On the performances of relay-aided FSO system over M distribution with pointing errors in presence of various weather conditions

Optics Communications 367 (2016) 59–67

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Invited Paper

On the performances of relay-aided FSO system over M distribution with pointing errors in presence of various weather conditions Ping Wang a,b,n, Ranran Wang a, Lixin Guo b, Tian Cao a, Yintang Yang c a

State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, Xi’an 710071, China School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China c Key Laboratory of the Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices, School of Microelectronics, Xidian University, Xi’an 710071, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 7 October 2015 Received in revised form 3 December 2015 Accepted 3 January 2016 Available online 4 February 2016

The average bit error rate (ABER) and outage performances of decode-and-forward (DF) based multi-hop parallel free-space optical (FSO) communication system with the combined effects of path loss, pointing errors (i.e., misalignment fading), and atmospheric turbulence-induced fading modeled by M distribution have been investigated in detail. Particularly, the end-to-end probability density function (PDF) and cumulative distribution function (CDF) over the aggregated fading channel are derived for the first time. Based on the binary phase-shift keying (BPSK) subcarrier intensity modulation scheme, the analytical expressions for the end-to-end ABER and outage probability are obtained, respectively. The ABER and outage performances of the present FSO system are then analyzed systematically with the effects of turbulence strengths, weather conditions, pointing errors, and structure parameters ( M and N ) taken into account. This study shows that the turbulent atmosphere, weather conditions and pointing errors can be mitigated by increasing the number of cooperative path (N ) over M fading channels. For the fixed hop length, the FSO system performance will be degraded with the increasing hop numbers (M ). But the performance will be improved with the increasing hop numbers (M ) when the total distance from the source to destination is fixed. Monte Carlo simulation is also provided to verify the correctness of the proposed ABER expression. & 2016 Elsevier B.V. All rights reserved.

Keywords: M distribution Path loss Pointing error Multi-hop parallel FSO system ABER Outage probability

1. Introduction Free-space optical communication (FSO) has received considerable attention for recent years as a cost-effective, licensefree and wide-bandwidth access technique and has been applied in high data rate wireless links [1–3]. However, the performance and availability of FSO links are limited by a number of atmospheric related issues. Among them, path loss is one of the major impairments, which normally takes place because of the adverse weather conditions (e.g. haze, fog) and results in a substantial loss of optical signal power in the communication path [4]. Another possible impairment over FSO links is the misalignment between transmitter and receiver due to building sway caused by weak earthquakes, dynamic wind loads and thermal expansion. It can induce vibrations of the transmitted beam, leading to further performance degradation by pointing errors [5]. In addition, atmospheric turbulence, which occurs as a result of the rapid n

Corresponding author. E-mail address: [email protected] (P. Wang).

http://dx.doi.org/10.1016/j.optcom.2016.01.004 0030-4018/& 2016 Elsevier B.V. All rights reserved.

fluctuations at the received signal and thereby increases the error rate, severely degrades the overall performance particularly over distances of 1 km or longer [6]. As an effective method to overcome such limitations, relaying transmission in FSO systems has gained increasing attention recently, and a lot of works have been reported on two more generic relay-aided schemes, namely, serial relaying (i.e., multi-hop transmission) and parallel relaying (i.e., cooperative diversity). In [7–9], the performance of multihop FSO system has been investigated. These studies show that multi-hop transmission is an effective method to broaden the coverage area for limited power transmitters, but it cannot provide diversity gains. Parallel relaying is another important scheme, which can achieve an artificial broadcasting with multilaser transmit apertures directed to the relay nodes [10,11]. However, its performance gains are less than those observed in serial relaying since parallel relaying (with only two hops) exploits distance-dependency of fading variance to a lesser extent. Very recently, a more practical FSO mesh network in which serial and parallel relaying are deployed together has been proposed by Kashani and Uysal in [12] with decode-and-forward (DF) protocol. It was shown that the outage performance of the so-called

60

P. Wang et al. / Optics Communications 367 (2016) 59–67

M-1 Relays

Relay

Relay

Source

Relay …

…

N paths

Relay

Destination

Fig. 1. The schematic of multi-hop parallel DF FSO cooperative communication system.

multi-hop parallel FSO system outperforms that of both standalone serial and parallel relaying schemes. In order to evaluate the performance of FSO system, an accurate mathematical model to describe the optical channel characteristics with respect to the atmospheric turbulence is required. Over the years, various statistical fading models have been presented, such as the log-normal (LN) and Gamma–Gamma (GG) distributions. As known, the scope of the LN model is restricted in weak turbulence regime. In [13], the outage probability for multi-hop FSO system with DF protocol has been analyzed based on the LN distribution. Compared with LN distribution, GG distribution can be used for all the turbulence regimes. The outage probability and the average symbol error rate (ASER) of the multi-hop DF FSO links over the GG turbulence channels considering the path loss and pointing errors have been studied systematically in [14]. Nevertheless, both LN and GG distributions are special cases of a more generic model named M distribution or Málaga distribution, which is proposed by Jurado-Navas et al. in [15]. Their study shows that M distribution unifies in an analytical expression most of the irradiance statistical models, such as LN, GG distributions and so on. Besides, this fading model could achieve excellent agreements with the experimental data under weak-to-strong turbulence conditions. Then, the outage performance of FSO system based on standalone serial and parallel relaying schemes over the M fading channels has been investigated by Wang et al. in [16]. However, to the best of our knowledge, no works on ABER and outage probability performances of the FSO system based on the multi-hop parallel relaying scheme with the joint effects of path loss, pointing errors and the M distributed turbulence have been reported. Motivated by the above analysis, the performances of a multihop parallel DF based FSO cooperative communication system over an aggregated channel, consisting of the path loss, pointing errors, and M-distributed atmosphere turbulence, have been studied deeply. Based on the best path selection scheme, the end-toend probability density function (PDF) and cumulative distribution function (CDF) are achieved with Meijer's G-function. After that, the ABER of identically and independently distributed (i.i.d.) FSO system is derived in terms of Gauss–Laguerre quadrature rule. Furthermore, the analytical expression of outage probability is also obtained. The ABER performance of the considered system has

