Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors

doi:10.1364/OE.20.016394

Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors

Antonio García-Zambrana, Carmen Castillo-Vázquez, Beatriz Castillo-Vázquez, and Rubén Boluda-Ruiz »View Author Affiliations

Optics Express, Vol. 20, Issue 15, pp. 16394-16409 (2012)
http://dx.doi.org/10.1364/OE.20.016394

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▸ Article Outline
– Abstract
1. Introduction
2. System and channel model
3. Error-rate performance analysis
4. Conclusions
– References and links

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Abstract

An unsuitable alignment between transmitter and receiver together with fluctuations in the irradiance of the transmitted optical beam due to the atmospheric turbulence can severely degrade the performance of free-space optical (FSO) systems. In this paper, cooperative FSO communications with decode-and-forward (DF) relaying and equal gain combining (EGC) reception over atmospheric turbulence and misalignment fading channels is analyzed in order to mitigate these impairments. Novel closed-form asymptotic bit error-rate (BER) expressions are derived for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2β^1/2 times the value of the pointing error displacement standard deviation at the receiver. It is contrasted that the available diversity order is strongly dependent on the relay location, achieving greater diversity gains when the diversity order is determined by β_AC + β_BC, where β_AC and β_BC are parameters corresponding to the turbulence of the source-destination and relay-destination links. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

1. Introduction

Atmospheric free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications, being an interesting alternative to consider for next generation broadband in order to support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1

V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006). [CrossRef]

]. Recently, the use of FSO transmission is being specially interesting to solve the “last mile” problem, as well as a supplement to radio-frequency (RF) links [2

L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE 97(6), 1109–1127 (2009). [CrossRef]

, 3

W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009). [CrossRef] [PubMed]

]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the link performance [4

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001). [CrossRef]

, 5

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). [CrossRef]

]. Additionally, since FSO systems are usually installed on high buildings, building sway causes vibrations in the transmitted beam, leading to an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [6

E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004). [CrossRef]

–12

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express 18(19), 445–454 (2010). [CrossRef]

]. The combined effect of atmospheric and misalignment fading is analyzed in the case of single-input/single-output (SISO) FSO channels in [13

S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003). [CrossRef] [PubMed]

]. In [14

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for SISO links. In [15

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011). [CrossRef]

] the error rate performance for coded FSO links over strong turbulence and misalignment fading channels is studied. In [16

H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009). [CrossRef]

, 17

W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010). [CrossRef]

], a wide range of turbulence conditions with gamma-gamma atmospheric turbulence and pointing errors is also considered on terrestrial FSO links, deriving closed-form expressions for the error-rate performance in terms of Meijer’s G-functions. In [18

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14),480–496 (2011). [CrossRef]

,19

A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20(3), 2096–2109 (2012). [CrossRef] [PubMed]

], comparing different diversity techniques, a significant improvement in terms of outage and error-rate performance is demonstrated when MIMO FSO links based on transmit laser selection are adopted in the context of wide range of turbulence conditions (weak to strong) with pointing errors, showing that better performance is achieved when increasing the number of transmit apertures instead of the number of receive apertures in order to guarantee a same diversity order.

An alternative approach to improving the performance in this turbulence FSO scenario is the employment of cooperative communications in order to overcome some limitations of MIMO structures. Cooperative transmission can significantly improve the performance by creating diversity using the transceivers available at the other nodes of the network. This is a well known technique employed in RF systems, wherein more attention has been paid to the concept of user cooperation as a new form of diversity for future wireless communication systems [20

A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003). [CrossRef]

–22

J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004). [CrossRef]

]. Recently, several works have investigated the adoption of this technique in the context of FSO systems [23

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008). [CrossRef]

–28

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012). [CrossRef]

]. In [23

M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008). [CrossRef]

] an artificial broadcasting through the use of multiple transmitter apertures directed to relay nodes is proposed as a parallel relaying transmission scheme as well as a serial transmission, evaluating the outage probability when amplify-and-forward (AF) and decode-and-forward (DF) relaying are considered. In [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

, 25

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010). [CrossRef]

] a 3-way FSO communication setup is proposed to implement a cooperative protocol in order to improve spatial diversity without much increase in hardware, being evaluated the error-rate performance by using the photon-count method as well as the outage performance for both AF and DF strategies. In [26

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011). [CrossRef]

] the error performance is evaluated for one-relay cooperative diversity scheme by using the photon-count method in the presence and absence of background radiation when lognormal and Rayleigh turbulence-induced fading channel models are assumed. In [27

C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol. 29(7), 1058 –1065 (2011). [CrossRef]

] the impact of the channel state information (CSI) available at the different nodes on the performance of cooperative FSO networks is investigated. In [28

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012). [CrossRef]

] a three-node cooperative DF FSO system under gamma-gamma fading channels using binary phase shift keying-subcarrier intensity modulation (BPSK-SIM) is analyzed, considering selective and perfect relay.

In this paper, this approach is extended to FSO communication systems using IM/DD over atmospheric turbulence and misalignment fading channels, considering cooperative FSO communications with decode-and-forward (DF) relaying and equal gain combining (EGC) reception. Novel closed-form asymptotic bit error-rate (BER) expressions are derived for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model, as in [14

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

, 15

H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011). [CrossRef]

], where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2β^1/2 times the value of the pointing error displacement standard deviation at the receiver. Moreover, it is contrasted that the available diversity order is strongly dependent on the relay location, achieving greater diversity gains when the diversity order is determined by β_AC + β_BC, where β_AC and β_BC are parameters corresponding to the turbulence of the source-destination and relay-destination links. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of signal-to-noise ratio (SNR) as the turbulence strength increases.

