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24, pp. 5639–5647, Dec. 2009. [7] C. Datsikas, K. Peppas, N. Sagias, and G. Tombras, “Serial free-space optical relaying communications over gamma-gamma ...
On The Average Channel Capacity Performance of Amplify-and-Forward MIMO/FSO Systems over Atmospheric Turbulence Channels and Misalignment Fading Duong Huu Ai†, ††, Duong Tuan Quang† †

Faculty of Electronics and Telecommunications, Vietnam Korea Friendship Information Technology College 136 Tran Dai Nghia Str, Ngu Hanh Son District, Danang, Viet Nam, [email protected] †† School of Electronics and Telecommunications, Hanoi University of Science and Technology No. 1, Dai Co Viet, Hai Ba Trung District, Hanoi, Viet Nam {aidh, quangdt}@viethanit.edu.vn

Abstract — This paper presents the theoretical analysis of average channel capacity performance of amplify-and-forward (AF) multiple-input multiple-output (MIMO) free-space optical (FSO) systems over atmospheric turbulence channels and misalignment fading. The AF - MIMO/FSO average channel capacity (ACC), which is expressed in terms of average spectral efficiency (ASE) is derived taking into account the atmospheric turbulence effects on the MIMO/FSO channel and number of relay stations and link distance. Atmospheric turbulence channels are modeled by log-normal and the gamma-gamma distributions for the cases of weak-to-strong turbulence conditions. The mathematical formulas of ACC for atmospheric turbulence cases are calculated and quantitatively discuss the influence of turbulence strength, different number of relay stations and different MIMO configurations on it. Keywords — Free-space optical; amplify-and-forward; average channel capacity; average spectral efficiency; atmospheric turbulence.

I. INTRODUCTION The necessity of a cost-effective, license-free, high security and high bandwidth access technique has lead to a continuous research and commercial interest in optical wireless communication systems [1]-[3]. One of major degradations to the performance of FSO communications is the influence of atmospheric turbulence caused by variations in the refractive index, pressure fluctuations in the air along the propagation path of the laser beam [4]. A recent study technology has been applied to increase the FSO system’s performance is the relaying transmission scheme [3]-[13]. For example, in [4] the error rate of multi-hop FSO fading channel has been deduced. In [5]-[8], the performance of FSO systems was concluded under two criteria, the average bit error probability and the outage probability. An optical relaying system for atmospheric channels has been indicated in [9]-[13].

Furthermore, the error performance of FSO system using sub-carrier modulation scheme has been extensively investigated in [14]-[24]. On the subject of SC-QAM systems, the average symbol error probability of the SISO FSO systems over atmospheric turbulence channels is presented in [17]. Recently, Hassan et al. [18] evaluated the sub-carrier intensity modulation in wireless optical communications. Bayaki et al. investigated the performance of MIMO FSO system under gamma-gamma fading model using series expansion of the modified Bessel function in [19]. In [20], Trung presented the ACC of MIMO FSO systems with different atmospheric turbulence conditions. However, the performance on average channel capacity of an AF FSO system using for both weak and strong atmospheric turbulence conditions has not yet mentioned in that study. This paper presents the theoretical analysis of average channel capacity performance of amplify-and-forward multiple-input multiple-output free-space optical over atmospheric turbulence channel and misalignment fading. In particular, theoretically derive and discuss the MIMO/FSO average channel capacity, which is expressed in terms of average spectral efficiency, under the impact of various channel conditions, number of relay stations, system parameters and configurations. The following part of the paper is structured as follows. The system model is described in details in Section II. In Section III, atmospheric turbulence models are presented. The next section, Section IV, will introduce the channel capacity of AF FSO links that is derived from different number of relaying nodes, link distance, and turbulence conditions. The last section, Section V, give the mathematical results of the systems’ ASE. The paper then draws a brief conclusion at the end.

Fig. 1. The source node, relaying node, and destination node of AF - MIMO/FSO system

II. SYSTEM MODEL General AF – MIMO/FSO system using SC-QAM signals with M transmitting lasers pointing toward an N aperture receiver as depicted in Fig 1. The MIMO/FSO channel can be modeled by M×N matrix of the turbulence channel, denoted as X   X mn m,n 1 . The

f X mn ( X ) 

2  X

1

2

(C  1)( A0 X l )

 ln(X/ Xl A0 )  a  1 b e  erfc   (4)  2 2 I  

a  0.5 I2   I2 ( 2  C)

where

b

2

  I2 ( 2

and

 C){1  (  C)}/ 2. 2

M ,N

electrical signal at the input of QAM demodulator can be expressed as follows [21] c M N c 2 i 1  re  t   Ps e(t )      X i 1 mn  Pi    ni (t ),  m 1 n 1 i  0  i 0