Table 1 Attenuation coefficients for different weather conditions. Weather condition Attenuation σ (dB/km) Very clear air Clear air/drizzle Haze Light fog

0.0647 0.2208 0.7360 4.2850

been analyzed systematically with different turbulence strengths, weather conditions, pointing errors, and structure parameters ( M and N ). Monte Carlo simulation is also provided to confirm the correctness of the analytical ABER expression.

2. System and channel models 2.1. System model A cooperative FSO communication system is depicted in Fig. 1. Assuming that there is a source node that communicates with a destination node via N + 1 paths, including N parallel paths and a direct link. Let M denote the total hops in each parallel path, that is, there are M − 1 relays in each path. Based on the symbol-wise DF relaying method, only one relay is allowed to decode and retransmit the signal to the next node at one time without using any forward error correction (FEC). A cooperative path is chosen based on the best path selection scheme to implement the transmission of the data from the source node to the destination node. The system under consideration uses intensity-modulation direct-detection employing BPSK modulation. Thus, the received signal at the j-th hop in the i-th path can be written as [17]

yi, j = Ii, j RPi, j X (t ) + n (t ), ∀ i ∈ ( 1, ... , N ), j ∈ ( 1, ... , M )

(1)

where Ii, j is the aggregated channel gain of the j-th hop in the i-th path, R is the receiver responsivity, Pi, j represents the average transmitted power in each link, X (t ) ∈ {−1, 1} is the source signal

P. Wang et al. / Optics Communications 367 (2016) 59–67

level, and n (t ) is the additive white Gaussian noise with zero mean and variance σn2 = N0/2 [18]. Here, N0 is the single sided power spectral density. Hence, the instantaneous signal to noise ratio (SNR) of j-th hop in the i-th path can be expressed as [19]

γi, j = γ¯i, j Ii2, j =

(RPi, j )2 2 Ii, j N0

(2)

models. For instance, setting ρ = 1, M distribution will reduce to the Gamma–Gamma (GG) model and it will reduce to the LN distribution with ρ = 0, Var [ UL ] = 0, μ → 0. Thus, the PDF of Ii, j can be obtained by

2.2. Atmospheric turbulence model fI

In this work, the aggregated channel model including the path loss Iil, j , the pointing error Iip, j and the turbulence-induced fading Iia, j has been considered. In this case, the channel gain Ii, j can be modeled as [20]

(3)

where Iil, j represents the deterministic path loss of j-th hop in the ith path, and it is determined by the Beer–Lambert’s law as

Iil, j = exp ( −σL i, j )

(4)

where σ is the attenuation coefficient, L i, j is the link length of j-th hop in the i-th path. The σ under different weather conditions at a wavelength of 1550 nm is chosen from [21] and the values adopted can be found in Table 1. Here, Iip, j in (3) denotes the pointing error. Considering the independent and identical Gaussian distributions for both horizontal and vertical sway, the PDF of Iip, j can be expressed as [22]

( )

f I p Iip. j = i .j

g2 A 0g

2

2 p g −1 , i, j

(I )

(5)

where A0 = [erf (v )] is the fraction of the collected optical power with v = ( π a) /( 2 wzi, j ), erf (⋅) is the error function, a denotes the radius of the receiver and wzi, j is the beamwidth at the distance of

L i, j . Moreover, g = wzeq/(2σs ), σs2 is the jitter variance, wzeq is the equivalent beamwidth and it can be obtained by

wzeq = wzi, j π erf (v ) /2v exp ( − v 2) . a Ii, j in (3) represents the turbulence fading, which is modeled by the M distribution, the PDF of Iia, j in each link is given as [15]

( )

f Iia, j Iia, j = A ∑ k=1

⎛ α+k −1 ak Iia, j 2 Kα − k ⎜⎜ 2

( )

⎝

a i, j Ii, j

( Ii,j Iia,j ) f I ( Iia,j ) dIia,j a i, j

a i, j Ii, j

(8)

the condition probability for a given turbuis defined as

⎛ I ⎞ ⎛ I ⎞g g2 i, j ⎟ ⎜ i, j ⎟ f I p ⎜⎜ = ⎜ a l ⎟ 2 l i, j I a I l ⎟ ⎝ i, j i, j ⎠ i, j A 0g Iia, j Iil, j ⎝ Ii, j Ii, j ⎠

2−1

( I I ) = I 1I a i, j i, j

a i, j

, 0 ≤ Ii, j ≤ A 0 Iia, j Iil, j

(9)

Substituting (6) and (9) into (8), the PDF of Ii, j can be expressed as fI

i, j

( Ii, j ) =

g 2A

(A I ) l 0 i, j

g2

( Ii, j )

β

g 2−1

∑

ak

k =1

∫I

∞

l i, j / Ii, j A0

(I ) a i, j

α +k −1 −g 2 2 K

⎛ ⎜

α −k ⎜ 2

⎝

αβIia, j ⎞ ⎟ a dI μβ + Ω′ ⎟ i, j ⎠

(10)

With the help of (07.34.21.0085.01) in [24] and (14) in [25], the final fIi, j (Ii, j ) can be obtained as β −1

fIi, j ( Ii, j ) = g 2A ( Ii, j )