2. System and channel model

Following the cooperative protocol presented in [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

,25

M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010). [CrossRef]

], we adopt a three-node cooperative system based on three separate full-duplex FSO links, assuming laser sources intensity-modulated and ideal noncoherent (direct-detection) receivers, as shown in Fig. 1, wherein nodes A and B are considered to be connected to the same source. For this 3-way FSO communication setup the cooperative protocol can be applied to achieve the spatial diversity without much increase in hardware. As in [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

], the cooperative strategy works in two phases or transmission frames. In the first phase, the nodes A and B send their own data to each other and the destination node C, i.e., the node A (B) transmits the same information to the nodes B (A) and C. In the second transmission frame, the node B (or A) sends the received data from its partner A (or B) in the first frame to the node C. The way that A and B send their partner’s data in the second frame is specified by the cooperative strategy here analyzed. Following the bit-detect-and-forward (BDF) cooperative protocol [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

], the relay (partner) node detects each code bit of the cooperative signal individually and forwards it to the destination, regardless of the channel coding. In this fashion, the relay node (A or B) detects each code bit to “0” or “1” and sends the bit with the new power to the destination node C. It must be noted the fact that the symmetry for nodes A and B assumed in this FSO communication setup implies that no rate reduction is applied, i.e., the same information rate can be considered at the destination node C compared to the direct transmission link without using any cooperative strategy. In contrast to the DF strategy considered in [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

] wherein it is assumed that all the bits received from the relay path A–B–C are detected correctly at B and are resended to C, or the DF strategy considered in [26

C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011). [CrossRef]

,28

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012). [CrossRef]

] wherein bits detected incorrectly are not resended, it is assumed in this paper that all the bits detected at the relay are always resended regardless of these bits are detected correctly or incorrectly. Next, bits received directly from A–C and from the relay A–B–C are detected at C following an EGC technique. This combining technique is conventionally adopted in FSO links because of its considerably lower implementation complexity even maintaining relevant performance [8

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009). [CrossRef]

, 9

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

]. In opinion of the authors, the detection and protocol here assumed is closer to the real scenario, being more easily implemented than that protocol wherein the relay node sends the information depending on the bit is correctly detected at B.

Fig. 1 Block diagram of the considered 3-way FSO communication system, where L_AC is the A–C link distance and (x_B, y_B) represents the location of the node B.

For each link of the three possible links in this three-node cooperative FSO system, the instantaneous current y_m(t) in the receiving photodetector corresponding to the information signal transmitted from the laser can be written as

y_{m} (t) = η i_{m} (t) x (t) + z_{m} (t)

(1)

where η is the detector responsivity, assumed hereinafter to be the unity, X ≜ x(t) represents the optical power supplied by the source and I_m ≜ i_m(t) the equivalent real-valued fading gain (irradiance) through the optical channel between the laser and the receive aperture. Z_m ≜ z_m(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance σ² = N₀/2, i.e. Z_m ∼ N(0, N₀/2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity, X must satisfy ∀tx(t) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of X is limited. The received electrical signal Y_m ≜ y_m(t), however, can assume negative amplitude values. We use Y_m, X, I_m and Z_m to denote random variables and y_m(t), x(t), i_m(t) and z_m(t) their corresponding realizations. Additionally, we consider on-off keying (OOK) formats with any pulse shape and reduced duty cycle, allowing the increase of the peak-to-average optical power ratio (PAOPR) parameter [11

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010). [CrossRef] [PubMed]

,18

,29

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010). [CrossRef]

The irradiance is susceptible to either atmospheric turbulence conditions and pointing error effects. In this case, it is considered to be a product of two independent random variables, i.e.

I_{m} = I_{m}^{(a)} I_{m}^{(p)}

, representing

I_{m}^{(a)}

and

I_{m}^{(p)}

the attenuation due to atmospheric turbulence and the attenuation due to geometric spread and pointing errors, respectively. As in [24

M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]

, 28

M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012). [CrossRef]

], for the sake of simplicity, link attenuation is not considered in this work since path loss is non-random in nature, not affecting the conclusions here obtained in relation to the diversity order analysis for the BDF cooperative protocol under study. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [30

D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (2009). [CrossRef]

]. To consider a wide range of turbulence conditions (weak to strong), the gamma-gamma turbulence model proposed in [4

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001). [CrossRef]

,31

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 8 (2001). [CrossRef]

] is here assumed, whose probability density function (PDF) is given by

f_{I_{m}^{(a)}} (i) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} i^{((α + β) / 2) - 1} K_{α - β} (2 \sqrt{α β i}), i \geq 0

(2)

where Γ(·) is the well-known Gamma function and K_ν(·) is the νth-order modified Bessel function of the second kind [32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (8.43)]. The parameters α and β can be selected to achieve a good agreement between Eq. (2) and measurement data [31

]. Alternatively, assuming plane wave propagation and negligible inner scale, α and β can be directly linked to physical parameters through the following expresions [31

,33

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010). [CrossRef]

α = {[exp (\frac{0.49 σ_{R}^{2}}{{(1 + 1.11 σ_{R}^{12 / 5})}^{7 / 6}}) - 1]}^{- 1}

(3)

β = {[exp (\frac{0.51 σ_{R}^{2}}{{(1 + 0.69 σ_{R}^{12 / 5})}^{7 / 6}}) - 1]}^{- 1}

(4)

where

σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6}

is the Rytov variance, which is a measure of optical turbulence strength. Here, k = 2π/λ is the optical wave number, λ is the wavelength and L is the link distance in meters.

