FSO links with diversity pointing errors and temporal broadening of the pulses over weak to strong atmospheric turbulence channels

Accepted Manuscript Title: FSO Links with Diversity Pointing Errors and Temporal Broadening of the Pulses Over Weak to Strong Atmospheric Turbulence Channels Author: G.K. Varotsos H.E. Nistazakis Ch.K. Volos G.S. Tombras PII: DOI: Reference:

S0030-4026(15)01969-5 http://dx.doi.org/doi:10.1016/j.ijleo.2015.12.060 IJLEO 56998

To appear in: Received date: Accepted date:

16-8-2015 10-12-2015

Please cite this article as: G.K. Varotsos, H.E. Nistazakis, Ch.K. Volos, G.S. Tombras, FSO Links with Diversity Pointing Errors and Temporal Broadening of the Pulses Over Weak to Strong Atmospheric Turbulence Channels, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.12.060 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

FSO Links with Diversity Pointing Errors and Temporal Broadening of the Pulses Over Weak to

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Strong Atmospheric Turbulence Channels

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G.K. Varotsos1, H.E. Nistazakis1,*, Ch.K. Volos2 and G.S. Tombras1 : Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, 15784, Greece

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e-mails: {georgevar; enistaz; gtombras}@phys.uoa.gr : Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

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e-mail: [email protected]

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*: Corresponding author, H.E. Nistazakis, [email protected]

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Abstract Free space optical (FSO) communication systems show a growing evolution in the commercial and the research field, due to the huge information capacity they provide

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along with the unlicensed and secure data transmission they can support. However, several destructive effects, such as the atmospheric turbulent and attenuation, the

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longitudinal pulse broadening due to the time dispersion, the spatial jitter which caused by the pointing errors between the trans-receivers, etc, mitigate their

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performance characteristics. Thus, for the improvement of the FSO communication

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systems’ performance, many techniques have been proposed, studied and used. In this work, we present a performance study and we derive closed form mathematical

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expressions for the estimation of the probability of fade for On-Off keying FSO communication systems which use Gaussian longitudinal pulses as information bit

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carriers, with reception diversity, over gamma gamma or gamma modeled,

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atmospheric turbulence channels taking into account the pointing errors and the group velocity dispersion (GVD) effect which, affects significantly the systems performance

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characteristics, especially for long link lengths and high data rate transmission. Finally, the derived mathematical expressions are used in order to present performance results, using common parameter values for channel’s and systems’ characteristics.

Key words: Free Space Optical Communications, Time Dispersion, Atmospheric Turbulence, Pointing Errors, Diversity Techniques, Probability of Fade.

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1. Introduction The terrestrial free space optical (FSO) communication systems have attracted significant research and commercial interest during the last years because of their low installation and operational cost, their very high data rate that they can transmit with

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high security level and the licence-free bandwidth use, [1]-[10]. However, due to their

nature, their performance depends strongly on the atmospheric conditions prevailing

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in the area were these systems are installed, the non-perfect alignment between the

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receiver and the transmitter of the optical link and the time dispersion effect which affects the form of the longitudinal pulses which transport the information signal, [9]-

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[16].

More specifically, a very significant phenomenon which decreases the performance

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of the FSO links is the atmospheric turbulence which attenuates the propagating signal and also causes the so-called scintillation effect which results in random

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fluctuations of the irradiance which arrives at the receiver’s side, analogous to fading

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in RF systems. Due to this effect, the systems’ performance could be significantly

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degraded, especially for the case of deep signal fades, [2], [3], [13], [17]. Many statistical models have been proposed and studied, in order to model these irradiance fluctuations, [6], [10], [12]-[30]. Here, we use the gamma gamma or the gamma distribution. The former, has been proven that models accurately, weak to strong turbulence conditions [18], [20], [28], [29], while the latter provides a less complicated model and additionally, it has been demonstrated that is accurate enough, especially for weak turbulence conditions cases, [24], [25], [30]. The pointing errors effect, which describes the non-perfect alignment between the transmitter and the receiver of the FSO link, represents another mitigation factor for the performance of the FSO communication systems. In particular, the transmitter and 3

Page 3 of 34

the receiver of the communication system rely on structures which are not remain motionless, because of the building sway, strong wind, earthquakes, etc. Hence, due to the fact that the transverse radius of the optical beam at the receiver is relatively

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small, these small movements of the trans-receiver systems can cause significant pointing errors [4], [9], [14], [18], [31], [32].

