New approximate and asymptotic closed-form expressions for the outage probability and the average BER of MIMO-FSO system with MRC diversity technique over Gamma–Gamma fading channels with generalized pointing errors

Optics Communications 456 (2020) 124633

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

New approximate and asymptotic closed-form expressions for the outage probability and the average BER of MIMO-FSO system with MRC diversity technique over Gamma–Gamma fading channels with generalized pointing errors Junrong Ding ∗, Siyuan Yu, Yulong Fu, Jing Ma, Liying Tan National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin 150001, China

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Keywords: Free-space optical communication MIMO system MRC Gamma–Gamma fading Beckmann distribution High SNR

ABSTRACT This paper studies the outage probability (OP) and the average bit-error rate (BER) of multiple-input multipleoutput (MIMO) free-space optical (FSO) system with maximal ratio combining (MRC) diversity technique over Gamma–Gamma (GG) fading channels with generalized pointing errors modeled by the Beckmann distribution. A new approximate closed-form probability density function (PDF) at high signal-to-noise ratio (SNR) for MRC scheme is derived by inverse Laplace transformation method. Based on the developed PDF, new approximate and asymptotic closed-form expressions for the OP and the average BER are derived, respectively. Then, the OP and the average BER performance of the MIMO-FSO system with MRC scheme at high SNR are analyzed with different parameters, including number of transmitters and receivers, receiver aperture sizes, nonzero boresight, and different jitters. Finally, numerical results and Monte Carlo simulations corroborate the accuracy of the derived approximate and asymptotic expressions over a wide range of SNR.

1. Introduction Free space optical (FSO) communications is one of the most potential alternative solutions for addressing the ’last mile’ access because of its merits of unregulated spectrum, low cost, easy and quick installation, high data rate and high security [1–3]. However, FSO link suffers highly from the atmospheric turbulence resulting in random fluctuations in the intensity and the phase of the received signal due to variations of refractive index along the laser beam propagation path [4]. To describe the turbulence-induced fading, various models have been put forward, e.g. log-normal (LN) distribution, negative exponential distribution, Gamma-Gamma (GG) distribution and so on [1,2,5,6]. Compared to the other two models, GG distribution has been widely used to investigate the behavior of the FSO links because it provides an excellent fit to the experimental data over wide range of atmospheric turbulence conditions (weak-tostrong). In order to satisfy the requirements for average BER, outage probability and capacity in FSO applications, multiple-input multipleoutput (MIMO) technology is effective solution to mitigate turbulenceinduced fading. In [4,7–12], the authors have studied the average bit-error rate (BER), ergodic capacity and outage probability (OP) in the case of multiple-input/single-output (MISO), multiple-input/singleoutput (SIMO) as well as MIMO-FSO systems with equal gain combining

(EGC) or maximal ratio combining (MRC) at the receiver. It is noticed that these literatures only consider the effects of the atmospheric turbulence. Apart from atmospheric turbulence, FSO link is also vulnerable to pointing errors caused by building sway. For modeling pointing errors, many statistical models have been suggested, including Rayleigh [13], Rician [14], Hoyt [15] and Beckmann distributions [16,17]. Beckmann distribution is more general of these models, because the models proposed above are its special cases. The use of MIMO technique can significantly improve the performance of FSO system, but pointing errors have the potential to eliminate the benefits of the MIMO technology. Therefore, the impact of pointing errors in the turbulence-induced fading channel needs to be considered in the actual study of MIMOFSO system. In [18–26], the joint effect of atmosphere turbulence and misalignment fading with zero and nonzero boresight pointing errors and identical jitters for horizontal and vertical displacement has been analyzed in terms of ergodic capacity, the average BER and OP in the situation of MISO, SIMO as well as MIMO-FSO systems with EGC or MRC at the receiver. However, when taking into account nonzero boresight pointing errors and different jitters for horizontal and vertical displacement (Beckmann distribution), determining the combined effect of pointing errors and atmospheric turbulence on the MIMO-FSO system becomes an exacting task, which is because the closed-form PDF and the cumulative density function (CDF) of the Beckmann random

∗ Corresponding author. E-mail address: [email protected] (J. Ding).

https://doi.org/10.1016/j.optcom.2019.124633 Received 1 June 2019; Received in revised form 22 September 2019; Accepted 24 September 2019 Available online 26 September 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633 −1 ( )−5∕6 ⎡ ⎛ ⎞ ⎤ 2 1 + 0.69𝜎 12∕5 0.51𝜎 𝑅 𝑅 ⎢ ⎜ ⎟ ⎥ 𝛽 = ⎢exp ⎜ ( (3b) )5∕6 ⎟ − 1⎥ , ⎢ ⎜ 1 + 0.9𝑑 2 + 0.62𝑑 2 𝜎 12∕5 ⎟ ⎥ ⎣ ⎦ 𝑅 ⎝ ⎠ √ 2 2 2 𝑘𝐷 ∕4𝐿 and 𝜎𝑅 = 0.5𝐶𝑛 𝑘7∕6 𝐿11∕6 . Here, 𝑘 = 2𝜋∕𝜆 where 𝑑 = is the wave number 𝐷 = 2𝑅 denotes the diameter of the receiver aperture and 𝐶𝑛2 refers to the refractive-index structure parameter varying from 10−17 m−2∕3 to 10−13 m−2∕3 for weak turbulence and strong turbulence regimes, respectively. When 𝐷 ≥ 𝜌sp holds, where 𝜌sp = ( )−3∕5 0.55𝐶𝑛2 𝑘2 𝐿 represents spherical wave coherence radius, aperture averaging needs to be considered. In relation to the impact of pointing errors, due to lack of perfect alignment between the transmitter and receiver, the pointing error effects cannot be neglected. The attenuation due to geometric spread and pointing errors can be approximated by [13, Eq. (9)] ) ( 2𝑟2 𝑝 , (4) ℎ𝑖,𝑗 (𝑟, 𝐿) ≈ 𝐴0 exp − 𝑤2𝐿𝑒𝑞 √ √ where 𝑟 represents 𝑣 = 𝜋𝑅∕ 2𝑤𝐿 , 𝑤𝐿 is the beam √ the√radial distance, /[ ( )] 𝑤2𝐿 𝜋erf (𝑣) 2𝑣 exp −𝑣2 is the equivalent beam waist, 𝑤𝐿eq =

variables (RVs) remain an open problem. This has attracted great attention to study the MIMO-FSO system performance at high SNR. Motivated by aforementioned facts, we put emphasis on the outage performance and the average BER performance of MIMO-FSO system with MRC scheme at high SNR over GG fading channels with generalized pointing errors. Our contribution in this study is summarized as follows: A new closed-form approximate PDF at high SNR for MRC scheme is derived by utilizing inverse Laplace transformation method. Based on the developed PDF, new approximate and asymptotic closed-form expressions of MRC for the outage probability and the average BER are derived as a simple elementary function, respectively. Moreover, the impacts of different number of transmitters and receivers, aperture averaging, nonzero boresight and different jitters for horizontal and vertical displacement on MIMO-FSO system are taken into account. The rest of the paper is organized as follows: In Section 2, the system and channel fading models are introduced; the approximate PDF for MRC scheme over GG fading channels with generalized pointing errors is developed. In Section 3, the closed-form expressions of MRC scheme for the outage probability and the average BER in the high SNR regime are derived. Numerical results are discussed and compared in Section 4. Finally, the main conclusions are drawn in Section 5.