C_{n}^{2}

stands for the altitude-dependent index of the refractive structure parameter and varies from 10⁻¹³ m^−2/3 for strong turbulence to 10⁻¹⁷ m^−2/3 for weak turbulence [4

L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001). [CrossRef]

]. It must be emphasized that parameters α and β cannot be arbitrarily chosen in FSO applications, being related through the Rytov variance. In this fashion, it can be shown that the relationship α > β always holds, and the parameter β is lower bounded above 1 as the Rytov variance approaches ∞ [33

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010). [CrossRef]

]. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [14

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. Assuming a Gaussian spatial intensity profile of beam waist radius, ω_z, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, the PDF of

I_{m}^{(p)}

is given by

f_{I_{m}^{(p)}} (i) = \frac{φ^{2}}{A_{0}^{φ^{2}}} i^{φ^{2} - 1}, 0 \leq i \leq A_{0}

(5)

where φ = ω_{z_eq}/2σ_s is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver,

ω_{z_{eq}}^{2} = ω_{z}^{2} \sqrt{π} erf (v) / 2 v exp (- v^{2})

v = \sqrt{π} r / \sqrt{2} ω_{z}

, A₀ = [erf(v)]² and erf(·) is the error function [32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (8.250)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being

σ_{s}^{2}

the jitter variance at the receiver. Using the previous PDFs for turbulence and misalignment fading, a closed-form expression of the combined PDF of I_m was derived in [16

] as

f_{I_{l m}} (i) = \frac{α β φ^{2}}{A_{0} Γ (α) Γ (β)} G_{1, 3}^{3, 0} (\frac{α β}{A_{0}} i | \begin{matrix} φ^{2} \\ φ^{2} - 1, α - 1, β - 1 \end{matrix}), i \geq 0

(6)

where

G_{p, q}^{m, n} [\cdot]

is the Meijer’s G-function [32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (9.301)]. Even though Meijer’s G-function can be expressed in terms of more familiar generalized hypergeometric functions, this PDF appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of MIMO FSO systems, leading to numerical solutions that obscure the impact of the basic system and channel parameters on performance. As in [18

, 19

], to overcome this inconvenience, the PDF in Eq. (6) is approximated by a single polynomial term as

f_{I_{m}} (i) \approx a_{m} i^{b_{m}},

(7)

based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin, i.e. f_{I_m} (i) at i → 0 determines high SNR performance [34

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

]. Hence, using the series expansion corresponding to the Meijer’s G-function [35

Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com.

, eqn. (07.34.06.0006.01)] and considering the fact that the two parameters α and β related to the atmospheric conditions verify that α > β, different expressions for a_m and b_m in Eq. (7), depending on the relation between the values of φ² and β, can be written as

a_{m} = {\begin{array}{l} a_{m}^{0} = \frac{φ^{2} {(α β)}^{β} Γ (α - β)}{A_{0}^{β} Γ (α) Γ (β) (φ^{2} - β)}, & φ^{2} > β \\ a_{m}^{1} = \frac{φ^{2} {(α β)}^{φ^{2}} Γ (α - φ^{2}) Γ (β - φ^{2})}{A_{0}^{φ^{2}} Γ (α) Γ (β)}, & φ^{2} < β \end{array}

(8a)

b_{m} = {\begin{array}{l} b_{m}^{0} = β - 1, & φ^{2} > β \\ b_{m}^{1} = φ^{2} - 1, & φ^{2} < β \end{array}

(8b)

From these expressions it can be observed that the superscripts 0 and 1 are used to indicate the way the impact of pointing errors is present in Eq. (7), being specially relevant how the power of the polynomial term maintains or not initial performance if compared to the similar FSO scenario where no pointing errors are present. In the following section, the fading coefficient I_m for the paths A–B, A–C and and B–C is indicated by I_AB, I_AC and I_BC, respectively.

3. Error-rate performance analysis

In this section, we can take advantage of these simpler asymptotic expressions in order to quantify the bit error probability at high SNR, showing that the asymptotic performance of this metric as a function of the average SNR is characterized by two parameters: the diversity and coding gains. For the sake of clarity, without loss of generality, we can consider node A as source and node B as its relay for the BER evaluation since similar results hold when node B is considered as the source and node A as its relay. Nonetheless, we later conclude the analysis in this paper by examining the inclusion of the symmetric scheme in order to maintain the same information rate at the destination node C as was explained in previous section. In addition to the performance evaluation of the BER corresponding to the cooperative protocol here proposed, we also consider the performance analysis for the direct path link (non-cooperative link A–C) to establish the baseline performance. Moreover, BER performance corresponding to the non-cooperative case with two transmitters following the transmit laser selection (TLS) scheme is also included as a benchmark of the FSO scenario when the diversity order is 2. In [11

,12

,18

,19

] it has been shown that the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO space-time codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). Here, it is assumed that the average optical power transmitted from each node is P_opt. In this way, according to Eq. (1) and the OOK signaling [18

, appendix], a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of

d = 2 P_{opt} \sqrt{T_{b} ξ}

, the statistical channel model corresponding to the A–B link can be written as

Y_{A B} = \frac{1}{2} X I_{A B} + Z_{A B}, X \in {0, d}, Z_{A B} ~ N (0, N_{0} / 2)

(9)

where the parameter T_b is the bit period and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, as explained in a greater detail in [18

, appendix]. Because of the fact that in the first phase of the cooperative protocol the node A transmits the same information to the nodes B and C, the division by 2 is considered so as to maintain the average optical power in the air at a constant level of P_opt, being transmitted by each laser an average optical power of P_opt/2. Assuming channel side information at the receiver, the conditional BER at the node B is given by

P_{b}^{A B} (E | I_{A B}) = Q (\sqrt{{(d / 2)}^{2} i^{2} / 2 N_{0}})

(10)

where Q(·) is the Gaussian-Q function defined as

Q (x) = \frac{1}{\sqrt{2 π}} \int_{x}^{\infty} e^{- \frac{t^{2}}{2}} d t

. Substituting the value of the Euclidean distance d gives

P_{b}^{A B} (E | I_{A B}) = Q (\sqrt{(γ / 2) ξ} i)

where

γ = P_{opt}^{2} T_{b} / N_{0}

represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats. Hence, the average BER,

P_{b}^{A B} (E)

, can be obtained by averaging

P_{b}^{A B} (E | I_{A B})

over the PDF as follows

P_{b}^{A B} (E) = \int_{0}^{\infty} Q (\sqrt{(γ / 2) ξ} i) f_{I_{A B}} (i) d i .

