Free-Space Optical channel turbulence analysis based on lognormal distribution and stochastic differential equation

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1 ISSN (Online) : ISSN (Print) J.Mech.Cont.& Math. Sci., Vol.-13, No.-5, November-December (018) Pages 16-4 Free-Space Optical channel turbulence analysis based on lognormal distribution and stochastic differential equation 1 TayyabaGul Tareen, Shahryar Shafique, 3 Mehr-e-Munir 1., 3 Department of Electrical Engineering, Iqra National University, Pakistan 1 tayyabatareen@gmail.com, its_shahryar@hotmail.com, 3 mehre.munir@inu.edu.pk *Corresponding author: Mehr-e-Munir, mehre.munir@inu.edu.pk Abstract An Optical wave propagating through a free-space optical channel may severely experience the intensity fluctuations that can result in channel gain fluctuations and fading. This paper provide a model that can analyze the influence of inevitable turbulence effect on a free-space channel which is based on the stochastic differential equation to synthesis lognormal distributed samples with a corresponding correlation time. The numerical analysis of theoretical model is presented and compared for performance evaluation. To examine the resemblance between numerical and theoretical analysis, two properties of free-space optical channel is considered including the probability density function and auto-covariance property. The model showed distinctive performance results when modelling typical channel situations. Keywords : Auto-covariance, Free-space optical, lognormal distribution, stochastic differential equation (SDE), Turbulence effects I. Introduction Free space optical (FSO) communication is one of the most advanced communication field. It is also known as optical wireless communication system or laser communication. It has become the most upcoming technology due to its many advantages such as low bit error rate (BE<10^-9) and high reliability. Free Space Optical Communication can be installed faster with low cost, due to these benefits, these systems are widely deployed in various events like in different disaster recovery and for military purposes. FSO channels are highly susceptible to severe weather conditions like fog channel turbulence etc. where laser beam are attenuated resulting in link loss. Even 16

2 temperatefog can create attenuation of 130 db/km, whereas dense oceanic fog can result in attenuations up to 480 db/km [I-II]. Similarly rain and snow fall attenuation also affects the availability of FSO system. It is also notice that on combination of different weather conditions like snowfall with fog or fog with rain, the performance of FSO system keeps varying [III-IV]. The received optical signal also practice fading due to the randomly changing temperature, pressure and turbulence along the propagating path is introduced. These changes also effects the Bit Error ate (BE). For treating different variations as noise in FSO, low pass filters are used. Stochastic Differential Equation (SDE) is powerful process used to model unstable exchange or systems subject to thermal fluctuations. This paper provides a model that can analyze the influence of inevitable turbulence effect on a free-space channel which is based on the stochastic differential equation Figure 1: SDE Generated Signal tosynthesis lognormal distributed samples with a corresponding correlation time. The technique was first presented in [XIII] with theoretical results only. In this paper the numerical analysis of theoretical model [XIII] is presented and compared for Performance evaluation. To examine the resemblance between numerical and theatrical analysis respectively. The paper is further explained in two important properties of freespace optical channelincluding the probability density function and auto-covariance. The paper well explained the typical performance results obtained from modeling a typical channel conditions. II. Background of Turbulence Theory Free-Space Optical Communication (FSO) is sternly influenced by the certain turbulence effect that can result in channel gain fluctuations and fading. The scintillation statistics describes such fluctuations. The scintillation index σ ξ represents fluctuations in received optical power measured by a point receiver and is defined as: [1]. 17

3 = (1) Where, bracket shows long-time average. A number of scintillation statistics models can be found in literature for various conditions [V-XIII]. For weak-turbulence circumstances, the lognormal distribution is usually engaged. The Probability Density Function (PDF) is given by equation: [XI] Figure The Probability density function of SDE-generated signal for various scintillation index σξ. (A) σξ=0.1, (B) σξ=0., (C) σξ=0.3 and (D) σξ =0.4. The ed line shows theoretical and black line shows simulated PDF 1 [ln( x) ln( )] () ( ;, ) exp{ } x Fx 18

