BER performance of FSOC system with differential signaling over partially correlated atmospheric turbulence fading and partially correlated pointing errors

Journal Pre-proof BER performance of FSOC system with differential signaling over partially correlated atmospheric turbulence fading and partially correlated pointing errors Li Xiaoyan, Zhao Xiaohui, Zhang Peng, Tong Shoufeng

PII: DOI: Reference:

S0030-4018(19)30815-6 https://doi.org/10.1016/j.optcom.2019.124545 OPTICS 124545

To appear in:

Optics Communications

Received date : 19 June 2019 Revised date : 7 September 2019 Accepted date : 8 September 2019 Please cite this article as: L. Xiaoyan, Z. Xiaohui, Z. Peng et al., BER performance of FSOC system with differential signaling over partially correlated atmospheric turbulence fading and partially correlated pointing errors, Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.124545. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Elsevier B.V. All rights reserved.

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BER performance of FSOC system with differential signaling over partially correlated atmospheric turbulence fading and partially correlated pointing errors

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Li Xiaoyan1,2, Zhao Xiaohui1*, Zhang Peng2, Tong Shoufeng 1 College of Communication Engineering, Jilin University, Changchun 130012,China National and Local Joint Engineering Research Center of Space Optoelectronics Technology, Changchun University of Science and Technology, Changchun 130022, China

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Abstract: Bit error rate (BER) performance of the free space optical communication (FSOC) systems using differential signaling over partially correlated atmospheric turbulence fading with and without partially correlated pointing errors is investigated. In order to do the BER performance analyses, we first derive a useful mathematical expression for the joint probability density function of the system under combined effects of the partially correlated atmospheric turbulence fading and partially correlated pointing errors. Then based on this obtained result, two mathematical BER expressions of the system affected by either the partially correlated atmospheric turbulence or both the partially correlated atmospheric turbulence fading and the partially correlated pointing errors are obtained. The accuracy of the derived BER expressions is demonstrated by the Monte Carlo simulations, and the effects of the system parameters on the BER performance are studied. From the simulations, we present some important and interesting analytical results of the BER performance with the system parameters.

1. Introduction

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Index Terms: Free space optical communication, BER performance, atmospheric turbulence fading, pointing errors, differential signaling.

Free space optical communication (FSOC) has gained wide attention due to its advantages of high data rate, large capacity, cost effective and abundant spectrum resources [1-4]. Considering the complexity of heterodyne detection, intensity modulation and direct detection (IM/DD) based on on-off keying (OOK) modulation is widely adopted in practical FSOC systems. However, this FSOC system faces two severe challenges: the atmospheric turbulence random fading and the pointing errors fading. To solve these problems, some previous works propose an aperture averaging scheme and a multiple transmitters/receivers method [2, 5], which are proved to be effective to improve the received signal intensity of the systems. Meanwhile, another problem of the irreducible error floor at high signal to noise ratio (SNR) puzzles the IM/DD FSOC systems with fixed threshold, which also results from the atmospheric turbulence and pointing errors random fading [6-7]. Attempts to overcome this problem, many researches concentrate on the adaptive thresholds detection. In [8], a maximum likelihood sequence detection (MLSD) scheme is used for the OOK modulation. However, this MLSD suffers from high computational complexity and requires the instantaneous channel state information (CSI). In order to eliminate the channel instantaneous CSI requirement unavailable in practical systems, a likelihood threshold detection scheme is proposed in [9]. Unfortunately, this method still requires prior information about the channel model, fading distribution, fading correlation and noise statistics. To avoid the need of the CSI and the prior information, a generalized likelihood ratio test based sequence detection scheme is studied in [7, 10]. Due to its complexity, this scheme faces implementation difficulties. For the complexity reduction, an electrical SNR optimized detection method without CSI is investigated in [6]. In this method, the electrical SNR optimized detection compares the received symbols with an adaptive threshold calculated periodically to recover the transmitted symbols. This adaptive threshold is obtained by solving an implicit equation with integrals and complex function.