(11)

To evaluate the integral in Eq. (11), we can use that the Q-function is related to the complementary error function erfc(·) by

erfc (x) = 2 Q (\sqrt{2} x)

[32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (6.287)] and the fact that

\int_{0}^{\infty} erfc (x) x^{a - 1} d x = Γ ((1 + a) / 2) / (π^{1 / 2} a)

[32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (6.281)], obtaining the corresponding closed-form asymptotic solution for the BER as can be seen in

P_{b}^{A B} (E) ≐ \frac{a_{A B} 2^{b_{A B}} Γ (\frac{b_{A B}}{2} + 1)}{\sqrt{π} (b_{A B} + 1)} {(γ ξ)}^{- \frac{1}{2} (1 + b_{A B})}

(12)

where the value of the parameters a_AB and b_AB depends on the relation between φ² and β as obtained in Eq. (8), corroborating that the diversity order corresponding to the A–B link is independent of the pointing error when the equivalent beam radius at the receiver at the node B is at least 2β^1/2 times the value of the pointing error displacement standard deviation, i.e. φ² > β. Once the error probability at the node B is known, two cases can be considered to evaluate the BER corresponding to the BDF cooperative protocol here proposed depending on the fact that the bit from the relay A–B–C is detected correctly or incorrectly. In this way, the statistical channel model corresponding to the BDF cooperative protocol, i.e. the bits received at C directly from A–C link and from the relay A–B–C can be written as

Y_{B D F} = \frac{1}{2} X I_{A C} + Z_{A C} + X^{*} I_{B C} + Z_{B C}, X \in {0, d}, Z_{A C}, A_{B C} ~ N (0, N_{0} / 2)

(13)

where X^* represents the random variable corresponding to the information detected at the node B and, hence, X^* = X when the bit has been detected correctly at B and X^* = d − X when the bit has been detected incorrectly. In this manner, considering that the bit is correctly detected at B, the statistical channel model for the BDF cooperative protocol can be expressed as

Y_{{B D F}_{0}} = X (\frac{1}{2} I_{A C} + I_{B C}) + Z_{E G C}, X \in {0, d}, Z_{E G C} ~ N (0, N_{0})

(14)

Assuming channel side information at the receiver, the conditional BER at the node C is given by

P_{b}^{{B D F}_{0}} (E | I_{A C}, I_{B C}) = Q (\sqrt{(γ / 4) ξ} (i_{1} + 2 i_{2}))

. Hence, the average BER,

P_{b}^{{B D F}_{0}} (E)

, can be obtained by averaging over the PDF as follows

P_{b}^{{B D F}_{0}} (E) = \int_{0}^{\infty} \int_{0}^{\infty} Q (\sqrt{(γ / 4) ξ} (i_{1} + 2 i_{2})) f_{I_{A C}} (i_{1}) f_{I_{B C}} (i_{2}) d i_{1} d i_{2} .

(15)

Since the variates are independent, knowing that the resulting PDF of their sum I_T = I_AC + 2I_BC can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms, approximate expression for the PDF, f_{I_T} (i), of the combined variate can be easily derived from Eq. (7) as

f_{I_{T}} (i) \approx \frac{a_{A C} a_{B C} 2^{- b_{B C} - 1} Γ (b_{A C} + 1) Γ (b_{B C} + 1)}{Γ (b_{A C} + b_{B C} + 2)} i^{b_{A C} + b_{B C} + 1} .

(16)

From this expression, the average BER,

P_{b}^{{B D F}_{0}} (E)

, can be determined as follows

P_{b}^{B D F_{0}} (E) = \int_{0}^{\infty} Q (\sqrt{(γ / 4) ξ} i) f_{I_{T}} (i) d i .

(17)

Evaluating this integral as in Eq. (11), we can obtain the corresponding closed-form asymptotic solution for the BER as follows

P_{b}^{{B D F}_{0}} (E) ≐ \frac{a_{A C} a_{B C} 2^{\frac{1}{2} (b_{A C} - b_{B C} - 2)} Γ (b_{A C} + 1) Γ (b_{B C} + 1)}{Γ (\frac{1}{2} (b_{A C} + b_{B C} + 4))} {(γ ξ)}^{- \frac{1}{2} (b_{A C} + b_{B C} + 2)}

(18)

Alternatively, considering now that the bit is incorrectly detected at B, the statistical channel model for the BDF cooperative protocol can be expressed as

Y_{{B D F}_{1}} = X (\frac{1}{2} I_{A C} - I_{B C}) + d \cdot I_{B C} + Z_{E G C}, X \in {0, d}, Z_{E G C} ~ N (0, N_{0})

(19)

Assuming channel side information at the receiver and given that the statistics corresponding to the term d · I_BC become irrelevant to the detection process, the conditional BER at the node C is given by

P_{b}^{{B D F}_{1}} (E | I_{A C}, I_{B C}) = Q (\sqrt{(γ / 4) ξ} (i_{1} - 2 i_{2}))

. Hence, the average BER,

P_{b}^{{B D F}_{1}} (E)

can be obtained by averaging over the PDF as follows

P_{b}^{{B D F}_{1}} (E) = \int_{0}^{\infty} \int_{0}^{\infty} Q (\sqrt{(γ / 4) ξ} (i_{1} - 2 i_{2})) f_{I_{A C}} (i_{1}) f_{I_{B C}} (i_{2}) d i_{1} d i_{2} .

(20)

Unfortunately, the asymptotic behavior of this expression cannot be determined by using the Eq. (7) as in previous integrals since the result in Eq. (20) is not dominated by the behavior of the PDF near the origin because of the argument of the Gaussian-Q function is not always positive [34

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

]. To overcome this inconvenience, we can use the expression Q(−x) = 1 − Q(x) to manipulate the negative values on the argument of the Gaussian-Q function in Eq. (20) together with the fact that Gaussian-Q function tends to 0 as γ → ∞, simplifying the integral in Eq. (20) as follows

P_{b}^{{B D F}_{1}} (E) ≐ \int_{0}^{\infty} \int_{0}^{2 i_{2}} f_{I_{A C}} (i_{1}) f_{I_{B C}} (i_{2}) d i_{1} d i_{2} .

