Abstract

The impact of relay placement on diversity order in adaptive selective decode-and-forward (DF) cooperative strategies is here investigated in the context of free-space optical (FSO) communications over atmospheric turbulence channels with pointing errors when line of sight is available. The irradiance of the transmitted optical beam here considered is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution together with a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Novel closed-form approximate bit error-rate (BER) expressions are obtained for a cooperative FSO communication setup with N relays, assuming that these relays are located in an area similar to an annulus around source or destination node. An analytical expression is here found that determines the best selection criterion based on the knowledge of the channel state information (CSI) of source-relay or relay-destination links in order to significantly increase the diversity order corresponding to the cooperative strategy under study. It is concluded that the highest diversity order is achieved when the relation βSRmin > βSD + βRminD is satisfied, wherein βSRmin, βRminD and βSD are parameters corresponding to the atmospheric turbulence conditions of source-relay and relay-destination link with the greatest scintillation index, and source-destination link, respectively.

© 2015 Optical Society of America

1. Introduction

Free-Space Optical (FSO) systems have achieved significant popularity providing higher data rates and transmission security for a large variety of applications, being an interesting alternative to consider for next generation broadband. FSO links can be used to setup a complete FSO network or as a supplement to conventional radio-frequency (RF) systems and fiber optics [1]. However, the major limitation of the FSO communication systems is the atmospheric turbulence, which produces fluctuations in the irradiance of the transmitted optical beam, as a result of random variations in the refractive index through the link [2]. Additionally, misalignment between the transmitter and receiver due to building sway causes pointing errors that limit the performance of FSO communication systems. Different techniques have been proposed in recent decades, such as the adoption of forward error correcting codes as well as spatial diversity based on multiple-input multiple-output (MIMO) configurations [39].

An alternative approach to improve the performance in this turbulent FSO scenario is based on the employment of cooperative transmission. In this context, cooperative communications has recently drawn significant attention in order to improve the diversity and coverage area by using the transceivers available at the other nodes of the network. Relay-assisted FSO communications have generated considerable recent research interest [1021], being recognized as a very promising solution for future ad-hoc optical wireless systems. It has been demonstrated that relay-assisted is an effective manner of fading mitigation in FSO links. In [21], a novel adaptive selective cooperative protocol with multiple relays using detect-and-forward (DF) over atmospheric turbulence channels with pointing errors was analyzed. Obtained results provide a better performance compared with similar cooperative protocols wherein the relay nodes are located in an area closer to the destination node than to the source node. Different atmospheric turbulence conditions and pointing errors were considered, being only required the knowledge of the channel side information (CSI) of the source-destination link, or source-relay links. A higher and robust diversity order gain, strongly dependent on the relay locations and number of relays, was here achieved when different values of pointing errors were considered. The authors concluded that relay placement plays an important role in performance of cooperative FSO communications, and even superior to the number of relays. In this sense, few researchers have addressed the problem of relay placement in order to obtain the optimal relay placement [22, 23], being this one of the most important aspects in the context of cooperative FSO systems. Optimal relay placement problem for serial and parallel relaying was investigated in [22], being evaluated the outage probability over log-normal fading channels wherein pointing error effects were not considered. In this parallel relaying scheme, the source node transmits the same signal to all relay nodes, and each relay retransmits the signal to the destination node only if the received signal-to-noise ratio (SNR) exceeds a given decoding threshold. The minimum outage probability is achieved for parallel relaying when all relays are located at the same place closer to the source node, being the exact location of this place dependent on the system and channel parameters, as in [23].

This paper focuses on the analysis of the impact of relay placement on diversity order in adaptive selective DF cooperative strategies in the context of FSO communications over atmospheric turbulence channels with pointing errors when line of sight is available. An analytical expression is here found that determines the best selection criterion based on the knowledge of the CSI of source-relay or relay-destination links in order to significantly increase the diversity order corresponding to the cooperative strategy under study. Obtained results show a better performance when the relay nodes are closer to the source node, as well as when the relay nodes are closer to the destination node, determining in each case which relay has to be selected to demonstrate the superiority of the adaptive DF cooperative scheme. In this way, novel closed-form approximate bit error-rate (BER) expressions are obtained for a cooperative FSO communications system with N relays by using the CSI of source-destination link and relay-destination links. The irradiance of the transmitted optical beam here considered is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma distribution of parameters α and β, or pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance are considered. The highest diversity order is achieved when the relation βSRmin > βSD + βRminD is satisfied, wherein βSRmin, βRminD and βSD are parameters corresponding to the atmospheric turbulence of source-relay and relay-destination link with the greatest scintillation index, and source-destination link, respectively. The diversity order gain analysis is performed as a function of the relay locations and number of relays under different atmospheric turbulence conditions and pointing errors. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