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Another attractive blind detection is the differential signaling scheme (DSS) desirable for practical implementation as it can recover the transmit data by a fix threshold, even the system is influenced by the atmospheric turbulence and pointing errors. Importantly, this scheme also possesses the advantage of low complexity. Therefore, a lot of research works focus on this topic. An interesting differential signaling scheme is proposed in [16] to reduce the negative effects of background noise. And an outdoor FSOC experimental set-up scheme using DSS is proposed in [17] where the benefits of DSS are verified through the experiments. In [18], the DSS is adopted to mitigate the pointing error induced fading, and the proposed approach is supported by the effective experimental investigation. For the theoretical performance analysis, the BER performance for the FSOC DSS systems under the independent and fully correlated atmospheric turbulence fading is studied in [19]. In order to improve the performance of the FSOC DDS systems, a quantized feedback is introduced to the above system in [20] under the independent atmospheric turbulence fading. In consideration of both the independent atmospheric turbulence fading and the independent pointing errors fading, the BER performance analyses are provided in [21]. Unlike previous works using OOK modulation, the authors in [22] employ pulse amplitude modulation to their FSOC DDS system and investigate its BER performance under fully correlated atmospheric turbulence fading. Recently, a new factor of the sampling clock offset is considered in the performance analysis for the FSOC DDS system in [23]. All above works do not consider the situations of either the partially correlated atmospheric turbulence fading or the partially correlated pointing error fading. Those researches are extended by [24] where the BER performance for the FSOC systems under weak partially correlated atmospheric turbulence fading is analyzed with a hardly obtained time varying threshold. In the previous works, the performance analyses about the effects of the partially correlated atmospheric turbulence over a wide range from weak to strong or the combined effects of the partially correlated atmospheric turbulence fading and the partially correlated pointing error fading are not presented. In our study, we will conduct those analyses. Our main contributions are as follows: 1) we obtain a joint probability density function (PDF) of the differential signal attentions under the combined effects of partially correlated atmospheric turbulence and the partially correlated pointing errors which is the basis of our BER performance analyses; 2) we develop two mathematical expressions for the BER of the FSOC DDS systems influenced by either the partially correlated atmospheric turbulence over a wide range of weak to strong or the partially correlated atmospheric turbulence fading and the partially correlated pointing errors; 3) we conduct the BER performance analyses on the effects of the turbulence fading correlation and the pointing error correlation in different turbulence fading conditions with varying pointing errors parameters.

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Interestingly, an EM-based blind detection scheme is studied in [11-12], and the adaptive threshold of this detection can be calculated by a simple closed-form expression. The problem of this scheme is the “error floor” in high SNR, and poor bit error rate (BER) performance with small length of observation window. Another blind detection based on the pilot-symbol assisted modulation is provided in [13], which shows that it can obtain good BER performance with reasonable frame length but reduces system throughput since it periodically inserts some symbols in the data frames. For the system throughput improvement, a new construction method for the adaptive threshold with assisted pilot symbol is proposed in [14]. With only 0.3% system throughput reduction, a small BER performance difference between the system using this threshold and that adopting the CSI knowledge is achieved. Recently, a new blind detection with BER performance similar to the detection with instantaneous CSI is proposed in [15] with analog to digital conversion (ADC). The main problem in this detection method is that the high speed ADC devices are difficult to obtain and very expensive. Therefore, it will produce higher cost or even is not easy to use in high-speed communication systems.

The rest of this paper is organized as follows. Section 2 introduces the scheme and the channel fading model of the FSOC DDS systems where the channel fading model is considered for the following two situations: partially correlated atmospheric turbulence fading, and partially correlated turbulence fading with partially correlated pointing errors. Under these two situations, section 3 provides our derived mathematical expressions for the average BER of the FSOC DDS systems. Section 4 discusses and analyzes the effects of the atmospheric

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2. System model 2.1 FSOC system with differential signaling The block diagram of the proposed FSOC DDS system is given in Fig.1. The OOK signal S  {0,1} and

of

its inverted version S {0,1} are used to modulate two optical sources with different wavelengths of 1 and

2 , respectively. Then the modulated optical signal pass through two transmission antennas, and be transmitted to the receiver. At the receiver, the received signals go into an optical filter and be separated into two optical

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signals which are converted to two electrical signals v1 and v2 by an optical detector respectively. Finally, we define a variable vt  v1  v2 used to recover the transmit data by comparing it with a fixed threshold. Similar to the previous works [16,17-18,20], the fixed threshold is set to 0, which has been proved to be reasonable in [17-18]. Laser  1

v1  1 y1

I1  SI Transmitting antenna

Modulater

Data Invert

S

Modulater

I 2  SI

Transmitting antenna

Laser 2

Ω1  n1

Detector

y1=h1I1

Receiving antenna

vt  v1  v2

Optical filter

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S S∈{0,1}

y2=h2I2

－ Detector

v2  2 y2

vt  0?

S∈{0,1}

Ω2  n2

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Fig. 1. Block diagram of FSOC system based on differential detection scheme The received electrical signals v1 and v2 can be expressed as

v1 

1h1SI  n1 1

(1)

v2 

 2 h2 SI  n2 2

(2)

where S and S are the transmitted signals and called differential signals, I is the transmitted optical intensity corresponding to the transmitted symbol ‘1’, n1 and n2 are the additive white Gaussian noises with zero mean and variance of N0/2, h1 and h2 denote the fluctuated channel fading on S and S , 1 and

2 are the optical-to-electrical conversion efficiencies, and 1 and  2 are the optical filter attenuations on the received optical signals y1 and y2 . For simplicity, without loss of generality, we make the assumptions that

1 = 2 =  and 1 =  2 =  [21]. With these conditions, we have

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turbulence conditions and the pointing errors on the average BER performance through the simulations. Finally, we draw the conclusions in section 5.

vt  v1  v2 

h1 SI





 h1 I  n1  n2       h2 I  n  n 1 2  

 h2 SI  n1  n2  S 1 S 0

(3)

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2.2 Channel model In this paper, we consider that the atmospheric turbulence fading follows the Gamma-Gamma fading which is widely accepted in virtue of its perfect agreement with experimental data over a wide range of turbulence conditions [25]. Therefore, the probability density functions (PDF) of hi , i=1,2, are

2 i i 

 i  i

 i  i

2

 i    i 

hi

2

1





Ki  i 2 i i hi

i=1, 2

(4)

of

f (hi ) 

where Kv   is the modified Bessel function of the second kind of order of v ,     is the Gamma function.