(1)

B. The gamma-gamma turbulence channel model of AF FSO communication systems The pdf of gamma-gamma fading channel, X mn  0, the pdf for an normalized irradiance with gamma-gamma, X mn  0 , is described as [24]

where, X mn denotes the stationary random process for the turbulence channel from the mth laser to the nth PD. When the equal gain combining (EGC) detector is employed at the destination node to the estimate the transmitted signal, the instantaneous electrical SNR can be expressed as a finite sum of sub-channels as 

C







2

 ( ) 2

fX ( X ) 

3,0 G1,3

C 1

(C  1)( A0 X l )( )(  )

2   (5) X    A0 X l  2  1,   1  C ,   1  C   

(2)

where A0  erf (v)2 is the fraction of the collected power at

where C is the number of relay station,  imn are RVs defined

radial distance 0, v is given by v   r /( 2z ) with r and  z respectively denote the aperture radius and the beam waist at the distance z and   zeq / 2 s , where the equivalent

M

N

    

 m 1 n 1 i  0

 imn 

as the instantaneous electrical SNR at the output of the nth PD caused by signal from the mth laser.  mn are given in the following 2

C  1   C  2i 1 Ps  X i 1 Pi  / N 0     X i 1   i 0  i 1  MN 

 mn  

beam radius can be calculated by  zeq   z (  erf(v) / 2v  exp(v 2 ))1/2

2

(3)

where X i 1 denotes the stationary random process of the turbulence channel from the ith AF to the AF (i+1)th AF,  is the photodiode (PD) responsivity, Ps denotes the average transmitted optical power per symbol at each hop,  (0    1) is the modulation index, Pi is the amplification power of the ith AF module, N 0 is the total noise variance. III. ATMOSPHERIC TURBULENCE MODELS A. The log-normal turbulence channel model of AF FSO communication systems In log-normal fading channel, the probability density function (pdf) for a normalized irradiance random variable (RV) with log-normal, X mn  0 , is described as [21]

where

1/2

z  0 1   ( L / 02 )2 

with

0

(6) is

the

transmitter beam waist radius at z  0 ,   (1  202 )/02 and

0  (0.55Cn2 k 2 L)3/5 is the coherence length, C n2 represents the refractive index structure, which depends on altitude [3] and it expressed by 2





10  v   h  Cn2 (h)  0.00594   105 h exp    27   1000   h   h  6 Cn2 (0) exp     2.7 10 exp   ,  1000   1500 

(7)

where, h is the altitude in meters, v is the wind speed in meters per second and Cn2 (0) is the value of C n2 at the ground in m-3/2 . C n2 varies from 1017 m-3/2 to 1013 m-2/3 for weak to strong turbulence cases, respectively.

IV. AVERAGE CHANNEL CAPACITY An optical wireless channel is a randomly time-variant channel and the received instantaneous electrical signal-tonoise ratio (SNR) is a random variable. Thus, the channel capacity must be considered as a random variable, and its average value, known as average channel capacity, C . The ACC can also be expressed in terms of average spectral efficiency (ASE) in bits/s/Hz if the frequency response of the channel is known. The average channel capacity is given by C   Blog 2 1     f Γ ()d , (bit/s/Hz) Γ

(8)

where B is the channel’s bandwidth and is the total channel



B. Capacity of AF FSO system using gamma-gamma distribution Substituting Eq. (10) into (8), the average channel capacity of AF FSO system can be given by

   C 1  B (C 1)( A0 X l )       2 ln  2   mn

   C 1  B (C 1)( A0 X l )       2 ln  2   mn    1,  2  G   2  A0 X l  1,  1,   1,   1  C ,   1  C 

in log-normal [21] and gamma-gamma distributions [24] as follows

f   mn  

2(C  1)( A0 X l )

2

0.5 2  mn

 0.5ln( mn /Xl2

erfc   

3,0 G1,3

C 1

1 )2

(9)

A02  mn )  a   

1

(C  1)( A0 X l )( )(  )

  ( mn /  mn  A0 X l 

e 

2 I

 ( ) 2

f ( mn ) 



b

2 mn

 (10)   2   1,   1  C ,   1  C 

2

V. NUMERICAL RESULTS Using the above closed mathematical forms as shown in (13) and (14) can estimate the average capacity of AF MIMO/FSO system over atmospheric turbulence channel. Using Eq (13) for the case of weak turbulence with log-normal distribution model, while (14) is used for strong case with the gamma-gamma distribution. In the analysis below, the average capacity is evaluated for three different values of turbulence strength, Cn2 , link distance L has been chosen to be moderate L = 2000 m, different values of number relay stations C , ( C  0, C  1, and C  2 ) and amplification gain PAF  2 dB. Assumes that, the relay stations are placed equidistant, turbulence conditions between relay of stations is the same. 6

k 1

xk ,0  x 1 k

(11)