waist, 𝐴0 = [erf (𝑣)]2 denotes the fraction of the collected power at 𝑟 = 0. Note that the approximation in Eq. (4) is extremely close to the exact value as long as 𝑤𝐿 > 6𝑅. After some straightforward algebraic manipulations in Eq. (4), 𝑟 can be expressed as √ ( 𝑝 ) √ √ 1 ℎ𝑖,𝑗 (𝑟; 𝐿) 𝑟 = √− 𝑤2𝐿 ln . (5) 2 𝑒𝑞 𝐴0

2. System and channel model The MIMO-FSO system considered here has M transmit and N receive apertures over a discrete-time ergodic channel with additive white Gaussian noise (AWGN). Assuming the spatial separation of the photodetectors is greater than the spatial coherence distance so that the channels are statistically independent and uncorrelated. We assume the system employs intensity-modulation and direct-detection (IM/DD) with On–Off keying (OOK) scheme. The received signal at the 𝑗th receive aperture is given as [27, Eq. (17)] 𝑦𝑗 = 𝜂𝑃𝑎𝑣𝑔 𝑥

𝑀 ∑

The radial displacement vector at the receiver plane can be expressed as 𝒓 = [𝑥 𝑦]𝑇 , where 𝑥 and 𝑦 is the elevation and horizontal displacements of the beam, respectively. Both 𝑥 )and 𝑦 are( modeled as ( ) independent Gaussian RVs , i.e. 𝑥 ∼  𝜇𝑥 , 𝜎𝑥 , 𝑦 ∼  𝜇𝑦 , 𝜎𝑦 , then √ 𝑟 = |𝒓| = 𝑥2 + 𝑦2 follows the well-known Beckmann distribution [30, Eq. (2.37)] ⎛ (𝑟 cos 𝜃 − 𝜇 )2 (𝑟 sin 𝜃 − 𝜇 )2 ⎞ 2𝜋 𝑦 ⎟ 𝑥 𝑟 𝑓𝑟 (𝑟) = exp ⎜− − d𝜃. (6) ⎜ ⎟ 2𝜋𝜎𝑥 𝜎𝑦 ∫0 2𝜎𝑥2 2𝜎𝑦2 ⎝ ⎠

(1)

ℎ𝑖,𝑗 + 𝑧𝑗 ,

𝑖=1

where 𝑥∈ {0, 1} denotes the information bits, 𝜂 is the optical-toelectrical conversion coefficient, 𝑃𝑎𝑣𝑔 represents is the transmitted average power, and 𝑧𝑗 is the signal-independent AWGN with zero mean and varianceN 0 /2. ℎ𝑖,𝑗 denotes the channel gain of the link between the 𝑖th transmitter and the 𝑗th receiver. ℎ𝑖,𝑗 is considered to be a product of three factors, i.e. ℎ𝑖,𝑗 = ℎ𝑙𝑖,𝑗 ℎ𝑎𝑖,𝑗 ℎ𝑝𝑖,𝑗 , where ℎ𝑙𝑖,𝑗 denotes the deterministic path loss, ℎ𝑎𝑖,𝑗 is the fading due to atmospheric turbulence and ℎ𝑝𝑖,𝑗 is the fading due to geometric spread and pointing errors [13]. ℎ𝑙𝑖,𝑗 is described by the Beer–Lambert law as ℎ𝑙𝑖,𝑗 (𝐿) = exp (−𝜎𝐿), where 𝐿 is the link distance between transmitter and receiver and 𝜎 is the attenuation coefficient dependent on the particle size distribution and the visibility, which is given by 𝜎 ≈ (3.91∕𝑉 ) (𝜆∕550)−𝑞(𝑉 ) , where V is the visibility in kilometers, 𝜆 is the operating wavelength in nanometers and 𝑞 (𝑉 ) is the particle size-related coefficient, being 𝑞 = 1.3 for average visibility (6 km<𝑉 <50 km), and 𝑞 = 0.16𝑉 + 0.34 for haze visibility (1 km<𝑉 <6 km) [28]. To characterize the fading gains ℎ𝑎𝑖,𝑗 over a wide range of turbulence scenarios (weak-to-strong), the GG turbulence model is adopted. The PDF of ℎ𝑎𝑖,𝑗 is given by [29, Eq. (1)] 𝑓

( ℎ𝑎𝑖,𝑗

ℎ𝑎𝑖,𝑗

)

) ( √ 2 (𝛼𝛽)(𝛼+𝛽)∕2 ( 𝑎 )(𝛼+𝛽)∕2−1 ℎ𝑖,𝑗 𝐾𝛼−𝛽 2 𝛼𝛽ℎ𝑎𝑖,𝑗 = 𝛤 (𝛼) 𝛤 (𝛽)

, ℎ𝑎𝑖,𝑗

For the MRC scheme, the gain of each sub-channel is proportional to the received signal intensity. The total transmitted power is normalized to unity, thus resulting in the combined signal [27, Eq. (23)] √ √ √𝑁 𝑥𝜂 √∑ 2 (7) 𝑦= √ √ ℎ𝑗 + 𝑧, 𝑀 𝑁 𝑗=1 ∑𝑀 where ℎ𝑗 = 𝑖=1 ℎ𝑖𝑗 is the combined signal at the jth receiver, 𝑧 denotes effective Gaussian noise at the receiver. Then the instantaneous electrical SNR of the combined signal at the receiver can be expressed as 𝛾=

> 0,

where 𝛼 and 𝛽 stand for the shaping parameters directly linked to the atmospheric conditions, 𝛤 (⋅) denotes the Gamma function and 𝐾𝑣 (⋅) is the modified Bessel function of the second kind of order 𝑣. Assuming the optical beam at the receiver is a spherical wave, which is widely adopted in the literature [4,22,28,29], the expressions corresponding to the GG parameters are respectively given by [29, Eq. (2)]

𝛾=

𝑁 𝛾̄ ∑ ( )2 ℎ𝑗 . 𝑀 2 𝑁 𝑗=1

(9)

By using a MGF approach, the PDF of written as (see the Appendix)

−1

,

(8)

The received electrical SNR from the / ith transmitter to the jth receiver can be defined as 𝛾𝑖,𝑗 = 𝜂 2 ℎ2𝑖,𝑗 𝑁0 , and the corresponding { [ ]}2 / average electrical SNR is 𝛾̄ = 𝛾̄𝑖,𝑗 = 𝜂 2 𝐸 ℎ𝑖,𝑗 𝑁0 . To ensure that the[ fading does not attenuate or amplify the average transmitted power, ] 𝐸 ℎ𝑖,𝑗 is normalized to unity. Then the electrical SNR of the combined signal at the receiver can be formulated as

(2)

⎡ ⎛ ⎞ ⎤ 2 0.49𝜎𝑅 ⎢ ⎜ ⎟ ⎥ 𝛼 = ⎢exp ⎜ ( − 1 ⎥ )7∕6 ⎟ 12∕5 ⎢ ⎜ 1 + 0.18𝑑 2 + 0.56𝜎 ⎟ ⎥ ⎣ ⎦ 𝑅 ⎝ ⎠

𝑁 ∑ ( )2 𝜂2 ℎ𝑗 . 𝑀 2 𝑁𝑁0 𝑗=1

(3a)

𝑓∑𝑁 𝑗=1

2

(∑

𝑀 𝑖=1 ℎ𝑖𝑗

)2

(ℎ) =

∑𝑁 (∑𝑀 𝑗=1

𝑖=1

ℎ𝑖𝑗

∞ 𝑁 𝑀𝑁 ∑ 𝑛+𝑁𝜏 𝑝𝑛 1 𝑐0 𝐶 ℎ 2 −1 . 2𝑁 (𝛤 (𝜏))𝑁 𝑛=0 𝛤 ((𝑛 + 𝑁𝜏) ∕2)