(21)

It can be noted that the asymptotic behavior of

P_{b}^{{B D F}_{1}} (E)

is independent of the SNR γ, resulting in a positive value that is upper bounded by 1. To evaluate the integral (21), we can use the Meijer’s G-function [32

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).

, eqn. (9.301)], available in standard scientific software packages such as Mathematica and Maple, in order to transform the integral expression to the form in [36

V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. on Symbolic and Algebraic Computation , 212–224 (Tokyo, Japan, 1990).

, eqn. (21)], expressing K_μ(·) [36

, eqn. (14)] in terms of Meijer’s G-function. In this way, a closed-form solution is derived as can be seen in

P_{b}^{{B D F}_{1}} (E) ≐ \frac{(φ_{A C}^{2} φ_{B C}^{2}) G_{5, 5}^{3, 4} (\frac{2 α_{A C} β_{A C} A_{0_{B C}}}{α_{B C} β_{B C} A_{0_{A C}}} | \begin{matrix} 1, 1 - α_{B C}, 1 - β_{B C}, 1 - φ_{B C}^{2}, φ_{A C}^{2} + 1 \\ φ_{A C}^{2}, α_{A C}, β_{A C}, - φ_{B C}^{2}, 0 \end{matrix})}{Γ (α_{A C}) Γ (α_{B C}) Γ (β_{A C}) Γ (β_{B C})} .

(22)

Nonetheless, it must be emphasized that the Meijer’s G-function has to be numerically calculated and, hence, the use of Monte Carlo integration to solve Eq. (21) may represent an alternative with less computational load. Once the error probability at the node B is known and considering these two cases, i.e. depending on the fact that the bit from the relay A–B–C is detected correctly or incorrectly, the BER corresponding to the BDF cooperative protocol here proposed is given by

P_{b}^{B D F} (E) = P_{b}^{{B D F}_{0}} (E) \cdot (1 - P_{b}^{A B} (E)) + P_{b}^{{B D F}_{1}} (E) \cdot P_{b}^{A B} (E) .

(23)

This expression can be simplified taking into account the asymptotic behavior previously obtained in Eq. (12) and Eq. (18) as follows

P_{b}^{B D F} (E) ≐ \frac{a_{A C} a_{B C} Γ (b_{A C} + 1) Γ (b_{B C} + 1) {(γ ξ)}^{- \frac{1}{2} (2 + b_{A C} + b_{B C})}}{2^{- \frac{1}{2} (- b_{A C} + b_{B C})} 2 Γ (\frac{1}{2} (b_{A C} + b_{B C} + 4))}, b_{A C} + b_{B C} + 1 < b_{A B}

(24a)

P_{b}^{B D F} (E) ≐ P_{b}^{{B D F}_{1}} (E) \cdot \frac{a_{A B} 2^{b_{A B}} Γ (\frac{b_{A B}}{2} + 1) {(γ ξ)}^{- \frac{1}{2} (1 + b_{A B})}}{\sqrt{π} (b_{A B} + 1)}, b_{A C} + b_{B C} + 1 > b_{A B}

(24b)

since it is easy to deduce from Eq. (23) that

P_{b}^{B D F} (E) ≐ P_{b}^{{B D F}_{0}} (E)

when b_AC + b_BC + 1 < b_AB and

P_{b}^{B D F} (E) ≐ P_{b}^{{B D F}_{1}} (E) \cdot P_{b}^{A B} (E)

when b_AC + b_BC + 1 > b_AB. It is straightforward to show that the average BER behaves asymptotically as (Λ_cγξ)^−Λ_d, where Λ_d and Λ_c denote diversity order and coding gain, respectively [34

Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]

]. At high SNR, if asymptotically the error probability behaves as (Λ_cγξ)^−Λ_d, the diversity order Λ_d determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain Λ_c (in decibels) determines the shift of the curve in SNR. Taking into account these expressions, the adoption of the BDF cooperative protocol here analyzed translates into a diversity order gain, G_d, relative to the non-cooperative link A–C of

G_{d} = min (2 + b_{A C} + b_{B C}, 1 + b_{A B}) / (1 + b_{A C})

(25)

Since the diversity order determines the slope of the BER performance, it must be noted that this parameter G_d quantifies the improvement in performance corresponding to the BDF cooperative protocol compared to the normal FSO system or direct path link (non-cooperative link A–C). From this asymptotic analysis, it can be deduced that the main aspect to consider in order to optimize the error-rate performance is the relation between φ² and β as obtained in Eq. (8b) for the links A–B, A–C and B–C, corroborating that the diversity order corresponding to each link is independent of the pointing error when the equivalent beam radius at each receiver is at least 2β^1/2 times the value of the pointing error displacement standard deviation, i.e. φ² > β. Once this condition is satisfied an analysis about how the Eq. (25) can be optimized is required, evaluating if the diversity order corresponding to the BDF cooperative protocol is determined by the source-destination and relay-destination links or by the source-relay link. For the better understanding of the impact of the configuration of the three-node cooperative FSO system under study, the diversity order gain G_d in Eq. (25) as a function of the horizontal displacement of the relay node, x_B, is depicted in Fig. 2 for a source-destination link distance L_AC= {3 km, 6 km} when different relay locations y_B={0.5 km, 1 km, 1.5 km, 2 km, 2.5 km} are assumed. Here, the parameters α and β are calculated from Eq. (3) and Eq. (4), and values of λ = 1550 nm and

C_{n}^{2} = 1.7 \times 10^{- 14} m^{- 2 / 3}

are adopted [9

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009). [CrossRef]

