Abstract

In this paper, a novel adaptive cooperative protocol with multiple relays using detect-and-forward (DF) over atmospheric turbulence channels with pointing errors is proposed. The adaptive DF cooperative protocol here analyzed is based on the selection of the optical path, source-destination or different source-relay links, with a greater value of fading gain or irradiance, maintaining a high diversity order. Closed-form asymptotic bit error-rate (BER) expressions are obtained for a cooperative free-space optical (FSO) communication system with Nr relays, when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions, 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. A greater robustness for different link distances and pointing errors is corroborated by the obtained results if compared with similar cooperative schemes or equivalent multiple-input multiple-output (MIMO) systems. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

© 2014 Optical Society of America

1. Introduction

Free-space optical (FSO) communications can provide high-speed links for a variety of applications. The most special characteristics are unlimited bandwidth, unlicensed spectrum, excellent security, and low cost [1]. The use of FSO communication systems is being specially interesting to solve the last mile problem when fiber-optic links are not practical, as well as a supplement to radio-frequency (RF) links. Among the most important disadvantages are the atmospheric propagation factors, such as haze, fog, rain and snow. However, the most serious problem is the atmospheric turbulence, which produces fluctuations in the irradiance of the transmitted optical beam, as a result of variations in the refractive index along the link [2]. Additionally, the high directivity of the transmitted beam in FSO systems, or building sway can produce an unsuitable alignment between transmitter and receiver and, hence, a greater deterioration in performance. To solve these inconveniences the adoption of forward error correcting codes as well as spatial diversity based on multiple-input multiple-output (MIMO) configurations is proposed. An alternative approach to improve the performance in this turbulent FSO scenario is based on the employment of cooperative communications in order to overcome some limitations of MIMO structures. Cooperative transmission can significantly enhance the performance, increasing diversity by using the transceivers available at the other nodes of the network.

Much research in recent years has focused on this technique in the context of FSO communications [314]. In [3] 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 [4, 5] a 3-way FSO system is proposed to implement a cooperative protocol in order to improve spatial diversity without much increase in hardware, being evaluated the error-rate performance, using the photon-count method, as well as the outage performance. In particular, an adaptive cooperative protocol is presented in [4] wherein the source node participates in cooperation only if the source-relay link is in an acceptable quality. In [6] two relay selection methods for a cooperative FSO system with DF relaying are introduced, being evaluated the outage probability by using the photon-count method with a log-normal turbulence-induced fading channel model. In [7], following the bit-detect-and-forward (BDF) cooperative protocol presented in [4], the analysis is extended over gamma-gamma and misalignment fading channels assuming equal gain combining (EGC) in reception. In [8] a relay selection protocol is investigated where the relay is operating in AF mode under log-normal fading, deriving outage probability of the system. In [10], two cooperative protocols based on relay selection for any number of relays are analyzed, proposing an alternative metric of selection obtaining high diversity order gain over gamma-gamma fading channels without pointing errors, by using the photon-count method in the presence and absence of background radiation. In [11] several transmission protocols are investigated for a cooperative FSO system over gamma-gamma fading channels, proposing alternative protocols to the all-active relaying scheme, which activate only a single relay in each transmission slot. In [12], DF based on FSO cooperative systems over gamma-gamma fading channels with relay selection are analyzed, wherein the source selects a relay on the basis of highest instantaneous signal-to-noise ratio (SNR) of the source-relay links. In [13], a cooperative FSO system is analyzed in which serial and parallel relaying are deployed together, being obtained the outage probability over log-normal fading channels. In [14], a cooperative FSO communication system is analyzed based on users selection, where the best user is selected, being evaluated the outage probability and bit error-rate (BER) over log-normal and gamma-gamma fading channels.

In this paper, a novel adaptive cooperative protocol with multiple relays is presented, wherein the relays are operating in DF mode over atmospheric turbulence channels with pointing errors. The adaptive-detect-and-forward (ADF) relaying scheme here proposed is based on the selection of the optical path, source-destination or different source-relay links, with a greater value of fading gain or irradiance, maintaining a high diversity order. Closed-form asymptotic BER expressions are obtained for a cooperative FSO communication system with Nr relays, when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, 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. The diversity order gain analysis is performed as a function of the number of relays and different atmospheric turbulence conditions. A greater robustness for different link distances and pointing errors is corroborated by the obtained results if compared with similar cooperative FSO schemes or equivalent MIMO systems with multiple transmitters. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

2. System and channel model

The system model under study is shown in Fig. 1. We consider a cooperative FSO system with multiple relays, consisting of one source node S, Nr relays denoted by Rk for k = {1,2,...,Nr}, and one destination node D. In this study, different relay-destination link distances within the range d′RD to dRD (d′RD < dRD) are considered in order to perform a more realistic analysis in the context of FSO cooperative systems, assuming that these relays are located in an area similar to an annullus. In particular, we adopt laser sources intensity-modulated and ideal non-coherent (direct-detection) receivers. The ADF relaying scheme with multiple relays here proposed selects between direct-transmission (DT) or BDF cooperative protocol presented in [7] on the basis of the value of the fading gain. When the irradiance of the source-destination link (ISD) is greater than the irradiances corresponding to the source-relay links (ISRk for k = {1, 2,...,Nr}), the FSO communication system is only based on the direct transmission to the destination node, obviating the cooperative mode. On contrary, the source node S performs cooperation using the relay Rk if the irradiance of the S-Rk link is greater than the corresponding irradiance of the S-D link and the irradiances S-Ri for ik, i.e. the remaining relays. This metric is able to select the best relay node with better possible conditions in order to detect correctly the received information from the source node, since a decode-and-forward relaying scheme is being considered. In contrast to [4], wherein the cooperative strategy is not considered when the source-relay links quality in terms of ISRk2 falls below a certain threshold, it is here assumed that cooperative communications are not established when the irradiance of S-D link takes the greater value if compared to the irradiances of the source-relay links. It must be noted that the selection criterion is here based on the value of the irradiance instead of the square magnitude, as previously commented when comparing to a certain threshold or when selecting the maximum value of the instantaneous received SNR at the relay, as usually is assumed in the literature [6, 8, 11, 12, 14]. It is taking into account that CSI is known not only at the receiver but also at the transmitter (CSIT). The knowledge of CSIT is feasible for FSO channels given that scintillation is a slow time varying process relative to the large symbol rate. In this sense, the receiver always knows if the cooperative protocol is being used. The ADF cooperative protocol works in two phases or transmission frames, as shown in Table 1. It can be observed that, for example, no cooperative transmission is being used in slot2 and slot5 since the irradiance of the S-D link is greater than the corresponding irradiances of the S-Rk links in these particular slots. It must be noted that one transmission overlapped implies that no rate reduction is applied and, hence, the same information rate can be considered at the destination node D compared to the direct transmission link without using any cooperative strategy. Here, it is assumed that all the bits detected at the relay node selected by the source node are always resended with the new power to the destination node D regardless of these bits detected correctly or incorrectly, unlike [10].