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The parameters  i and  i can be expressed by [26-27]

    0.49 i 2    1  i  exp   1  0.18d 2  0.56 12/5 7/ 6    i i   

1

i=1, 2

(5)

i=1, 2

(6)

1

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  0.51 2 1+0.69 12/5 5/ 6   i    1 i  exp  2 2 12/5 5/ 6     1  0.9di  0.62di  i   

ki D 2 , i=1, 2, D is the receiver aperture diameter, L is the communication distance between the 4L

where d i 

transmitter and the receiver, ki , i=1, 2, is the optical wave number defined by ki  2 / i , and i , i=1, 2, is

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the operational wavelength. For the spherical wave, the parameter  i 2  0.5Cn2i ki 7/6 L11/6 , i=1, 2, where Cn2i , i=1,2, is the refractive index structure parameter.

In our analysis, we consider the horizontal communication links in which two optical communication paths from the transmitter to the receiver are close. In this case, we find that

  2  1  / 1

  2  1  / 1

<0.015 and

<0.03 with 2  1 <20 nm from our previous works [19]. Therefore, 1   2 and 1   2

can be satisfied when the chosen wavelengths 1 and 2 are close enough. This useful conclusion is also used by the previous works [16, 20-21] for their FSOC system performance analyses. According to this conclusion,

 i , i=1, 2, and  i , i=1, 2, are all denoted by  and  respectively in the following sections. Then, from (4), the PDF of hi , i  1, 2 , can be written as

f (hi ) 

2  

 

 

2

      

hi

2

1



K   2  hi



i=1, 2

In addition, we define an instantaneous SNR and an average SNR as  i =  hi I 

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(7)

2

N 0 , i=1, 2, and

 i =  E  hi  I  N0 , i=1,2, respectively, where E[.] denotes the expectation [16]. Considering E  hi   1 , i=1, 2, 2

since hi , i=1, 2, are normalized to unity, we can obtain  1   2   . Then we derive hi   i  i , i=1, 2, by using E  hi   1 and the definitions of  i and  i . Based on this result, the joint PDF of  1 and  2 , which is essential to the evaluation of the BER performance, can be expressed in the following forms under different channel fading.

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2.2.1 Under partially correlated atmospheric turbulence fading In this case, the joint PDF of  1 and  2 is given by [28] 

A



2 +K1 +K 2 +2  2

1

  1 2       

2

 1- 1     1+   1  

2      +K1 +K 2    i     i 1 



 + +K 2 2

1

 1     K1  0 K 2  0  1+ 1 

   



  K   +Ki  2 A i      +K i  K i !

K1 +K 2

   

(8)

of

f  1 , 2   1 ,  2  

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where A   1- 1 , 1  0,1 denotes the atmospheric turbulence fading correlation coefficient between

 1 and  2 , which takes the form of [17]

  d 5/3  r     0  

1  exp   

(9)

wave propagation model, 0 is given by [29]

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where d r is the distance between the propagation axes of two optical communication paths, and for aspheric

  

2  2    L    

3/5

0   0.55Cn2 

(10)

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2.2.2. Under combined effects of partially correlated turbulence fading and partially correlated pointing errors fading Considering the combined effects of the partially correlated atmospheric turbulence and the partially correlated pointing errors, h1 and h2 can be expressed as [30]

and

h1  hp1ha1

(11)

h2  hp 2 ha 2

(12)

where ha1 and ha 2 are the atmospheric turbulence fading on S and S , and h p1 and hp 2 are the random pointing errors fading on S and S respectively. Under this channel model, the joint PDF for h1 and h2 is not obtained by previous works. However, it is essential for our performance analysis. To get this PDF, we first present the joint PDF of ha1 and ha 2 and that of h p1 and hp 2 which are the key to the derivations of the joint PDF of h1 and h2 . The joint PDF for ha1 and ha 2 can be expressed as [28] f ha1 , ha 2 (ha1 , ha 2 ) 

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where A 

22

      

2



 1- 1     1+   1   2

  +K1 +K 2   hai i 1

 1     K1  0 K 2  0  1+ 1

 + +Ki 2



1





   

K1 +K 2

K   +Ki 2 Ahai   +K i  K i !