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Another effect which affects the performance of the FSO links, especially for the

very high data rate, long link length, FSO communications systems which are using

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very narrow longitudinal pulses as information carriers, is the group velocity

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dispersion (GVD) due to the atmospheric propagation of the laser beam. In fact, the widening or narrowing of the shape of the propagating longitudinal pulse-envelope

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which entails the bit carrier, is caused by the GVD, and changes its characteristics, along the propagation path and thus affects the signal’s detection capability at the

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receiver. Consequently, its influence on the performance of the whole FSO system

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can be either detrimental or beneficial, under specific circumstances, [11], [12], [15]. As mentioned above, due to the propagation of the optical signal through the

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atmosphere, there are many effects which degrade significantly the performance of the FSO links. In order to improve it, a technique which is very effective, concerns the reception diversity schemes which are very popular in wireless radio, as well, [33][42]. In principle, the use of diversity refers to the consideration of multiple copies of the propagated signals in an attempt to overcome a poor transmission media state and enhance the communications systems’ reliability and performance. The reception diversity in FSO links can, mainly, be realized in space, in time or in wavelength, [33]-[41]. For the spatial reception diversity scheme, the FSO system includes one transmitter and multiple receivers at different places. The transmitter is sending to each receiver, a copy of the same information signal, resulting to a decreased 4

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probability of error, [33], [34], [41]. In time diversity schemes, the FSO link uses a single transmitter – receiver pair, but the copies of the information signal is retransmitting at different moments and this, results in decreasing the total effective

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bit rate of the link, [38], [39]. Finally, the wavelength diversity scheme uses a composite transmitter and the signal is transmitted at the same time at different

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wavelengths towards a number of receivers, each one of them detects the only the

signal which has been transmitted at the suitable (for each receiver) wavelength, [35]-

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[37], [40].

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In this work, we investigate the influence of the atmospheric turbulence, modelled with the gamma or the gamma gamma distribution, along with the GVD and the

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pointing errors, or spatial jitter, effects, at the availability of the FSO communication systems with reception diversity. More specifically, by assuming that the longitudinal

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pulses, which are the information bits carriers, are chirped Gaussian, we investigate

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the influence of each one of the above mentioned effects, at the probability of fade of the whole FSO link. Thus, we derive closed form mathematical expressions, for the

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estimation of this availability metric, for FSO links either with or without the reception diversity techniques, under the action of atmospheric turbulence, GVD and spatial jitter effects. Our mathematical expressions and the associated numerical results provide an efficient way for the estimation of the reliability of FSO links and therefore, the proposed analysis is useful to the system design engineer for performance evaluation purposes. The remainder of this work is organized as follows. In section 2, the channel model is presented while, in section 3, the extraction of the performance metrics for the single point to point FSO pink is described. Next, in section 4, the reception diversity scheme is studied and in section 5 the numerical results are presented for common 5

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parameter values which are using in practical FSO links. Finally, section 6 concludes

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2. The Channel Model

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with a summary of the main results.

