scale scintillations .This feature of the gamma-gamma distribution makes it particularly attractive for predicting detection and fades probabilities associated with a given atmospheric channel.
The normalized irradiance arising from a Gaussian beam propagation through strong atmospheric turbulence for the case of a point receiver can be expressed as the product of two independent gamma distributed random variables, I = X∙Y, where X and Y result from large-scale and small-scale atmospheric effects, respectively. Thus, the summed output of the array can be reasonably approximated by a Gamma-Gamma distribution , and the resulting normalized PDF is then
Based on the extended Rytov theory with a modified spatial filter function, the large scale log-irradiance variance,, of a partially coherent Gaussian beam propagation through non-Kolmogorov strong turbulence is [29,30]31]
The small-scale log-irradiance fluctuation is 
In the above expressions,, and are the effective beam parameters at the receiver plane. qc = L/(klc2), where lc is the spatial coherence radius of the source. Ns = 1 + 4qc/Λ0is the number of speckle cells . The complementary parameter.Θ = 1 + L/R is the curvature parameter and R is the phase front radius of curvature, whereas Λ = 2L/kW is the Fresnel ratio and W is the spot size of a Gaussian-beam at the output plane. The output plane parameters Θ and Λ are related to the input plane beam parameters Θ0and Λ0by Θ = Θ0/(Θ02 + Λ02),Λ = Λ0/(Θ02 + Λ02).Θ0 = 1-L/R0is the curvature parameter and R0isthe phase front radius of curvature,whereasΛ0 = 2L/kW0is the Fresnel ratio and W0is the spot size of a Gaussian-beam at the input plane.
3. Spatial diversity and combining gain
The sum of multiple independent identically distributed gamma-gamma random variables are expressed as the sum of the product of two independent gamma random variables .
Because scintillation is caused primarily by small-scale in homogeneities in strong turbulence, we argue now that the small-scale scintillation index associated with the summed output of the array is roughly the small-scale scintillation index of a single aperture output divided by the number of the apertures. The two resulting gamma random variables for the summed output have variances ofσX2/S andσY2/S, respectively, resulting in αS = Sα1and𝛽S = S𝛽1. Using nonlinear regression, the error term can be closely approximated to be 
Therefore, for MIMO free space optical communication system with M transmitting apertures and N receiving apertures, the equivalent Gamma-Gamma parameters related to large scale and small scale irradiance fluctuations for partially coherent Gaussian beams propagation through non-Kolmogorov strong turbulence are
The large scale and small scale log-irradiance variance of multiple partially coherent Gaussian beams propagation through non-Kolmogorov strong turbulence are expressed by
The total longitudinal component of scintillation index for multiple partially coherent Gaussian beams in non-Kolmogorov moderate-strong turbulence is expressed as
Thus, the probability density function (PDF) of the summed output Is can be approximated by the PDF of a single Gamma-Gamma variate,
4. Average capacity of MIMO FSO links for multiple partially coherent beams
Considering that perfect channel-state information possessing additive white Gaussian noise with zero mean is available at the receiver and the transmitter, the average (ergodic) capacity of a single FSO communication channel by using the intensity modulation/direct detection scheme can be defined as. Assuming that the transmitted signal vector is comprised of M statistically independent, equal power components, each with a Gaussian distribution, and the received signal vector consists of N statistically independent, equal gain combination, each receiver branch with the uncorrelated noise, the general expression for channel capacity of MIMO FSO links in bps/Hz(or bits/cycle) is 
Further analysis of the MIMO channel capacity given is obtained by diagonalizing the product matrix . By using singular value decomposition, the matrix product is written as, where U and V are unitary matrices of left and right singular vectors respectively, and Σ is a diagonal matrix with singular values on the main diagonal .All elements on the diagonal are zero except for the first k elements. The number of non-zero singular values k equals the rank of the channel matrix. The capacity is a lower bound on the MIMO channel capacity .Using singular value decomposition, the MIMO channel capacity can be written as 
With the fact that the determinant of a unitary matrix is equal to 1 and ,MIMO capacity can be expressed respectively as
The capacity of the MIMO fading channel is a function of the distribution of the singular values of the random channel matrix. By Jensen’s inequality , we obtain the bounding capacity that35].
