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Performance of FSO Links under Exponentiated Weibull Turbulence with Misalignment Errors

P. K. Sharma†,1, A. Bansal‡,2, P. Garg‡,3, T. A. Tsiftsis††,4 and R. Barrios†‡,5 †Department of EECE, ITM University, Gurgaon, India ‡ Division of ECE, Netaji Subhas Institute of Technology, New Delhi, India †† Dept. of Electrical Engineering, Technological Educational Institute of Central Greece, Lamia, Greece †‡ Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Wessling, Germany [email protected], [email protected], parul [email protected], [email protected], [email protected]

Abstract—The exponentiated Weibull (EW) distribution has link in the receiver side. The placement of the collecting been recently proposed for the modelling of free-space optical aperture integrates the more light in the receiver plane and (FSO) links in the presence of finite sized receiver aperture. thus a greater portion of the incoming wavefront can be In this paper, the performance of FSO communication sys- tems over EW is studied. Specifically, the probability density concentrated into the photodetector of the point receiver. function (PDF) and cumulative distribution function (CDF) The characterization of the atmospheric turbulence induced of the instantaneous signal-to-noise ratio (SNR), over EW fading is an important step in the performance analysis of an turbulence fading, are studied. The derived statistics of the SNR FSO communication systems. The existing channel models is utilized to analyse the performance of an FSO communication such as log-normal and Gamma-Gamma can not be used in system over a generalized communication environment with turbulence induced fading, misalignment errors and path loss. all turbulence situations, and even sometimes they do not New expression for the outage probability is obtained, and exact provide a good fit for experimental and simulated data [2], expressions for the average bit error rate (BER) are derived [5]. Recently, Barrios et. al. in [6], [7] proposed a new dis- for various binary schemes. Finally, the obtained tribution, known as exponentiated-Weibull (EW) distribution, analytical results are verified via Monte Carlo simulations. to model the atmospheric turbulence induced fluctuations in the irradiance received at the finite size aperture-averaged I.INTRODUCTION receiver. The EW distribution captures the effect of aperture- Free-space optical (FSO) communication has emerged averaging through its constituent parameters as these pa- recently as an efficient solution to match the larger bandwidth rameters depend on the scintillation index of the received and high data rates requirement of the upcoming irradiance. communication systems. However, the major challenge in The performance of the FSO communication system over establishing a wireless link at optical frequencies is atmo- EW channels has been analysed in [8]-[12]. The approximate spheric turbulence experienced by the laser beam when it expressions for the bit error rate (BER) are derived in propagates through the medium [1]. The atmospheric turbu- [8]-[11]. In [8], [11], the BER is obtained using Gauss- lence causes various perturbations in the propagating beam Laguerre quadrature rule, while Gauss-Hermite quadrature like scintillation which represents the random fluctuations approximation is used in [9]. The average capacity of the in the beam irradiance, beam wander which stands for the optical wireless communication systems over EW distribution random movement of the instantaneous centre of the beam turbulence channels is derived in [12]. However, the analysis at receiving aperture, and beam spreading that indicates the in all these works utilized the probability density function spreading beyond the diffraction limit of the beam radius [2]. (PDF) of the fading channel, and none of them derived and The receivers in FSO systems generally have the point ge- utilized the PDF of the signal-to-noise ratio (SNR). The sta- ometry which further worsens their performance in presence tistical analysis through SNR based approach is less complex of the atmospheric turbulence. To improve the performance of and more general as can be extended to different modulation a communication system in atmospheric turbulence, several techniques directly. So in this paper we, first derive the techniques have been proposed in the literature [3]-[4]. The distribution of the instantaneous SNR, and apply the derived larger dimensions of the apertures in the receiver may help in statistics to analyze the outage and error performance of the collecting the higher irradiance flux, but it will place a hard- FSO communication systems. Specifically, expression for the ware constraint on the receiver structures. Aperture averaging outage probability using the cumulative distribution function is one of the most widely used alternative technique due to its (CDF) of the instantaneous SNR is obtained. Finally, the simplicity and lower cost. In aperture averaging, to mitigate expression for the average bit error rate (BER) is derived, the adverse effect of the atmospheric turbulence induced which is applicable for various binary modulation schemes fading, a collecting aperture is placed at the end of the FSO such as coherent binary frequency shift keying (BFSK), non-