A



2 +K1 +K 2 +2  2

(13)

 , the definitions of the parameters  ,  , and 1 refer to (5), (6) and (9) respectively. For 1  1

the joint PDF of h p1 and hp 2 , there is no ready-made result available. In order to calculate this PDF, the

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attenuation due to pointing errors should be given first, which can be expressed as [30]

 2r 2 hpi  A0 exp   2i  w zeq 

  , ri  0 , 

i=1, 2

(14)

where 2 wzeq   wz2  erf  v  2v exp  v 2 



(15)

of

is the equivalent beam width, wz is the beam waist radius, v   a



2wz , a is the receive aperture,

A0  erf  v   , and ri , i=1,2, are the radial distances of the pointing errors [28-29]. Like previous works

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2

[30-31], ri , i=1,2, can be modeled by the Rayleigh distribution. Then the joint PDF for r1 and r2 can be expressed as [32, Eq.(118)]

f r1 , r2  r1 , r2  

  r12 r22    2 r1r2  1 exp   2   I0    2 2  2  12 22 1  22   2 1   2    1  2    1   2  1  2  r1r2

(16)

re-

where I 0  x  is the zero order modified Bessel function of the first kind [31, Eq.(8.431.2)],  i , i=1, 2, are the pointing error displacement standard deviations (jitter) at the receiver, 2  0,1 denotes the correlation coefficient between r1 and r2 . The expression for 2 is not given by the previous works, but the pointing errors fading correlation coefficient is obtained by the experiment in [18]. In our work, we also do not seek the mathematical expression for 2 , instead, like previous work [21], we derive the average BER expressions for

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FSOC DDS system with given 2 . Without loss of generality, we consider  1 =  2 =  . With the above obtained joint PDF of r1 and r2 , we can further derive the joint PDF for hp1 and hp2. From (14) we know that

2 ri  wzeq ln  A0 hpi  2, hpi  A0 ,

i=1,2.

(17)

Then, according to the theorem 3.9.5 in [33], the joint PDF for h p1 and hp 2 can be derived from (16) with a transformation of

and

as

f hp1 , hp 2 (hp1 , hp 2 ) 

where

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   

2 r1  wzeq ln  A0 hp1  2

(18)

2 r2  wzeq ln  A0 hp 2  2

(19)

r1 r2 fr ,r hp1 hp 2 1 2

denotes the derivative, and the terms



r1 hp1



2 2 wzeq ln  A0 hp1  2, wzeq ln  A0 hp 2  2 .

and

r2 hp 2

can be derived by

2 2 wzeq hpi A0 wzeq ri   , 2 2 hpi 4 w2 ln  A h  2 A0 hpi 4hpi wzeq ln  A0 hpi  2 zeq 0 pi

Substituting (16) and (21) into (20), yields

(20)

i=1, 2.

(21)

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2 2 wzeq ln  A0 hp1  2 wzeq ln  A0 hp 2  2   2   

  I0  2 2  1   2 

of

   w2 ln  A h  2 w2 ln  A h  2 2 2 wzeq wzeq zeq 0 p1 zeq 0 p2   f hp1 , hp 2 (hp1 , hp 2 )   4 2    2 2  1   2  4h w ln A h 2  4h w ln A h 2  p1 zeq  0 p1   p 2 zeq  0 p 2   2 2     wzeq wzeq exp  2 ln A h exp  ln  A0 hp 2      0 p1  2 2 2  4 1   2    4 1   2  

(22)

    w w ln  A0 hp1   and exp  2 ln  A0 hp 2   calculated by With the terms exp  2 2 2  4 1   2  4 1   2   2 zeq





2 zeq



pro



2 2 2 wzeq wzeq   wzeq     2 2  2 4  1   4  1 exp  2 ln  A0 hpi    exp ln  A0 hpi   2     A0 hpi    22  , i=1, 2, 2  4 1  2    

we further transform (22) into

t 4  hp1hp 2 



(1  22 ) A0

2t 2

 2  2 wzeq ln  A0 hp1  ln  A0 hp 2  I  1 22  0  1  22 2 2  





f hp1 , hp 2 (hp1 , hp 2 ) 

where t  wzeq

re-

t 2 1 22 1

 2  .



1 2

   , hp1 , hp 2  A0  

(23)

(24)

With this joint PDF, we can calculate the joint PDF of h1 and h2 . We start with the joint PDF of h p1 ,



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

hp 2 , ha1 and ha 2 denoted by f hp1 , hp 2 , ha1 , ha 2 hp1 , hp 2 , ha1 , ha 2 . Since ha1 and ha 2 are independent to h p1 and



hp 2 [30], we can present the f hp1 , hp 2 , ha1 , ha 2 hp1 , hp 2 , ha1 , ha 2



as

f hp1 , hp 2 , ha1 , ha 2  hp1 , hp 2 , ha1 , ha 2   f hp1 , hp 2  hp1 , hp 2  f ha1 , ha 2  ha1 , ha 2  .