The ASE of a log-normal of AF - FSO channels capacity can be expressed as  C  2  eb  2 c 1 (1) k 1    2 2 B k k 1 2 ln 2  (C  1)( A0 X l )  0.5 mn

 t 2  2 )  exp  exp(  2k  2)erfc(   2k  2  2 I   1 t (  I2 ( 2  2k  2) 2 erfc( I ( 2  2k  2)  ) 2 2 2 I  1

where c  a  ln( A02 X l2 ) , t  0.5ln   c.

ASE (b/s/Hz)

ln(1  x)   (1)

k 1

c C = 0, 1, 2, P AF = 2 dB

5

A. Capacity of AF FSO system using log-normal distribution Using Eqs. (8) and (9), and use Taylo expansion 

4

3

2

C2n = 310-14 m-2/3 C2n = 910-15 m-2/3

1

C2n = 110-15 m-2/3

L = 2000 m,  = 1500 nm 0

(12)

(14)

5,1 3,5

each element  imn . The pdfs of SNR are respectively described

1

C 1

2

MIMO atmospheric turbulence channels. The joint p.d.f f Γ () can be reduced to a product of the firt-order p.d.f of

0.5 2 1

mn



(13) 2   ( /  )0.5  G   d  mn A0 X l  2  1,   1  C ,   1  C   The ASE of AF – MIMO/FSO channel with gammagamma distribution model can be given by



 mn

 ln 1   

3,0 1,3

SNR and    imn , n  1,..., N , m  1,..., M is the matrix of the

2

C 1

2

0

10

20

30

40

50

60

SNR (dB) Fig. 2. ASE versus the average SNR of FSO channels with different atmospheric turbulence strengths, Cn2 under link distance L=2000 m

Figure 2 illustrate the ASE of different number of relay stations (i.e., C  0, C  1, C  2 ) with respect to  , for three values of the turbulence strength C n2 , with link distances

L = 2000 m, and amplification gain PAF  2 dB. It can be also seen that the ASE strongly depends on the atmospheric turbulence strength, the influence of atmospheric turbulence becomes stronger. Obviously, the ASE under the weak turbulence conditions is higher than in the cases of moderate and strong turbulence, especially with longer link distance L. It has been that observed that with increase in the value of C , capacity performance of the system deteriorates. 10 SISO M=N=2 M=N=4

8

C=0 7

C=1 6

C=2

strength C n2 , for two values of the relay stations C , ( C  0, C  1 ) and with link distances L = 2000 m. It can be also seen that the ASE strongly depends on the atmospheric turbulence strength, number of relay station, the influence of atmospheric turbulence becomes stronger. Obviously, the ASE under the weak turbulence conditions is higher than in the cases of moderate and strong turbulence. On the other hand, as expected, the ASE could be improved by approximately 2 (b/s/Hz) when the system is upgraded from SISO/FSO to 2×2 MIMO/FSO or from 2×2 MIMO/FSO to 4×4 MIMO/FSO for number of relay stations C  0, approximately 1 (b/s/Hz) for C  1.

5 4

6 C2n=310-14 m-2/3

3

0

C2n=910-15 m-2/3

5

2 1

0

L = 2000 m, P AF = 2, C2n=310-14 m-2/3 dB, 30 40 50

10

20 Average electrical SNR (dB)

60

Fig. 3. ASE versus the average electrical SNR of various AF - MIMO/FSO channels and different number of relay station for strong atmospheric turbulence strengths under link distance L = 2000 m

In Figure 3 illustrate the ASE of different AF MIMO/FSO channels (i.e., SISO, 2×2 and 4×4 MIMO/ FSO channels) with respect to  for three values of the relay stations C , ( C  0, C  1, and C  2 ) with link distances L = 2000 m, amplification gain PAF  2 dB, for strong 14

2/3

atmospheric turbulence, C  3  10 m . It can be also seen that the ASE strongly depends on the number relay stations, MIMO configurations. The impact of SNR on the ASE of systems is more significant in low regions than in high regions. 2 n

10

M=N=4

9

M=N=2 Average spectral efficiency (b/s/Hz)

Figure 4 illustrate the ASE of different AF - MIMO/FSO channels with respect to  , for three values of the turbulence

8 7 6 5

Average spectral efficiency (b/s/Hz)

Average spectral efficiency (b/s/Hz)

9

Fig. 4. ASE versus the average electrical SNR of different AF - MIMO/FSO channels for weak to strong atmospheric turbulence strengths under link distance L = 2000 m, number of relay stations C = 0.