]. In any case, the condition φ² > β is satisfied for each link and, hence, these results are independent of pointing errors. These curves are corresponding to the intersection of two profiles related to the expressions (β_AC +β_BC)/β_AC and β_AB/β_AC, as deduced from Eq. (25). It can be easily contrasted from Eq. (4) that the symmetric quasi-Gaussian shape is related to the expression β_AB/β_AC, scenario in which the diversity order is determined by the source-relay link A–B since β_AB < (β_AC +β_BC). Hence, it can be concluded that the available diversity order is strongly dependent on the relay location, achieving greater diversity gains when the diversity order is determined by β_AC + β_BC, corresponding to the turbulence of the source-destination and relay-destination links. The results corresponding to this asymptotic analysis with rectangular pulse shapes and ξ = 1 are illustrated in the Fig. 3, when different relay locations for source-destination link distances L_AC= {3 km, 6 km} are assumed together with values of normalized beamwidth and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (10, 2). Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the derived results. Due to the long simulation time involved, simulation results only up to BER=10⁻⁹ are included. Simulation results corroborate that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases. Additionally, we also consider the performance analysis for the direct path link (non-cooperative link A–C) to establish the baseline performance as well as BER performance corresponding to the non-cooperative case with two transmitters following the transmit laser selection scheme as a benchmark of the FSO scenario when the diversity order is 2. From [19

] and using the notation here assumed, the asymptotic BER performance corresponding to the TLS scheme with M transmit lasers can be rewritten as

P_{b}^{T L S} (E) ≐ \frac{{(\frac{a_{A C}}{b_{A C} + 1})}^{M} Γ (\frac{1}{2} (b_{A C} M + M + 1))}{2 \sqrt{π}} γ^{- \frac{1}{2} (b_{A C} + 1) M} .

(26)

It is noted that Eq. (26) can be also used for bounding BER performance corresponding to the direct path link (non-cooperative link A–C) case when M is set to 1. Nonetheless, although BER performance corresponding to the TLS scheme is here considered as a benchmark of the FSO scenario with diversity order of 2, it must be commented that this is not a fair comparison since changes in hardware are required compared to the 3-way FSO communication setup, wherein the BDF protocol is applied to achieve the spatial diversity without any demand for extra hardware. As expected, it can be corroborated that these BER results are in excellent agreement with previous results shown in Fig. 2 in relation to the diversity order gain achieved for this 3-way FSO communication setup. In this way, it can be seen diversity gains of 2.42 and 1.3 when L_AC=3 km and relay locations of (x_B=1 km; y_B=0.5 km) and (x_B=2 km; y_B=1 km), respectively, or diversity gains of 2 and 1.18 when L_AC=6 km and relay locations of (x_B=0.5 km; y_B=1 km) and (x_B=3.5 km; y_B=1.5 km), respectively. Both cases for L_AC=3 km and L_AC=6 km represent the two possible scenarios considered in Eq. (24), being Eq. (24a) the bound corresponding to the configuration of the three-node cooperative FSO system in which greater diversity gains are achieved, i.e. β_AC +β_BC < β_AB.

Fig. 2 Diversity order gain G_d for a 3-way FSO communication setup with BDF relaying and EGC reception for a source-destination link distance of (a) L_AC = 3 km and (b) L_AC = 6 km when different relay locations of y_B={0.5 km, 1 km, 1.5 km, 2 km, 2.5 km} are assumed, once the condition φ² > β is satisfied for each link.

Fig. 3 BER performance for a 3-way FSO communication setup with BDF relaying and EGC reception over atmospheric turbulence and misalignment fading channels, when different relay locations for source-destination link distances of (a) L_AC = 3 km and (b) L_AC = 6 km are assumed together with values of normalized beamwidth and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (10, 2).

From previous results, it can be deduced that a greater diversity gain is achieved as the source-relay link distance is shorter and, hence, β_AB is greater. This is also concluded in Fig. 4a for a vertical displacement of the relay node of y_B=0.2 km and a source-destination link distance of L_AC= 2 km. In this configuration, a normalized beamwidth of ω_z/r = 7 and different values of normalized jitter σ_s/r = {1, 1.5, 1.75, 2, 3} are assumed in order to contrast the impact of pointing errors when the condition φ² > β is or not satisfied for each link. It can be observed that diversity gains even greater than 3 are achieved when (ω_z/r, σ_s/r) = (7, 1), not being affected by pointing errors. However, the maximum values of G_d corresponding to this configuration are significantly decreased as the normalized jitter increases and, hence, the condition φ² > β is not satisfied. These conclusions are contrasted in Fig. 4b, wherein BER performance for a source-destination link distance of L_AC = 2 km and a relay location of (x_B=0.8 km; y_B=0.2 km) when values of normalized beamwidth of ω_z/r = 7 and normalized jitter of σ_s/r = {1, 2, 3} are assumed. As before, we also consider the performance analysis for the direct path link (non-cooperative link A–C) to establish the baseline performance as well as BER performance corresponding to the non-cooperative case with two transmitters following the TLS scheme as a benchmark of the FSO scenario when the diversity order is 2. These BER results are in excellent agreement with previous results shown in Fig. 4a in relation to the diversity order gain achieved for this 3-way FSO communication setup when pointing errors are present. In this way, it can be seen diversity gains of 3, 1.38 and 1 when values of normalized jitter of σ_s/r = {1, 2, 3} are assumed, respectively. These results show that the impact of pointing errors is more severe for the BDF protocol compared to other diversity techniques like the TLS scheme here considered, being the adoption of transmitters with accurate control of their beamwidth especially important to satisfy the condition φ² > β in order to maximize the diversity order gain. Once this condition is satisfied, it can be convenient to compare with the BER performance obtained in a similar context when misalignment fading is not present. Taking into account the BDF FSO setup configuration in which greater diversity gains are achieved, i.e. β_AC + β_BC < β_AB, and knowing that the impact of pointing errors in our analysis can be suppressed by assuming A₀ → 1 and φ² → ∞ [14