∑ k=1

⎡ 2 ⎤ ak − α + k 3,0 ⎢ Ii, j 1 + g ⎥ B 2 G1,3 B ⎢⎣ A 0 Iil, j g 2 , α, k ⎥⎦ 2

(11)

where B = αβ /(μβ + Ω′), Gpm, q, n [⋅] is the Meijer's G-function. And then, the PDF with regard to γi, j can be derived by the relationship ⎛ γ ⎞ 1 f (γi, j ) = 2 γ γ¯ f ⎜ i, j ⎟ as i, j i, j ⎝ γ¯i, j ⎠ ⎡ β γi, j 1 + g 2 ⎤ α+k −1 a 3,0 ⎢ B ⎥ f γi, j = γi, j g 2A ∑ k B− 2 G1,3 ⎢⎣ A 0 Iil, j γ¯i, j g 2 , α, k ⎥⎦ 4 k=1 (12)

( ) ( )

0 ≤ Iip, j ≤ A 0

2

β

∫ fI

fIi, j ( Ii, j ) =

where fIi, j I a (Ii, j Iia, j ) is i, j lence state Iia, j , which

where γ¯i, j = (RPi, j )2 /N0 is the average SNR of each link.

Ii, j = Iil, j Iip, j Iia, j,

61

⎞ ⎟ μβ + Ω′ ⎟⎠ αβIia, j

(6)

β

( )

F γi, j = g 2A ∑ k=1

⎡ γi, j 1, 1 + g 2 ⎤ ak − α + k 3,1 ⎢ B ⎥ B 2 G2,4 ⎢⎣ A 0 Iil, j γ¯i, j g 2 , α, k, 0⎥⎦ 2

(13)

3. Performances analysis of subcarrier coherent modulations For the multi-hop parallel DF based FSO cooperative system, the equivalent instantaneous SNR, γeqi , of the i-th path can be obtained as follows [26]

γequi = min (γi,1, γi,2, ... , γi, M )

where α ⎧ β+ α 2α 2 ⎛ μβ ⎞ 2 ⎪A= ⎜ ⎟ α ⎪ μ1 + 2 Γ(α ) ⎝ μβ + Ω′ ⎠ ⎪ ⎨ k k ⎪ ⎛ β − 1⎞ ( μβ + Ω′)1 − 2 ⎛ Ω′ ⎞k − 1⎛ α ⎞ 2 ⎪ ak = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎝ k − 1⎠ ( k − 1) ! ⎝ μ ⎠ ⎝ β ⎠ ⎩

Thus, the CDF in terms of γi, j can be given as

(14)

It is assumed that all the hops have the same average SNR, i.e., γ¯i, j = γ¯s, d = γ¯ , here γ¯s, d is the average SNR for the direct link between the source and destination. Considering the identically and independently distributed (i.i.d.) FSO system, the CDF of γequi can be written as

(7)

Additionally, α is a positive parameter related to the effective number of large-scale cells of the scattering process, and β is a natural number, which denotes the amount of fading parameter [23]. Kν (⋅) is the second kind of modified Bessel function of order ν . Γ (⋅) is the Gamma function. Here, μ = ρμ0 , μ0 is the average power of the classic scattering components received by the off axis, p (0 ≤ ρ ≤ 1)) represents the amount of scattering power coupled to the line-of-sight (LOS) component (UL ). The parameter Ω′ denotes the average power from the coherent contributions. As we know, the M distribution covers several commonly used fading

⎡ ⎤M F γequ (γ ) = 1 − ⎣ 1 − F γi, j (γ ) ⎦ i ⎡ ⎡ β a − α + k 3,1 ⎢ B ⎢ = 1 − ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎢ l 2 ⎢⎣ k =1 ⎣ A 0 I i, j

M 2 ⎤⎤ γ 1, 1 + g ⎥ ⎥ ⎥ ⎥ 2 γ¯ g , α , k , 0 ⎥ ⎦⎦

(15)

where the Fγi, j (γ ) is the CDF of the j-th hop in the i-th path. Based on the best path selection scheme, the signal with the largest SNR of the γequi (i = 1, ... , N ) and γs, d (the instantaneous SNR of the direct link between the source and the destination) is selected [27]. Thus, the equivalent output SNR γequ at the destination can be achieved as

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P. Wang et al. / Optics Communications 367 (2016) 59–67

γequ = max (γs, d, γ ′equ )

(16)

where γ′equ = max γequi , and the CDF of γ′equ is

3.1. The end-to-end ABER

⎡ ⎤N F γ ′equ (γ ) = ⎣ F γequ (γ ) ⎦ i

For this cooperative system, the end-to-end ABER at the destination can be calculated by the conditional error rate Pe , which can be expressed as

N ⎡ ⎡ ⎤M⎤ = ⎢ 1 − ⎣ 1 − F γ i, j (γ ) ⎦ ⎥ ⎣ ⎦

P¯e =

i = 1,..., N

⎧ ⎡ ⎡ β ⎪ a − α + k 3,1 ⎢ B ⎢ = ⎨ 1 − ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎢ A Il 2 ⎪ ⎢⎣ k=1 ⎣ 0 i, j ⎩

⎤ ⎤M⎫ γ 1, 1 + g 2 ⎥ ⎥ ⎪ ⎬ γ¯ g 2, α , k , 0 ⎥ ⎥⎥ ⎪ ⎦⎦ ⎭

N

(17)

The PDF can be obtained by differentiating (17) with regard to γ as [24, (07.34.20.0001.01)]