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coherent BFSK, coherent binary phase shift keying (BPSK), particles given by and differential BPSK. 1.6 for V > 50 km The rest of this paper is organized as follows: In section II 1.3 for 6

h = hI hmhℓ. (1) Lemma 1: The PDF of the instantaneous SNR γ can be given as

The atmospheric turbulence induced fading is modelled ∞ 2 ρ − β under EW distribution. The PDF of the coefficient hI (hI > f (γ) = B Ψ(j)γ 2 1Γ τ,B (j)γ 2 , (6) ), is [6] γ 1 2 0 j=0 ∑ [ ] − j 2 β−1 β β α 1 ( 1) Γ(α) αρ αβ hI hI hI where Ψ(j) = − ρ2 , B1 = ρ2 , f h , 1− 2(hℓηA0√γ¯0) hI( I )= exp − 1 − exp − j!Γ(α j)(1+j) β η η η η 2 ( ) [ ( ) ]{ [ ( ) ]} ρ − 1+j (2) τ = 1 − , and B2(j) = β , Γ(·) is the gamma β (hℓηA0√γ¯0) where β > 0 and α> 0 are the shape parameters, and η > 0 function, and Γ(·, ·) is the upper incomplete gamma function is the scale parameter. The parameter α,β depends on the [17]. scintillation index of the irradiance [13, Eq. 20,21], and the Proof: In (1), the terms hI and hm are random variables (RVs) whose PDFs are given in (2) and (3), respectively, and parameter η depends on the mean value of the irradiance. The the term hℓ is deterministic. In order to obtain the PDF of shape parameter α not characterizes the fading conditions of h, first we derive the PDF of RV X = hI hm. The PDF of the link but also it along with parameter β captures the effect X can be written as, of aperture averaging. [13]. ∞ fX (x) = fhm (x|hI )fhI (hI )dhI . (7) Assuming a Gaussian spatial intensity profile for the beam x ∫ A0 waist, the misalignment fading hm is statistically charac- terized in [14]. For independent but identical Gaussian Using (2) and (3), (7) can be written as, ∞ β−1 β distributed horizontal sway and elevation, the radial displace- 2 2 2 ρ ρ −1 −ρ αβ hI hI fX (x)= 2 x hI exp − ment at the receiver follows a Rayleigh distribution. Now ρ A x η η η 0 ∫ A0 ( ) [ ( ) ] using these facts the PDF of hm is given, as in [12], as − β α 1 hI 2 × 1 − exp − dh . (8) ρ 2 I ρ 1 { [ η ]} fhm (hm)= 2 hm − , 0 ≤ hm ≤ A0 (3) ( ) A0 A1 where A0 is the fraction of the collected optical power To solve the integration| in (8), we{z expand the }A1 using ω the Newton’s generalized binomial theorem i.e. (1 + y)t = and ρ = 2σ , ω is the equivalent beam width at the s Γ(t+1)yj 2 ∞ receiver, and σs is the variance of pointing error displacement j=0 Γ(t j+1)j! . After some mathematical manipulations characterized by the horizontal sway and elevation. (8)∑ can be− rewritten as, ∞ For the path loss factor we consider the Kim’s model 2 2 j αβρ ρ −1 (−1) Γ(α) proposed in [15]. The deterministic path loss factor for a fX (x) = 2 x β ρ j!Γ(α − j) η A0 j=0 channel of length L is given by hℓ = exp(−AL), here A is ∞ ∑ 2 the attenuation constant defined as β−1−ρ −(j + 1) β × hI exp hI dhI . (9) x ηβ ψ V ∫ A0 ( ) 13 λ × 109 ( ) A = , (4) The integral term in (9) can be solved using [17, Eq. 381.3], V [ 550 ] as ∞ 2 2 2 β where V is the visibility, λ is operating wavelength, and αρ ρ −1 ρ (1 + j)x fX (x) = 2 x Ψ(j)Γ 1 − , .(10) ρ β is the coefficient related to the size of atmospheric (ηA ) β (ηA0) ψ(V ) 0 j=0 ∑ [ ]