For the optical wireless link with diversity, we assume that the transmitter emits M

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copies of the same part of the information signal towards M receivers, for the cases of wavelength or spatial diversity schemes, or in M different time moments at the same

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receiver, for the time diversity configuration. Each copy arrives at the receiver(s) and remains at the buffer of the system until all the M copies been received. Thus, each

M

copy of the information signal emitted from the transmitter, propagates through different or the same path(s) but with different channel’s characteristics and the total

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information signal at the receiver’s buffer is composed by M different versions of the

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same, initially, signal, [35]. Consequently, the wireless optical communication

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system, under consideration, can be emulated as a single input multiple output (SIMO) scheme, with one transmitter and multiple receivers, for all the above mentioned diversity schemes, [33]-[35]. Each copy of the optical information signal propagates through a turbulent channel

with additive white Gaussian noise (AWGN) and pointing errors effect [14], [31], [32], [39]. The channel, between the transmitter and each the receiver is assumed to be memoryless, stationary and ergodic with independent and identically distributed (i.i.d.) intensity fast fading statistics. Additionally, we assume that the transmitter of the intensity modulation direct detection (IM/DD) and On-Off Keying (OOK) modulation FSO system, emits longitudinal Gaussian optical pulses as information bit 6

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carriers, [12], [15], [43], [44]. Thus, the statistical channel model is given as, [33]-

m  1,..., M

(1)

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y m   m xI r ,m  n,

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[35]:

where ym represents each of the M signals’ copy at the receiver, ηm is the effective

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photo-current conversion ratio of each receiver, I r ,m stands for the normalized

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received irradiance of m-th copy of the optical signal, x is the modulated signal which takes the binary values “1” or “0”, while n represents the additive white

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Gaussian noise (AGWN) with zero mean and variance equal to N 0 / 2 , [29]. Assuming that the normalized irradiance, Ir,m, which arrives at the receiver’s side,

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fluctuates due to the atmospheric turbulence and the pointing errors effects, we

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conclude that it can be expressed as, [45]:

I r ,m  I a , m I p ,m

(2)

where Ia,m and Ip,m represent the dependence of the normalized irradiance value due to the atmospheric turbulence and the pointing errors effects, respectively, for each one of the M signals’ copies. Many statistical models have been proposed and accurately represent the irradiance fluctuations due to the atmospheric turbulence effect. Here, we present results using the gamma gamma or the gamma distribution. The former, has been proven that 7

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models accurately, the irradiance fluctuations for the cases of weak to strong turbulence conditions [18], [20], [22], [28], [35], while the latter provides a much less complicated model, compared to gamma gamma and additionally, it has been

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demonstrated that is very accurate, especially for weak turbulence conditions cases, [24], [25], [30].

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The probability density function (PDF), f GG , I a ,m I a ,m  , of the gamma gamma

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 m m

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distribution as a function of Ia,m, is given as [28]:

m 1 2 m  m  2 f GG , I a ,m ( I a ,m )  I a ,m2 K m   m 2  m  m I a ,m  m  m 





(3)

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 

d

where Kν(.) stands for the modified Bessel function of the second kind of order ν, Γ(.)

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is the gamma function while αm and βm can be directly related to link’s parameters as,

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[35]:

  0.49 m2  m  exp 2 12 / 5   1  0.18d m  0.56 m







    1 7/ 6   



1

  0.51 2 1  0.69 12 / 5 5 / 6   m m   1  m  exp 56   1  0.9d m2  0.62 d m2  m12 / 5  





1

(4)

where d m  10 Dm 5m1 z m1 , Dm stands for the receiver’s aperture diameter (in meters) while the parameters of wavelength, λm, and link length, zm, are expressing in 8

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μm and km respectively [12], [15]. Additionally, the parameter δm2 stands for the Rytov variance which is given as [12], [15]:

 m2  0.5Cn2 m7 / 6 zm 103 

11/ 6

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(5)

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with κm=2×106π/λm being the optical wave number and Cn2 is a parameter which is proportional to the atmospheric turbulence strength and varies between 10-17 m-2/3 and

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10-13 m-2/3 for weak to strong atmospheric turbulence conditions, respectively, [13]. Next, the PDF, f G , I a ,m I a , m  , of the much less complex, gamma distribution as a

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function of Ia, is given as [24], [25], [30]:

d

I a,mm1 m m e  m I a

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f G , I a , m ( I a, m ) 