N −1 ⎧ ⎡ ⎤M ⎫ ⎡ β ⎪ ⎪ 2⎤ ⎪ ⎢ ak − α + k 3,1⎢ B γ 1, 1 + g ⎥ ⎥ ⎪ 2 ⎨ ⎬ B 2 G = MN 1 − ⎢ 1 − g A ∑ ⎥⎥ 2,4 ⎢ A I l 2 ⎪ ⎢ ⎢⎣ 0 i, j γ¯ g 2, α, k , 0⎥⎦ ⎥⎦ ⎪ k =1 ⎪ ⎣ ⎪ ⎩ ⎭

(18)

Then, the CDF of γs, d can be expressed as

⎡ 2⎤ γ 1, 1 + g ⎥ ak − α + k 3,1 ⎢ B B 2 G2,4 ⎢⎣ A 0 Isl , d γ¯ g 2 , α, k, 0⎥⎦ 2

β k=1

⎡ 2 ⎤ γ 1+g ⎥ ak − α + k 3,0 ⎢ B B 2 G1,3 ⎢⎣ A 0 Isl , d γ¯ g 2 , α, k ⎥⎦ 4

(19)

(20)

With the fact that Fγequ = Fγ ′equ × Fγs, d , the CDF of γequ , Fγequ , can be given as N ⎧ ⎤M ⎫ ⎡ ⎡ β ⎪ 2⎤ ⎪ ⎪ ⎢ a − α + k 3,1 ⎢ B γ 1, 1 + g ⎥ ⎥ ⎪ ⎬ Fγequ (γ ) = ⎨ 1 − ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎥⎥ ⎢ l 2 2 ⎪ ⎢⎣ ⎢⎣ A 0 Ii, j γ¯ g , α , k , 0 ⎥⎦ ⎥⎦ ⎪ k=1 ⎪ ⎪ ⎭ ⎩ ⎡ ⎤ β 2 ak − α + k 3,1 ⎢ B γ 1, 1 + g ⎥ 2 B 2 G 2,4 ⎢ ×g A ∑ ⎥ l 2 ⎢⎣ A 0 Is , d γ¯ g 2, α , k , 0 ⎥⎦ k=1

⎛ ⎡ β a − α + k 3,1 ⎢ B ⎜ × ⎜ g 2A ∑ k B 2 G 2,4 ⎢ A Il 2 k=1 ⎣ 0 s, d ⎝

1 2 π

∫0

β

∑

⎤⎞ γ 1 + g 2 ⎥⎟ 2 γ¯ g , α , k ⎥⎦ ⎟ ⎠

(26)

β ⎧ ⎡ ⎪ α +k 1 a 3,1 γ − 2 e−γ ⎨ 1 − ⎢ 1 − g 2A ∑ k B − 2 G 2,4 ⎪ ⎣⎢ k =1 2 ⎩

k =1

N

⎡ ak − α + k 3,1⎢ B B 2 G 2,4 ⎢ A Il 2 ⎣ 0 s, d

2 ⎤ γ 1, 1 + g ⎥ dγ 2 γ¯ g , α, k, 0 ⎥ ⎦

Here, (27) is in the form of

1

∞ a x e−xf 0

∫

t

⎧ ⎪

⎡

⎩

⎢⎣

β

∑ p = 1 Wp⎨⎪ 1 − ⎢ 1 − g 2A ∑

2 π

⎡ B ×⎢ ⎢ A Il ⎣ 0 i, j

∑ k =1

(27)

(x ) dx and it can be further

xp γ¯

k =1

M⎫ ⎤⎤ ⎪ 1, 1 + g 2 ⎥⎥ ⎪ ⎬ ⎥⎥ g 2, α , k , 0 ⎦⎥⎦ ⎪ ⎪ ⎭

⎡ ak − α + k 3,1⎢ B B 2 G 2,4 ⎢ A Il 2 ⎣ 0 s, d

ak − α + k 3,1 B 2 G 2,4 2

N

⎤ x p 1, 1 + g 2 ⎥ 2 γ¯ g , α , k , 0 ⎥ ⎦

(28)

where xp is the p-th root of the generalized Laguerre polynomial

Lt(−1/2) (x ), and the weight can be calculated by [29]

⎤⎞ γ 1 + g 2 ⎥⎟ γ¯ g 2, α , k ⎥ ⎟⎟ ⎦⎠

⎤ ⎤M⎫ γ 1, 1 + g 2 ⎥ ⎥ ⎪ ⎬ γ¯ g 2, α , k , 0 ⎥ ⎥⎥ ⎪ ⎦⎦ ⎭

(25)

⎤M ⎫ 2 ⎤ ⎪ γ 1, 1 + g ⎥⎥ ⎪ ⎬ γ¯ g 2 , α, k, 0 ⎥⎥⎥ ⎪ ⎦⎦ ⎪ ⎭

⎡ B ×⎢ ⎢ A 0 Il ⎣ × g 2A

∞

β

Wp =

⎤⎞ γ 1, 1 + g 2 ⎥ ⎟ γ¯ g 2, α , k , 0 ⎥⎦ ⎟ ⎠

⎛ ⎡ β a − α + k 3,0 ⎢ B ⎜ × ⎜ γ −1g 2A ∑ k B 2 G1,3 ⎢ A Il 4 k=1 ⎣ 0 s, d ⎝

P¯e =

× g 2A

N −1 ⎤ ⎤M⎫ γ 1, 1 + g 2 ⎥ ⎥ ⎪ ⎬ γ¯ g 2, α , k , 0 ⎥ ⎥⎥ ⎪ ⎦⎦ ⎭

⎤ ⎤M−1 γ 1, 1 + g 2 ⎥ ⎥ γ¯ g 2, α , k , 0 ⎥ ⎥⎥ ⎦⎦

⎧ ⎡ ⎡ β ⎪ a − α + k 3,1 ⎢ B ⎢ + ⎨ 1 − ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎢ l 2 ⎪ ⎢⎣ k=1 ⎣ A 0 I i, j ⎩