 figure: Fig. 1

Fig. 1 Block diagram of the cooperative FSO system under study, where dSD is the source-destination link distance, Rk are the relays nodes for k = {1, 2,...,Nr}, and (dRkD,ϕRk) represents the relay location using polar coordinates.

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Tables Icon

Table 1. ADF cooperative protocol with multiple relays when the message s = {s1, s2,..., sn} is transmitted, being Rmax the relay selected by the source node.

For each link of this cooperative FSO communications system, the instantaneous current ym(t) in the receiving photodetector corresponding to the information signal transmitted from each laser can be written as

ym(t)=ηim(t)xm(t)+zm(t),
where η is the detector responsivity, assumed hereinafter to be the unity, Xxm(t) represents the optical power supplied by the source, Imim(t) the equivalent real-value fading gain (irradiance) through the optical channel between the laser and the receive aperture. Zmzm(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. It can usually be modeled to high accuracy as AWGN with zero mean and variance σ2 = N0/2, i.e. ZmN(0, N0/2), independent of the on/off state of the received bit. We use X, Ym, Im and Zm to denote random variables and xm(t), ym(t), im(t) and zm(t) their corresponding realizations. The irradiance is considered to be a product of three factors i.e., Im=LmIm(a)Im(p) where Lm is the deterministic propagation loss, Im(a) is the attenuation due to atmospheric turbulence and Im(p) the attenuation due to geometric spread and pointing errors. Lm is determined by the exponential Beers-Lambert law as Lm = e−Φd, where d is the link distance and Φ is the atmospheric attenuation coefficient. It is given by Φ = (3.91/V (km))(λ (nm)/550)q where V is the visibility in kilometers, λ is the wavelength in nanometers and q is the size distribution of the scattering particles, being q = 1.3 for average visibility (6 km < V < 50 km), and q = 0.16V + 0.34 for haze visibility (1 km < V < 6 km) [15]. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [16]. To consider a wide range of turbulence conditions, the gamma-gamma turbulence model proposed in [2] is here assumed. Regarding to the impact of pointing errors, we use the general model of misalignment fading given in [17] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. A closed-form expression of the combined probability density function (PDF) of Im was derived in [18] as
fIlm(i)=αmβmφm2A0LmΓ(αm)Γ(βm)G1,33,0(αmβmA0Lmi|φm2φm21,αm1,βm1),i0
where Gp,qm,n[] is the Meijer’s G-function [19, eqn. (9.301)] and Γ(·) is the well-known Gamma function. Assuming plane wave propagation, α and β can be directly linked to physical parameters through the following expresions [20]:
α=[exp(0.49σR2/(1+1.11σR12/5)7/6)1]1
β=[exp(0.51σR2/(1+0.69σR12/5)5/6)1]1
where σR2=1.23Cn2κ7/6d11/6 is the Rytov variance, which is a measure of optical turbulence strength. Here, κ = 2π/λ is the optical wave number and d is the link distance in meters. Cn2 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 [2]. It must be emphasized that parameters α and β cannot be arbitrarily chosen in FSO applications, being related through the Rytov variance. 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 ∞ [21]. In relation to the impact of pointing errors [17], assuming a Gaussian spatial intensity profile of beam waist radius, ωz, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, φ = ωzeq/2σs is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver, ωzeq2=ωz2πerf(v)/2vexp(v2), v=πr/2ωz, A0 = [erf(v)]2 and erf(·) is the error function [19, eqn. (8.250)].

Nonetheless, the PDF in Eq. (2) appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of FSO communication systems. To overcome this inconvenience, the PDF is approximated by a single polynomial term as fIm(i) ≈ amibm−1, based on the fact that the asymptotic behavior of the system performance is dominated by the behavior of the PDF near the origin [7]. Different expressions for fIm(i), depending on the relation between the values of φ2 and β, can be written as

fIm(i)amibm1=φm2(αmβm)βmΓ(αmβm)(A0Lm)βmΓ(αm)Γ(βm)(φm2βm)iβm1,φm2>βm
fIm(i)amibm1=φm2(αmβm)φm2Γ(αmφm2)Γ(βmφm2)(A0Lm)φm2Γ(αm)Γ(βm)iφm21,φm2<βm

In the following section, the fading coefficient Im for the paths S-D, S-Rk and Rk-D is indicated by ISD, ISRk and IRkD, respectively, for k = {1, 2,...,Nr}. The subscript k is used to represent the different relays, which can be selected by the source node in each transmission frame. Here, we assume that all coefficients are independent statistically.

