## 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 2*β*^{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*, where

_{BC}*β*and

_{AC}*β*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.

_{BC}© 2012 OSA

## 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]. 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, 3]. 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, 5]. 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 [6–12]. The combined effect of atmospheric and misalignment fading is analyzed in the case of single-input/single-output (SISO) FSO channels in [13]. In [14], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for SISO links. In [15] the error rate performance for coded FSO links over strong turbulence and misalignment fading channels is studied. In [16, 17], 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 [18,19], 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 [20–22]. Recently, several works have investigated the adoption of this technique in the context of FSO systems [23–28]. In [23] 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, 25] 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] 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] the impact of the channel state information (CSI) available at the different nodes on the performance of cooperative FSO networks is investigated. In [28] 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, 15], 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 2*β*^{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*, where

_{BC}*β*and

_{AC}*β*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.

_{BC}## 2. System and channel model

Following the cooperative protocol presented in [24,25], 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], 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], 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] 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 [26,28] 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, 9]. 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.

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

*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

*σ*

^{2}=

*N*

_{0}/2, i.e.

*Z*∼

_{m}*N*(0,

*N*

_{0}/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*and

_{m}*Z*to denote random variables and

_{m}*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 [11,18,29].

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, 28], 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]. To consider a wide range of turbulence conditions (weak to strong), the gamma-gamma turbulence model proposed in [4,31] is here assumed, whose probability density function (PDF) is given by

*·*) is the well-known Gamma function and

*K*(

_{ν}*·*) is the

*ν*th-order modified Bessel function of the second kind [32, 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]:

*k*= 2

*π*/

*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]. 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]. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [14] 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*, on the receiver plane at distance

_{z}*z*from the transmitter and a circular receive aperture of radius

*r*, the PDF of ${I}_{m}^{(p)}$ is given by

*φ*=

*ω*

_{zeq}/2

*σ*is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver, ${\omega}_{{z}_{\mathit{eq}}}^{2}={\omega}_{z}^{2}\sqrt{\pi}\text{erf}(v)/2v\text{exp}\left(-{v}^{2}\right)$, $v=\sqrt{\pi}r/\sqrt{2}{\omega}_{z}$,

_{s}*A*

_{0}= [erf(

*v*)]

^{2}and erf(·) is the error function [32, eqn. (8.250)]. Here, independent identical Gaussian distributions for the elevation and the horizontal displacement (sway) are considered, being ${\sigma}_{s}^{2}$ 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*was derived in [16] as

_{m}*f*

_{Im}(

*i*) at

*i*→ 0 determines high SNR performance [34]. Hence, using the series expansion corresponding to the Meijer’s G-function [35, 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*and

_{m}*b*in Eq. (7), depending on the relation between the values of

_{m}*φ*

^{2}and

*β*, can be written as

*I*for the paths A–B, A–C and and B–C is indicated by

_{m}*I*,

_{AB}*I*and

_{AC}*I*, respectively.

_{BC}## 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
$d=2{P}_{\text{opt}}\sqrt{{T}_{b}\xi}$, the statistical channel model corresponding to the A–B link can be written as

*T*is the bit period and

_{b}*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 where

*Q*(·) is the Gaussian-

*Q*function defined as $Q(x)=\frac{1}{\sqrt{2\pi}}{\int}_{x}^{\infty}{e}^{-\frac{{t}^{2}}{2}}dt$. Substituting the value of the Euclidean distance

*d*gives ${P}_{b}^{AB}\left(E|{I}_{AB}\right)=Q\left(\sqrt{\left(\gamma /2\right)\xi}i\right)$ where $\gamma ={P}_{\text{opt}}^{2}{T}_{b}/{N}_{0}$ represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats. Hence, the average BER, ${P}_{b}^{AB}(E)$, can be obtained by averaging ${P}_{b}^{AB}\left(E|{I}_{AB}\right)$ over the PDF as follows

*·*) by $\text{erfc}(x)=2Q\left(\sqrt{2}x\right)$ [32, eqn. (6.287)] and the fact that ${\int}_{0}^{\infty}\text{erfc}(x){x}^{a-1}dx=\mathrm{\Gamma}\left(\left(1+a\right)/2\right)/\left({\pi}^{1/2}a\right)$ [32, eqn. (6.281)], obtaining the corresponding closed-form asymptotic solution for the BER as can be seen in

*a*and

_{AB}*b*depends on the relation between

_{AB}*φ*

^{2}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 2

*β*

^{1/2}times the value of the pointing error displacement standard deviation, i.e.