(24)

1 − 1 −γ γ 2e 2 π

(21)

fγequ (γ ) = fγ ′equ (γ ) F γs , d (γ ) + F γ ′equ (γ ) fγ (γ ) s, d ⎧ ⎡ ⎡ β ⎪ a − α + k 3,1 ⎢ B ⎢ = MN ⎨ 1 − ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎢ l 2 ⎪ ⎢⎣ k=1 ⎣ A 0 Ii, j ⎩

Pe′(γ ) Fγequ (γ ) dγ

1 erfc ( γ ) 2

Pe′(γ ) = −

P¯e =

fγequ , can be achieved as

⎛ ⎡ β a − α + k 3,0 ⎢ B ⎜ × ⎜ γ −1g 2A ∑ k B 2 G1,3 ⎢ l ⎜ 4 k=1 ⎣ A 0 I i, j ⎝

∞

approximated by using the generalized Gauss–Laguerre quadrature rule [28] as

Taking the derivative of Fγequ with respect to γ , the PDF of γequ ,

⎡ ⎡ β a − α + k 3,1 ⎢ B ⎢ × ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎢ l 2 ⎢⎣ k=1 ⎣ A 0 Ii, j

∫0

P¯e = −

Thus, substituting (26) and (21) into (24), the end-to-end ABER at the destination can be written as

where Isl , d is path loss of the direct link between the source and the destination. The PDF of γs, d can be obtained as

fγs, d (γ ) = (γ )−1g 2A ∑

(23)

where erfc (x ) is the complementary error function. By taking the derivative of Pe (γ ), Pe′(γ ) can be given by

⎛ ⎞ ⎡ β 2⎤ ⎜ a − α + k 3,0 ⎢ B γ 1 + g ⎥⎟ × ⎜ γ−1g 2A ∑ k B 2 G1,3 ⎢ ⎥⎟ l 2 4 ⎜ ⎢⎣ A 0 Ii, j γ¯ g , α, k ⎥⎦ ⎟ k =1 ⎝ ⎠

k=1

Pe (γ ) fγequ (γ ) dγ

Using the method of integration by parts, (23) can be simplified with the help of the CDF of the variable γequ as

Pe (γ ) =

⎡ ⎤M − 1 ⎡ β 2⎤ ⎢ a − α + k 3,1⎢ B γ 1, 1 + g ⎥ ⎥ × ⎢ 1 − g 2A ∑ k B 2 G 2,4 ⎥ ⎢ ⎥ 2 ⎢ ⎢⎣ A 0 Iil, j γ¯ g 2, α, k , 0⎥⎦ ⎥⎦ k =1 ⎣

β

∞

where Pe′(γ ) is the first order derivative of Pe (γ ) with respect to γ . For BPSK modulation, the conditional BER over the AWGN channel can be obtained as

⎤N − 1 ⎡ ⎤M − 1 ⎡ ⎤M ⎡ fγ ′equ (γ ) = MN ⎢ 1 − ⎣⎢ 1 − Fγ (γ ) ⎦⎥ ⎥ × ⎣⎢ 1 − Fγ i, j (γ ) ⎦⎥ × fγ (γ ) i, j ⎥⎦ ⎢⎣ i, j

Fγs, d (γ ) = g 2A ∑

∫0

Γ (t + 1/2) xp 1/2) ⎤2 t!(t + 1)2 ⎡⎣ L t(− + 1 (xp ) ⎦

(29)

3.2. Outage probability

N

The outage probability is defined as the probability that the endto-end output SNR falls below a specified threshold γth [30]. Then, the outage probability of this FSO system can be expressed as

(22)

(

) ∫

Pout = Pr γequ < γth =

0

γth

fγequ (γ ) dγ

(30)

P. Wang et al. / Optics Communications 367 (2016) 59–67

Substituting (21) into (30), the final outage probability can be obtained as ⎧ M ⎫N ⎡ ⎡ β 2 ⎤⎤ ⎪ ak − α + k 3,1⎢ B γth 1, 1 + g ⎥ ⎥ ⎪ ⎢ 2 Pout = ⎨ 1 − ⎢ 1 − g A ∑ B 2 G 2,4 ⎢ ⎥⎥ ⎬ l 2 γ 2 ⎪ ⎢⎣ ⎣ A 0 Ii, j ¯ g , α, k, 0⎦ ⎥⎦ ⎪ k =1 ⎪ ⎪ ⎩ ⎭ ⎡ 2⎤ ak − α + k 3,1⎢ B γth 1, 1 + g ⎥ B 2 G 2,4 ⎢ l γ¯ g 2, α, k, 0⎥ 2 A I 0 k =1 ⎦ ⎣ s, d β

× g 2A

∑

(31)