3. Error-rate performance analysis

In this section, the ADF cooperative protocol with multiple relays is analyzed in order to obtain asymptotic expressions for 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. In addition to the performance evaluation of the BER corresponding to the ADF cooperative protocol here proposed, we also consider the performance analysis for the direct transmission (non-cooperative S-D link) to establish the baseline performance. The average optical power transmitted from each node is Popt, being adopted an on-off keying (OOK) signaling based on a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of dE=2PoptTbξ, where the parameter Tb 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 peak-to-average optical power ratio (PAOPR) [7]. According to Eq. (1), the statistical channel model corresponding to the source-destination link without cooperative communication can be written as

YSD=XISD+ZSD,X{0,dE},ZSD~N(0,N0/2).
Assuming channel side information at the receiver, the conditional BER at the destination node is given by
PbSD(E|ISD)=Q(dE2i2/2N0)=Q(2γξi),
where Q(·) is the Gaussian Q-function and γ=Popt2Tb/N0 represents the received electrical SNR in absence of turbulence. Hence, the average BER, PbSD(E), can be obtained by averaging PbSD(E|ISD) over the PDF as follows
PbSD(E)=0Q(2γξi)fISD(i)di.
To evaluate the integral in Eq. (7), we can use that the Q-function is related to the complementary error function erfc(·) by erfc(x)=2Q(2x) [19, eqn. (6.287)] and the fact that 0erfc(x)xa1dx=Γ((1+a)/2)/(π1/2a) [19, eqn. (6.281)], obtaining the corresponding closed-form asymptotic solution for the BER as can be seen in
PbSD(E)aSDΓ((bSD+1)/2)2bSDπ(γξ)bSD/2,
where the value of the parameters aSD and bSD depends on the relation between φSD2 and βSD, as shown in Eq. (4), corroborating that the diversity order corresponding to the source-destination link is independent of the pointing errors when φSD2>βSD. The statistical channel model corresponding to the ADF cooperative protocol using Nr relays can be written as
YADF0=12XISD+ZSD+X*IRmaxD+ZRmaxD,ISRmax>ISD
YADF1=XISD+ZSD,ISRmax<ISD
being Rmax the relay node where the value of the fading gain corresponding to the S-Rmax link is maximum, denoted as ISRmax, i.e. ISRmax = max{ISR1, ISR2,...,ISRN}, X ∈ {0, dE} and ZSD, ZRmaxDN(0, N0/2). X* represents the random variable corresponding to the information detected at the node Rmax and, hence, X* = X when the bit has been detected correctly, and X* = dEX when the bit has been detected incorrectly at Rmax. The Eq. (9a) is the statistical channel model corresponding to the BDF cooperative protocol analyzed in [7] and, hence, depending on the fact that the bit from the relay S-Rmax-D is detected correctly or incorrectly, it can be expressed as
YADF0=(1/2)X(ISD+2IRMaxD)+ZSD+ZRMaxD,X*=X
YADF0=(1/2)X(ISD2IRMaxD)+dEIRMaxD+ZSD+ZRMaxD,X*=dEX
The Eq. (9b) is the statistical channel corresponding to the direct transmission without cooperative communication. Assuming channel side information at the receiver and transmitter, the conditional BER at the node D, PbADF(E|ISD,ISRk,IRkD) for k = {1, 2,...,Nr}, can be written as
PbADF(E|ISD,ISRk,IRkD)=k=1NrPbBDFk(E|ISD,ISRk,IRkD)FISD(ISRk)j=1jkNrFISRj(ISRk)+PbSD(E|ISD)j=1NrFISRj(ISD),
being FIm(i) the cumulative density function (CDF) of the random variable Im, which is given by FIm(i) = Prob(Imi). Knowing the fact that the irradiances are statistically independent, the probability corresponding to ISD > ISRj is computed by using FISRj (ISD). In the same way, the probability corresponding to ISRj > ISD can be computed by using FISD(ISRj). For the sake of simplicity, in spite of the fact that this CDF can be expressed in terms of generalized hypergeometric functions, an approximated expression by a single polynomial term as FIm(i) ≈ (am/bm)ibm is here assumed, as can easily be deduced from Eq. (4). PbBDFk(E|ISD,ISRk,IRkD) is the conditional BER corresponding at the node D when the relay node Rk is selected by the source node, as follows