)2

can be

(10)

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633

Using the Taylor series expansion of 𝑥𝑛 ∕ (1 − 𝑥), we can simplify the summation term in Eq. (16) and obtain an upper bound of the truncation error as { } 1 𝜀out (𝐽 ) ≤ (√ 𝑢out (𝑛) (18) ) (√ )𝐽 max 𝑛>𝐽 / / 𝛾̄ 𝑀 2 𝑁𝛾th − 1 𝛾̄ 𝑀 2 𝑁𝛾th

The corresponding cumulative distribution function (CDF) is derived by taking the integration of 𝑓∑𝑁 (∑𝑀 )2 (ℎ) as 𝑗=1

(∑

𝐹∑𝑁 𝑗=1

)2 𝑀 𝑖=1 ℎ𝑖𝑗

(ℎ) =

𝑖=1 ℎ𝑖𝑗

∞ 𝑁 𝑀𝑁 ∑ 𝑛+𝑁𝜏 𝑝𝑛 1 𝑐0 𝐶 ℎ 2 . 𝑁 𝑁 2 (𝛤 (𝜏)) 𝑛=0 𝛤 ((𝑛 + 𝑁𝜏) ∕2+1)

(11)

3. Performance analysis of MRC FSO links

After examining the first term in Eq. (17), we note that 𝑢out (𝑛) approaches zero when n approaches ∞; hence the truncation error 𝜀out (𝐽 ) diminishes with increasing J. Furthermore, we observe from Eq. (18) that the truncation error decreases rapidly with increase of electrical SNR 𝛾̄ . This observation shows that our series solution is very accurate at high SNR. Hence, we can use it to perform outage probability analysis.

3.1. Outage probability analysis In this section, the outage performance of MIMO-FSO systems with MRC at high SNR is analyzed over GG atmospheric channels with generalized pointing errors. The OP is defined as the probability that the instantaneous electrical SNR of the combined signal 𝛾 is lower than a specified threshold 𝛾th , that is 𝑃out

( ) = Pr 𝛾 < 𝛾th =

3.2. Average BER analysis

𝛾th

∫0

(12)

𝑓𝛾 (ℎ) 𝑑ℎ.

In this section, the average BER of MIMO-FSO systems with MRC at high SNR is analyzed over GG atmospheric channels with generalized pointing errors. For OOK modulation, the conditional BER over an AWGN channel is obtained as [10,11] ) ( √ 𝜂 1 𝑃𝑒|ℎ = erf c ℎ , (19) √ 2 2𝑀 𝑁𝑁0

Substituting Eq. (52) into Eq. (12), the outage probability can be expressed as √ / ) ( 𝑀 2 𝑁𝛾th 𝛾̄ 𝛾̄ ℎ (13) < 𝛾th = 𝑓∑𝑁 (∑𝑀 )2 (ℎ) 𝑑ℎ. 𝑃out = Pr ∫0 𝑀 2𝑁 𝑖=1 ℎ𝑖𝑗 𝑗=1 Using Eqs. (8) and (53), the approximate closed-form expression for the outage probability of MIMO MRC is obtained as 𝑃out =

𝑁 𝑀𝑁 1 𝑐0 𝐶 𝑁 2 (𝛤 (𝜏))𝑁

( ) 𝑛+𝑁𝜏 ∞ ∑ 𝑝𝑛 𝑀 2 𝑁𝛾th 2 𝑛=0

𝛤 ((𝑛 + 𝑁𝜏) ∕2+1)

𝛾̄ −

𝑛+𝑁𝜏 2

where erf c (⋅) is the complementary error function. Hence, the average BER for MIMO MRC can be obtained by averaging 𝑃𝑒|ℎ over the PDF of ℎ as follows

(14)

.

∞

𝑃𝑒 =

Further, the exponent of 𝛾̄ is given by −(𝑛 + 𝑁𝜏)∕2. When 𝛾̄ approaches ∞, the term with the largest exponent of 𝛾̄ in Eq. (14) becomes the dominant term. Therefore, the asymptotic closed-form solution for the OP of MIMO MRC is ( ) 𝑁𝜏 𝑝0 𝑐0𝑁 𝐶 𝑀𝑁 𝑀 2 𝑁𝛾th 2 − 𝑁𝜏 𝛾̄ 2 . (15) 𝑃out = 2𝑁 𝛤 (𝑁𝜏∕2+1) (𝛤 (𝜏))𝑁

𝑃𝑒 =

)2 𝑀 𝑖=1 ℎ𝑖𝑗

(∑

(ℎ) dℎ.

(20)

∞ 𝑁 𝑀𝑁 ∑ 𝑝𝑛 1 𝑐0 𝐶 𝑁 𝑁 2 (𝛤 (𝜏)) 𝑛=0 𝛤 ((𝑛 + 𝑁𝜏) ∕2) ( ) ∞ 𝜂 × erf c ℎ ℎ𝑛+𝑁𝜏−1 dℎ. √ ∫0 2𝑀 𝑁𝑁

(21)

Following substitution of 𝑥 = 𝑦2 in Eq. (21) and then using Eq. (22) ∞

∫0

𝑦𝑎−1 erf c (𝑐𝑦) 𝑑𝑦 =

1 √

𝑎𝑐 𝑎

( 𝜋

) 𝑎+1 . 2

(22)

The approximate closed-form solution for the average BER of MIMO MRC can be derived: 𝑐 𝑁 𝐶 𝑀𝑁 𝑃𝑒 = √ 0 𝜋 (𝛤 (𝜏))𝑁 ( √ )𝑛+𝑁𝜏 ⎛ ⎞ ∞ 𝑝𝑛 𝑀 𝑁 2𝑛+𝑁𝜏−𝑁 ( ) ∑ ⎜ ⎟ 𝑛 + 𝑁𝜏+1 − 𝑛+𝑁𝜏 2 𝛤 𝛾̄ × ⎜ ⎟. + 𝑁𝜏) 𝛤 + 𝑁𝜏) ∕2) 2 (𝑛 ((𝑛 ⎟ 𝑛=0 ⎜ ⎠ ⎝

(23)

Similarly, when 𝛾̄ approaches ∞, the average BER in Eq. (23) is dominated by the leading term of the series at high SNR. Then, the asymptotic closed-form expression for the average BER of MIMO MRC is ( √ )𝑁𝜏 𝑝0 𝑐0𝑁 𝐶 𝑀𝑁 2𝑀 𝑁 2𝑁𝜏−𝑁 ( ) 𝑁𝜏+1 − 𝑁𝜏 𝑃𝑒 = 𝛤 𝛾̄ 2 . (24) √ 𝑁 2 𝜋𝑁𝜏 (𝛤 (𝜏)) 𝛤 (𝑁𝜏∕2)

𝑛

(16)

where 𝑢out (𝑛) is defined as ⎛ ⎞ 𝑁 𝑀𝑁 𝑝𝑛 ⎜ ⎟ 1 𝑐 𝐶 1 𝑢out (𝑛) = 𝑁 0 ⎜√ / ⎟ 2 (𝛤 (𝜏))𝑁 𝛤 ((𝑛 + 𝑁𝜏) ∕2+1) ⎜ 2 𝛾̄ 𝑀 𝑁𝛾th ⎟ ⎝ ⎠

𝑗=1

0

3.1.1. Truncation error analysis of outage probability To evaluate the truncation error caused by eliminating the infinite terms after the first 𝐽 +1 term in Eq. (14), we first define the truncation error as ⎛ ⎞ ⎜ ⎟ 1 𝜀out (𝐽 ) = 𝑢out (𝑛) ⎜ √ / ⎟ 2 ⎜ 𝛾̄ 𝑀 𝑁𝛾th ⎟ 𝑛=𝐽 +1 ⎝ ⎠

𝑃𝑒|ℎ 𝑓∑𝑁

Substituting Eqs. (19) and (52) into Eq. (20) gives

It is obvious to show that the outage probability behaves asymptotically ( )−𝑂𝑑 , where 𝑂𝑑 and 𝑂𝑐 represent diversity gain and coding as 𝑂𝑐 𝛾̄ gain, respectively [31]. For large values of 𝛾̄ , the outage diversity gain determines the negative slope of the outage probability versus average SNR curve on a log–log plot and the coding gain (in decibels) determines the shift of the curve in SNR. It must be noted that the outage diversity gain is (𝑛 + 𝑁𝜏)∕2 when the effect of the GG atmospheric turbulence is the dominant effect compared with generalized pointing errors at high SNR. When generalized pointing errors are dominant, it is conjectured that outage diversity gain will hinge on 𝜑2𝑥 and 𝜑2𝑦 and nonzero boresight errors, however, this case is mathematically intractable in practice to derive given the Beckmann distribution.