Thus, the MIMO capacity is
For SISO case
For SIMO case
For MISO case
For MIMO case M = N
Assuming the underlying channel of MIMO FSO system are independent and identically distributed, the PDF of combined signal can be approximated by the PDF of a single Gamma-Gamma variate with parameter (,,). The average capacity of the MIMO FSO system is equivalent to the average capacity of a SISO system operating over the Gamma-Gamma turbulence model with parameter ().Eq. (21), we obtain the following pdf with respect to for the gamma-gamma distribution modelEq. (34) to Eq. (33) and interchanging the order of integration, the maximum average capacity of MIMO FSO links is obtained as
The integral can be solved using Meijer’s G functions and their properties. Hence, by substituting the PDF of the Gamma-Gamma distribution, expressing the Kv(.) integrands in terms of Meijer’s G-function , the closed-formsolution of average capacity yields as follow,
5. Numerical results
Based on the above system model, the average capacity performance of MIMO FSO links for multiple partially coherent Gaussian beams propagating through non-Kolmogorov strong turbulence is generated in the conditions of various transmit/receive apertures diversity, beam spatial coherence and turbulence parameters. We consider that all collimated laser beams (Θ0 = 1) at the transmitter have a beam width of w0∇1 cm and a wavelength of λ = 1550 nm. The initial spatial coherence length of the transmitted laser beam is varied from lc = 0. 01 cm to lc = 1 m. The power law of non-Kolmogorov turbulence spectrum α is between 3 and 4.The propagation distance L is varied from 1km to 10 km, and the turbulence structure parameterCn2takes values in the set (4 × 10−14,8 × 10−14,1 × 10−13,4 × 10−13)m3-α.
5.1 Average capacity versus electrical SNR
The plots in Fig. 1 show average channel capacity versus average SNR for MIMOFSO IM/DD links with multiple partially coherent beams in non-Kolmogorov strong atmospheric turbulence. We have plotted the average capacity for several values of (a) the number of transmit and receive apertures M × N, (b) the spatial coherence length lc of the beam source with different diversity configurations, (c) the power law 𝛼of non-Kolmogorov turbulence with different combiner branches and (d) the power law 𝛼 with various spatial coherence length. In all cases, the propagation distance is fixed at L = 1 km with a constant structure parameter Cn2 = 1 × 10−13m3-α.
As it is evident in Fig. 1(a), the average capacity increases significantly with the number of transmit or receive apertures M × N compared to the SISO deployment. Moreover, the greater improvement in average capacity is obtained as MIMO configuration goes from 1 × 2 to 2 × 2, in comparison with diversity number going from 2 × 2 to 2 × 4. This can be explained by the Eq. (37) that a noticeable increase in capacity in the former case mainly arises from an increasing rank of MIMO channel matrix Nm from 1 to 2, while in the latter case with a constant Nm, the capacity improvement is because of the reduction in scintillation index with the increasing diversity apertures. Considering the increase in capacity at the cost of larger number of apertures, the MIMO 2 × 2 case is the most efficient configuration for capacity improvement.
Figure 1(b) considers the effect of multiple partially coherent beams on average capacity of MIMO FSO links for the 2 × 2 and 2 × 4cases. We observe that in all cases, the average capacity increases apparently with decreasing the spatial coherence length of beam source, and the dependence on initial coherence length gets weaker at larger number of diversity order, i.e. M × N = 2 × 4. It is deduced from Fig. 1(c) that, as the power law α increases from 3.1 to 3.8, the average capacity reduces slightly at power law α of 3.2 and then grows up sharply. The effect of power law 𝛼 on capacity is lessened as M and N increase. Fig. 1(d) shows the influence of power law and coherence length on average capacity of MIMO FSO links for the 2 × 2case. It can be seen that, for most values of spectrum index 𝛼,the average capacity of partially coherent beams is larger than that of coherent beams, whereas for power law 𝛼 = 3.2 the dependence of capacity on spatial coherence length is reversed.