Γ ( m )

(6)

where ζm stands for the parameter of the gamma distribution and is connected with the parameters of the FSO link and the Cn2, as, [24], [30]:

 1 1 1  m       m  m  m  m 

1

(7)

In order to investigate the behavior of all the parameters of Eq. (2), the irradiance variations due to the pointing errors should be modeled. Thus the PDF for the

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normalized irradiance fluctuations due to the spatial jitter effect, is given as, [31], [32],

 m2  1 I p ,m A0,m 2 m

f I p , m ( I p ,m ) 

with

0  I p ,m  A0,m

(8)

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2 m

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[45], [46]:

where ξm stands for the ratio between the equivalent beam radius at the receiver and

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the pointing error displacement standard deviation at the receiver and is given as

 m  Wz , eq, m 2 s , m , σs,m is the pointing error displacement at the receiver, the Wr,eq,m is





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the equivalent beam width and is given as Wz2, eq, m   erf vm Wz2,m 2vm exp  vm2 ,

A0, m  erf vm  , vm   rm 2

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2Wz , m , and erf(.) is the error function while rm =Dm/2

[45], [46].

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and Wz,m represents the waist of the beam’s Gaussian spatial intensity profile, [32],

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As mentioned above, the atmospheric turbulence and the spatial jitter effects affect

simultaneously the irradiance which arrives at the receiver of the FSO link. Thus, the combined PDF, f comb , I r ,m ( I r ,m ) , as a function of the Ir,m, which includes both effects is

given through the following integral, [32], [45]:

f comb , I r ,m ( I r ,m )   f I r ,m |I a ,m I r ,m | I a , m  f I a ,m I a ,m dI a ,m

(9)

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where f I r ,m |I t I r , m | I a ,m  stands for the conditional probability given Ia,m and is obtained from Eq. (8), [32], [45], [46]. Thus, by substituting (3) and (8) into (9), we conclude to the following integral for the estimation of the combined PDF for gamma gamma

2 m

 m2 I r,m1 2

( m )(  m ) A0,mm

  m  m 2  m 1 2 a ,m I r ,m A0 ,m

I





K am   m 2  m  m I a ,m dI a ,m (10)

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am   m 2

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f comb,GG , Ir ,m ( I r , m ) 

2 m  m 

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modeled turbulence along with pointing errors effect, [32]:

while, the corresponding integral for PDF estimation of the gamma modeled

 m2 m I r,m1  ( m ) A0,m I m

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f comb,G , Ir ,m ( I r ,m ) 

d

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turbulence with pointing errors, is given as:

2 m



I

2 m

e

 m I a ,m

dI a ,m

(11)

A0 ,m

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r ,m

 m  m2 1 a ,m

By substituting the functions exp(.) and Kv(.), into the integral (10), with the

appropriate Meijer functions, [47], and using the expressions presented in Ref. [47], we result in the following closed form mathematical expression for the combined PDF for the case of the gamma gamma modeled turbulence, [32]:

f comb, GG, I r , m ( I r , m ) 

 m m I r , m   m  m m2  m2  (12) G13,,30  2        1 ,  1 ,  1 0, m( m )(  m ) m m m 0, m   11

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  m Ir,m   m2 m  m2  G12, 2,0  2       1 ,  1 0, m( m ) m m  0, m 

(13)

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f comb, G , I r , m ( I r , m ) 

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while the corresponding combined PDF for the gamma modeled turbulence is given as:

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where G m,n  stands for the Meijer function that is a standard built in function which p,q 

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can be evaluated with most of the well known mathematical software packages and additionally, can be transformed to the more familiar hypergeometric functions, [47].