A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

], the corresponding asymptotic expression can be easily derived from Eq. (24a) as follows

P_{b}^{B D F} (E) ≐ \frac{a_{A C}^{n p e} a_{B C}^{n p e} Γ (β_{A C}) Γ (β_{B C}) {(γ ξ)}^{- \frac{1}{2} (β_{A C} + β_{B C})}}{2^{- \frac{1}{2} (- β_{A C} + β_{B C})} 2 Γ (\frac{1}{2} (β_{A C} + β_{B C} + 2))},

(27)

where the parameters

a_{A C}^{n p e}

and

a_{B C}^{n p e}

are obtained from Eq. (8a) when no pointing errors are present as follows

a_{m}^{n p e} = \frac{{(α_{m} β_{m})}_{m}^{β} Γ (α_{m} - β_{m})}{Γ (α_{m}) Γ (β_{m})}

(28)

In Fig. 4b, BER performance in the same FSO context without pointing errors is also displayed. From this asymptotic analysis, taking into account the coding gain in Eq. (24a), the impact of the pointing error effects translates into a coding gain disadvantage, D_pe[dB], relative to this 3-way FSO communication setup without misalignment fading given by

D_{p e} [d B] ≜ \frac{20}{β_{A C} + β_{B C}} {log}_{10} (\frac{φ_{A C}^{2} φ_{B C}^{2}}{A_{0}^{β_{A C} + β_{B C}} (φ_{A C}^{2} - β_{A C}) (φ_{B C}^{2} - β_{B C})}) .

(29)

According to this expression, it can be observed in Fig. 4b that a coding gain disadvantage of 28.8 decibels is achieved for a value of (ω_z/r, σ_s/r) = (7, 1) in the three-node cooperative FSO system under study.

Fig. 4 (a) Diversity order gain G_d for a source-destination link distance of L_AC = 2 km and vertical displacement of the relay node of y_B={0.2 km} when values of normalized beamwidth of ω_z/r = 7 and normalized jitter of σ_s/r = {1, 1.5, 1.75, 2, 3} are assumed. (b) BER performance is depicted for the same source-destination link distance and a relay location of (x_B=0.8 km; y_B=0.2 km) when values of normalized beamwidth of ω_z/r = 7 and normalized jitter of σ_s/r = {1, 2, 3} are assumed as well as when no pointing errors are considered.

Finally, we conclude the analysis in this paper by examining the inclusion of the symmetric scheme in order to maintain the same information rate at the destination node C as previously commented, since the symmetry for nodes A and B assumed in this FSO communication setup implies that no rate reduction is applied, i.e., the same information rate can be considered at the destination node C compared to the direct transmission link without using any cooperative strategy. Taking into account the FSO scenario more favorable to achieve greater diversity gains, i.e. β_AC +β_BC < β_AB, it can be deduced from Eq. (24a) that the interchange of roles of source and relay for the nodes A and B only affects to the BER performance in relation to the division by

2^{\frac{1}{2} (- b_{A C} + b_{B C})}

. In this way, the impact of considering both schemes simultaneously operating, i.e. A(source)-B(relay) and B(source)-A(relay), translates into a coding gain disadvantage, D_sym[dB], relative to the scheme A(source)-B(relay), previously analyzed in this paper, given by

D_{sym} [d B] ≜ \frac{20}{β_{A C} + β_{B C}} {log}_{10} (2^{- β_{A C} + β_{B C} - 1} + \frac{1}{2}) .

(30)

According to this expression, coding gain disadvantages of 1.35, 0.58 and 0.02 decibels are achieved in the three-node cooperative FSO system under study for the configurations shown in previous figures, corresponding to a source-destination link distance and relay location of (L_AC=2 km; x_B=0.8 km; y_B=0.2 km), (L_AC=3 km; x_B=1 km; y_B=0.5 km) and (L_AC=6 km; x_B=0.5 km; y_B=1 km), respectively. From here, it is corroborated that similar results are achieved when the symetric scheme is considered in order to maintain the code rate at the destination node C.

4. Conclusions

In this paper, cooperative FSO communications with DF relaying and EGC reception using IM/DD over atmospheric turbulence channels with pointing errors are analyzed. Novel closed-form asymptotic BER expressions are derived for a 3-way FSO communication setup when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (weak to strong), following a gamma-gamma distribution of parameters α and β, or pointing errors, following a misalignment fading model, where the effect of beam width, detector size and jitter variance is considered. Obtained results provide significant insight into the impact of various system and channel parameters, showing that the diversity order is independent of the pointing error when the equivalent beam radius at the receiver is at least 2β^1/2 times the value of the pointing error displacement standard deviation at the receiver. Moreover, it is contrasted that the available diversity order is strongly dependent on the relay location, achieving greater diversity gains when the diversity order is determined by β_AC +β_BC, where β_AC and β_BC are parameters corresponding to the turbulence of the source-destination and relay-destination links. Additionally, as previously reported by the authors [18

], a relevant improvement in performance must be noted as a consequence of the pulse shape used, providing an increment in the average SNR of 10log₁₀ ξ decibels. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability that get tighter over a wider range of SNR as the turbulence strength increases. At last, it is verified that cooperative FSO communications with DF relaying and EGC reception can be applied to achieve spatial diversity without much increase in hardware or rate reduction at the destination node. From the relevant results here obtained, investigating the impact of the path loss on the coding gain Λ_c for different FSO setups as well as the incorporation of physics-based models (like a wave optics based approach) for representative FSO scenarios are interesting topics for future research in order to extend the analysis in this paper.

Acknowledgments

The authors are grateful for financial support from the Junta de Andalucía (research group “Communications Engineering (TIC-0102)”).