5.2 Average capacity versus power law
In order to explore the effect of non-Kolmogorov turbulence, the average channel capacity of MIMOFSO links using multiple partially coherent beams is depicted in Fig. 2 as a function of spectrum power index𝛼for various values of the number of transmit/ receive apertures M × N, the spatial coherence length lc and turbulence strength parameter Cn2. In all cases, the propagation distance is fixed at L = 1 km with a constant SNR = 10 dB. As can be seen in Fig. 2(a), for all the values of power law𝛼, average capacity is monotonically improved as the number of transmit and receive apertures M × N increases. Furthermore, we observe that the average capacity falls down slightly to the minimum value at 𝛼 = 3.2, and then rises up rapidly as the power law 𝛼increases from 3 to 4.Thereason for this phenomenon can be deduced from Fig. 2(b) where the scintillation index as a function of α corresponding to links in Fig. 2(a) is depicted. The scintillation index of MIMO FSO links has an opposite characteristics dependence on the power law 𝛼, which leads to a gain in average capacity.
Specifically, as indicated by Fig. 2(c), for alpha values larger than 3.5, or close to 3, the average capacity improves apparently with the decrease in spatial coherence length of transmit beams. On the other hand, when power law 𝛼 is between 3.1 and 3.5 (around 𝛼 = 10/3), the average capacity decreases with the decreasing spatial coherence length. It is obvious in Fig. 2(d) that the influence of alpha value on the average capacity becomes stronger as the turbulence strength gets stronger. The physical reason can be deduced from the relationship between alpha values and atmospheric turbulence layers: turbulence tends to vanish for alpha approaching 3, 𝛼 = 11/3 corresponds to the boundary layer, α= 10/3 corresponds to the free troposphere layer, and alpha approaching4 represents lower stratosphere layer under the condition of stable stratification. Thus, as laser beams propagate through the free troposphere layer (around 𝛼 = 10/3), the turbulence effect gets stronger, and the dependence of channel capacity on spatial coherence length gets stronger.
5.3 Average capacity versus propagation distance
Figure 3 plots the average channel capacity of MIMOFSO links using multiple partially coherent beams as a function of propagation distance L for various values of the number of transmit/ receive apertures M × N, the spatial coherence length lc and power law 𝛼. In all cases, the turbulence structure parameter is fixed at Cn2 = 1 × 10−13m3-αwith a constant initial SNR = 10 dB. It is clearly depicted in Fig. 3(a) that, for all the values of L, average capacity is significantly improved as M and N increase. For all the diversity cases, average capacity is initially reduced as the propagation distance L increases under weak-moderate turbulence, and then reaches the minimum values and changes the slope at a distance of about 2000 m in the focusing regime, lastly saturates and increases slightly with the increasing distance in the strong irradiance fluctuation. This phenomenon can be explained by Fig. 3(b) that the reduction and saturation of scintillation index as diversity order M × N and propagation distance L increase. As it is indicated in Fig. 3(c), considering various coherence length values and diversity deployment, for shorter distance under weak-moderate turbulence, average capacity increases with decreasing spatial coherence length. However, for longer distance in strong turbulence, average capacity decreases with spatial coherence length. Furthermore, the dependence of average capacity on coherence length and power law becomes less significant at larger number of transmit and receive apertures.
5.4 Average capacity versus spatial coherence length
The plots in Fig. 4 show the effect of spatial coherence length of transmit beams on the average channel capacity of MIMOFSO links in non-Kolmogorov strong turbulence, for various values of the number of transmit/ receive apertures M × N, spectrum power index 𝛼 and turbulence strength parameter Cn2. In all cases, the propagation distance is fixed at L = 1 km with a constant SNR = 10 dB.
As it is illustrated in Fig. 4(a), for all values of spatial coherence length, average capacity is significantly improved as diversity order M and N increase. Moreover, we observe in Fig. 4(b) that as the spatial coherence length decreases, average capacity increases significantly at first, then reaches the maximum value at a certain coherence length, at last saturates and decreases slightly as it approaches incoherent beams. In the current case, the optimum coherence length corresponds to about lc = 0.002 m = w0/5.This phenomenon can be explained by the fact that for laser beams with lower coherence length, optical energy is delivered by mutually independent coherent modes that propagate through statistically independent regions of turbulence. As a result, turbulence induced scintillations of the modes are relatively uncorrelated, and the averaged scintillation of partially coherent beams is decreased. Consequently, the average capacity of partially coherent beams is improved. As all the coherent modes are mutually independent with the decreasing coherent length, the reduction of scintillation and improvement in capacity goes into saturated regions.