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As mentioned above, due to the fact that the optical pulses, which are carrying the information signal, propagate through the atmosphere, their longitudinal form can be

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affected by the group velocity dispersion effect (GVD), especially for the cases where

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very narrow pulses are used, i.e high data rate FSO links, and the propagation distance is relatively, long [11], [12], [15]. Therefore, by taking into account that the

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constitution of the atmosphere does not remain invariable, for low altitudes, the atmospheric temperature and pressure can be accurately expressed as [11], [15], [48]:

288.19  6.49h  10 3  ( h )         P(h)   2.23  10 6 44.41  h  10 3



   5.256 



(14)

where h, Θ(h) and P(h) stand for the altitude (in meters), the atmospheric temperature (in Kelvin degrees) and the atmospheric pressure (in millibars), respectively. These

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variations, clearly affect the atmosphere’s refractive index, n(λ), and the GVD parameter, β2, [11], [15], [49], which is given as, [15]:

135.8  Ph   9 255.3  Ph   6 3.5Ph   10    10 2  1015  2   ch    ch  2c h  2

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2

cr

(15)

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where the β2 is measured in ps2/km, while υ stands for the propagation speed (in m/sec) of each spectral component of the signal, c represents the speed of light in

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vacuum and ω is the angular frequency which is connected with the operational wavelength through the expression, ω=2πυ/λ, [11], [12].

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Using the expressions (14) and (15), the influence of the GVD effect at the longitudinal shape of the optical pulse, can be estimated according the specific

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characteristics of each optical link installation. Here, we investigate FSO links which

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are using chirped Gaussian longitudinal pulses as information carriers [11], [12], [43],

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[44]. In this case, the normalized irradiance Ir,m, for each one of the M transmissions of the diversity scheme, at the receiver’s side is given as [12], [15]:

Ir,m 

2

u ( z, T ) m 2

U ( z, T ) m

(16)

with T being the retarded time (measured in ps) and is defined through the usual way as in many areas of electromagnetic propagation and physics, as well, [49]-[52]. Additionally, the expected and the instantaneous normalized and dimensionless 13

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intensities of the longitudinal pulses at the receiver are obtained through the norms |U(z,T)|m2 and |u(z,T)|m2, and the value of the former, which depends on the GVD





(17)

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



  

cr

 T2 exp  2 2 2 2 2 2  T0  2  2 z mC  T0  2 z m C  1 U ( z, T ) m  1  2T0 2  2 z mC  T0 4  22 zm2 C 2  1

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effect, for a longitudinal chirped Gaussian pulse envelope, is given as [12]:

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with T0 and C being the half width at e-1 of maximum intensity and the chirp parameter, respectively, [12], [49]. From Eq. (17), it can be seen that the quantity

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which is inside the square root is almost equal to one for small values of the GVD parameter, β2, and link length, zm, and large values of pulsewidth, T0. Moreover, from

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(17), concludes that for chirped pulses with C>0, i.e. upshifted chirped pulses, the

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value of |U(z,T)|m2, for specific value of T, decreases with the propagation distance, while, for C<0, i.e. downshifted chirped pulses, increases with the propagation

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distance, up to a specific distance zmax, and its maximum value is achieved at zm=zth.

The values of these critical distances are depending on the atmospheric and pulses characteristics, [12].

The value of |U(z,T)|m2, in (17), depends on the time parameter, T, which

determines its amplitude for different time moments. Thus, in order to obtain better reliability characteristics, the pulse’s detection should be done at the centre of the chirped Gaussian longitudinal pulse, i.e. for T=0ps, where its amplitude is larger, [12]. Hence, by substituting (17) into (16), the normalized irradiance detecting at T=0ps at

the receiver of the FSO link is given as, [15]: 14

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



I r , m  u  z,0 m 1  2T0 2  2 z mC  T04  22 z m2 C 2  1 (18)

The Performance Metrics for the Single Point to Point FSO Link

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3.