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13.	S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett. 28(2), 129–131 (2003). [CrossRef] [PubMed]
14.	A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]
15.	H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun. 59(3), 669–674 (2011). [CrossRef]
16.	H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27(20), 4440–4445 (2009). [CrossRef]
17.	W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett. 14(5), 468–470 (2010). [CrossRef]
18.	A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14),480–496 (2011). [CrossRef]
19.	A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express 20(3), 2096–2109 (2012). [CrossRef] [PubMed]
20.	A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun. 51(11), 1927 – 1938 (2003). [CrossRef]
21.	A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun. 51(11), 1939 – 1948 (2003). [CrossRef]
22.	J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory 50(12), 3062 – 3080 (2004). [CrossRef]
23.	M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008). [CrossRef]
24.	M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol. 27(24), 5639 –5647 (2009). [CrossRef]
25.	M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423 –1432 (2010). [CrossRef]
26.	C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun. 59(3), 658 –663 (2011). [CrossRef]
27.	C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol. 29(7), 1058 –1065 (2011). [CrossRef]
28.	M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett. 24(7), 545 –547 (2012). [CrossRef]
29.	A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express 18(24),422–440 (2010). [CrossRef]
30.	D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27(18), 3965–3973 (2009). [CrossRef]
31.	M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 8 (2001). [CrossRef]
32.	I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products , 7th ed. (Academic Press Inc., 2007).
33.	N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express 18(12), 824–831 (2010). [CrossRef]
34.	Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51(8), 1389–1398 (2003). [CrossRef]
35.	Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com.
36.	V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. on Symbolic and Algebraic Computation , 212–224 (Tokyo, Japan, 1990).

OCIS Codes
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(060.4510) Fiber optics and optical communications : Optical communications
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: May 14, 2012
Revised Manuscript: June 27, 2012
Manuscript Accepted: June 27, 2012
Published: July 3, 2012

Citation
Antonio García-Zambrana, Carmen Castillo-Vázquez, Beatriz Castillo-Vázquez, and Rubén Boluda-Ruiz, "Bit detect and forward relaying for FSO links using equal gain combining over gamma-gamma atmospheric turbulence channels with pointing errors," Opt. Express 20, 16394-16409 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16394

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References

V. W. S. Chan, “Free-Space Optical Communications,” J. Lightwave Technol.24(12), 4750–4762 (2006). [CrossRef]
L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid optical RF airborne communications,” Proc. IEEE97(6), 1109–1127 (2009). [CrossRef]
W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under gamma-gamma turbulence channels,” Opt. Express17(6), 4479–4484 (2009). [CrossRef] [PubMed]
L. Andrews, R. Phillips, and C. Hopen, Laser beam scintillation with applications (Bellingham, WA: SPIE Press, 2001). [CrossRef]
X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun.50(8), 1293–1300 (2002). [CrossRef]
E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun.22(9), 1896–1906 (2004). [CrossRef]
I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol.26(5), 478–487 (2008). [CrossRef]
T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun.8(2), 951–957 (2009). [CrossRef]
E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun.57(11), 3415–3424 (2009). [CrossRef]
E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun.58(1), 58–62 (2010). [CrossRef]
A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express18(6), 5356–5366 (2010). [CrossRef] [PubMed]
A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Average capacity of FSO links with transmit laser selection using non-uniform OOK signaling over exponential atmospheric turbulence channels,” Opt. Express18(19), 445–454 (2010). [CrossRef]
S. Arnon, “Effects of atmospheric turbulence and building sway on optical wireless-communication systems,” Opt. Lett.28(2), 129–131 (2003). [CrossRef] [PubMed]
A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol.25(7), 1702–1710 (2007). [CrossRef]
H. G. Sandalidis, “Coded free-space optical links over strong turbulence and misalignment fading channels,” IEEE Trans. Commun.59(3), 669–674 (2011). [CrossRef]
H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol.27(20), 4440–4445 (2009). [CrossRef]
W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett.14(5), 468–470 (2010). [CrossRef]
A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express19(14),480–496 (2011). [CrossRef]
A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Asymptotic error-rate analysis of FSO links using transmit laser selection over gamma-gamma atmospheric turbulence channels with pointing errors,” Opt. Express20(3), 2096–2109 (2012). [CrossRef] [PubMed]
A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part I. System description,” IEEE Trans. Commun.51(11), 1927 – 1938 (2003). [CrossRef]
A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity. Part II. Implementation aspects and performance analysis,” IEEE Trans. Commun.51(11), 1939 – 1948 (2003). [CrossRef]
J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory50(12), 3062 – 3080 (2004). [CrossRef]
M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun.7(12), 5441–5449 (2008). [CrossRef]
M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-wave Technol.27(24), 5639 –5647 (2009). [CrossRef]
M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications4(12), 1423 –1432 (2010). [CrossRef]
C. Abou-Rjeily and A. Slim, “Cooperative diversity for free-space optical communications: transceiver design and performance analysis,” IEEE Trans. Commun.59(3), 658 –663 (2011). [CrossRef]
C. Abou-Rjeily and S. Haddad, “Cooperative FSO systems: performance analysis and optimal power allocation,” J. Lightwave Technol.29(7), 1058 –1065 (2011). [CrossRef]
M. Bhatnagar, “Performance analysis of decode-and-forward relaying in gamma-gamma fading channels,” IEEE Photon. Technol. Lett.24(7), 545 –547 (2012). [CrossRef]
A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Rate-adaptive FSO links over atmospheric turbulence channels by jointly using repetition coding and silence periods,” Opt. Express18(24),422–440 (2010). [CrossRef]
D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol.27(18), 3965–3973 (2009). [CrossRef]
M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng.40, 8 (2001). [CrossRef]
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products, 7th ed. (Academic Press Inc., 2007).
N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model.” Opt. Express18(12), 824–831 (2010). [CrossRef]
Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun.51(8), 1389–1398 (2003). [CrossRef]
Wolfram Research Inc., “The Wolfram functions site,” URL http://functions.wolfram.com .
V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system,” in Proc. Int. Conf. on Symbolic and Algebraic Computation, 212–224 (Tokyo, Japan, 1990).

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