PbBDFk(E|ISD,ISRk,IRkD)=PbBDFk0(E|ISD,IRkD)(1PbSRk(E|ISRk))+PbBDFk1(E|ISD,IRkD)PbSRk(E|ISRk)=Q((γ/4)ξ(iSD+2iRkD))(1Q((γ/2)ξiSRk))+Q((γ/4)ξ(iSD2iRkD))Q((γ/2)ξiSRk).
The division by 2 in PbSRk(E|ISRk) is considered so as to maintain the average optical power in the air at a constant level of Popt, being transmitted by each laser an average optical power Popt/2 in the first phase when only relay Rk is selected by the source node. Hence, the average BER, PbADF(E), can be obtained by averaging PbADF(E|ISD,ISRk,IRkD) over the PDFs as follows
PbADF(E)=k=1Nr(PbBDFk0(E)PbSRkADF0(E)+PbBDFk1(E)PbSRkADF1(E))+PbSDADF(E)=k=1NrPbBDFk0(E)0[1PbSRk(E|ISRk)]FISD(i)j=1jkNrFISRk(i)fISRk(i)di.+k=1NrPbBDFk1(E)0PbSRk(E|ISRk)FISD(i)j=1jkNrFISRj(i)fISRk(i)di.+0PbSD(E|ISD)j=1NrFISRj(i)fISD(i)di.
Next, the following generic expression is computed for the BER corresponding to the S-Rk and S-D links as follows
0Q(cγξi)n=1nmNrFIn(i)fIm(i)di,c+
In particular, the parameter c is set to 2 in Eq. (14) for the BER corresponding to the S-D link. This expression, corresponding to a direct transmission without cooperation, i.e. PbSDADF(E), is given by
PbSDADF(E)=0PbSD(E|ISD)j=1NrFISRj(i)fISD(i)di.
In a similar way, when the parameter c is set to 1/2 in Eq. (14), the BER corresponding to the S-Rk link, i.e. PbSRkADF1(E), is computed as follows
PbSRkADF1(E)=0PbSRk(E|ISRk)FISD(i)j=1jkNrFISRj(i)fISRk(i)di.
Evaluating the Eq. (15) and Eq. (16) as in Eq. (7), we can obtain the corresponding closed-form asymptotic solution for the BER as can be seen in
PbSDADF(E)aSDΠi=1NraSRiΓ((1+bSD+i=1NrbSRi)/2)(γξ)(bSD+i=1NrbSRi)/2Πi=1NrbSRi2π(bSD+i=1NrbSRi)
and
PbSRkADF1(E)bSRkbSD2(bSD+i=1NrbSRi)PbSDADF(E),
respectively. Finally, the probability when the bit is detected correctly at the node Rk, i.e. PbSRkADF0(E), is computed as
PbSRkADF0(E)=0[1PbSRk(E|ISRk)]FISD(i)j=1jkNrFISRj(i)fISRk(i)di.
Here, we can use that de Gaussian Q-function tends to 0 as γ → ∞, simplifying the integral in Eq. (19) as follows
PbSRkADF0(E)0j=1jkNrFISRj(i)FISD(i)fISRk(i)di.
It can be noted that the asymptotic behavior of PbSRkADF0(E) is independent of the SNR γ, resulting in a positive value that is upper bounded by 1. Here, the Eq. (20) was obtained by using the Monte Carlo integration, being analytically intractable. The corresponding closed-form asymptotic solution for the BER corresponding to the BDF relaying scheme was derived in [7, Eq. (18) and Eq. (21)], i.e. PbBDFk0(E) and PbBDFk1(E), respectively. Using the notation here assumed, these expressions are here rewritten as
PbBDFk0(E)aSDaRkDΓ(bSD)Γ(bRkD)(γξ)(bSD+bRkD)/22(bSD+bRkD)/22Γ((bSD+bRkD+2)/2),
PbBDFk1(E)002i2fISD(i1)fIRkD(i2)di1di2.
Therefore, taking into account the asymptotic behavior previously obtained, the expression in Eq. (13) can be simplified as follows
PbADF(E)PbBDFmin0(E)PbSRminADF0(E),bRminD<i=1NrbSRi
PbADF(E)(1=2(bSD+i=1NrbSRi)bSDk=1NrbSRkPbBDFk1(E))PbSDADF(E),bRminD>i=1NrbSRi
being bRminD=min{bR1D,...,bRNrD}. Taking into account these expressions, the adoption of the ADF cooperative protocol using Nr relays here analyzed translates into a diversity order gain, Gd(Nr), relative to the non-cooperative link S-D of
Gd(Nr)=1+min(bRminD,i=1NrbSRi)bSD.
For the better understanding of the impact of the cooperative FSO communications system here analyzed, the diversity order gain Gd(Nr) in Eq. (23) as a function of the radius of a circumference, dRD, whose center is the node D, is depicted in Fig. 2 for a S-D link distance dSD=3 km when different number of relays Nr = {1, 2, 3, 4} is considered, as shown in Fig. 1. Furthermore, different scenarios are considered in order to evaluate the performance when different relay-destination link distances are assumed. In this figure, the performance analysis is evaluated for dRD = d′RD, dRDd′RD = 0.1dSD and dRDd′RD = 0.2dSD. Here, the parameters α and β are calculated from Eq. (3a) and Eq. (3b) respectively, with λ =1550 nm. Different weather conditions are here adopted, assuming haze visibility of 4 km with Cn2=2×1014m2/3 and clear visibility of 16 km with Cn2=8×1014m2/3, corresponding to moderate and strong atmospheric turbulence conditions, respectively.