∞ ∑

∫0

It is also obvious to show that the average BER varies asymptotically ( )−𝑂𝑑 as 𝑂𝑐 𝛾̄ , where 𝑂𝑑 and 𝑂𝑐 represent diversity gain and coding gain, respectively [31]. Hence, the corresponding diversity gain is (𝑛 + 𝑁𝜏)∕2.

𝑁𝜏

(17)

3

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633

3.2.1. Truncation error analysis of average BER To evaluate the truncation error caused by eliminating the infinite terms after the first 𝐽 +1 term in Eq. (23), we first define the truncation error as )𝑛 ( ∞ ∑ 1 𝜀𝑒 (𝐽 ) = 𝑢𝑒 (𝑛) √ (25) 𝛾̄ ∕4𝑀 2 𝑁 𝑛=𝐽 +1 where 𝑢𝑒 (𝑛) is defined as 𝑐 𝑁 𝐶 𝑀𝑁 𝑝𝑛 2−𝑁 𝑢𝑒 (𝑛) = √ 0 𝑁 (𝑛 + 𝑁𝜏) 𝛤 ((𝑛 + 𝑁𝜏) ∕2) 𝜋 (𝛤 (𝜏)) ( )𝑁𝜏 ( ) 𝑛 + 𝑁𝜏+1 1 ×𝛤 √ 2 𝛾̄ ∕4𝑀 2 𝑁

(26)

Using the Taylor series expansion of 𝑥𝑛 ∕ (1 − 𝑥), we can simplify the summation term in Eq. (25) and obtain an upper bound of the truncation error as { } 1 𝜀𝑒 (𝐽 ) ≤ (√ 𝑢𝑒 (𝑛) (27) ) (√ )𝐽 max 𝑛>𝐽 𝛾̄ ∕4𝑀 2 𝑁 − 1 𝛾̄ ∕4𝑀 2 𝑁 After examining the first term in Eq. (26), we note that 𝑢out (𝑛) approaches zero when n approaches ∞; hence the truncation error 𝜀out (𝐽 ) decreases with increasing J. Furthermore, we observe from Eq. (27) that the truncation error decreases rapidly with increase of electrical SNR 𝛾̄ . This observation suggests that our series solution is pretty accurate in the large SNR regime. Hence, we can use it to perform average error rate analysis. 4. Numerical results In this section, some analytical results of the derived previously expressions for the outage probability and average BER of MIMO MRC are presented. In order to verify the analytical solutions, Monte Carlo (MC) simulations through MATLAB have been performed. In order to avoid a long time of simulation, 108 i.i.d. random samples are generated resulting in simulation results only up to 10−8 . For 𝛾̄ → ∞, higher powers of 𝛾̄ in Eq. (14) and (23) soon get smaller, so we can truncate the series after a finite number of terms for obtaining the analytical results. In this simulation, the power series is truncated at 𝑛 = 49. The asymptotic outage probability and BER of the MRC scheme are Eqs. (15) and (24), respectively. The system configuration parameters under consideration in all simulations are as the followings: optical wavelength 𝜆 = 1550 nm, link distance 𝐿 = 5 km, the receiver aperture diameter 𝐷 = 100 mm and 𝐷 = 200 mm, refractive-index structure parameter 𝐶𝑛2 = 1×10−13 m−2∕3 , transmit divergence 𝜃𝐿 = 0.66 mrad at 1∕𝑒2 jitter angle 𝜃𝐿 = 0.11 mrad at 1∕𝑒2 , and boresight angle 𝜃𝑏 = 0.06 mrad at 1∕𝑒2 . In this case, to evaluate the effect of generalized pointing errors on the MIMO-FSO system with MRC scheme, different ( ) ( ) jitter standard deviation 𝜎𝑥 , 𝜎𝑦 together with boresight mean 𝜇𝑥 , 𝜇𝑦 ( ) ( ) are considered. i.e., 𝜎𝑥 , 𝜎𝑦 = ((15, 5) , (55, 55)) cm and 𝜇𝑥 , 𝜇𝑦 = (15, 30) cm. In addition, the effect of aperture averaging is analyzed because of 𝐷 > 𝜌𝑠𝑝 = 6.4 mm. Fig. 1 portrays the simulated, approximate (Eq. (14) ) and asymptotic (Eq. (15)) outage probability of MRC as a function of normalized electrical SNR 𝛾̄ with 𝐷 = 100 mm for different values of M and N. It is seen that the approximate results and the asymptotic results are close to the exact outage probability for average-to-large values of 𝛾̄ and for large values of 𝛾̄ , respectively. Compare to Eqs. (15), (14) provides a faster convergence towards the exact outage probability for lower values of 𝛾̄ . For instance, for 2 × 3 systems, the approximate expression can accurately predict the outage performance for values of 𝛾̄ exceeding 80 dB. In addition, we also notice that the outage performance of MRC scheme is greatly improved as M and N get larger, compared to the SISO deployment. Indeed, at a target 𝑃out = 10−9 , for ( ) 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, the SNR improvements with respect to the SISO scheme are about 29.18 dB, 43.70 dB and 49.10 dB for 1 × 2, 2 × 2

Fig. 1. The outage probability versus normalized electrical SNR for MRC spatial ( ) diversity with 𝐷 = 100 mm under various jitter standard deviations (𝑎) 𝜎𝑥 , 𝜎𝑦 = ( ) (15, 5) cm, (𝑏) 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm.

( ) and 2 × 3 systems, respectively, and for 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm, they are (29.18 dB, 43.67 dB, 49.17 dB). This can be explained due to the rise of the number of sub-channels. Moreover, the effect of generalized pointing errors can be observed from figure (compare Fig. 1(𝑎) and (𝑏)); as expected, the outage probability of MRC scheme increases with increasing jitters values. For instance, at 𝑃out = 10−9 , the SNR loss of (𝑀 × 𝑁) = (1 × 2, 2(× 2, 2 )× 3) systems is approximately ( ) 1.20 dB, 1.24 dB and 1.13 dB at 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm than at 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm. Fig. 2 illustrates the simulated, approximate (Eq. (14) ) and asymptotic (Eq. (15)) outage probability of MRC versus normalized electrical SNR 𝛾̄ with 𝐷 = 200 mm for various values of M and N. Note that the parameters of Figs. 1 and 2 are nearly the same except for the aperture size: the former is 100 mm, whereas the latter is 200 mm, so we put more emphasis on the differences between them. Compared with Fig. 1, the outage probability decreases with the ( increase ) of the receiver aperture size For example, at 𝑃out = 10−9 , for 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, the SNR difference of (𝑀 × 𝑁) = (1 × 2, 2 × 2, 2 × 3) systems between 𝐷 = 200 mm( and 𝐷 ) = 100 mm is about (14.85 dB, 13.64 dB, 13.21 dB), and for 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm, it is (14.74 dB, 13.68 dB, 13.24 dB). This phenomenon shows that aperture averaging can significantly improve the outage performance of MRC FSO system. In Fig. 3, we compare the exact average BER with the approximate and asymptotic expressions given in Eqs. (23) and (24) with 𝐷 = 100 4

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633

Fig. 2. The outage probability against normalized electrical SNR for MRC spatial ( ) diversity with 𝐷 = 200 mm under various jitter standard deviations (𝑎) 𝜎𝑥 , 𝜎𝑦 = ( ) (15, 5) cm, (𝑏) 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm.