2. System model

In this paper, we adopt a cooperative FSO communications system with multiple relays, consisting of one source node S, N relays denoted by Rk for k = {1,2,...,N}, and one destination node D. Furthermore, different source-relay link distances within the range d′SR to dSR (d′SR < dSR) are considered in order to perform a more realistic approach in the context of cooperative FSO systems, assuming that these relays are located in an area similar to an annulus. The system model under study is shown in Fig. 1. In particular, we consider laser sources intensity-modulated and ideal non-coherent (direct-detection) receivers. The adaptive selective relaying scheme with N relay nodes by using the CSI of source-relay links, i.e. ADF-SR cooperative protocol, was presented in [21]. In order to analyze the impact of different relays placement we must here assume that relay nodes may be also located closer to the source node. In this way, the adaptive selective relaying with N relay nodes by using the CSI of relay-destination links, i.e. ADF-RD cooperative protocol, is also here considered. In this sense, ADF-RD cooperative protocol selects between direct transmission or bit-detect-and-forward (BDF) cooperative protocol analyzed in [13] on the basis of the value of the fading gain. When the irradiance of the source-destination link (ISD) is greater than the irradiance values corresponding to the relay-destination links (IRkD for k = {1, 2,...,N}), the cooperative FSO 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 Rk-D link is greater than the irradiance corresponding to the S-D link and those corresponding to the Ri-D links for ik, i.e. the remaining relays. It must be emphasized that only one relay node is selected by the source node in order to avoid synchronization problems at the destination node when more than one is selected by the source node. It should be noted that the selection criterion is here based on the value of the irradiance instead of its square magnitude, as usually assumed in the literature [1618, 24], not being required the square magnitude due to the fact that the fading gain in FSO systems using IM/DD is always positive. 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. This has been considered for FSO links from the point of view of information theory [25, 26]. The CSI can be acquired by using the training sequence at the receiver side and feedback the CSI back to transmitter. An interesting approach to acquire the CSI at the transmitter was proposed in [27], wherein the CSI is estimated at the receiver side and feed this channel estimate back to the transmitter using an RF feedback channel. Because the atmospheric turbulence changes slowly, with correlation time ranging from 10 μs to 10 ms, this is a plausible scenario for FSO channels with data rates in the order of Gb/s, implementing a continuous feedback from the receiver to maintain a specific performance level. In this sense, the receiver always knows if the cooperative protocol is being used. The ADF-RD cooperative protocol works in two phases or transmission frames, as in [13, 21]. 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 path link without using any cooperative strategy. For the sake of simplicity and an easier practical implementation, it should be emphasized that it is here assumed that all the bits detected at the relay node selected by the source node are always forwarded with the new power to the destination node D regardless of these bits are detected correctly or incorrectly.

 figure: Fig. 1

Fig. 1 Diagram showing the cooperative FSO communications system, where dSD is the source-destination link distance, Rk are the relay nodes for k = {1,2,...,N}, and (dSRk,θRk) represents the relay placement using polar coordinates.

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3. Channel model

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−Φdm, where dm 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) [28]. Although the effects of turbulence and pointing are not strictly independent, for smaller jitter values they can be approximated as independent [29]. 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 [30], 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 [31] in terms of the Meijer’s G-function [32, eqn. (9.301)]. Here, an approximate expression by using the first two terms of the Taylor expansion at i = 0 as fIm(i)=amibm1+cmibm+O(ibm+1) is adopted in order to obtain simpler closed-form expressions, which can provide a deeper insight on how atmospheric turbulence and pointing errors deteriorate the performance in adaptive selective DF relay-assisted FSO communications. As proposed in [33], the PDF is approximated as follows
fIm(i)amibm1eicmam.
Different expressions for fIm (i), depending on the relation between the jitter variance and turbulence conditions [9], can be written as
fIm(i){φm2(αmβm)βmΓ(αmβm)(A0Lm)βmΓ(αm)Γ(βm)(φm2βm)iβm1eiαmβm(φm2βm)(A0Lm)1(αmβm1)(βmβmφm2+1),φm2>βmφm2(αmβm)φm2Γ(αmφm2)Γ(βmφm2)(A0Lm)φm2Γ(αm)Γ(βm)iφm21,φm2<βm
where Γ(·) is the well-known Gamma function, and α and β can be directly linked to physical parameters through the following expressions [34]:
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 ∞ [35]. In relation to the impact of pointing errors [30], 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 [32, eqn. (8.250)]. It must be noted that the parameter cm is equal to 0 when the diversity order gain is independent of the pointing error, i.e. when the relation φm2>βm is satisfied.