(25)

Then using the theorem 3.9.5 in [33], and with a transformation of hp1  h1 ha1 and hp 2  h2 ha 2 , the above PDF is converted to the joint PDF of h1 , h2 , ha1 and ha 2 , which is given by

f h1 , h2 , ha1 , ha 2  h1 , h2 , ha1 , ha 2  

Finally the joint PDF for

h1

and

h2

1 f h , h  h1 ha1 , h2 ha 2  f ha1 , ha 2  ha1 , ha 2  . ha1ha 2 p1 p 2

(26)

can be derived by calculating the marginal PDF of

f h1 , h2 , ha1 , ha 2  h1 , h2 , ha1 , ha 2  as



f (h1 , h2 )=  h1

Jo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

A0





h2 A0

1 f h , h  h1 / ha1 , h2 / ha 2  f ha1 , ha 2 (ha1 , ha 2 )dha1dha 2 ha1ha 2 p1 p 2

(27)

Substituting (13) and (24) into (27), then using [34, Eq. (8.447.1)] 

I0  z     z 2 P 0

and making some algebraic manipulations, we have

2P

 P !

2

(28)

Journal Pre-proof



 1   1+  1 

   

K1 +K 2





h1 A0

h2 A0

 

t2



 1-1  f (h1 , h2 )=   2 2        (1   2 )  1+1 

122 

 h1h2 

P

 h1   h2   ln   ln  h A h a1 0   a 2 A0  

P





2 2P



2

P  0 K1  0 K 2  0

 + +Ki

2

h i 1

2 +K1 +K 2 +2  2

 t    +K +K  A   1     P !   +K    +K  K ! K

-1

2

T 1

ai

1

2 2P 2

2





2

1

2

1

2

!

.(29)

K   +Ki 2 Ahai dha1dha 2

Using [34, Eq.(9.343)]

of

 2   1 x  K v  x   G0202   4 v , v  2   2 2  and [35, Eq.(8.4.6.4)] P

 1 1 ;  H ( x  1)  P !GP0 1PP 11  x  ;  0 0    

(31)

x 1 1 H ( x  1)=  , 0 0  x 1 

(32)

where



2t 2



 1-1  f (h1 , h2 )=  2 2         (1   2 )  1+1  

0

0

 

t2

 h1h2 

122 

-1

 t    1    





P  0 K1  0 K 2  0

2 2P

2

2 2P 2

  +K1 +K 2  A

2 +K1 +K 2 +2  2

 1    +K1    +K 2  K1 ! K 2 !  1+ 1

  

K1 +K 2

    + +Ki  h 1 1 ;  0 P 1  h2 1 1 ;  2 T 1 20  dh dh 2 GP0 1PP 11  1 GP 1P 1  h G Ah      + K    + K ai 02  ai a1 a2 i i   ha1 A0 ;  0 0    ha 2 A0 ;  0 0    ,  i  1       2 2  

urn al P



re-

we get t 4  A0  1 22 

(30)

pro

 ln x 

.

(33)

Calculating above equation using [36, Eq.(21)], we derive

f (h1 , h2 ) 



 1- 1  1   2 2         (1   2 )  1+ 1  h1h2 t4

  t 2  2 P  1   +K1 +K 2   2      2   +K1    +K 2  K1 ! K 2 !  1+ 1 P  0 K1  0 K 2  0  1   2   





   

K1 +K 2

    t2 t2 t2 t2 ;1  , ,1  ;1  , ,1      2 2 2 2 1  2 1   2  P  30  Ah2 1  2 1  2  Ah GPP130P  3  1 G P 1 P  3  A A0 t2 t2  t2 t 2   0  +K1 ,  ,   , , ;  + K ,  , , , ;  2   1   22 1   22  1   22 1   22    

.(34)

Finally the joint PDF of  1 and  2 can be obtained from (34) with a transformation of h1 =  1  and

h2 =  2 

as



 1- 1  1      2 t 2    +K1 +K 2      2 2 2  4       (1   2 )  1+ 1   1 2 P  0 K1  0 K2  0  1   2    +K1    +K 2  K1 ! K 2 !     t2 t2 t2 t2 K1 +K 2 ;1  , ,1  ;1  , ,1      2 2 2 2  1  1  2 1   2  P  30  A  2 1  2 1  2  A 1  P  30 GP 1 P  3  GP 1 P  3     1+   A  A  t2 t2  t2 t2  1    0   0  + K ,  , , , ;  +K 2 ,  , , , ; 1 2 2  2   1  2 1  2  1  2 1   22    (35) f ( 1 ,  2 ) 

t4

Jo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2t 2

4t 4  A0  1 22 

2P

Journal Pre-proof

3. Average BER First we give the instantaneous BER since it is indispensable for the evaluation of the average BER. In our previous work [19], we have obtained its expression as  hI 1 erfc  1  2N  4 0  