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The PDF of the channel coefficient h = Xhℓ can be obtained 1 as 0.08 100 summation terms ∞ 2 2 10 summation terms αρ 2− ρ 1+ j ρ 1 β 0.07 α =2.1, β = 3.8, η = 0.65 fh(h)= 2 Ψ(j)h Γ 1− , h ,h ≥ 0. ρ β β α =2.1, β = 4.6,η = 0.89 (hℓηA0) j=0 [ (hℓηA0) ] ∑ 0.06 α = 2.65, β = 3.8, η = 0.89 (11) 0.05 )

Now the PDF of instantaneous SNR γ defined can easily be γ ( γ {α, β, η, ρ} = {2.1,3.8,0.89,1.45} derived using [16, Eq. 5-8]. 0.04 Observation 1: The expression of the PDF in (6) consists PDF, f 0.03 of infinite summation which results from the use of Newton’s generalized binomial expansion, however the for the given 0.02 values of the parameters α, β, and η, the infinite summation 0.01 is convergent and ten to fifteen terms are sufficient for this 0 series to converge [10] and can be verified analytically 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 using MATLAB or MATHEMATICA. Further this can γ also be verified from Fig. 1, where we plotted the PDF Fig. 1. The PDF of the instatntaneous SNR γ for aperture-averaged receivers under the influence of atmospheric turbulence induced fading, misalignment for j = 0 to 10, and j = 0 to 100 terms, and it can be fading and path loss. observed that the plots of the PDF in both the cases are same.

0.016 γ¯0 = 0 dB Observation 2: The effect of channel parameters α, β, γ¯0 = 2 dB ¯ = 4 dB 0.014 γ0 and η on the SNR can also be observed from Fig. 1. The γ¯0 = 6 dB increase in the parameters α and β spreads the distribution 0.012 as they are shape parameters and they control the steepness of the tail of the PDF. On the other hand, the height of 0.01 ) γ ( the distribution reduces with an increase in the value of γ 0.008 the parameter η because it is the scale parameter and PDF, f it determines the scale or the height of the distribution. 0.006 Thus any change in the channel parameters affects the tail probabilities which are of maximum importance in 0.004 the analysis of any communication system as the tail of 0.002 distribution defines the error performance significantly. 0 0 1 2 3 4 5 6 γ Observation 3: To analyse the effect of γ¯0, we plotted the PDF for different values of average SNR γ¯0 in Fig. 2. The effect of average SNR γ¯0 on the PDF of the instantaneous SNR γ. Fig. 2. The plots suggest that the higher the γ¯0, wider will be the distribution and accordingly its height is scaled down. This integral can be solved using [18, Eq. Lemma 2: The CDF of the instantaneous SNR γ can be (07.34.21.0084.01)]. given as 2 IV. PERFORMANCE ANALYSIS ∞ 2 ρ ρ β 1 − , 1 2B1 2 2,1 2 β Fγ (z) = Ψ(j)z G2,3 B2(j)z 2 , (12) In this section we apply the derived statistics of the β  ,τ, ρ  j=0 0 − β ∑ SNR to analyse the performance of an FSO communication m,n   where G · · is the Meijer’s G function [18, Eq. system model. We consider a source and an aperture-averaged pq (07.34.02.0001.01)].( · ) destination. The source has one transmit antenna aperture Proof: The CDF of the SNR is defined as, to send its message, and the destination has a finite sized z receive antenna aperture for receiving from the source. The F z f γ dγ, γ ( ) = γ ( ) channel between the source and the destination is represented ∫0 ∞ z 2 ρ − β by the coefficient h and is defined in (1). In the following 2 1 2 = B1 Ψ(j) γ Γ τ,B2(j)γ . (13) subsections we will derive the expressions of the outage j=0 0 ∑ ∫ [ ] probability and the average BER. On replacing the upper incomplete gamma function in (13) with its Meijers-G equivalent [18, Eq. (06.06.26.0005.01)] A. Outage Probability β 2 and substituting γ = t we get, The outage probability Pout is defined as the probability β ∞ 2 2 of instantaneous SNR γ over a channel h falling below a z ρ 2B1 −1 2,0 1 F (z)= Ψ(j) t β G B (j)t dt.(14) threshold value. We assume γth to be the threshold SNR, so γ β 1,2 2 0,τ j=0 ∫0 ( ) the corresponding outage probability is, ∑ γ 1 th The derived PDF fh(h) in (11) is a valid PDF as it is non-negative (see Table I), and area under this PDF is unity (see appendix for proof). Pout = Pr{γ<γth} = fγ (γ)dγ. (15) ∫0