C2n=110-15 m-2/3

M = N =4 4

M = N =2 3

SISO 2

1

L = 2000 m,  = 1500 nm 0

0

1

2 3 4 Number relay stations N

5

6

Fig. 5. ASE versus the number of relay stations of AF - MIMO/FSO channels for weak to strong atmospheric turbulence strengths under link distance L = 2000 m

Figure 5 illustrate the ASE of different AF - MIMO/FSO channels versus number of relay stations for three values of the turbulence strength under free-space link distance L = 2000 m. It is clearly shown that the ASE performance is improved significantly with the increase of number of lasers and receivers and the ASE performance is decreases significantly with the increase of number of relay stations. In addition, as expected, ASE increases as number of lasers M and receiver N increase from SISO to 2×2 MIMO and 4×4 MIMO. It can be also seen that the ASE strongly depends on the atmospheric turbulence strength, MIMO configurations.

4

VI. CONCLUSION

3 C2n=310-14 m-2/3

SISO

2

C2n=910-15 m-2/3

1

C = 0, L = 2000 m,  = 1500 nm 0

0

10

20 30 40 Average electrical SNR (dB)

C2n=110-15 m-2/3 50

60

This paper, have theoretically analyze on the average channel capacity performance of AF - MIMO/FSO systems over atmospheric turbulence channels and misalignment fading. The log-normal and gamma-gamma distribution models were used to describe the fluctuation of the optical propagating over atmospheric turbulence channels. The results

present the theoretical expressions for ASE performance of SISO and MIMO systems taking into account the number of AF relay stations, MIMO configurations and turbulence conditions. The numerical results showed that, with the similar link distance and turbulence strength, regardless the link distance and turbulence condition. The ASE strongly depends on the atmospheric turbulence strength, MIMO configurations, and number of relay stations. The numerical results proved that, regardless the link distance and turbulence condition, the ASE could be deteriorated by approximately 1 (b/s/Hz) when the number relay of stations is upgraded from C  0 to C  1 or approximately 0.7 (b/s/Hz) from C  1 to C  2 to achieve the SNR of 20 dB. The ASE could be improved by approximately 0.8 (b/s/Hz) when the amplification gain is upgraded from PAF  2 to PAF  6 to achieve the number of relay stations, C  1 or approximately 0.5 (b/s/Hz) to achieve the number of relay stations, C  2. REFERENCES A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” L. Opt. Fiber Commun. Rep., vol. 2, pp. 345–396, 2005. [2] S. Arnon, “Optical wireless communications,” in Encyclopedia of OpticalEngineering. New York: Marcel Dekker, 2003, pp. 1866–1886. [3] D. Kedar and S. Arnon, “Urban optical wireless communication networks: The main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no.5, pp.2-7, May 2004. [4] J. Akella, M. Yuksel, and S. Kalyanaraman, “ Error analysis of multihop free-space optical communication,” in Proc. IEEE Int. Conf. on Communications (ICC), Seoul, South Korea, May 2005. [5] M. Safari and M. Uysal, “ Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5441–5449, Dec. 2008. [6] M. Kamiri and N. Nasiri-Kerari, “ BER analysis of cooperative systems in free-space optical networks,” J. Lightwave Technol., vol. 27, no. 24, pp. 5639–5647, Dec. 2009. [7] C. Datsikas, K. Peppas, N. Sagias, and G. Tombras, “Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels,” J. Opt. Commun. Netw., vol. 2, pp. 576–586, Aug. 2010. [8] M. Kamiri and N. Nasiri-Kerari, “Free-space optical communications via optical amplify-and-forward relaying,”J. Lightwave Technol., vol. 29, no. 2, pp. 242–248, Jan. 2011. [9] M. Safari, M. M. Rad, and M. Uysal, “Multi-hop relaying over the atmospheric Poisson channel: Outage analysis and optimization,”IEEE Trans. Commun., vol. 60, no. 3, pp. 817–829, Mar. 2012. [10] M. A. Kashani, M. M. Rad, M. Safari, and M. Uysal, “ All-optical amplify-and-forward relaying system for atmospheric channels,” IEEE Commun. Lett., vol. 16, no. 10, pp. 1684–1687, Oct. 2012. [1]

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