 figure: Fig. 2

Fig. 2 Diversity order gain Gd for the ADF cooperative protocol for a source-destination link distance of dSD=3 km when different weather conditions and relay locations are assumed. (a) (ωz/r,σs/r)=(5,1) for each link and (b) (ωz/r,σs/r)=(5,1) for the S-Rk and Rk-D links, and (ωz/r,σs/r)=(5,3) for the S-D link.

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Hereinafter, φRkD2φRD2 and φSRk2φSR2 are assumed for k = {1, 2,...,Nr}. In Fig. 2(a), the conditions φ2 > β is satisfied in both figures for each link and, hence, these results are independent of pointing errors with regards to the diversity order due to the diversity gain only depends of the atmospheric turbulence when this relation is satisfied. Firstly, it can be concluded that the adaptive DF cooperative protocol using relay selection provides a diversity order gain greater than 1 in any case. By other hand, the diversity order gain is a consequence of the inverse relation between the number of relays and the relay-destination link distance. In this way, it is corroborated that the available diversity order is strongly dependent of the relay-destination link distances and the number of relays, achieving a greater diversity gain as the number of relays increases and the relay-destination link distances decrease. Moreover, the diversity order gain presents a maximum when the relation βRminD=i=1NrβSRi holds. Diversity order gain increases as the relay-destination link distance decreases, being initially dominated by bRminD, i.e., the relay node whose parameter bRkD is the smallest. Once bRminD>i=1NrbSRi is satisfied, the diversity order gain will be determined by the number of relays. As can be seen in Fig. 2 when dRDd′RD = 0.1dSD or dRDd′RD = 0.2dSD, only one relay is around of the circumference of radius dRD (outer ring) and the remaining relays are around of the circumference of the radius d′RD (inner ring), being these scenarios representative enough in order to evaluate the diversity order gain when different relay-destination link distances are considered, modelling the worst case. The comparison of these different locations of relays corroborate that changes in diversity performance are negligible, specially over strong atmospheric turbulence or when the link distance is increased, tending to an idealized situation wherein all relays are at the same distance from the destination node. In addition to previous scenario, different values of normalized beam width and normalized jitter of (ωz/r,σs/r)=(5,3) for the S-D link are considered in Fig. 2(b). As a result, a greater robustness to the pointing errors for the ADF cooperative protocol is corroborated, increasing the diversity order when pointing errors for the S-D link are more severe compared to the S-Rk links. The results corresponding to this asymptotic analysis with rectangular pulse shapes and ξ =1 are illustrated in Fig 3, for a source-destination link distance dSD=3 km, together with values of normalized beam width and normalized jitter of (ωz/r,σs/r)=(5,1) and (ωz/r,σs/r)=(10,2). For the sake of simplicity, it is assumed that dRD = d′RD as a consequence of previous conclusions corroborated in Fig. 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. Additionally, we also consider the performance analysis for the direct transmission (non-cooperative link S-D) to establish the baseline performance. 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 the ADF cooperative protocol when different number of relays Nr = {1, 2, 3} are assumed. In this sense, we can see diversity gains of 2.06, 2.18 and 2.18 for a R-D link distance dRD = 0.8dSD, or diversity gains of 2.08, 3.17 and 4.37 for a R-D link distance dRD = 0.3dSD, when the number of relays is 1, 2 and 3, respectively. Both cases are considered in Eq. (22), being Eq. (22a) and Eq. (22b) the bounds corresponding to each FSO scenario, respectively. Next, Gd is depicted in Fig. 4 for a source-destination link distance of dSD=3 km and dRD = d′RD, when normalized beam width of ωz/r=5 and different values of normalized jitter σs/r={1, 2, 3} are assumed, in order to contrast the impact of pointing errors when the condition φ2 > β is not satisfied for each link. It can be observed that diversity gains even greater than 2, 3 and 4 for Nr set to 1, 2 and 3, respectively, are achieved when (ωz/r, σs/r) = (5, 1), not being affected by pointing errors regardless of the weather conditions. Moreover, in Fig. 4(b) and Fig. 4(c), for values of normalized beam width and normalized jitter of (ωz/r,σs/r)=(5,2), the condition φ2 > β is not satisfied for the relay-destination link when the relation φRD2=βRkD holds. In this case, the diversity gain is limited mainly by the pointing errors of the relay-destination links. Under strong atmospheric turbulence conditions the ADF relaying scheme presents a greater robustness to the pointing errors, achieving the maximum diversity gain when dRD ≈ 0.4dSD instead of dRD ≈ 0.85dSD assuming moderate atmospheric turbulence. Finally, the FSO scenario in which the condition φ2 > β is not satisfied for each link is depicted in Fig. 4(c) for values of normalized beam width and jitter of (ωz/r,σs/r)=(5,3) and, hence, the diversity gain is saturated to the value of 2, as can be deduced from Eq. (23) as follows

Gd(Nr)=1+min(φRD2,NrφSR2)φSD2=1+min(φ2,Nrφ2)φ2=2,
being an independent value of the number of relays or the relay locations. These conclusions are contrasted in Fig. 5, wherein BER performance is displayed for a source-destination link distance of dSD=3 km when R-D link distance dRD = 0.2dSD and the number of relays is Nr={1, 2, 3}, together with values of normalized beam width of ωz/r= 5 and normalized jitter of σs/r={1, 2, 3}. As before, the performance analysis for the direct transmission is considered to establish the baseline performance. These BER results are in excellent agreement with previous results shown in Fig. 4 in relation to the diversity order gain achieved for this cooperative FSO communication system with multiple relays where pointing errors are presented. These results show that the impact of pointing errors is less severe for the ADF cooperative protocol compared with other similar cooperative schemes or equivalent multiple-input single-output (MISO) systems [22]. In this sense, it can be seen that diversity gains of 2.01, 2.01 and 2 for a number of relays of Nr = 1, 3.03, 2.5 and 2 for a number of relays of Nr = 2, and 4.06, 2.5 and 2 for a number of relays of Nr = 3 are achieved with values of normalized jitter of σs/r = {1, 2, 3}, respectively. As also displayed in Fig. 3, 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. Finally, we conclude the paper by evaluating that the impact of the pointing errors effects translates into a coding gain disadvantage. Knowing that the average BER behaves asymptotically as (Gcγξ)Gd, where Gd and Gc denote diversity order and coding gain, respectively. Once the condition φ2 > β is satisfied for each link, it can be convenient to compare with the BER performance obtained in a similar context without pointing errors. Taking into account that the greatest diversity order gain is achieved when the relation βRminD>i=1NrβSRi holds, and knowing that the impact of pointing errors in our analysis can be suppressed by assuming A0 → 1 and φ2 → ∞ [17], the corresponding asymptotic expression can be easily derived from Eq. (22b) as follows
PbADF(E)(1+2(βSD+i=1NrβSRi)βSDk=1NrβSRkPbBDFk1(E))PbSDADFnpe(E),
As can be numerically corroborated, the effects of the pointing errors on PbBDFk1(E) can be considered negligible in this FSO scenario and, hence, the first factor in Eq. (22b) and Eq. (25) can be suppressed for the computation of the coding gain disadvantage. PbSDADFnpe(E) is obtained from Eq. (17) when misalignment fading is not present, wherein the corresponding parameter am is given by
amnpe=(αmβm)βmΓ(αmβm)LmβmΓ(αm)Γ(βm).
In Fig. 5, BER performance in the same FSO context without pointing errors is also displayed. From this asymptotic analysis and taking into account the coding gain in Eq. (22b), the impact of the pointing errors translates into a coding gain disadvantage, Dpe[dB], as follows
Dpe[dB]20βSD+i=1NrβSRilog10(φSD2A0βSD(φSD2βSD)i=1NrφSRi2A0βSRi(φSRi2βSRi)).
According to this expression, it can be observed in Fig. 5 that a coding gain disadvantage of 23.75 decibels is achieved for a value of (ωz/r, σs/r) = (5, 1) regardless of the number of relays. This result can be justified from the fact that βSDβSR1 ≈ ··· ≈ βSRNr holds in this scenario and, hence, Eq. (27) can be simplified as follows
Dpe[dB]20(Nr+1)βSDlog10(A0βSDφSD2φSD2βSD)Nr+1=20βSDlog10(A0βSDφSD2φSD2βSD),
obtaining the coding gain disadvantage corresponding to the direct transmission based on the non-cooperative source-destination link.

 figure: Fig. 3

Fig. 3 BER performance when different number of relays Nr={1, 2, 3} and source-destination link distance of dSD=3 km are assumed. Different relay locations of (a) dRD=0.3dSD and (b) dRD=0.8dSD are assumed together with values of normalized beam width and normalized jitter of (ωz/r, σs/r) = {(5, 1), (10, 2)}.