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0 10 0 10 Simulation Analytical −1 10 L = 500 m −1 10 −2 out 10

out Increasing γ th {6 dB, 8 dB, 10 dB, 12 dB} γ = 2 dB −3 −2 10 10 th

−4 10 Outage Probability, P Outage Probability, P −3 10 {ρ, α, β, η} = {1.25, 2.65, 3.88, 0.35} Decreasing Lengths L −5 {ρ, α, β, η} = {1.25, 1.5, 4.2, 0.76} {50 m , 100 m, 200 m, 10 300 m } {ρ, α, β, η} = {2.1, 2.65, 3.88, 0.35} {ρ, α, β, η} = {2.1, 1.5, 4.2, 0.76}

−4 −6 Simulations 10 10 0 10 20 30 40 50 5 10 15 20 25 30 35 40 45 50 Average SNR [dB] Average SNR, [dB]

Fig. 3. Outage Probability vs av. SNR for different values of threshold SNR Fig. 4. Effect of misalignment fading : Outage probability vs av. SNR for γth, and channel length L. different values of parameters {α,β,η}.

The outage probability P in (15) can be obtained using out 0 10 (12) as Pout = Fγ (γth).

−1 B. Average BER 10

The average BER for binary modulation schemes can be ρ = 1.45 −2 given as [19] 10

∞ −3 P (e)= P (e|γ) fγ (γ)dγ, (16) 10 ∫0 −4 Non−coherent BFSK where P (e|γ) is the conditional error probability and for 10

Average BER Differential BPSK the given it can be written as Γ(µ1,µ2γ) , the γ P (e|γ) = 2Γ(µ1) Coherent BPSK −5 parameters µ1,µ2 are modulation dependent constants and 10 Coherent BFSK their values are : µ1 = µ2 = 0.5 for coherent BPSK, µ1 = −6 ρ = 1.95 0.5,µ2 = 1 for coherent BFSK, µ1 = 1,µ2 = 0.5 for non- 10 coherent BFSK, and µ1 = µ2 = 1 for differential BPSK. −7 Now using [18, Eq. (06.06.26.0005.01)], and (6), followed 10 0 5 10 15 20 25 30 35 40 by using [18, Eq. (07.34.21.0013.01)], the average BER in Average SNR, [dB] (16) can be obtained as, ∞ 2 2 Fig. 5. Average BER for the different binary modulation schemes. B kτ−1/2lµ1+(ρ −3)/2µ −ρ /2 P e 1 2 ( )= l+k−2 2Γ(µ1)(2π) j=0 ∑ ∆(k, 1), ∆(l,ε) , ∆(l,ε − µ ) values of threshold SNR plots the link length is considered ×G2k,2l Ω(j) 1 (17) 2l+k,2k+l ∆(k, 0) , ∆(k,τ) , ∆(l,ε) to be L = 500 m, and for various values of link length ( ) plots the threshold SNR is taken as γ = 2 dB. It can l th where l and k are integer constants so that β = k , ∆(m, n)= k l be observed that the outage performance of the system , (B2(j)) l (n/m)((n + 1)/m)....((n + m − 1)/m) Ω(j) = µ2lkk deteriorates as the value of the target or threshold SNR ρ2 increases. Further, as the separation between transmitter and and ε = 1 − 2 . The expressions of the average BER for various modu- receiver is increased, the outage probability increases and lation schemes can be found on substituting the values of hence the system performance worsens, intuitively. parameters µ1 and µ2 accordingly. In Fig. 4, the effect of misalignment errors is observed on the outage performance with A0 = 0.85, γth = 5 dB, V. NUMERICAL RESULTS V = 1.2 km, L = 500 m. In this figure the effects of the In this section, numerical results are presented for the variations in the values of the parameters α, β, and η can be outage performance of the considered system. In Fig. 3, the observed. Further, it can be seen from this figure that as the outage probability is plotted for various values of threshold value of ρ increases the outage probability decreases. This SNR γth, and link length L considering A0 = 0.35, ρ = 1, happens due the parameter ρ is inversely proportional to the V = 1.2 km, α = 2.18, β = 4.0, η = 0.5. For various pointing error jitter at the receiver side [14], thus increased