Fig. 3. The average BER versus electrical SNR for MRC spatial diversity with 𝐷 = 100 ( ) ( ) mm under different jitter standard deviations (𝑎) 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, (𝑏) 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm.

mm for different values of M and N, respectively. We observe that the approximate expressions and the asymptotic expressions match the corresponding MC simulations perfectly for average-to-large values of 𝛾̄ and for large values of 𝛾̄ , respectively. While lower values of 𝛾̄ affect the convergence rate of the asymptotic expressions towards the exact average BER, particularly for (𝑀 × 𝑁) = (2 × 3) system, the approximate expression exhibit a faster convergence rate where, for instance, it can correctly predict the BER performance of (𝑀 × 𝑁) = (1 × 2, 2 × 2, 2 × 3) systems for values of 𝛾̄ exceeding 105 dB. In addition, we also observed that the BER performance of MRC is significantly improved as M and N increases in( comparison with the SISO deployment. Particularly, at ) 𝑃𝑒 = 10−9 , for 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, the improvements in SNR are about 25.90 dB, 37.54 dB and 41.55 dB than the ( SISO) system for 1 × 2, 2 × 2 and 2 × 3 systems, respectively, and for 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm, they are about 25.72 dB, 37.39 dB and 41.59 dB. This can be interpreted due to the increase of the number of sub-channels. It is also seen from this figure (compare Fig. 3(𝑎) and (𝑏)) that pointing errors can lead to a degradation in the BER performance. For example, at 𝑃𝑒 = 10−9 , the SNR penalty of (𝑀 × 𝑁) = (1 × 2, 2(× 2, 2 × ) 3) systems is approximately ((1.24 dB ) , 1.20 dB, 1.02 dB) at 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm than that at 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm. Fig. 4, we compare the exact average BER with the approximate and asymptotic expressions given in Eqs. (23) and (24) with 𝐷 = 200 mm

for different values of M and N, respectively. Notice that parameters of Figs. 3 and 4 are nearly the same, only aperture size is different: the former is 100 mm, whereas the latter is 200 mm, so we placed greater emphasis on the differences between them. Compared with Fig. 3, the average BER descends when the receiver aperture size increases. For example, at 𝑃𝑒 = 10−9 , when the aperture size varies from 100 mm to ( ) 200 mm, for 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, the SNR improvement of (𝑀 × 𝑁) = (1 × 2, 2 × 2, 2 × 3) systems of 𝐷 = 200 mm is about (14.37 dB, 13.35 dB, ( ) 12.99 dB) than 𝐷 = 100 mm, and for 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm, it is (14.45 dB, 13.39 dB, 12.95 dB). This phenomenon indicates that aperture averaging can significantly improve the average BER performance of MRC FSO system. 5. Conclusion In this paper, we have studied the outage probability and the average bit-error rate (BER) of MIMO-FSO system with MRC scheme over GG fading channels with generalized pointing errors modeled as the Beckmann distribution. A new approximate closed-form probability density function (PDF) at high signal-to-noise ratio (SNR) for MRC scheme is derived in terms of th by inverse Laplace transformation method. Based on the developed PDF, new approximate and asymptotic 5

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633

( )| | Using Eqs. (5), (6) and the technique 𝑓𝑌 (𝑦) = 𝑓𝑋 𝑔 −1 (𝑦) | 𝑑𝑥 | for per| 𝑑𝑦 | ( ) forming the transformation of continuous random variables, 𝑓ℎ𝑝 ℎ𝑝𝑖,𝑗 𝑖,𝑗 can be written as 𝑓ℎ𝑝

𝑖,𝑗

( ) ℎ𝑝𝑖,𝑗 =

𝑟 2𝜋𝜎𝑥 𝜎𝑦 ∫0 ×

2𝜋

⎛ (𝑟 cos 𝜃 − 𝜇 )2 (𝑟 sin 𝜃 − 𝜇 )2 ⎞ 𝑦 ⎟ 𝑥 exp ⎜− − d𝜃 ⎜ ⎟ 2𝜎𝑥2 2𝜎𝑦2 ⎝ ⎠

1 , | 𝑝 | ′ ℎ | 𝑖,𝑗 (𝑟, 𝐿)| | | (30)

where ′

ℎ𝑝𝑖,𝑗 =

−4𝑟ℎ𝑝𝑖,𝑗 𝑤2𝐿

(31)

.

𝑒𝑞

Substituting Eqs. (29) and (30) into Eq. (28), Eq. (28) can be computed as ( )(𝛼+𝛽)∕2 −1 𝐴0 2𝜋 ℎ𝑖,𝑗 ( ) 2(𝛼𝛽)(𝛼+𝛽)∕2 1 𝑓ℎ𝑖,𝑗 ℎ𝑖,𝑗 = 𝛤 (𝛼) 𝛤 (𝛽) ∫0 ∫0 ℎ𝑙 ℎ𝑝 ℎ𝑙𝑖,𝑗 ℎ𝑝𝑖,𝑗 𝑖,𝑗 𝑖,𝑗 √ √ ( 𝑝 )| | √ ⎛ √ √ ℎ𝑖,𝑗 || √ ℎ𝑖,𝑗 𝛼𝛽 ⎞ 2 || 𝑝 √ 1 2 √ ⎜ ⎟ 𝑤 × 𝐾𝛼−𝛽 2 − ln 𝑤 ∕ −4ℎ | | 𝑝 ⎟ 𝐿𝑒𝑞 | 𝑖,𝑗 𝐿𝑒𝑞 𝑙 ⎜ 2 𝐴0 || ℎ𝑖,𝑗 ℎ𝑖,𝑗 | ⎝ ⎠ | | √ ( 𝑝 ) √ √ 1 ℎ𝑖,𝑗 × √− 𝑤2𝐿 ln ∕2𝜋𝜎𝑥 𝜎𝑦 2 𝑒𝑞 𝐴0 2 ( 𝑝 ) ⎛ ⎛√ √ ⎞ ℎ𝑖,𝑗 ⎜ ⎜√ 1 2 √ − 𝑤𝐿 ln cos 𝜃 − 𝜇𝑥 ⎟ ∕2𝜎𝑥2 × exp ⎜− ⎟ 2 𝑒𝑞 𝐴0 ⎜ ⎜⎝ ⎠ ⎝

√ 2 ( 𝑝 ) ⎞ ⎛√ √ 1 ℎ𝑖,𝑗 √ 2 ⎟ ⎜ sin 𝜃 − 𝜇𝑦 ∕2𝜎𝑦2 − − 𝑤𝐿 ln ⎟ ⎜ 2 𝑒𝑞 𝐴0 ⎠ ⎝

⎞ ⎟ 𝑝 ⎟ 𝑑𝜃𝑑ℎ𝑖,𝑗 . ⎟ ⎠ (32)