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 figure: Fig. 4

Fig. 4 Diversity order gain Gd for the Adaptive-DF cooperative protocol for a source-destination link distance of dSD=3 km when (a) 1, (b) 2 and (c) 3 relays are assumed together with different values of normalized beam width of ωz/r= 5 and normalized jitter of σs/r={1, 2, 3}.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 BER performance when different number of relays Nr={1, 2, 3} and source-destination link distance of dSD=3 km are assumed when dRD=0.2dSD together with a normalized beam width of ωz/r= 5 and normalized jitter of σs/r={1, 2, 3} as well as the FSO scenario without pointing errors.

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4. Conclusions

In this paper, a novel adaptive cooperative protocol with multiple relays using detect-and-forward over atmospheric turbulence channels with pointing errors is proposed. The adaptive DF cooperative protocol here analyzed is based on the selection of the optical path, source-destination or different source-relay links, with a greater value of fading gain or irradiance, maintaining a high diversity order. Closed-form asymptotic expressions are obtained for this cooperative protocol using relay selection when the irradiance of the transmitted optical beam is susceptible to either a wide range of turbulence conditions (moderate 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. The superiority of this cooperative protocol is related to the significant improvement in diversity gain, being this dependent of the relay-destination link distance and number of relays. Regarding the more severe impact of pointing errors, it is concluded that the diversity gain is saturated to 2 when the parameter φ2 < 1 since the condition φ2 > β is not satisfied for any link, regardless of the relay location and number of relays. As a result, the ADF relaying scheme here analyzed presents a greater order diversity gain if compared with similar cooperative schemes or equivalent MIMO systems. Finally, simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions.

Acknowledgments

The authors wish to acknowledge the financial support given by Spanish MINECO Project TEC2012-32606.

References and links

1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006). [CrossRef]  

2. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). [CrossRef]  

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

4. 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]  

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

6. Y. Celik and N. Odabasioglu, “On relay selection for cooperative free-space optical communication,” in Networks and Optical Communications (NOC), 2012 17th European Conference on, pp. 1–5 (IEEE, 2012). [CrossRef]  

7. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and R. 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(15), 16394–16409 (2012). [CrossRef]  

8. S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013). [CrossRef]  

9. C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013). [CrossRef]  

10. C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013). [CrossRef]  

11. N. D. Chatzidiamantis, D. S. Michalopoulos, E. E. Kriezis, G. K. Karagiannidis, and R. Schober, “Relay selection protocols for relay-assisted free-space optical systems,” J. Opt. Commun. Netw. 5(1), 92–103 (2013). [CrossRef]  

12. M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in IEEE International Conference on Communications (ICC), pp. 3142–3147 (2013).

13. M. A. Kashani and M. Uysal, “Outage performance and diversity gain analysis of free-space optical multi-hop parallel relaying,” J. Opt. Commun. Netw. 5(8), 901–909 (2013). [CrossRef]  

14. L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014). [CrossRef]  

15. I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

16. 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]  

17. 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]  

18. 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]  

19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

20. 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]  

21. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010). [CrossRef]   [PubMed]  

22. 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]  

References

  • View by:

  1. V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006).
    [Crossref]
  2. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
    [Crossref]
  3. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
    [Crossref]
  4. 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]
  5. M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010).
    [Crossref]
  6. Y. Celik and N. Odabasioglu, “On relay selection for cooperative free-space optical communication,” in Networks and Optical Communications (NOC), 2012 17th European Conference on, pp. 1–5 (IEEE, 2012).
    [Crossref]
  7. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and R. 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(15), 16394–16409 (2012).
    [Crossref]
  8. S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013).
    [Crossref]
  9. C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013).
    [Crossref]
  10. C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013).
    [Crossref]
  11. N. D. Chatzidiamantis, D. S. Michalopoulos, E. E. Kriezis, G. K. Karagiannidis, and R. Schober, “Relay selection protocols for relay-assisted free-space optical systems,” J. Opt. Commun. Netw. 5(1), 92–103 (2013).
    [Crossref]
  12. M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in IEEE International Conference on Communications (ICC), pp. 3142–3147 (2013).
  13. M. A. Kashani and M. Uysal, “Outage performance and diversity gain analysis of free-space optical multi-hop parallel relaying,” J. Opt. Commun. Netw. 5(8), 901–909 (2013).
    [Crossref]
  14. L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
    [Crossref]
  15. I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).
  16. 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]
  17. 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]
  18. 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]
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).
  20. 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]
  21. N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010).
    [Crossref] [PubMed]
  22. 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]

2014 (1)

L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

2013 (4)

2012 (2)

2010 (2)

N. Wang and J. Cheng, “Moment-based estimation for the shape parameters of the gamma-gamma atmospheric turbulence model,” Opt. Express 18(12), 12824–12831 (2010).
[Crossref] [PubMed]

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

2009 (3)

2008 (1)

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

2007 (1)

2006 (1)

2001 (1)

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]

Abdallah, M. M.

S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013).
[Crossref]

Abou-Rjeily, C.

C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013).
[Crossref]

C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013).
[Crossref]

Al-Habash, M. A.

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]

Alouini, M.-S.

L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

Andrews, L.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

Andrews, L. C.

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]

Bhatnagar, M. R.

M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in IEEE International Conference on Communications (ICC), pp. 3142–3147 (2013).

Boluda-Ruiz, R.

Borah, D. K.

Castillo-Vazquez, B.

Castillo-Vazquez, C.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Celik, Y.