4. Results and discussion In this section, the analytical results of the end-to-end ABER and outage probability of the present FSO system have been achieved from (28) and (31), respectively. When computing the generalized Gauss–Laguerre approximations, parameter t is chosen to be 30. The acceptance/rejection method and inverse transform method are adopted in MC simulation to generate random numbers of M distribution and pointing errors, respectively [31]. The structure parameters (N = 2, M = 3), (N = 2, M = 6) and (N = 4, M = 3) have been selected to avoid entanglement. Without loss of generality, the equal distance in each hop is assumed. Details of the system parameters employed are summarized in Table 2. Under moderate turbulence condition, the end-to-end ABER against SNR for different weather conditions is illustrated in Fig. 2 with normalized beamwidth Wz /a = 20 and normalized jitter σs/a = 3. It can be found from the figure that the analytical results have excellent agreements with MC simulation, which confirms the accuracy of our model. The value of ABER is the least under very clear weather condition and increases significantly in the presence of haze and light fog for fixed N and M . For example, in Fig. 2(a), when the SNR equals 80 dB, the ABER of N = 2, M = 3 is about 1 × 10−6 for very clear weather condition, while it is about 2 × 10−4 for light fog weather condition. It can be also found from Fig. 2 that for the same weather condition, the ABER values increase with the increasing M for fixed hop length. For instance, to achieve the ABER of 10−6 under light fog weather condition, about 93 dB of SNR is required for N = 2, M = 3 in Fig. 2(a) while about 97 dB is needed for N = 2, M = 6 in Fig. 2(b). Actually, this phenomenon is contrary to the case when the total distance from the source to destination is fixed (Please see Appendix for reference). Besides, the ABER performacne will be improved with increasing N . For example, to achieve the ABER of 10−6 , the SNR difference between N = 4, M = 3 in Fig. 2(c) and N = 2, M = 3 in Fig. 2(a) is about 13 dB for the clear air/drizzle weather condition. The end-to-end ABER versus SNR under strong turbulence and very clear weather condition with different normalized beamwidth ( Wz /a = 10 and Wz /a = 15) and fixed normalized jitter ( σs/a = 3) is shown in Fig. 3. For comparison, the performance of direct link is also presented. It can be seen that the performance of multi-hop parallel FSO system outperforms that of the direct link FSO regardless of the selected parameters. The end-to-end ABER

63

increases with the increasing normalized beamwidth at given values of N and M [32]. This is because the collected power at the receiver will decrease with the increasing beamwidth, which will deteriorate the received SNR. Therefore, the ABER performance is degraded. For example, under strong turbulence and very clear weather conditions, to achieve the ABER of 10−6 , the SNR difference between Wz /a = 10 and Wz /a = 15 is about 5 dB for N = 2, M = 3. The end-to-end ABER against electrical SNR under moderate turbulence and haze weather conditions with different normalized jitter (σs/a = 4 and σs/a = 5) for Wz /a = 15 is illustrated in Fig. 4. It can be seen that the higher value of σs/a is, the larger ABER will be for fixed N and M . This is because that the effect of misalignment between the transmitter and the receiver will become stronger when the jitter increases, leading to a degradation of the ABER. For example, under moderate turbulence and haze weather conditions, to achieve the ABER of 10−6 , the SNR of σs/a = 4 is about 2 dB less than that of σs/a = 5 for N = 2, M = 3. Besides, it can be found from Figs. 3 and 4 that the ABER performance improves as the value of M decreases for fixed hop length. For instance, under strong turbulence and very clear weather conditions, to achieve the ABER of 10−6 , the SNR difference between N = 2, M = 3 and N = 2, M = 6 is about 5 dB for Wz /a = 10 in Fig. 3. Furthermore, it is observed that the degradation of ABER performance caused by pointing errors can be mitigated by higher value of N . For example, to achieve the ABER of 10−6 , in the presence of moderate turbulence and haze weather conditions, the SNR difference is about 10 dB between N = 2, M = 3 and N = 4, M = 3 for σs/a = 4 in Fig. 4. The end-to-end ABER of multi-hop parallel FSO cooperative communication system with different structure parameters in the presence of clear air/drizzle and light fog weather conditions is presented in Fig. 5. Under weak, moderate and strong turbulence conditions, the corresponding scintillation index σI2 equals 0.36, 0.8360 and 1.466, respectively. The pointing error normalized jitter σs/a and beamwidth Wz /a are 3 and 20, respectively. It can be seen that the analytical results match very well with the MC simulation, which verifies the validity of the proposed model. The ABER of this system increases with the increase of the strength of atmosphere turbulence. For example, under clear air/drizzle weather conditions, to achieve P¯e = 10−6 , about 67, 81 and 92 dB are required for N = 2, M = 3 in weak-to-strong turbulence regimes, respectively. Furthermore, it can be found from Fig. 5(a) that the ABER increases with the increasing number of hops ( M ) in each path for fixed hop length. For instance, under moderate turbulence and clear air/drizzle weather conditions, when SNR is equal to 81 dB, the ABER for N = 2, M = 3 is about 1 × 10−6 , while it is about 0.3 × 10−5 for N = 2, M = 6 . Additionally, as shown in Fig. 5(b), the ABER performance improves with the increasing number of paths ( N ). And this effect becomes more apparent with the increasing turbulence strength. For example, under light fog weather condition, at a given ABER of 10−6 , the SNR difference between N = 2, M = 3 and N = 4, M = 3 is about 5 dB under weak turbulence condition, but 9 dB under the moderate turbulence condition.

Table 2 System parameters. Parameter

Symbol

Value

Receiver responsivity Noise standard deviation

R σn

0.5 A/W

The link length of j-th hop in the i-th path

L i, j

10−7 A/Hz 1.5 km

Weak turbulence Moderate turbulence Strong turbulence

(α , β , ρ ) (α , β , ρ ) (α , β , ρ )

(10,5,0.95) (4.3,3,0.75) (2.3,2,0.5)

64

P. Wang et al. / Optics Communications 367 (2016) 59–67

100 End-to-end ABER

-1

10

10-2 10-3 10-4 10-5 10-6 30

End-to-end ABER

100 10-1 10-2

very clear air clear air/drizzle haze light fog MC simulations 40 50 60 70 SNR(dB)

80

90

100

Moderate turbulence N=2,M=6 σs/a=3,Wz/a=20

Fig. 3. End-to-end ABER performance of multi-hop parallel cooperative system versus SNR with normalized beamwidth Wz /a = 10, 15 and fixed jitter σs/a = 3 under strong turbulence and very clear weather condition.