The PDF in Eq. (32) does not lend itself to closed-form solution. Hence, for the sake of obtaining the closed-from expression of PDF of the channel gain, we resort to the approximate PDF near the origin as follows [16, Eq. (21)] ( ) ( ) 𝑓ℎ𝑖,𝑗 ℎ𝑖,𝑗 ≈ 𝐶𝑓𝛾 ℎ𝑖,𝑗 , 𝑏, 𝐴 , (33) ( ) ( 𝑏 ) 𝑏−1 −ℎ∕𝐴 ∕ 𝐴 𝛤 (𝑏) denotes the gamma PDF where 𝑓𝛾 ℎ𝑖,𝑗 , 𝑏, 𝐴 = ℎ𝑖,𝑗 𝑒 with shape parameter 𝑏 and scale parameter 𝐴 and where [16, Eq. (22)]

Fig. 4. The average BER against electrical SNR for MRC spatial diversity with 𝐷 = 200 ( ) ( ) mm under different jitter standard deviations (𝑎) 𝜎𝑥 , 𝜎𝑦 = (15, 5) cm, (𝑏) 𝜎𝑥 , 𝜎𝑦 = (55, 55) cm.

closed-form expressions for the outage probability and the average BER are derived as infinite power series and a simple elementary function, respectively. MC simulations justify that our derived results can accurately forecast the performance of the MIMO-FSO system with MRC scheme in the high SNR regime. Finally, the MIMO-FSO system performance with MRC scheme can be improved with the increase of the receiver aperture size and/or electrical SNR.

𝑎𝑐 , 𝑐𝑐

(34a)

𝐶 = 𝑎𝑐 𝐴𝑏 𝛤 (𝑏) ,

(34b)

𝐴=−

with [16, Eq. (13), Eq. (18), Eq. (20)]

Acknowledgment

𝑏 = min (𝛼, 𝛽) ,

This work was supported by an excellent team at the Harbin Institute of Technology.

( )−𝑏 𝜑𝑥 𝜑𝑦 𝛤 (|𝛼 − 𝛽|) (𝛼𝛽)𝑏 ℎ𝑙𝑖,𝑗 𝐴0 exp √

𝑎𝑐 =

Appendix

𝛤 (𝛼) 𝛤 (𝛽)

The joint distribution of the channel gain ℎ𝑖,𝑗 can be expressed as ( ) 𝑓ℎ𝑖,𝑗 ℎ𝑖,𝑗 =

𝐴0

∫0

𝑓ℎ

| 𝑝 𝑖,𝑗 ||ℎ𝑖,𝑗

(

| ℎ𝑖,𝑗 |ℎ𝑝𝑖,𝑗 |

)

𝑓ℎ𝑝

𝑖,𝑗

( ) ℎ𝑝𝑖,𝑗 dℎ𝑝𝑖,𝑗 ,

(

𝜑2𝑥 − 𝑏

(

𝑏𝜇𝑥2 2𝜎𝑥2 (𝜑2𝑥 −𝑏)

𝑐𝑐 =

( ) | where 𝑓ℎ ||ℎ𝑝 ℎ𝑖,𝑗 |ℎ𝑝𝑖,𝑗 is the conditional probability given ℎ𝑝𝑖,𝑗 state | 𝑖,𝑗 | 𝑖,𝑗 and can be written as ( ) ( ) ℎ𝑖,𝑗 1 | 𝑝 𝑓ℎ ||ℎ𝑝 ℎ𝑖,𝑗 |ℎ𝑖,𝑗 = 𝑙 𝑓ℎ𝑎 . (29) | 𝑖,𝑗 | 𝑖,𝑗 ℎ𝑖,𝑗 ⋅ ℎ𝑝𝑖,𝑗 𝑖,𝑗 ℎ𝑙𝑖,𝑗 ⋅ ℎ𝑝𝑖,𝑗

𝛤 (𝛼) 𝛤 (𝛽) (1 − |𝛼 − 𝛽|)

√ (

(35a)

)

( ) 2𝜎𝑦2 𝜑2𝑦 −𝑏

,

) )( 𝜑2𝑦 − 𝑏 (

( )−𝑏−1 𝜑𝑥 𝜑𝑦 𝛤 (|𝛼 − 𝛽|) (𝛼𝛽)𝑏+1 ℎ𝑙𝑖,𝑗 𝐴0 exp

(28)

+

𝑏𝜇𝑦2

𝜇𝑥2 (𝑏+1)

2𝜎𝑥2 (𝜑2𝑥 −(𝑏+1))

+

(35b)

𝜇2 (𝑏+1) (𝑦 ) 2𝜎𝑦2 𝜑2𝑦 −(𝑏+1)

) )( 𝜑2𝑥 − (𝑏 + 1) 𝜑2𝑦 − (𝑏 + 1)

) ,

(35c) where 𝜑𝑥 = 𝑤𝐿𝑒𝑞 ∕2𝜎𝑥 and 𝜑𝑦 = 𝑤𝐿𝑒𝑞 ∕2𝜎𝑦 are the ratios between the equivalent beam radius and the corresponding jitter standard deviation 6

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633

∞

𝑀ℎ𝑖,𝑗 (𝑠) =

∫0

𝑒

−𝑠ℎ𝑖,𝑗

𝑓ℎ𝑖,𝑗

( ) ℎ𝑖,𝑗 dℎ𝑖,𝑗 .

𝑗=1

𝑖=1 ℎ𝑖,𝑗

𝑀 ∏

(38)

Substituting Eq. (37) into Eq. (38) gives 𝑖=1 ℎ𝑖,𝑗

(∑

𝑀∑ 𝑁 where

(39)

Expanding Eq. (39) for large values of s ( ( )) 1 1 1 𝑀∑𝑀 ℎ (𝑠) = 𝐶 𝑀 − 𝑀𝑏 +𝑜 . 𝑀𝑏 𝑀𝑏+1 𝑀𝑏+2 𝑖=1 𝑖,𝑗 𝑠 (𝐴𝑠) (𝐴𝑠) which can be written in the following form )]) ([ ( 𝑐0 𝑐1 1 𝑀∑𝑀 ℎ (𝑠) = 𝐶 𝑀 . + + 𝑜 𝑖=1 𝑖,𝑗 𝑠𝜏 𝑠𝜏+1 𝑠𝜏+2

𝑞𝑚 = (40)

(41)

𝑐0 = 𝐴−𝜏 ,

(42b)

𝑐1 = −𝑐0 𝜏𝐴−1 .

(42c)

𝑖=1 ℎ𝑖,𝑗

(ℎ) =

(45b)

𝐾 𝛬𝜏 𝛤

∞ ∑ (−1)𝑛 ℎ𝑛+𝜏−1 . 𝑛!𝛬𝑛 (𝜏) 𝑛=0

(46)

Following from Eq. (46), the PDF of 𝑛+𝜏

𝑓(∑𝑀

𝑖=1 ℎ𝑖,𝑗

)2

(ℎ) =

𝑀 ∞ 1 𝑐0 𝐶 ∑ (−1)𝑛 ℎ 2 2 𝛤 (𝜏) 𝑛=0 𝑛!𝛬𝑛

(∑ 𝑀 𝑖=1

ℎ𝑖,𝑗

)2

will be

−1

.