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2

A significant performance metric concerning the availability of the FSO links is the

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probability of fade, PF,m, as a function of the normalized irradiance, Ir,m, at the

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receiver’s input. More specifically, it shows the probability that the Ir,m value falls below each receiver’s threshold, Ir,m,th, and is given through the following

d

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mathematical expression [12], [53]:

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PF , m I r , m,th   Pr I r , m  I r , m, th   FI r , m I r , m,th  

I r , m , th

 f I Ir, m

r , m ,th

dI

r,m

(19)

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0

where FI r , m (I r , m,th ) stands for the cumulative distribution function (CDF) which can be estimated by integrating the corresponding PDF. Thus, in order to estimate the probability of fade for the gamma gamma modelled turbulence with pointing errors we should estimate the combined CDF of the PDF of (12), through the following integral:

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Page 15 of 34

PF , m, comb,GG I r , m,th   Fcomb,GG, Ir , m I r , m, th  

I r , m , th

f

comb, GG, I r , m

( I r , m )dI r , m 

0

I r , m , th

 0

  m m I r , m  m2 dI G13,,30  2  m  1,  m  1,  m  1 r , m  0

(20)

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 m  m m2  0,m( m )( m )

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Using the mathematical expressions of Ref. [47], we conclude to the following expression for the probability of fade for the gamma gamma modeled turbulence with

  m  m I r , m,th  m2 1,  m2  1  G23,,14   m2 ,  m ,  m , 0  ( m )(  m )  0 , m

(21)

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PF , m, comb, GG I r , m, th  

an

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pointing errors:

d

Similarly, for the estimation of the probability of fade for the gamma modelled

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turbulence with pointing errors, we substitute the combined PDF of (13) into (19) and

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using the using the mathematical forms of Ref. [47], we obtain the following expression:

PF , m, comb,G I r , m, th  

  m I r , m, th 1,  m2  1   m2  G22,,31  2   ,  , 0 ( m )  m m m 0 ,  

(22)

Next, in order to take into account the influence of the GVD effect at the performance of the single link, we substitute the equation (18) into (21), for the case of the gamma gamma turbulence, and into (22), for the gamma distribution. Thus, the 16

Page 16 of 34

total probability of fade, for each one of the M copies of the FSO diversity scheme, under gamma gamma modelled turbulence with pointing errors, as a function of the threshold of the instantaneous normalized intensity at the receiver’s side, is given

an

us



   uz,0 2  m  m2 1,  m2  1  m,th 3,1  m m G2,4  (23)  m2 ,  m ,  m , 0  ( m )( m ) 0,m  

cr



PF , m, comb, GG u z,0 m,th

ip t

through the following closed form mathematical expression:

with  m  m2  1  mC  and m  T02  2 zm . 2

M

Furthermore, following the procedure presented above for the gamma gamma distribution model, we conclude to the following mathematical expression for the

d

estimation of the probability of fade for each one of the M information signals, under

Ac ce p

te

the action of GVD, gamma modelled atmospheric turbulence and pointing errors:



PF , m, comb,G u z,0 m,th

4.



  u z,0 2  m  m2 1,  m2  1  m, th 2,1  m  G2,3  (24) m2 ,  m , 0  ( m ) 0 , m   

The Reception Diversity Scheme

The closed form mathematical expressions (23) and (24) which have been obtained above are presenting the probability of fade of the optical link, without any diversity 17

Page 17 of 34

scheme, under various turbulence conditions and pointing errors. However, as mentioned before, the diversity schemes could improve the systems availability, taking into account that the information signal is sent in multiple copies in different

ip t

moments which correspond in different channels states. More specifically, the probability of fade for the diversity scheme which is sending each part of the

us

cr

information signal in M copies, is given as [35]:

PF ,total   PF , m , comb , X   Pr I r , m  I r , m , th    FI r , m I r , m , th  M

M

m 1

m 1

m 1

(25)

an

M

M

where the parameter X which appears at the subscript of the probability of fade, corresponds to the specific distribution which has been used in order to model the

te

d

atmospheric turbulence effect, i.e. the gamma gamma or the gamma distribution. By substituting (23) into (25), is obtained the close form mathematical expression