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,...,N}. 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.

4. Error-rate performance analysis

In this section, approximate expressions are obtained in order to quantify the bit error probability for this cooperative FSO system in the range from low to high SNR, taking advantage of the simpler expressions in Eq. (3). Here, it is assumed that 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 ADF-RD cooperative protocol can be written as

Y0=XISD+ZSD,X{0,dE},ZSD~N(0,N0/2)IRkD<ISD
Y1=12XISD+X*IRkD+ZSD+ZRkD,ZRkD~N(0,N0/2)IRkD>ISD
being Rk the relay node whose value of the fading gain corresponding to the Rk-D link is maximum, denoted as IRkD, i.e. IRkD = max{IR1D,IR2D,...,IRND}. X* represents the random variable corresponding to the information detected at the relay node Rk and, hence, X* = X when the bit has been detected correctly, and X* = dEX when the bit has been detected incorrectly at the relay node Rk. In this way, the Eq. (5a) is the statistical channel model corresponding to the direct transmission without cooperative communication. Thus, assuming channel side information at the receiver and transmitter (CSIRT), the conditional BER under the assumption that ISD > IRkD at the destination node D is given by
PbSDRD(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, PbSDRD, can be obtained by averaging PbSDRD(ISD) over the PDF as follows
PbSDRD0Q(2γξi)j=1NFIRjD(i)fISD(i)di.
Knowing the fact that the irradiance values are statistically independent, the probability corresponding to ISD > IRkD is computed by using j=1NFIRjD(ISD), being FIm (i) the cumulative density function (CDF) of the random variable Im, which is given by FIm (i) = Prob(Imi). For the sake of simplicity, in spite of the fact that this CDF can be expressed in terms of the Meijer’s G-function as in [31, eqn. (15)], an approximate expression can easily be deduced from Eq. (2) as follows
FIm(i)ambmibmeicmbmam(bm+1),
where the value of the parameters am, bm and cm depends on the relation between φ2 and β as obtained in Eq. (3). 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) [32, eqn. (6.287)] and the fact that 0erfc(cx)xα1epxdx can be found in [36, eqn. (2.8.5.2)]. Therefore, a approximate closed-form solution for the BER, PbSDRD, can be expressed after some algebraic manipulations as follows
PbSDRDaRDTΓ((bRDT+1)/2)(γξ)bRDT/22bRDTπF22(bRDT2,bRDT+12;bRDT+22,12;cRDT24aRDT2γξ)+cRDTΓ((bRDT+2)/2)(γξ)(bRDT+1)/22(bRDT+1)πF22(bRDT+12,bRDT+22;bRDT+32,32;cRDT24αRDT2γξ),
where 2F2(a1,a2;b1,b2;z) is the generalized hypergeometric function whose coefficients p and q are equal to 2, i.e. Fqp(a1,,ap;b1,,bq;z)=k=0j=1p(aj)kzkj=1q(bj)kk! [32, eqn. (9.14.1)]. The notation (·)k is called the Pochhammer symbol and is defined as (a)k = Γ(a + k)/Γ(a). Hence, corresponding expressions for coefficients aRDT, bRDT and cRDT are given by
aRDT=aSDi=1NaRiDi=1NbRiD,bRDT=bSD+i=1NbRiD,cRDTaRDT=cSDaSD+i=1NcRiDbRiDaRiD(bRiD+1).
In a similar way as in Eq. (7), we can obtain the closed-form solution corresponding to the source-destination link without cooperative communication in order to establish the baseline performance. Therefore, the average BER for the direct path link (DL), PbDL, can be determined as follows
PbDL=0Q(2γξi)fISD(i)di.
Evaluating this integral as in Eq. (7), we can obtain the approximate closed-form solution for the BER as follows
PbDLaSDΓ((bSD+1)/2)(γξ)bSD/22bSDπF22(bSD2,bSD+12;bSD+22,12;cSD24aSD2γξ)+cSDΓ((bSD+2)/2)(γξ)(bSD+1)/2(bSD+1)πF22(bSD+12,bSD+22;bSD+32,32;cSD24αSD2γξ).
By other hand, the Eq. (5b) is the statistical channel model corresponding to the detect-and-forward (DF) cooperative protocol analyzed in [13], but taking into account that this relaying scheme is only used under the assumption that IRkD > ISD. Hence, two cases can be considered depending on the fact that the bit from the relay S-Rk-D is detected correctly or incorrectly. In this way, considering that the bit has been detected correctly at relay the node Rk, the statistical channel model corresponding to this case can be expressed as
Y1=12X(ISD+2IRkD)+ZSD+ZRkD,X*=X
Assuming CSIRT, the conditional BER at the destination node D when the bit is detected correctly ( DFk0) is given by
PbDFk0(ISD,IRkD)=Q(γξ4(i1+2i2)),
Hence, the average BER, PbDFk0, can be obtained by averaging over the PDFs as follows