(47)

By taking the Laplace transformation of Eq. (47), the MGF of (∑ )2 𝑀 can be obtained as 𝑖=1 ℎ𝑖,𝑗 ( ) 𝑛+𝜏 𝑛 ∞ 𝑐0 𝐶 𝑀 ∑ (−1) 𝛤 2 1 )2 (𝑠) = 𝑀( ∑ 𝑀 . 𝑛+𝜏 2 𝛤 (𝜏) 𝑛=0 𝑖=1 ℎ𝑖,𝑗 𝑛!𝛬𝑛 𝑠 2

∞ ∑

𝑐0𝑁 𝐶 𝑀𝑁 (𝛤

𝑛 − 𝑝𝑛 𝑠 2 , 𝑁𝜏 (𝜏))𝑁 𝑠 2 𝑛=0

(50)

) (51a)

,

(∑ 𝑀

𝑖=1 ℎ𝑖𝑗

)2

(ℎ) =

∞ 𝑁 𝑀𝑁 ∑ 𝑛+𝑁𝜏 𝑝𝑛 1 𝑐0 𝐶 ℎ 2 −1 . 𝑁 𝑁 2 (𝛤 (𝜏)) 𝑛=0 𝛤 ((𝑛 + 𝑁𝜏) ∕2)

(52)

)2 𝑀 𝑖=1 ℎ𝑖𝑗

(∑

(ℎ) =

𝑖=1 ℎ𝑖𝑗

∞ 𝑁 𝑀𝑁 ∑ 1 𝑐0 𝐶 2𝑁 (𝛤 (𝜏))𝑁 𝑛=0

𝑛+𝑁𝜏 𝑝𝑛 ℎ 2 . 𝛤 ((𝑛 + 𝑁𝜏) ∕2+1)

(53)

[1] Z. Ghassemlooy, W. Popoola, S. Rajbhandari, Optical Wireless Communications: System and Channel Modelling with Matlab® , CRC press, 2017. [2] L.C. Andrews, R.L. Phillips, Laser Beam Propagation Through Random Media, Vol. 152, SPIE press Bellingham, WA, 2005, URL https://doi.org/10.1117/3. 626196. [3] A.K. Majumdar, Advanced Free Space Optics (FSO): A Systems Approach, Vol. 186, Springer, 2014, URL https://doi.org/10.1007/978-1-4939-0918-6. [4] E. Bayaki, R. Schober, R.K. Mallik, Performance analysis of MIMO free-space optical systems in gamma-gamma fading, IEEE Trans. Commun. 57 (11) (2009) 3415–3424, http://dx.doi.org/10.1109/TCOMM.2009.11.080168. [5] T.A. Tsiftsis, Performance of heterodyne wireless optical communication systems over gamma-gamma atmospheric turbulence channels, Electron. Lett. 44 (5) (2008) 372–373, http://dx.doi.org/10.1049/el:20083028. [6] X. Zhu, J.M. Kahn, Free-space optical communication through atmospheric turbulence channels, IEEE Trans. Commun. 50 (8) (2002) 1293–1300, http: //dx.doi.org/10.1109/TCOMM.2002.800829. [7] J. Zhang, L. Dai, Y. Han, Y. Zhang, Z. Wang, On the ergodic capacity of MIMO free-space optical systems over turbulence channels, IEEE J. Sel. Areas Commun. 33 (9) (2015) 1925–1934, http://dx.doi.org/10.1109/JSAC.2015.2452631. [8] X. Yi, M. Yao, X. Wang, MIMO FSO communication using subcarrier intensity modulation over double generalized gamma fading, Opt. Commun. 382 (2017) 64–72, http://dx.doi.org/10.1016/j.optcom.2016.07.064, URL http: //www.sciencedirect.com/science/article/pii/S0030401816306526. [9] S.G. Wilson, M. Brandt-Pearce, Qianling Cao, J.H. Leveque, Free-space optical MIMO transmission with q-ary PPM, IEEE Trans. Commun. 53 (8) (2005) 1402–1412, http://dx.doi.org/10.1109/TCOMM.2005.852836. [10] T.A. Tsiftsis, H.G. Sandalidis, G.K. Karagiannidis, M. Uysal, Optical wireless links with spatial diversity over strong atmospheric turbulence channels, IEEE Trans. Wireless Commun. 8 (2) (2009) 951–957, http://dx.doi.org/10.1109/TWC.2009. 071318. [11] S.M. Navidpour, M. Uysal, M. Kavehrad, BER performance of free-space optical transmission with spatial diversity, IEEE Trans. wirel. commun. 6 (8) (2007) 2813–2819, http://dx.doi.org/10.1109/twc.2007.06109. [12] C. Abou-Rjeily, Performance analysis of FSO communications with diversity methods: Add more relays or more apertures? IEEE J. Sel. Areas Commun. 33 (9) (2015) 1890–1902, http://dx.doi.org/10.1109/jsac.2015.2432526.

Expanding exp (−ℎ∕𝛬) in the form of infinite series, we get 𝑓∑𝑀

1 2𝑁

References

(45a)

𝐾 = 𝑐0 𝛬𝜏 𝐶 𝑀 .

(

=

By taking the inverse Laplace transformation of Eq. (50), the PDF )2 ∑𝑁 (∑𝑀 can be written as 𝑗=1 𝑖=1 ℎ𝑖𝑗

𝑗=1

(44)

𝑐0 𝜏 , 𝑐1

(𝑠)

(51c)

𝐹∑𝑁

where 𝛬=−

)2

𝑗=1

Based on the approximation methodology in [32], i.e., = 𝑎𝑥𝑚−1 𝑒𝑥𝑐∕𝑎 as 𝑥 → 0+ , then the PDF in Eq. (43) can be approximated after some straightforward algebraic manipulations by: 𝐾ℎ𝜏−1 𝑒−ℎ∕𝛬 , 𝛬𝑏 𝛤 (𝜏)

)2

The corresponding cumulative distribution function (CDF) is derived by taking the integration of 𝑓∑𝑁 (∑𝑀 )2 (ℎ) as

𝑎𝑥𝑚−1 +𝑐𝑥𝑚

(ℎ) =

𝑖=1 ℎ𝑖,𝑗

(51b)

of

By taking the inverse Laplace transformation of Eq. (41), the PDF ∑ of 𝑀 𝑖=1 ℎ𝑖,𝑗 can be expressed as ] [ ( ) 𝑐 ℎ𝜏−1 𝑐 1 ℎ𝜏 (43) 𝑓∑𝑀 ℎ (ℎ) = 𝐶 𝑀 0 + + 𝑂 ℎ𝜏+1 . 𝑖=1 𝑖,𝑗 𝛤 (𝜏) 𝛤 (𝜏 + 1)

𝑖=1 ℎ𝑖,𝑗

𝑀(∑𝑀

=

𝑛 1 ∑ 𝑝𝑛 = (𝑚𝑁 − 𝑛 + 𝑚)𝑞𝑚 𝑝𝑛−𝑚 . 𝑛𝑞0 𝑚=1

𝑗=1

(42a)

)𝑁

𝑝0 = 𝑞0𝑁 ,

𝑓∑𝑁

𝜏 = 𝑀𝑏,

)2

𝑚+𝜏 2 𝑚!𝛬𝑚

(−1)𝑚 𝛤

where

𝑓∑𝑀

𝑀 𝑖=1 ℎ𝑖𝑗

𝑗=1

(𝑠) = 𝐶 𝑀 (1 + 𝐴𝑠)−𝑀𝑏 .

is given by

𝑁 ( ) (49) ⎛∞ ⎞ (−1)𝑛 𝛤 𝑛+𝜏 𝑛⎟ ∑ ⎜ 2 − 1 2 = 𝑁 𝑠 ⎟ . 𝑁𝜏 ⎜ 𝑛!𝛬𝑛 2 ⎟ (𝛤 (𝜏))𝑁 𝑠 2 ⎜⎝𝑛=0 ⎠ (∑∞ ) ∑∞ 𝑛 𝑘 𝑘 Using the fact = 𝑘=0 𝑐𝑘 𝑥 as mentioned in [33, Eq. 𝑘=0 𝑎𝑘 𝑥 (0.314)], we can rewrite the expression in Eq. (49) as follows