Ac ce p

that follows, for the estimation of the probability of fade of the whole diverse FSO link with GVD, gamma gamma turbulence and pointing errors:

M

PF , GG , total   PF , m , comb , GG  m 1

    m  m u  z ,0  2  m 2  1,  m2  1   m ,th 3 , 1 m G2 , 4     m2 ,  m ,  m , 0    0,m m 1   ( m )  (  m )    M

(26)

18

Page 18 of 34

For the specific case where the time diversity scheme is used, the parameters of (26), are the same for all the M-copies of the information signal, i.e. α=α1, …, αΜ, β=β1, …, βΜ, ξ=ξ1, …, ξΜ, Ξ=Ξ1, …, ΞΜ, Α0=Α0,1, …, Α0,Μ, |u(z,0)|th=|u(z,0)|1,th, …,

cr

M

(27)

an

us

PF , GG , total

 2   u  z ,0  2  1,  2  1    3,1  th  G2 , 4    2 ,  ,  , 0  0   ( )(  )   

ip t

|u(z,0)|M,th, and consequently the above expression, is taking the following form:

Additionally, the probability of fade of the FSO with reception diversity, taking into

M

account the GVD, the gamma modeled turbulence and the pointing errors, has the

d

following form and is derived by the Eqs (24) and (25):

PF , G , total   PF , m , comb , G

M

Ac ce p

m 1

 2   u z ,0  2  m  m 1,  m2  1  m ,th 2 ,1  m   (28) G2 , 3   m2 ,  m , 0   0,m m 1   ( m )   

te

M

Similarly to the previous case, for the limiting case of time diversity schemes, the

parameters of the expression (28) are equal for all the M copies of the information

signal, and the probability of fade is given as:

PF , G , total

 2   u  z ,0  2  1,  2  1  2 ,1  th   G2 , 3    2 ,  , 0  0   ( )   

M

(29)

19

Page 19 of 34

cr

5. Numerical Results

ip t

where ζ=ζ1, …, ζΜ.

In this section we present numerical results which are illustrate the influence of

us

GVD, atmospheric turbulence and spatial jitter effects at the performance of the FSO

an

links, with or without reception diversity schemes, by means of the probability of fade estimation. More specifically, using the derived above expressions (23), (24) and

M

(26)-(29), we investigate the system’s performance for common parameter values. Taking into account that the gamma gamma distribution is suitable for weak to strong

d

turbulence conditions, while the gamma, for weak, the results that we present below

te

have been obtained using the former distribution model for larger values of Cn2, i.e. 1×10-15 m-2/3 and 1×10-13 m-2/3, while the latter has been used for weak turbulence

Ac ce p

conditions, i.e. 1×10-17 m-2/3. Moreover, the results that we present here have been

obtained only for a time diversity scheme with M=2 and 3, link length 6 km and the

whole system is assumed to be established at h=30 m above the sea’s surface. Moreover, the operational wavelength of the system, λ, has been fixed at 1.55μm

while the receiver’s aperture diameter, D, is set at 0.01 m. It is clear that using the expressions (27) and (29), one, can easily obtain numerical results for the availability of the FSO system for others reception diversity schemes, e.g. wavelength, spatial diversity, etc, according the requirements of each FSO communication system.

20

Page 20 of 34

ip t cr us an

d

M

Figure 1: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively narrow longitudinal pulsewidth, i.e. T0=4ps, and weak atmospheric turbulence conditions, i.e. Cn2=1×10-17 m-2/3.

te

In order to investigate the influence of the GVD and the pointing errors effects, we

Ac ce p

have used two values for the longitudinal pulsewidth of the chirped Gaussian pulses, i.e. T0=4 ps and 7 ps, two values for the chirp parameter C, i.e. -20 and +20, while the

ratio σs/r is set at 0.5 and the Wz/r is taking the values 1.5 and 4, for weak to strong pointing errors effects, respectively. Finally, the receiver’s dimensionless intensity threshold is varying between the values 0.01 and 0.1, [15].