Then using  1  h12 I 2 N 0 and  2  h2 2 I 2 N 0

  h I   erfc  2   2N  0  

(36)

defined in section 2, we obtain the expression for the

instantaneous BER as

 1 1 erfc   4  2  

  2   erfc   2   

    

pro

Pe 

   

of

Pe 

(37)

3.1. Average BER under partially correlated atmospheric turbulence fading The average BER can be expressed by Pe  



0





0

f   1 ,  2  Pe d 1d 2

Pe





 + +K1



 0 0  i 1

2

i 

  2

  +K1 +K 2 





urn al P

By transforming  1 and  2 into 1 and  2 with 1   1 /  Eq. 8.4.14.1]

(38)

K1 +K 2

2 +K1 +K 2 +2   1  2 A       K1  0 K 2  0   +K1    +K 2  K1 K 2  1+ 1  (39)  + +Ki    2 1   2     2   erfc    d d K   +Ki  2 A  i /   erfc     2  2   1 2      

 1-1   2  4        1+1 



re-

Substituting (8) and (37) into (38), yields

erfc  x  

and 2   2 /  , then using (30) and [35,

 ;1  G1202  x 2 1  ,  0, ;     2  

1

(40)

we can transform (39) into Pe 



 1-1   2  4          1+1  1

  +K1 +K 2 

 1     K1  0 K 2  0   +K1    +K 2  K1 ! K 2 !  1+ 1 



     2 ;1   + +K1    2 1G 20  A  G 20     2     + K    + K 1 d1  02  1 1 1 1  12  0 1 2 0, ;  ,     2  2 2        + +K 2 1  20   0 2 2 G0 2  A2    +K 2 ,     +K 2 d2 + 2 2  

   

K1 +K 2

A

2 +K1 +K 2 +2  2

.(41)

   20  d  G A     + K    + K 02  1 1 0 1 1  ,   2 2       2 ;1   + +K 2 1  20   20    2  0 2 2 G0 2  A2    +K 2 ,     +K 2  G1 2  2 2 0, 1 ; d2 2  2 2    

 + +K1

1

2

Jo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

1

Calculating (41) by [36, Eq. 21] and [36, Eq. 24], we finally obtain the closed form expression for the average BER under the correlated atmospheric turbulence fading as

Journal Pre-proof



 1-1    3 1+1   2         2    4

  +K1 +K 2   1    K1 ! K 2 !  1+ 1 K1  0 K 2  0 



   

K1 +K 2

  1    Ki 2    Ki 1   2    , , , ;1   2  2 2 Ki 2 2 2 2 24 8    G5 2  2 1  i 1   +Ki   A  0, ;    2   

of

(42)

The summation in (42) is calculated for Ki =0, 1, 2,…, M, i=1, 2. The convergence of this infinite series of (42) is given in Table 1 when  =40 dB. We can find that the number of the required terms depends highly on

pro

1 . Higher 1 requires larger number of terms. In addition, the atmospheric turbulence strength slightly affects the required number of terms.

Although above equation is derived for the partially correlated atmospheric turbulence fading, we can also present a simple expression for the BER expression for the uncorrelated atmospheric turbulence fading by setting

1  0 . In this case, only the terms corresponding to K1  K2  0 contribute the BER analysis. Based on these remarks, the BER over the uncorrelated turbulence fading is presented by

re-

 1 2  1  2    , , , ;1  82 2 2 2 2 . Pe  3 G  2 2 1     2 0, ;           2   2   3

24 52

(43)

urn al P

This expression is in accordance with our previously obtained expression (12) in [19] which shows the BER under the uncorrelated channel fading. Therefore, the analysis in our previous work [19] is only a special case of this work. TABLE 1 Maximum M of every Ki in (33) at accuracy at 5th significant digit

(  ,  )= (2.296, 1.822) (  ,  )= (2.064, 1.342)

1 =0.1

1 =0.3

1 =0.5

1 =0.7

9 10

17 18

30 32

55 57

3.2 Average BER under combined effects of partially correlated atmospheric turbulence and partially correlated pointing errors Considering the combined effects of the atmospheric turbulence and pointing errors under the partially correlated conditions, we can get the average BER by substituting (34) and (37) into (38) as

 1 2 2  16       (1   2 )  1+        1  0 0 erfc   2   erfc         Pe 

t4

Jo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Pe 



1   1 

2 2

 2t 2      2  P  0 K1  0 K 2  0  1   2  





    2 1 P  30  A    GP 1 P  3  A   i 1  i  0  

  +K1 +K 2 

K1 +K 2

 1      +K1    +K 2  K1 ! K 2 !  1+ 1   t2 t2 ;1  , ,1   2 2 1  2 1  2  i d 1d 2  t2 t 2    +K i ,  , , , ; 1   22 1   22  

2P

. (44)

Substituting (40) into (44), and transforming  1 and  2 into 1 and  2 with 1   1 / 

and

2   2 /  , then using [36, Eq. 21] and [36, Eq. 24], we derive the closed form expression for the average