Y. Celik and N. Odabasioglu, “On relay selection for cooperative free-space optical communication,” in Networks and Optical Communications (NOC), 2012 17th European Conference on, pp. 1–5 (IEEE, 2012).
[Crossref]

Chan, V. W. S.

Chatzidiamantis, N. D.

Cheng, J.

Farid, A. A.

Gao, X.

L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

Garcia-Zambrana, A.

García-Zambrana, A.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

Hranilovic, S.

Hussain, S. I.

S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013).
[Crossref]

Karagiannidis, G. K.

Karimi, M.

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

Kashani, M. A.

Kim, I. I.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Korevaar, E. J.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Kriezis, E. E.

McArthur, B.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

Michalopoulos, D. S.

Nasiri-Kenari, M.

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

Odabasioglu, N.

Y. Celik and N. Odabasioglu, “On relay selection for cooperative free-space optical communication,” in Networks and Optical Communications (NOC), 2012 17th European Conference on, pp. 1–5 (IEEE, 2012).
[Crossref]

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

Phillips, R. L.

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]

Qaraqe, K. A.

S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

Safari, M.

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

Sandalidis, H. G.

Schober, R.

Tsiftsis, T. A.

Uysal, M.

M. A. Kashani and M. Uysal, “Outage performance and diversity gain analysis of free-space optical multi-hop parallel relaying,” J. Opt. Commun. Netw. 5(8), 901–909 (2013).
[Crossref]

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

Voelz, D. G.

Wang, N.

Yang, L.

L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

IEEE Photonics J. (1)

L. Yang, X. Gao, and M.-S. Alouini, “Performance Analysis of Free-Space Optical Communication Systems with Multiuser Diversity Over Atmospheric Turbulence Channels,” IEEE Photonics J. 6(2), 7901217 (2014).
[Crossref]

IEEE Trans. Commun. (1)

C. Abou-Rjeily, “Achievable Diversity Orders of Decode-and-Forward Cooperative Protocols over Gamma-Gamma Fading FSO Links,” IEEE Trans. Commun. 61(9), 3919–3930 (2013).
[Crossref]

IEEE Trans. Wireless Commun. (1)

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

IET Communications (1)

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

J. Light-wave Technol. (1)

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]

J. Lightwave Technol. (5)

J. Opt. Commun. Netw. (2)

Opt. Eng. (1)

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]

Opt. Express (3)

Other (6)

S. I. Hussain, M. M. Abdallah, and K. A. Qaraqe, “Power optimization and k th order selective relaying in free space optical networks,” in GCC Conference and Exhibition (GCC), 2013 7th IEEE, pp. 330–333 (IEEE, 2013).
[Crossref]

Y. Celik and N. Odabasioglu, “On relay selection for cooperative free-space optical communication,” in Networks and Optical Communications (NOC), 2012 17th European Conference on, pp. 1–5 (IEEE, 2012).
[Crossref]

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
[Crossref]

M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in IEEE International Conference on Communications (ICC), pp. 3142–3147 (2013).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Information Technologies 2000, pp. 26–37 (International Society for Optics and Photonics, 2001).

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Figures (5)

Fig. 1
Fig. 1 Block diagram of the cooperative FSO system under study, where dSD is the source-destination link distance, Rk are the relays nodes for k = {1, 2,...,Nr}, and (dRkD,ϕRk) represents the relay location using polar coordinates.
Fig. 2
Fig. 2 Diversity order gain Gd for the ADF cooperative protocol for a source-destination link distance of dSD=3 km when different weather conditions and relay locations are assumed. (a) (ωz/r,σs/r)=(5,1) for each link and (b) (ωz/r,σs/r)=(5,1) for the S-Rk and Rk-D links, and (ωz/r,σs/r)=(5,3) for the S-D link.
Fig. 3
Fig. 3 BER performance when different number of relays Nr={1, 2, 3} and source-destination link distance of dSD=3 km are assumed. Different relay locations of (a) dRD=0.3dSD and (b) dRD=0.8dSD are assumed together with values of normalized beam width and normalized jitter of (ωz/r, σs/r) = {(5, 1), (10, 2)}.
Fig. 4
Fig. 4 Diversity order gain Gd for the Adaptive-DF cooperative protocol for a source-destination link distance of dSD=3 km when (a) 1, (b) 2 and (c) 3 relays are assumed together with different values of normalized beam width of ωz/r= 5 and normalized jitter of σs/r={1, 2, 3}.
Fig. 5
Fig. 5 BER performance when different number of relays Nr={1, 2, 3} and source-destination link distance of dSD=3 km are assumed when dRD=0.2dSD together with a normalized beam width of ωz/r= 5 and normalized jitter of σs/r={1, 2, 3} as well as the FSO scenario without pointing errors.

Tables (1)

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Table 1 ADF cooperative protocol with multiple relays when the message s = {s1, s2,..., sn} is transmitted, being Rmax the relay selected by the source node.

Equations (34)