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value of the parameter ρ signifies the reduced level of the On substituting hβ = r in (20), and using [17, Eq. magnitude of the misalignment error. (6.455.1)], we get ∞ Fig. 5 analytically plots the average BER of the various Ψ(j) ρ2 S = 2 2F1 1, 1; +1;0 , (21) modulation schemes with respect to average SNR for differ- ρ β j=0 (1 + j) β ent values of pointing error parameter ρ. The coherent BFSK, ∑ ( ) non-coherent BFSK, coherent BPSK, and Differential BPSK where 2F1 (·, ·; ·; ·) is Gauss hypergeometric function modulation schemes are considered. The values of other [18, Eq. (07.23.02.0001.01)]. Now using [18, Eq. parameters are assumed as follows: A0 = 0.35, V = 1.2 (07.23.03.0001.01)] and (19) we get S = 1. km, α = 2.18, β = 3.8, η = 0.75. The main observation from this plot is that for the given value of ρ the average REFERENCES BER for coherent BFSK and differential BPSK is almost [1] L.C. Andrews, R.L. Phillips, and C.Y. Hopen, “Laser beam scintillation same. Further, the complexity of the coherent detection can with applications,” Bellingham, Washington: SPIE Press, 20011. [2] L.C. Andrews, and R.L. Phillips, “Laser beam propagation through be justified as the coherent BPSK outperforms the other random media”, 2nd ed. Bellingham, Washington: SPIE Press, 2005. detection techniques and provides the minimum gain of [3] F. Xu, M.A. Khalighi, P. Causse´, and S. Bourennane, “Channel coding approximately 2.5 dB, when compared to differential BPSK, and time-diversity for optical wireless links,” Optics Express, vol. 17, no. 2, pp. 872-887, Jan. 2009. and 4 dB when compared to non-coherent BFSK. [4] M.A. Khalighi, N. Schwartz, N. Aitamer, and S. Bourennane, “Fad- ing reduction by aperture averaging and spatial diversity in optical ONCLUSIONS VI.C wireless systems,” IEEE/OSA Journal of Optical Communications and In this paper, a finite sized receiver aperture based commu- Networking, vol.1, no.6, pp. 580-593, Nov. 2009. [5] B. Epple, “Simplified channel model for simulation of free-space opti- nication system is analyzed over EW distributed FSO chan- cal communications”, IEEE/OSA Journal of Optical Communications nels. Initially, the PDF of the composite channel coefficient and Networking, vol.2, no.5, pp. 293-304, 2010. has been derived with turbulence induced fading, distance [6] R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Optics Express dependent path loss, and misalignment fading. Using the PDF vol. 20, issue 12, pp. 13055-13064, 2012. of the composite channel coefficient the statistics of the SNR [7] R. Barrios and F. Dios, “Reply to comment on The exponentiated over EW distribution were obtained. Finally, new closed form Weibull distribution family under aperture averaging for Gaussian beam waves” Optics Express vol. 20, issue 12, pp. 20684-20687, 2012. expressions for the outage probability and average BER were [8] P. Wang, L. Zhang, L. Guo, F. Huang, T. Shang, R. Wang, and Y. derived. Yang, “Average BER of subcarrier intensity modulated free space APPENDIX optical systems over the exponentiated Weibull fading channels,” Optics Express vol. 22, issue 17, pp. 20828-20841, 2014. [9] M.M. Ahmed, and