PbDFk0=00Q(γξ4(ii+2i2))fISD(i1)FISD(i2)j=1jkNFIRjD(i2)fIRkD(i2)di1di2.
Before evaluating the integral in Eq. (15) we define IDFT, which represents the sum of variates IDFT=ISD+2IRkD, under the assumption that IRkD > ISD, as determined by the terms FISD (i2) and FIRjD (i2) for jk. Knowing that ISD and IRkD are independent statistically, the resulting PDF of their sum IDFT can be determined by using the moment generating function of their corresponding PDFs, obtained via single-sided Laplace and its inverse transforms. Firstly, we obtain the corresponding PDF of IRkD = max{IR1D,IR2D,...,IRND} under the assumption that IRkD > ISD, i.e. fIRDmax(i)=FISD(i)jkNFIRjD(i)fIRkD(i) and, then, its corresponding approximate expression, fIRDmax(i)aRDmaxibRDmax1ei(cRDmax/aRDmax), can be written as
fIRDmax(i)aSDbRkDj=1NaRjDbSDj=1NbRjDibSD+j=1NbRjD1ei(cRkDaRkD+cSDbSDaSD(bSD+1)+j=1jkNcRjDbRjDaRjD(bRjD+1)).
, of combined variates can be easily derived as
fIDFTaSDaRDmaxΓ(bSD)Γ(bRDmax)2bRDmaxΓ(bSD+bRDmax)ibSD+bRDmax1ei(2aRDmaxbSDcSD+aSDbRDmaxcRDmax2aSDaRDmax(bSD+bRDmax)).
From this expression, the average BER, PbDFk0, can be determined as follows
PbDFk0=0Q(γξ4i)fIDFT(i)di.
Evaluating this integral as in Eq. (7), we can obtain the approximate closed-form solution for the BER, PbDFk0 as follows
PbDFk0aDFTΓ((bDFT+1)/2)(γξ)bDFT/22(23bDFT)/2bDFTπF22(bDFT2,bDFT+12;bDFT+22,12;2cDFT2aDFT2γξ)+cDFTΓ((bDFT+2)/2)(γξ)(bDFT+1)/22(3bDFT+1)/2(bDFT+1)πF22(bDFT+12,bDFT+22;bDFT+32,32;2cDFT2aDFT2γξ).
Alternatively, considering now that the bit has been detected incorrectly at the relay node Rk, the statistical channel model for this case can be expressed as
Y1=12X(ISD2IRkD)+dEIRkD+ZSD+ZRkD,X*=dEX
Assuming CSIRT and given that the statistics corresponding to the term dE · IRkD become irrelevant to the detection process, the conditional BER at the destination node D when the bit is detected incorrectly ( DFk1) is given by
PbDFk1(ISD,IRkD)=Q(γξ4(i12i2)).
Hence, the average BER, PbDFk1, can be obtained by averaging over the PDFs as follows
PbDFk1=00Q(γξ4(i12i2))fISD(i1)FISD(i2)j=1jkNFIRjD(i2)fIRkD(i2)di1di2.
Unfortunately, in Eq. (23) cannot be applied the approximation of the PDF as in previous equations due to the argument of the Gaussian-Q function is not always positive. 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. (23) together with the fact that Gaussian-Q function tends to 0 as γ → ∞, simplifying the integral in Eq. (23) as follows
PbDFk1=˙002i2fISD(i1)FISD(i2)j=1jkNFIRjD(i2)fIRkD(i2)di1di2.
It must be noted that the asymptotic behavior in Eq. (24) is independent of the SNR γ, resulting in a positive value that is upper bounded by 1. Here, the Eq. (24) was obtained by using the Monte Carlo integration, being analytically intractable. Finally, in a similar way as in Eq. (7), we can obtain the approximate closed-form solution for the BER corresponding to the S-Rk link, as follows
PbSRkRDΓ((bSRk+1)/2)(γξ)bSRk/2(aSRk)12(1bSRk)bSRkπF22(bSRk2,bSRk+12;bSRk+22,12;cSRk2aSRk2γξ)+Γ((bSRk+2)/2)(γξ)(bSRk+1)/2(cSRk)12bSRk(bSRk+1)πF22(bSRk+12,bSRk+22;bSRk+32,32;cSRk2aSRk2γξ),
wherein the division by 2 has been considered 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. The BER corresponding to the ADF-RD cooperative protocol here proposed is given by
PbRD=PbSDRD+k=1NPbDFk0(1PbSRkRD)+k=1NPbDFk1PbSRkRD.
Considering now that the PDF in Eq. (3) is approximated by using the first term of the Taylor expansion, i.e. assuming a value of cm = 0, it is straightforward to show that the average BER behaves asymptotically as (Gcγξ) Gd due to 2F2(a1,a2;b1,b2;0) = 1, where Gd and Gc denote diversity order and coding gain, respectively. Therefore, the expression in Eq. (26) can be simplified taking into account the asymptotic behavior previously obtained in Eq. (9) and Eq. (25) as follows
PbRD=˙{PbDFmin1PbSRminRD,bSRmin<bSD+i=1NbRiDPbSDRD,bSRmin>bSD+i=1NbRiD
being bSRmin=min{bSR1,...,bSRN } and bSD+i=1NbRiD the diversity order gain corresponding to PbSRminRD and PbSDRD, respectively. It can be observed that the Eq. (19) has not been contemplated in Eq. (27) due to the fact that it is negligible compared with the other two terms of this equation. Taking into account these expressions, the adoption of the ADF-RD cooperative protocol here proposed translates into a diversity order gain, GdRD, relative to the non-cooperative link S-D of
GdRD=min(bSRmin,bSD+i=1NbRiD)/bSD.