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y m ( t ) = η i m ( t ) x m ( t ) + z m ( t ) ,
f I l m ( i ) = α m β m φ m 2 A 0 L m Γ ( α m ) Γ ( β m ) G 1 , 3 3 , 0 ( α m β m A 0 L m i | φ m 2 φ m 2 1 , α m 1 , β m 1 ) , i 0
α = [ exp ( 0.49 σ R 2 / ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ) 1 ] 1
β = [ exp ( 0.51 σ R 2 / ( 1 + 0.69 σ R 12 / 5 ) 5 / 6 ) 1 ] 1
f I m ( i ) a m i b m 1 = φ m 2 ( α m β m ) β m Γ ( α m β m ) ( A 0 L m ) β m Γ ( α m ) Γ ( β m ) ( φ m 2 β m ) i β m 1 , φ m 2 > β m
f I m ( i ) a m i b m 1 = φ m 2 ( α m β m ) φ m 2 Γ ( α m φ m 2 ) Γ ( β m φ m 2 ) ( A 0 L m ) φ m 2 Γ ( α m ) Γ ( β m ) i φ m 2 1 , φ m 2 < β m
Y SD = X I SD + Z SD , X { 0 , d E } , Z SD ~ N ( 0 , N 0 / 2 ) .
P b SD ( E | I SD ) = Q ( d E 2 i 2 / 2 N 0 ) = Q ( 2 γ ξ i ) ,
P b SD ( E ) = 0 Q ( 2 γ ξ i ) f I SD ( i ) d i .
P b SD ( E ) a SD Γ ( ( b SD + 1 ) / 2 ) 2 b SD π ( γ ξ ) b SD / 2 ,
Y ADF 0 = 1 2 X I SD + Z SD + X * I R max D + Z R max D , I SR max > I SD
Y ADF 1 = X I SD + Z SD , I SR max < I SD
Y ADF 0 = ( 1 / 2 ) X ( I SD + 2 I R Max D ) + Z SD + Z R Max D , X * = X
Y ADF 0 = ( 1 / 2 ) X ( I SD 2 I R Max D ) + d E I R Max D + Z SD + Z R Max D , X * = d E X
P b ADF ( E | I SD , I SR k , I R k D ) = k = 1 N r P b BDF k ( E | I SD , I SR k , I R k D ) F I SD ( I SR k ) j = 1 j k N r F I SR j ( I SR k ) + P b SD ( E | I SD ) j = 1 N r F I SR j ( I SD ) ,
P b BDF k ( E | I SD , I SR k , I R k D ) = P b BDF k 0 ( E | I SD , I R k D ) ( 1 P b SR k ( E | I SR k ) ) + P b BDF k 1 ( E | I SD , I R k D ) P b SR k ( E | I SR k ) = Q ( ( γ / 4 ) ξ ( i SD + 2 i R k D ) ) ( 1 Q ( ( γ / 2 ) ξ i SR k ) ) + Q ( ( γ / 4 ) ξ ( i SD 2 i R k D ) ) Q ( ( γ / 2 ) ξ i SR k ) .
P b ADF ( E ) = k = 1 N r ( P b BDF k 0 ( E ) P b SR k ADF 0 ( E ) + P b BDF k 1 ( E ) P b SR k ADF 1 ( E ) ) + P b SD ADF ( E ) = k = 1 N r P b BDF k 0 ( E ) 0 [ 1 P b SR k ( E | I SR k ) ] F I SD ( i ) j = 1 j k N r F I SR k ( i ) f I SR k ( i ) d i . + k = 1 N r P b BDF k 1 ( E ) 0 P b SR k ( E | I SR k ) F I SD ( i ) j = 1 j k N r F I SR j ( i ) f I SR k ( i ) d i . + 0 P b SD ( E | I SD ) j = 1 N r F I SR j ( i ) f I SD ( i ) d i .
0 Q ( c γ ξ i ) n = 1 n m N r F I n ( i ) f I m ( i ) d i , c +
P b SD ADF ( E ) = 0 P b SD ( E | I SD ) j = 1 N r F I SR j ( i ) f I SD ( i ) d i .
P b SR k ADF 1 ( E ) = 0 P b SR k ( E | I SR k ) F I SD ( i ) j = 1 j k N r F I SR j ( i ) f I SR k ( i ) d i .
P b SD ADF ( E ) a SD Π i = 1 N r a SR i Γ ( ( 1 + b SD + i = 1 N r b SR i ) / 2 ) ( γ ξ ) ( b SD + i = 1 N r b SR i ) / 2 Π i = 1 N r b SR i 2 π ( b SD + i = 1 N r b SR i )
P b SR k ADF 1 ( E ) b SR k b SD 2 ( b SD + i = 1 N r b SR i ) P b SD ADF ( E ) ,
P b SR k ADF 0 ( E ) = 0 [ 1 P b SR k ( E | I SR k ) ] F I SD ( i ) j = 1 j k N r F I SR j ( i ) f I SR k ( i ) d i .
P b SR k ADF 0 ( E ) 0 j = 1 j k N r F I SR j ( i ) F I SD ( i ) f I SR k ( i ) d i .
P b BDF k 0 ( E ) a SD a R k D Γ ( b SD ) Γ ( b R k D ) ( γ ξ ) ( b SD + b R k D ) / 2 2 ( b SD + b R k D ) / 2 2 Γ ( ( b SD + b R k D + 2 ) / 2 ) ,
P b BDF k 1 ( E ) 0 0 2 i 2 f I SD ( i 1 ) f I R k D ( i 2 ) d i 1 d i 2 .
P b ADF ( E ) P b BDF min 0 ( E ) P b SR min ADF 0 ( E ) , b R min D < i = 1 N r b SR i
P b ADF ( E ) ( 1 = 2 ( b SD + i = 1 N r b SR i ) b SD k = 1 N r b SR k P b BDF k 1 ( E ) ) P b SD ADF ( E ) , b R min D > i = 1 N r b SR i
G d ( N r ) = 1 + min ( b R min D , i = 1 N r b SR i ) b SD .
G d ( N r ) = 1 + min ( φ RD 2 , N r φ SR 2 ) φ SD 2 = 1 + min ( φ 2 , N r φ 2 ) φ 2 = 2 ,
P b ADF ( E ) ( 1 + 2 ( β SD + i = 1 N r β SR i ) β SD k = 1 N r β SR k P b BDF k 1 ( E ) ) P b SD ADF npe ( E ) ,
a m npe = ( α m β m ) β m Γ ( α m β m ) L m β m Γ ( α m ) Γ ( β m ) .
D pe [ d B ] 20 β SD + i = 1 N r β SR i log 10 ( φ SD 2 A 0 β SD ( φ SD 2 β SD ) i = 1 N r φ SR i 2 A 0 β SR i ( φ SR i 2 β SR i ) ) .
D pe [ d B ] 20 ( N r + 1 ) β SD log 10 ( A 0 β SD φ SD 2 φ SD 2 β SD ) N r + 1 = 20 β SD log 10 ( A 0 β SD φ SD 2 φ SD 2 β SD ) ,

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