21

Page 21 of 34

ip t cr us an

d

M

Figure 2: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively wide longitudinal pulsewidth, i.e. T0= 7ps, and weak atmospheric turbulence conditions, i.e. Cn2=1×10-17 m-2/3.

te

Thus, in Figure (1), we present the availability results for the case of weak turbulence conditions, i.e. using the gamma distribution model, for longitudinal

Ac ce p

upshifted and downshifted Gaussian pulses with pulsewidth equal to 4 ps, for weak and strong pointing errors effect with and without time diversity scheme, i.e. M=1, 2

and 3. The positive influence of the negative chirp is clearly visible in this Figure, as well the better performance results which are obtained by increasing the time diversity value M. However, it should be mentioned here that the larger value of M increases significantly the system’s availability but, on the other hand, decreases its data rate due to the fact that the same information signal should be sent (and received) by the same transmitter (and receiver) more than once is a specific time domain, [39].

22

Page 22 of 34

ip t cr us an

d

M

Figure 3: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively narrow longitudinal pulsewidth, i.e. T0=4ps, and moderate atmospheric turbulence conditions, i.e. Cn2=1×10-15 m-2/3.

te

In Figure (2), we present the corresponding results with those of Figure (1), but

Ac ce p

with larger pulsewidth for the longitudinal Gaussian pulses. By comparing the results of Figure (1) with those of Figure (2), it is visible that we obtain the same qualitatively results but, the influence of the positive or the negative chirp value is much more weak. This conclusion was somehow expected because, as mentioned above, the influence of GVD effect is strong enough for narrow longitudinal pulses and is getting weaker as their pulsewidth increases.

23

Page 23 of 34

ip t cr us an

d

M

Figure 4: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively wide longitudinal pulsewidth, i.e. T0= 7ps, and moderate atmospheric turbulence conditions, i.e. Cn2=1×10-15 m-2/3.

te

Next, in Figures (3) and (4), we present outcomes with similar parameters with

Ac ce p

those of Figures (1) and (2) but, for moderate atmospheric turbulence conditions which are modeled with the gamma gamma distribution model. Here, the stronger atmospheric turbulence results in worse availability results but, similarly to the previous results, the diversity scheme improves significantly the performance. Additionally, as in the previous cases, it is clear that the negative chirp upgrades the performance, but its influence is significant only for relatively narrow longitudinal Gaussian pulses which are used for the cases of communications systems with very high data rates.

24

Page 24 of 34

ip t cr us an

d

M

Figure 5: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively narrow longitudinal pulsewidth, i.e. T0=4ps, and strong atmospheric turbulence conditions, i.e. Cn2=1×10-13 m-2/3.

te

Finally, in Figures (5) and (6), we present the corresponding results with the

Ac ce p

previous cases but for strong turbulence conditions. By observing them, it is clear that the use of a diversity scheme, the time diversity here, is necessary due to the fact that the performance of the FSO link is very low without such a technique, even for the cases of narrow longitudinal pulses where the negative chirp can improve, slightly the system’s availability.

25

Page 25 of 34

ip t cr us an M

Ac ce p

te

d

Figure 6: The probability of fade is shown, as a function of the normalized receiver’s threshold, for relatively wide longitudinal pulsewidth, i.e. T0= 7ps, and strong atmospheric turbulence conditions, i.e. Cn2=1×10-13 m-2/3.

6. Conclusions

In this work, we studied the influence of the atmospheric turbulence, modelled

either with the gamma gamma or the gamma distribution, the GVD and the spatial jitter, effects at the performance of an FSO link with reception diversity. We extracted closed form mathematical expressions for the estimation of this communication system’s performance and availability, by means of its probability of fade, either with or without reception diversity. Using the extracted expressions we presented numerical results which are shown the influence of each effect at the system’s performance, depending on the system’s parameter values.

26

Page 26 of 34

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