5. Numerical results and discussions

For the better understanding of the impact of relay placement on diversity order in this cooperative FSO system here analyzed, the diversity order gain GdRD corresponding to the ADF cooperative protocol by using the CSI of relay-destination links is compared with the diversity order gain corresponding to the ADF cooperative protocol by using the CSI of source-relay links presented in [21, eqn. (23)]. This expression is here rewritten as follows

GdSR=1+min(bRminD,i=1NbSRi)/bSD.
being bRminD=min{bR1D,...,bRND}. The diversity order gain dependent on relay placement is defined as the maximum between GdRD and GdSR, and it can be written as
Gd={GdRD,bSRmin>bSD+bRminDGdSR,bSRmin<bSD+bRminD
It is straightforward to show that the knowledge of the CSI for all links, i.e. source-relay and relay-destination links, is not required in the cooperative FSO system under study due to the relay placement and pointing error effects determine the better choice of cooperative strategy through of the relation between bSRmin and bSD + bRminD, i.e. ADF-SR or ADF-RD cooperative protocol.

5.1. Diversity order gain analysis

From this analysis, it can be deduced that the main aspect to consider in order to optimize the diversity order is the relation between φ2 and β for the source-destination link, and the source-relay and relay-destination links, corroborating that the diversity order is independent of the pointing error when the relation φ2 > β is satisfied. In this way, the diversity order gain Gd in Eq. (30) as a function of the radius of a circumference, dSR, whose center is the source node, is depicted in Fig. 2 for a source-destination link distance dSD=3 km when different number of relays N = {1,2,3,4} are considered, as shown in Fig. 1. Moreover, different scenarios are assumed in order to evaluate the performance when different source-relay link distances are considered in this cooperative FSO system, wherein all relays are located in an area similar to an annulus. Here, the parameters α and β are calculated from Eq. (4a) and Eq. (4b) respectively, with λ =1550 nm. Different weather conditions are here adopted, assuming haze visibility of 4 km with Cn2=1.7×104m2/3 and clear visibility of 16 km with Cn2=8×1014m2/3, corresponding to moderate and strong atmospheric turbulence conditions, respectively. In these figures, the performance analysis is evaluated for dSR = d′SR, dSRd′SR = 0.3 km and dSRd′SR = 0.6 km. Hereinafter, φSRk2φSR2 and φRkD2φRD2 are assumed for k = {1,2,...,N}. Moreover, the relay locations assumed in this paper are (θR1,θR3) = (π/12,π/18) and (θR1,θR3) = (π/18, π/36) for moderate and strong turbulence, respectively. In Fig. 2, the conditions φ2 > β holds 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 on the atmospheric turbulence when this relation is satisfied. Firstly, it can be concluded that the available diversity order gain is always greater than 2 regardless of the number of relays and relay locations, achieving a greater robustness for different source-relay link distances. The diversity order gain shows two clear maximums dependent on the relay locations and source-destination link distance: one closer to the source node, when the relation bSRmin=bSD+i=1NbRiD holds, while the other closer to the destination node, when the relation bRminD=i=1NbSRi holds. This behavior is due to the fact that the use of knowledge of the CSI for the source-relay and relay-destination links is being optimized in order to achieve the best possible performance. Furthermore, the lowest diversity order gain is provided when the relation bSRmin = bSD + bRminD is satisfied regardless of the number of relays. The intersection of the two diversity order gains provides the choice of the