10-3 10-4 10-5 10-6 30

100 -1

10 End-to-end ABER

Moderate turbulence N=2,M=3 σs/a=3,Wz/a=20

10-2

very clear air clear air/dizzle haze light fog MC simulations 40 50 60 70 SNR(dB)

80

90

100

Moderate turbulence N=4,M=2 σs/a=3,Wz/a=20 Fig. 4. End-to-end ABER performance of multi-hop parallel cooperative system versus the electrical SNR with normalized jitter σs/a = 4, 5 and fixed beamwidth Wz /a ¼15 under moderate turbulence and haze weather conditions.

10-3 10-4 10-5 10-6 30

very clear air clear air/drizzle haze light fog MC simulations 40 50 60 70 SNR(dB)

80

90

100

Fig. 2. End-to-end ABER performance of multi-hop parallel cooperative system versus SNR under moderate turbulence and different weather conditions. The structure of (a) is N = 2, M = 3. The structure of (b) is N = 2, M = 6 . The structure of (c) is N = 4, M = 3.

Fig. 6 plots the outage probability versus SNR for this multi-hop parallel FSO system with different structure parameters. The very clear and light fog weather conditions have been taken into account for both moderate and strong turbulence conditions. It can be seen that the weather condition has less effect on outage probability under moderate turbulence condition than that of strong turbulence condition. For example, to achieve the outage

probability of 10−6 , the SNR differences between the very clear and light fog weather conditions are about 11 dB and 14 dB for moderate and strong turbulence conditions with N = 2, M = 3, respectively. Fig. 7 shows the the outage probability with different normalized beamwidths and jitters for clear air/drizzle weather condition under weak turbulence condition. It can be found that the outage probability increases with the increase of normalized beamwidth Wz /aand jitter σs/a when the values of structure parameters ( M and N ) are fixed. For instance, in Fig. 6(a), for the structure parameters N = 2, M = 3, to achieve the outage probability of 10−6 , approximate 60 dB and 63 dB are required for Wz /a = 10 and Wz /a = 15, respectively, under clear air/drizzle weather condition. In Fig. 7(b), to achieve Pout = 10−6 , about 65 dB and 70 dB are required for σs/a = 4 and σs/a = 5, respectively, under the same weather condition. In addition, it can be also found from both Figs. 6 and 7 that the outage probability increases with the increasing M for fixed hop length. For instance, in Fig. 6(a), to achieve the outage probability of 10−6 , about 65 dB and 68 dB are needed for N = 2, M = 3 and N = 2, M = 6, respectively, in the

P. Wang et al. / Optics Communications 367 (2016) 59–67

100

100 σ /a=3,Wz/a=20

-1

10

clear air/drizzle

10-2

10-4 10-5 10-6 30

N=2,M=3,weak N=2,M=3,moderate N=2,M=3,strong N=2,M=6,weak N=2,M=6,moderate N=2,M=6,strong Direct,weak Direct,moderate Direct,strong MC simulations

40

50

60 70 SNR(dB)

N=2,M=3,very clear air N=2,M=3,light fog N=2,M=6,very clear air N=2,M=6,light fog N=4,M=3,very clear air N=4,M=3,light fog Direct,very clear air Direct,light fog

-1

Outage Probability

End-to-end ABER

10

10-3

65

10-2 10-3 10-4

Moderate turbulence σs/a=3,Wz/a=10

10-5

80

90

10-6 30

100

40

50

60

70 80 SNR(dB)

σ /a=3,Wz/a=20 light fog

-1

10-2

10-5 10-6 30

N=2,M=3,weak N=2,M=3,moderate N=2,M=3,strong N=4,M=3,weak N=4,M=3,moderate N=4,M=3,strong Direct,weak Direct,moderate Direct,strong MC simulations

40

50

60 70 SNR(dB)

Strong turbulence

10-1

Outage Probability

End-to-end ABER

10

10-4

100 110 120

100

100

10-3

90

σs/a=3,Wz/a=10

10-2 10-3 N=2,M=3,very clear air N=2,M=3,light fog N=2,M=6,very clear air N=2,M=6,light fog N=4,M=3,very clear air N=4,M=3,light fog Direct,very clear air Direct,light fog

10-4 10-5

80

90

100

Fig. 5. End-to-end ABER of the present system with different structure parameters in weak-to-strong turbulence regimes in the presence of (a) clear air/drizzle and (b) light fog weather conditions.

presence of very clear weather condition. In Fig. 7(a), to achieve the outage probability of 10−6 , about 60 dB and 63 dB are required for N = 2, M = 3 and N = 2, M = 6, respectively, with Wz /a = 10. Furthermore, it can be found that the outage probability decreases with the increase of the number of paths ( N ). For example, in Fig. 6 (b), under light fog weather condition, when SNR is equal to 93 dB, the outage probability decreases from 3 × 10−4 to 1 × 10−6 as the value of N increases from 2 to 4. In Fig. 7(b), when SNR is equal to 60 dB, the outage probability for σs/a = 4 decreases from 2 × 10−4 to 1 × 10−6 as N increases from 2 to 4. The end-to-end ABER of the present FSO system for different parameters ( α , β , ρ ) of M distribution with the same turbulence strength (σI2 = 0.36) in light fog weather condition has been shown in Fig. 8. The normalized jitter σs/a and beamwidth Wz /a are assumed to be 3 and 10, respectively. It can be observed that all the curves with the same value of N tend to a same diversity order

10-6 30

40

50

60

70 80 SNR(dB)

90

100 110 120

Fig. 6. Outage probability versus SNR for multi-hop parallel FSO system with different structure parameters under very clear and light fog weather conditions in (a) moderate and (b) strong turbulence regimes.