(37)

( )𝑀 . 𝑀ℎ𝑖,𝑗 (𝑠) = 𝑀ℎ𝑖,𝑗 (𝑠)

)2

𝑐0𝑁 𝐶 𝑀𝑁

𝑖=1

𝑀∑ 𝑀

𝑖=1 ℎ𝑖𝑗

𝑗=1

Since ℎ𝑖,𝑗 are independent and identically distributed (i.i.d.) random ∑ variables (RVs), the MGF of 𝑀 𝑖=1 ℎ𝑖,𝑗 is expressed as (𝑠) =

(∑ 𝑀

𝑀∑ 𝑁

Substituting Eq. (33) into Eq. (36), the MGF of ℎ𝑖,𝑗 can be written as

𝑀∑ 𝑀

𝑖=1 ℎ𝑖𝑗

(

(36)

𝑀ℎ𝑖,𝑗 (𝑠) = 𝐶 (1 + 𝐴𝑠)−𝑏 .

(∑ 𝑀

The MGF of 𝑀∑𝑁

at the receiver, respectively. The moment generating function (MGF) of ℎ𝑖,𝑗 is given as

(48)

7

J. Ding, S. Yu, Y. Fu et al.

Optics Communications 456 (2020) 124633 [23] A.A. Farid, S. Hranilovic, Diversity gain and outage probability for MIMO freespace optical links with misalignment, IEEE Trans. Commun. 60 (2) (2012) 479–487, http://dx.doi.org/10.1109/TCOMM.2011.121611.110181. [24] R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, C. Castillo-Vázquez, Impact of nonzero boresight pointing error on ergodic capacity of MIMO FSO communication systems, Opt. Express 24 (4) (2016) 3513–3534, http:// dx.doi.org/10.1364/OE.24.003513, URL http://www.opticsexpress.org/abstract. cfm?URI=oe-24-4-3513. [25] R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, C. Castillo-Vázquez, On the capacity of MISO FSO systems over gamma-gamma and misalignment fading channels, Opt. Express 23 (17) (2015) 22371–22385, http://dx.doi.org/10.1364/ OE.23.022371, URL http://www.opticsexpress.org/abstract.cfm?URI=oe-23-1722371. [26] M.R. Bhatnagar, Z. Ghassemlooy, Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors, J. Lightwave Technol. 34 (9) (2016) 2158–2169, http://dx.doi.org/10.1109/JLT.2016.2526053. [27] T.V. Pham, T.C. Thang, A.T. Pham, Average achievable rate of spatial diversity MIMO-FSO over correlated gamma-gamma fading channels, J. Opt. Commun. Netw. 10 (8) (2018) 662–674, http://dx.doi.org/10.1364/JOCN.10.000662, URL http://jocn.osa.org/abstract.cfm?URI=jocn-10-8-662. [28] R. Boluda-Ruiz, A. García-Zambrana, C. Castillo-Vázquez, B. Castillo-Vázquez, Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels, Opt. Express 22 (13) (2014) 16629–16644, http://dx.doi.org/10.1364/OE.22.016629, URL http: //www.opticsexpress.org/abstract.cfm?URI=oe-22-13-16629. [29] M. Uysal, J. Li, M. Yu, Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels, IEEE Trans. Wireless Commun. 5 (6) (2006) 1229–1233, http://dx.doi.org/10.1109/TWC. 2006.1638639. [30] M.K. Simon, M.-S. Alouini, Digital Communication Over Fading Channels, Vol. 95, John Wiley & Sons, 2005, URL https://doi.org/10.1002/0471715220. [31] Z. Wang, G.B. Giannakis, A simple and general parameterization quantifying performance in fading channels, IEEE Trans. Commun. 51 (8) (2003) 1389–1398. [32] Y. Dhungana, C. Tellambura, New simple approximations for error probability and outage in fading, IEEE Commun. Lett. 16 (11) (2012) 1760–1763, http: //dx.doi.org/10.1109/LCOMM.2012.092112.121280. [33] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic press, 2014, http://dx.doi.org/10.1016/c2010-0-64839-5.

[13] A.A. Farid, S. Hranilovic, Outage capacity optimization for free-space optical links with pointing errors, J. Lightwave Technol. 25 (7) (2007) 1702–1710, http://dx.doi.org/10.1109/jlt.2007.899174, URL http://jlt.osa.org/abstract.cfm? URI=jlt-25-7-1702. [14] F. Yang, J. Cheng, T.A. Tsiftsis, Free-space optical communication with nonzero boresight pointing errors, IEEE Trans. Commun. 62 (2) (2014) 713–725, http: //dx.doi.org/10.1109/TCOMM.2014.010914.130249. [15] W. Gappmair, S. Hranilovic, E. Leitgeb, OOK performance for terrestrial FSO links in turbulent atmosphere with pointing errors modeled by Hoyt distributions, IEEE Commun. Lett. 15 (8) (2011) 875–877, http://dx.doi.org/10.1109/LCOMM.2011. 062911.102083. [16] S. Yu, J. Ding, Y. Fu, J. Ma, L. Tan, L. Wang, Novel approximate and asymptotic expressions of the outage probability and BER in gamma–gamma fading FSO links with generalized pointing errors, Opt. Commun. 435 (2019) 289–296, http://dx.doi.org/10.1016/j.optcom.2018.11.021, URL http://www. sciencedirect.com/science/article/pii/S0030401818309726. [17] H. AlQuwaiee, H. Yang, M. Alouini, On the asymptotic capacity of dualaperture FSO systems with generalized pointing error model, IEEE Trans. Wireless Commun. 15 (9) (2016) 6502–6512, http://dx.doi.org/10.1109/TWC. 2016.2585486. [18] L. Yang, M.O. Hasna, X. Gao, Asymptotic BER analysis of FSO with multiple receive apertures over M-distributed turbulence channels with pointing errors, Opt. Express 22 (15) (2014) 18238–18245, http://dx.doi.org/10.1364/OE.22. 018238, URL http://www.opticsexpress.org/abstract.cfm?URI=oe-22-15-18238. [19] K. Prabu, D.S. Kumar, BER analysis of DPSK-SIM over MIMO free space optical links with misalignment, Optik 125 (18) (2014) 5176–5180, http://dx.doi.org/ 10.1016/j.ijleo.2014.05.012, URL http://www.sciencedirect.com/science/article/ pii/S0030402614005798. [20] K. Prabu, D.S. Kumar, MIMO free-space optical communication employing coherent BPOLSK modulation in atmospheric optical turbulence channel with pointing errors, Opt. Commun. 343 (2015) 188–194, http://dx.doi.org/10. 1016/j.optcom.2015.01.012, URL http://www.sciencedirect.com/science/article/ pii/S0030401815000176. [21] A. García-Zambrana, B. Castillo-Vázquez, C. Castillo-Vázquez, Asymptotic errorrate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors, Opt. Express 20 (3) (2012) 2096–2109, http://dx.doi.org/10.1364/OE.20.002096, URL http://www. opticsexpress.org/abstract.cfm?URI=oe-20-3-2096. [22] A. Farid, S. Hranilovic, Outage capacity for MISO intensity-modulated free-space optical links with misalignment, IEEE/OSA J. Opt. Commun. Networking 3 (10) (2011) 780–789, http://dx.doi.org/10.1364/JOCN.3.000780.

8