It can be seen in Fig. 4(c), for most alpha values, the average capacity of partially coherent beams increases with smaller spatial coherence length, except for the case 𝛼 = 10/3. That is because the power law of 10/3 corresponds to the free troposphere layer, in which case the partially coherent beams will experience much stronger atmospheric turbulence. We can also observe in Fig. 4(d) that, for weak to moderate atmospheric turbulence (the cases: Cn2 = 4 × 10−14, 8 × 10−14 and 1 × 10−13m3-α, Rytov variance σR< 1.5), average capacity increases with decreasing spatial coherence length. However, for strong atmospheric turbulence (the cases: Cn2 = 4 × 10−13m3-α, Rytov variance σR> 1.5), average capacity decreases as the spatial coherence length decreases. The reason for the phenomenon can be deduced from Fig. 4(e) and Fig. 4(f) in which the scintillation and large-scale and small-scale log-irradiance variance corresponding to links in Fig. 4(d) are plotted.
The scintillation in Fig. 4(e) takes an opposite dependence on spatial coherent length compared with the average capacity in Fig. 4(d). Furthermore, we observe in Fig. 4(f) that the large-scale log-irradiance variance decreases with the spatial coherence length except for the case of strong turbulence (Cn2 = 4 × 10−13m3-α), whereas the small-scale log-irradiance variance decreases with the coherence length for all the turbulence strength. Thus, for partially coherent beams in strong turbulence, the increase in large-scale log-irradiance variance is the dominating factor for the decrease in average capacity. The physical reason is that the large scale size of turbulent eddy becomes shorter than the spatial coherence length as turbulence gets stronger. In this case, the refraction effect of large scale turbulent eddy is stronger at smaller coherence length. In addition, turbulent eddy sizes bounded below by the spatial coherence radius and above by the scattering disk radius contribute little to scintillation under strong fluctuations.
This paper analyses average capacity performance for MIMO FSO communication systems using multiple partially coherent beams propagation through non-Kolmogorov strong turbulence. The analytic expressions and statistical models of the scintillation index and average capacity for multiple partially coherent beams in MIMO FSO links are derived, assuming equal gain combining diversity configuration and the sum of multiple gamma-gamma random variables for multiple, independent partially coherent beams in atmospheric turbulence. The models allow for the effects of the number of independent diversity branches, power law α, refractive-index structure parameter, propagation distance and spatial coherence length of source beams on scintillations and average capacity.
In this scenario of MIMO FSO links over non-Kolmogorov strong turbulence, obtained results show that the average capacity increases more significantly with the increase in the rank of MIMO channel matrix Nm rather than the diversity order M × N. The MIMO case 2 × 2 is the most efficient configuration for capacity improvement at the cost of larger number of apertures. The diversity order is independent of the effects of power law, propagation distance, turbulence strength parameter and spatial coherence length, whereas these effects on average capacity are gradually mitigated as the diversity order increases. Furthermore, as the spatial coherence length decreases, average capacity increases significantly and then reaches the maximum value at coherence length of w0/5, at last saturates slightly as it approaches incoherent beams. However, the dependence of average capacity on spatial coherence length are changed, as alpha value approaches10/3 that corresponds to the free troposphere layer, or turbulence strength is stronger with Rytov variance larger than 1.5. The results would provide a useful approach for the optimization of diversity configuration and spatial coherence length to maximize channel capacity of MIMO FSO links over a variety of atmospheric turbulence conditions.
This research was financially supported by National Natural Science Foundation of China(NSFC) (No. 61077058, No. 61275081).The authors are grateful for financial support from the program of China Scholarships Council (No.2011616097) and the Pennsylvania State University CICTR, through US National Science foundation (NSF) ECCS Directorate for partially supporting this work under award (1201636).
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