*φ*

^{2}>

*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

*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

*I*=

_{T}*I*+ 2

_{AC}*I*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,

_{BC}*f*

_{IT}(

*i*), of the combined variate can be easily derived from Eq. (7) as

*d · I*become irrelevant to the detection process, the conditional BER at the node C is given by ${P}_{b}^{{BDF}_{1}}\left(E|{I}_{AC},{I}_{BC}\right)=Q\left(\sqrt{\left(\gamma /4\right)\xi}\left({i}_{1}-2{i}_{2}\right)\right)$. Hence, the average BER, ${P}_{b}^{{BDF}_{1}}(E)$ can be obtained by averaging over the PDF as follows

_{BC}*Q*function is not always positive [34]. 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

*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, 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, eqn. (21)], expressing

*K*(·) [36, eqn. (14)] in terms of Meijer’s G-function. In this way, a closed-form solution is derived as can be seen in

_{μ}*b*+

_{AC}*b*+ 1

_{BC}*< b*and ${P}_{b}^{BDF}(E)\doteq {P}_{b}^{{BDF}_{1}}(E)\cdot {P}_{b}^{AB}(E)$ when

_{AB}*b*+

_{AC}*b*+ 1 >

_{BC}*b*. It is straightforward to show that the average BER behaves asymptotically as (Λ

_{AB}*)*

_{c}γξ^{−Λd}, where Λ

*and Λ*

_{d}*denote diversity order and coding gain, respectively [34]. At high SNR, if asymptotically the error probability behaves as (Λ*

_{c}*)*

_{c}γξ^{−Λd}, the diversity order Λ

*determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain Λ*

_{d}*(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,*

_{c}*G*, relative to the non-cooperative link A–C of Since the diversity order determines the slope of the BER performance, it must be noted that this parameter

_{d}*G*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

_{d}*φ*

^{2}and

*β*as obtained in Eq. (8

*b*) 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 2

*β*

^{1/2}times the value of the pointing error displacement standard deviation, i.e.

*φ*

^{2}>

*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*in Eq. (25) as a function of the horizontal displacement of the relay node,

_{d}*x*, is depicted in Fig. 2 for a source-destination link distance

_{B}*L*= {3 km, 6 km} when different relay locations

_{AC}*y*={0.5 km, 1 km, 1.5 km, 2 km, 2.5 km} are assumed. Here, the parameters

_{B}*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]. In any case, the condition

*φ*

^{2}>

*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*and

_{AC}*β*/

_{AB}*β*, as deduced from Eq. (25). It can be easily contrasted from Eq. (4) that the symmetric quasi-Gaussian shape is related to the expression

_{AC}*β*/

_{AB}*β*, scenario in which the diversity order is determined by the source-relay link A–B since

_{AC}*β*< (

_{AB}*β*+

_{AC}*β*). 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

_{BC}*β*+

_{AC}*β*, corresponding to the turbulence of the source-destination and relay-destination links. The results corresponding to this asymptotic analysis with rectangular pulse shapes and

_{BC}*ξ*= 1 are illustrated in the Fig. 3, when different relay locations for source-destination link distances

*L*= {3 km, 6 km} are assumed together with values of normalized beamwidth and normalized jitter of (

_{AC}*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

*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*=3 km and relay locations of (

_{AC}*x*=1 km;

_{B}*y*=0.5 km) and (

_{B}*x*=2 km;

_{B}*y*=1 km), respectively, or diversity gains of 2 and 1.18 when

_{B}*L*=6 km and relay locations of (

_{AC}*x*=0.5 km;

_{B}*y*=1 km) and (

_{B}*x*=3.5 km;

_{B}*y*=1.5 km), respectively. Both cases for

_{B}*L*=3 km and

_{AC}*L*=6 km represent the two possible scenarios considered in Eq. (24), being Eq. (24

_{AC}*a*) 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}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*=0.2 km and a source-destination link distance of

_{B}*L*= 2 km. In this configuration, a normalized beamwidth of

_{AC}*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

*φ*

^{2}>

*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*corresponding to this configuration are significantly decreased as the normalized jitter increases and, hence, the condition

_{d}*φ*

^{2}>

*β*is not satisfied. These conclusions are contrasted in Fig. 4b, wherein BER performance for a source-destination link distance of

*L*= 2 km and a relay location of (

_{AC}*x*=0.8 km;

_{B}*y*=0.2 km) when values of normalized beamwidth of

_{B}*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

*φ*

^{2}>

*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*, and knowing that the impact of pointing errors in our analysis can be suppressed by assuming

_{AB}*A*

_{0}→ 1 and

*φ*

^{2}→ ∞ [14], the corresponding asymptotic expression can be easily derived from Eq. (24

*a*) as follows

*a*) when no pointing errors are present as follows

*a*), 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

*ω*/

_{z}*r*,

*σ*/

_{s}*r*) = (7, 1) in the three-node cooperative FSO system under study.

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*, it can be deduced from Eq. (24

_{AB}*a*) 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 ${2}^{\frac{1}{2}\left(-{b}_{AC}+{b}_{BC}\right)}$. 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

*L*=2 km;

_{AC}*x*=0.8 km;

_{B}*y*=0.2 km), (

_{B}*L*=3 km;

_{AC}*x*=1 km;

_{B}*y*=0.5 km) and (

_{B}*L*=6 km;

_{AC}*x*=0.5 km;

_{B}*y*=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.

_{B}## 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 2*β*^{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*, where

_{BC}*β*and

_{AC}*β*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 10log

_{BC}_{10}

*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

*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.*

_{c}## Acknowledgments

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

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