4 Where μ shows the average received power. The normalized temporal covariance function is given by equation: [1] () () (3) ( 0) Where Ψ ξ (τ) shows temporal covariance function. The correlation time τ c is used for characterization of covariance function. The value of τ c is given by solving the equation (3) as: ( c) exp( 1) (4) The stochastic differential equation technique is used to synthesis lognormal distribution, ( ), with predefined correlation time τ c by a first-order stochastic differential equation: [7] dx f ( x) g( x) ( t), (5) dt The drift f( ) and diffusion g( ) functions are given as: [7] M d f ( x) log qx( x) (6) dx g( x) M Drift and diffusion functions are independent of time and the resulting solution will be distribution ( ), which can be seen from Fokker-Plank equation given as: [7] (7) qx x [ f ( x) qx x] [ g( x) q x x] t t x (8) The above equation provides a link between SDE and Fokker-Plank equation. By applying the above Technique for the lognormal distribution given in equation (), resulting SDE is given by equation: M f ( x) [ln( x / )] (9) 0 x g( x) M (10) 19

5 Figure 1:The Probability density function of the SDE-generated signal for various values of average power μ. (A) μ =0., (B) μ =0.4, (C) μ =0.6 and (D) μ =0.8. The ed line shows theoretical and black line shows simulated PDF Where M is given by equation: III. exp( )[exp( ) 1] 0 M (11) c esults and Discussions The SDE solution was computed by Matlab 013b for numerical solutions of SDE [I] using Euler method [XIII].The results has been successfully taken by the comparison between the analytical and the experimental results during a communication analysis based on the theory explained in former section An example of SDE generated signal, shown in the Fig., with ξ0=0.85, σξ=0.1, τc=17 ms and sampling frequency fs = 0 khz respectively with varying length of signals used from 0-5 sec range to understand the turbulent channels property. This phenomena has been explained in the below section. 0

6 III.a Probability Density Function Statistics The probability density function of gain samples is the most important property of free-space optical channel. In order to verify this particular property, we have generated eight signals with two different tuning parameters to show effects. In first four signals (Fig.) the tuning parameter, scintillation index, σξ, is varied from 0.1 to 0.4 (ξ0=0.85 and τc=17 ms) and the other parameter, μ, average received power remains unchanged. The results in Fig. show strong resemblance between theoretical and numerical (simulated) results except for higher values of x at the right tail of PDF. In Fig.3, the scintillation index, σξ,remains unchanged and by varying the average received power μ, again a close resemblance between theoretical and numerical (simulated) results except for higher values of x at the right tail of PDF is noticed III.b. Auto-covariance Statistics Auto-covariance is another very important property of free-space optical channel and we have utilized the correlation time metric. The normalized auto-covariance with five stochastically generated signals with their mean is presented in Fig Among two parameters, one is varied and other remains unchanged. In Fig.4, the scintillation index, σξ, is varied from 0.1 to 0.4 (ξ0=0.85 and τc=17 ms) and average received power μ remains unchanged while in Fig.5 the scintillation index, σξ, remains unchanged and μ is varied from 0. to 0.8. The results in both figures (Fig.4-5) illustrate considerable match between the covariance function and approximated correlation time. The auto-covariance of stochastically created signals is very comparable to that of experimental measurements. IV. Conclusion The stochastic differential equation provides mathematically elegant and easy ways to implement solution for turbulence channel model. The SDE simulation results show a significant similarity between the generated states of channel and the theoretical predictions. The presented model was based on the exponential covariance function In which Two properties of channels are evaluated for resemblance and model effectively showed the significant resemblance between numerical and simulated results. The model is tested for two channel properties but in future themodel can be tested for other channel properties. 1

7 Figure : Auto-covariance of the stochastically generated and experimental measured signals for various scintillation index σξ. (A) σξ=0.1, (B) σξ=0., (C) σξ=0.3 and (D) σξ =0.4. The ed line shows mean, black line shows original signal while gray line.

8 Figure 3: Auto-covariance of the stochastically generated and experimental measured signals for various average power μ. (A) μ =0., (B) μ =0.4, (C) μ =0.6 and (D) μ =0.8. The ed line shows mean, black line shows original signal. V. Acknowledgement The authors would acknowledge to Iqra National University Electronics Lab for providing sufficient environment and guidance. 3

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