Journal Pre-proof



 1- 1  Pe  3    1+   1   2        (1   22 )  2 +  K1  P 5 t 4

T    +K1 +K 2   1         P  0 K  0 K  0   +K1  K1 ! K 2 !  1+ 1  





1

2

P 1

2  T  ;1  2    

(45)

t2 . The summation in (45) is performed for P =0, 1, 2,…, M1, and Ki =0, 1, 2,…, M, i =1, 2. In 1   22

pro

where T 

,

of

  1    Ki 2    Ki 1   2   2  T , , , , , 2 2  2  2 Ki 2 2 2 2 2 2 P  5 8 A0    GP  6 P  3  2 1 T T  i 1   +Ki   A 0, ;  , ,    2 2 2  

K1 +K 2

Table 2, the convergence of the infinite series in (45) for Ki , i=1, 2, is presented with  =40 dB, wz a =0.5,

a  =5, 2 =0 and M1=0. The maximum M1 of P is illustrated in Table 3 where  =40 dB, (  ,  ) = (2.064, 1.342), 1 =0 and M=0. We discover that higher 1 requires large number of terms. And the same conclusion

re-

can be obtained for the required number of P. In addition, the atmospheric turbulence and pointing errors slightly affect the required number of terms. By adopting similar derivation method with (43), the special case of the BER over the uncorrelated Gamma-Gamma turbulence fading and the uncorrelated pointing error fading is rewritten as

 1    Ki 2    Ki 1   2   2  T  , , , , ;1  8 A 2 2  2 2 2 2 2 0  Pe  3 G  2 1 T  A  2 0, ;   2          2 2   25 63

urn al P

2 +  K1  P  4 t 4

(46)

TABLE 2 Maximum M of every Ki in (42) at 5th significant digit accuracy when 2 =0

(  ,  )= (2.296, 1.822) (  ,  )= (2.064, 1.342)

1 =0.1

1 =0.3

1 =0.5

1 =0.7

11 11

20 20

33 34

61 65

TABLE 3

Maximum M1 of P in (42) at 5th significant digit accuracy when 1 =0

2 =0.1

2 =0.3

2 =0.5

2 =0.7

( wz a , a  )= (0.5, 2)

4

6

10

18

( wz a , a  )= (0.5, 5)

4

6

10

18

( wz a , a  )= (2.0, 5)

4

6

10

18

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BER under the partially correlated fading channel in consideration of the combined effects of the atmospheric turbulence and pointing errors as

4. Simulation results and analysis In this section, the effects of the turbulence fading conditions and the pointing errors parameters on the BER performance of the FSOC DDS systems are investigated. Analytical BER results over the atmospheric turbulence fading without and with pointing errors can be calculated by using (42) and (45) respectively. Their power series based expressions are truncated to a finite number of terms. Referring to previous works [37-38], we have two pairs of atmospheric turbulence fading parameters of  and  as follows: (  ,  ) = (2.296, 1.822) for moderate turbulent conditions and (  ,  ) =(2.064, 1.342) for strong turbulent conditions. In addition,

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if there is no special specified setting, the values of  and  are set to 1, and the values of 1 and

2 are 0. 4.1 Effects of 1 and 2 on average BER In Fig. 4, we give the average BER in terms of 1 for the different atmospheric conditions (moderate and strong) and the different pointing error conditions ( a   2 , wz a  0.5 and a   3 , wz a  3 ). Fig. 5

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presents the average BER with regarding to 2 for the same parameters as in Fig. 4. The Monte Carlo simulation results presented in those figures verify the accuracy of our derived analytical results (42) and (45). Besides, we can observe that the average BERs keep constant when 1 or 2 varies. This implies that the

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BERs are not sensitive to the correlation of the atmospheric turbulence and the pointing errors conditions. In other words, they have no relationship with these conditions. This is different from the conclusion given by the previous work in [24] where the BER varies with the correlation of atmospheric turbulence fading, since our decision threshold is different. Precisely, our threshold is 0, but a time varying threshold is adopted in [22]. The construction of this varying threshold requires the knowledge of instantaneous atmospheric turbulence fading which is difficult to obtain in reality. On the contrary, our threshold requires no extra knowledge, which is more desirable for practical implementation. The rationality of the FSOC DDS systems with zero threshold has also been proved by the works in [17-18, 20].

Fig. 4 Average BER verses 1

Fig. 5 Average BER verses 2

4.2 Effects of the atmospheric turbulence conditions and pointing errors Considering the FSOC system with and without the influence of the pointing errors, we show the average BER in Fig. 6 and Fig. 7 in terms of wz a for  =70 dB with different (  ,  ) and a  = {0.2, 0.3…2, 3, 4, 5}. We can observe from these two figures that the BERs go almost monopoly upward with the increase of

wz a when a  is larger than 2. And these BERs fluctuate at other values of a  with a peak in small wz a where wz a  1 . For example, in Fig. 6 when a  =0.6, the BER with wz a =1.2 is 0.06 higher than 6.3×10-4 with wz a =5 by almost 2 orders of magnitude.