[33]. Besides, it can be found that the diversity order increases with the increase of N . Accordingly, the system performance will be improved with a higher value of N .

5. Conclusions In this paper, the performances of the multi-hop parallel FSO cooperative communication system over an aggregated fading channel considering the path loss, M distributed turbulence and pointing errors have been studied. The PDF and CDF of the aggregated channel have been achieved in terms of Meijer’s G-function with the best path selection. After that, the analytical expressions of end-to-end ABER and outage probability of BPSK subcarrier intensity modulation with i.i.d links are derived, respectively. Furthermore, the ABER and outage performances with different weather conditions, pointing errors, turbulence strengths

66

P. Wang et al. / Optics Communications 367 (2016) 59–67

100

Outage Probability

10-1 10

-2

Weak turbulence σs/a=3 clear air/drizzle

10-3 10

-4

10-5 10-6 30

N=2,M=3,Wz/a=10 N=2,M=3,Wz/a=15 N=2,M=6,Wz/a=10 N=2,M=6,Wz/a=15 N=4,M=3,Wz/a=10 N=4,M=3,Wz/a=15 Direct,Wz/a=10 Direct,Wz/a=15

40

50 SNR(dB)

60

70 Fig. 8. End-to-end ABER for multi-hop parallel FSO system with different parameters ( α , β , ρ ) of M distribution assuming the same turbulence intensity ( σI2 = 0.36 ).

100

Outage Probability

10-1 10-2

Weak turbulence Wz/a=15 clear air/drizzle

(Grant no. 2014JM8340), the China Postdoctoral Science Special Foundation (Grant no. 201104659), the China Post-doctoral Science Foundation (Grant no. 20100481322), the Foundation of State Key Lab on Integrated Service Networks (Grant no. ISN1003006), the Fundamental Research Funds for the Central Universities (Grant no. K50511010023) and this work is also partly supported by 111 Project of China (Grant no. B08038).

N=2,M=3,σs/a=4

10

-3

10

-4

N=2,M=3,σs/a=5 N=2,M=6,σs/a=4 N=2,M=6,σs/a=5 N=4,M=3,σs/a=4

10-5 10-6 30

N=4,M=3,σs/a=5

Appendix A. Performance analysis of the present system under the same total link distance between source and destination

Direct,σs/a=4 Direct,σs/a=5

40

50 60 SNR(dB)

70

80

Fig. 7. Outage probability versus SNR for multi-hop parallel FSO system with different normalized (a) beamwidths and (b) jitters under weak turbulence and clear air/drizzle weather conditions.

and structure parameters ( M and N ) have been analyzed systematically. MC simulation is offered to verify and evaluate the ABER performance of this FSO system. This study shows that the performance of this FSO system can be enhanced with increasing N or decreasing M for fixed hop length regardless of the selected weather condition, pointing error, and turbulence strength under M turbulence condition. However, the FSO system performance can be improved with increasing M when the total link distance between source and destination is fixed. Besides, the ABER curves with the same value of N tend to a same diversity order, which increases with the increase of N . This work is of good help for the design of multi-hop parallel system.

Acknowledgments This work has been supported by supported by Nature Science Basic Research Plan in Shaanxi Province of China

In this appendix, Figs. A1–A2 show the ABER and outage performances of multi-hop parallel FSO communication system. The link distance between source and destination equals 4 km and the length of the direct link between source and destination is 3 km. It is assumed that the relays are evenly distributed along the path from the source to the destination. For different number of hops in each path, M = 1, 2, 3, the corresponding parameters ( α , β , ρ ) of M distribution are ( 3, 2, 0.6 ), ( 8, 5, 0.8), ( 12, 10, 0.95), respectively. The parameters ( αsd, βsd , ρsd ) for the direct link are ( 4, 2, 0.6) [15]. Fig. A1 presents the ABER performance of multi-hop parallel FSO communication system under light fog weather condition with different number of hops in each path. The pointing error normalized jitter σs/a and beamwidth Wz /a are 3 and 20, respectively. As seen, the analytical results have good agreement with MC simulation, which verifies the accuracy of our ABER model. Additionally, the ABER is improved with the increase of M . For example, when SNR value is equal to 94 dB, the ABER of N = 2, M = 1 is about 4 × 10−4 while that of N = 2, M = 2 is approximately 10  6. Fig. A2 depicts the outage performance of the present system with different number of hops in each path in the presence of light fog weather condition. The results show that the outage performance can be enhanced for higher value of M . For instance, to achieve the outage probability of 10−6 , about 97 dB of SNR is needed for N = 2, M = 3 while only 75 dB is required for N = 2, M = 2.

P. Wang et al. / Optics Communications 367 (2016) 59–67

[9]

[10]

[11]

[12]

[13]

[14]

[15] Fig. A1. ABER versus SNR for multi-hop parallel systems with different hops in each path when σs/a = 3, Wz /a = 20 in the presence of light fog weather . [16]

10

0

α=3,β=2,ρ=0.6 10

[17]

-1

Outage Probability

[18]

10-2 10-3

σs/a=3,Wz/a=20 light fog αsd=4,β sd=2,ρsd=0.6

[20]

α=12,β=10,ρ=0.95

10-4

N=2,M=1 N=2,M=2 N=2,M=3

10-5 10-6 30

[19]

[21]

α=8,β=5,ρ=0.8 [22]

[23]

40

50

60

70

80

90

100

SNR(dB) Fig. A2. Outage probability versus SNR for multi-hop parallel systems with different hops in each path when σs/a = 3, Wz /a = 20 in the present of light fog weather.

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