K.T. Ahmmed, “An expression for average bit error TABLE I rate of free space optical links over Exponentiated Weibull atmospheric h 0 0.1 0.2 0.3 0.4 0.5 0.6 turbulence channel,” in proc. 2014 International Conference on Infor- matics, Electronics & Vision (ICIEV), pp.1-6, 23-24 May 2014. fh(h) 0 .0284 .0609 .0948 .1279 .1547 .1656 h 0.7 0.8 0.9 1.0 1.1 1.2 1.3 [10] R. Barrios and F. Dios, “Probability of fade and BER performance f (h) .1513 .1124 .0645 .0273 .0082 .0017 .002 of FSO links over the exponentiated Weibull fading channel under h aperture averaging,” in Unmanned/Unattended Sensors and Sensor Networks IX, ser. Proc. SPIE, vol. 8540, p. 85400D, 2012. For the expression in (11) to be a valid PDF it must be [11] X. Yi, Z. Liu, and P. Yue, “Average BER of free-space optical sys- non-negative and area under this PDF should be unity. The tems in turbulent atmosphere with exponentiated Weibull distribution,” non-negativity of PDF f (h) in (11) can be observed from Optics Letters, vol. 37, no. 24, pp. 5142-5144, 2012. h [12] M. Cheng, Y. Zhang, J. Gao, F. Wang, and F. Zhao, “Average ca- Table I as the range of fh(h) in the possible domain of h pacity for optical wireless communication systems over exponentiated is positive. Following is the proof of unit area under PDF Weibull distribution non-Kolmogorov turbulent channels,” Applied Optics, vol. 53, issue 18, pp. 4011-4017, 2014. fh(h) : As the PDF given in (2) is a valid PDF [6], area under this [13] R. Barrios, F. Dios, “Exponentiated Weibull model for the irradiance probability density function of a laser beam propagating through ∞ PDF should be one i.e. 0 fhI (hI )dhI = 1. Rewriting the atmospheric turbulence”,Optics & Laser Technology, Vol. 45, pp. 13- PDF fhI (hI ) using Newton’s∫ generalized binomial theorem, 20, February 2013. we get [14] A. A. Farid, and S. Hranilovic, “Outage capacity optimization for free- space optical links with pointing errors,” J. of Light. Tech., vol. 25, no. ∞ ∞ β−1 β αβ (−1)j Γ(α) h h 7, pp. 1702–1710, July 2007. I exp −(1 + j) I dh =1. η j!Γ(α − j) η η I [15] I. Kim, B. McArthur, and E. Korevaar, “Comparison of laser beam j=0 0 [ ] ∑ ∫ ( ) ( ) propagation at 785 nm and 1550 nm in fog and haze for optical wireless (18) communications”, Optical Wireless Communications III - Proceedings After solving this integral, followed by some mathematical of SPIE vol. 4214, p. 26-37, Feb. 2001. rearrangements (18) reduces to [16] A. Papoulis and S. U. Pillai, ”Probability, random variables and ∞ stochastic processes”,4th edition , Tata McGraw Hill, New Delhi, 2002. j (−1) Γ(α) 1 Table of integrals, series and = . (19) [17] I. S. Gradshetyn and I. M. Ryzhic, j!Γ(α − j)(1 + j) α products, (U.S.A.: Academic Press, 6th ed.), 2000. j=0 ∑ [18] I. Wolfram Research, Mathematica Edition: Version 8.0, Champaign, Illinois:Wolfram Research Inc., 2010. The area under the PDF is ∞ . Using fh(h) S = 0 fh(h)dh [19] A. H. Wojnar, “Unknown bounds on performance in Nakagami chan- (11) we get, ∫ nels,” IEEE Transactions on Communications, vol. 34, no. 1, pp. 22-24, ∞ 2 ∞ 2 Jan. 1986. αρ 2− ρ 1+ j S = Ψ(j) hρ 1Γ 1− , hβ dh. ρ2 β β (hℓηA0) j=0 0 [ (hℓηA0) ] ∑ ∫ (20)

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