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Moreover, the FSO systems with pointing errors and without pointing errors have similar BER performance on the condition that wz a is smaller than 0.4. We can also observe that the increase of a has a positive effect on the BER performance, and this positive effect is not obvious when a  is larger than 3. Specifically, the system BER decreases by 2 orders of magnitude if a  changes from 0.2 with corresponding BER=0.11 to 1 with corresponding BER=1.02×10-3 when wz a =5 under strong turbulence, whereas, the BER is nearly constant when

a

increases from 3 with corresponding BER=9.9×10 -4 to 5 with corresponding

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Fig. 6. Average BER verses wz a under moderate turbulence conditions

Fig. 7. Average BER verses wz a under strong turbulence conditions

4.3 Optimum pointing error parameters

In general, wz a is larger than 1 in practical application. From the results presented in Fig. 6 and Fig. 7,

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we know that there is an optimum wz a which gives the minimum BER for a  smaller than 2. To obtain this optimum wz a , we calculate the BERs for wz a =1~25 in discrete steps 0.1, and adopt the wz a

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corresponding to the minimum BER as the optimum wz a .

Fig. 8. Optimum beamwidth against  a

In Fig. 8, the optimum wz a versus  a for  = 30~90 dB in discrete scale of 10 dB with different turbulence conditions is illustrated. We find that the optimum wz a is 1 for some values of  a , such as

 a =0.5 for the case of  =70 dB and (  ,  )= (2.296, 1.822). This can be illustrated by the results presented in Fig. 6 from which we know that the BER shows an upward tendency all the time with the increase of wz a

in the above average SNR and the turbulence conditions. Therefore, the smaller the wz a is, the lower the BER will be. Moreover, the minimum wz a used in the simulation is 1, then the optimum wz a obtained in

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BER=9.7×10-4.

the simulation is also 1.

Besides, when  a is larger than 1.1, we find that the optimum wz a follows nearly linear performance with slope depending on  , and start value depending on  and the turbulence conditions. More specifically, the slopes of this linear performance for  =30 dB and 70 dB are 1 and 3 respectively under the moderate turbulence conditions. Moreover this slope is also 3 for the strong turbulence condition with  =70 dB. For the

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start values of  a , they are 0.7 and 0.8 respectively under the moderate and the strong turbulence conditions for  =70 dB, and is 1.1 for  =30 dB under the moderate turbulence condition. 4.4. BER performance under optimum pointing errors Fig. 9 and Fig. 10 illustrate the BER with optimum pointing error parameters and without pointing errors under both the moderate and strong turbulence conditions where  a =0.5, 1, 3, 5, and 7 respectively. The

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wz a adopted here is the corresponding optimum wz a which gives the minimum BER obtained by the method in section 4.3, and the number of cumulative terms for ki, i=1, 2 and P are M=0 and M1 =0 respectively. In these figures, we find that the BER for the system with  =70 dB under the moderate turbulence with the pointing errors (  a =1) and without pointing errors are 1.02×10-4 and 1.15×10-6 respectively. Whereas, in the

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case of the strong turbulence, the BER with pointing errors (  a =1) and without pointing errors are 5.6×10 -4

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and 2.3×10-5 respectively. These results illustrate that the BER with the pointing errors is higher than that without pointing errors by 2 orders of magnitude under the moderate turbulence, whereas it is 1 order of magnitude under the strong turbulence. Therefore we can obtain that compared to the strong turbulence, the pointing errors have greater impacts on the BER performance under the moderate turbulence.

Fig. 9 BER with optimum pointing errors under moderate turbulence 5. Conclusions

Fig. 10 BER with optimum pointing errors under strong turbulence

We have investigated the BER performance of the FSOC DDS transmission system over partially correlated atmospheric turbulence fading under both partially correlated pointing errors and without pointing errors conditions. Our results show that the BER performance is not affected by the correlation coefficients 1 and

2 . And the increase of the receiver aperture has positive effects on the BER performance, but those effects are not obvious when a  is larger than 3. Thus a  =3 is a good choice to reach a compromise between the BER performance and the practical implementation. We also find that the smaller the wz a is, the better the BER performance will be when a   2 . For the case of a   2 , there is an optimum wz a which gives a minimum BER with linear performance. In addition, the BER with pointing errors (  a =1) is higher than that

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without pointing errors by 2 orders of magnitude under the moderate turbulence and 1 order of magnitude under the strong turbulence. Therefore, the pointing errors under moderate turbulence have greater impacts on the BER performance. Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.61705019，61571209), the Foundation for Young Scientists of Jilin Province (Grant No. 20170520161JH) and the excellent young

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scientist lift project of Jilin Province (2017-2018). References

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