cooperative protocol. From Eq. (28) and Eq. (29), it is easy to deduce that ADF-RD cooperative protocol is superior to the ADF-SR cooperative protocol when the relation bSRmin > bSD + bRminD holds. By other hand, the ADFSR cooperative protocol presents a better performance when the relation bSRmin <bSD + bRminD holds, corroborating the results obtained in [21]. It should be noted that the parameters bSRmin and bRminD do not necessarily have to be related to the same relay node. According to Fig. 2 when dSRd′SR = 0.3 km or dSRd′SR = 0.6 km, only one relay is around of the circumference of radius dSR (outer ring) and the remaining relays are around of the circumference of the radius d′SR (inner ring), being these scenarios representative enough in order to evaluate the diversity order gain when different source-relay link distances are considered, modeling the worst case. The comparison of these different relay locations corroborate that changes in diversity performance are negligible, specially over more severe atmospheric turbulence or when the link distance is increased, tending to an idealized situation wherein all relays are at the same distance from the source node.

 figure: Fig. 2

Fig. 2 Diversity order gain for a source-destination link distance of dSD=3 km when different weather conditions are assumed. Different relay locations are assumed together with values of normalized beam width and normalized jitter of (ωz/r,σs/r) = (7,1).

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Next, the maximum source-relay link distance allowed in order to fully exploit the ADF-RD cooperative protocol when the relay node is closer to the source node than to the destination node is depicted in Fig. 3 when dSR = d′SR. This maximum source-relay link distance, dSRmax, is depicted as a function of source-destination link distance when different maximum relay-destination link distances are assumed under different weather conditions. Taking into account the relay locations assumed, this maximum source-relay link distance is determined in this cooperative FSO system when the relation βSRmin = βSD + βRminD holds. In Eq. (31), we present the corresponding closed-form expression for the maximum source-relay link distance allowed in this FSO scenario when the condition φ2 > β is satisfied for each link. Substituting βSRmin by its corresponding expression in Eq. (4b), we can obtain the closed-form expression for dSRmax through some algebraic manipulations as follows

dSRmax(km)=(11.23Cn2κ7/6)6/11(ln6/11(1+1βSD+βRminD)(0.440.69ln6/5(1+1βSD+βRminD))5/11)×103,
where βSD and βRminD are parameters corresponding to the atmospheric turbulence conditions of source-destination and maximum relay-destination links. As can be seen in Fig. 3, the maximum source-relay link distance is increased as the refractive structure parameter, Cn2, decreases. In other words, the ADF-RD cooperative protocol here proposed is more appropriate when the turbulence conditions are less severe due to the fact that the area similar to annulus around the source node wherein the relay nodes must be located is bigger and, hence, a greater freedom to choose potential relay nodes is provided. From this expression, we can also obtain the limit of the maximum source-relay link distance when βSD → 1 and βRminD → 1 and, hence, the corresponding limit is approximately 2.2 km and 1 km for moderate and strong turbulence, respectively. The results corresponding to this performance analysis with rectangular pulse shapes and ξ =1 are illustrated in Fig. 4 for a source-destination link distance dSD=3 km, together with values of normalized beam width and normalized jitter of (ωz/r,σs/r)=(7,1) and (ωz/r,σs/r)=(10,1). For the sake of simplicity, hereinafter it is assumed that dSR = d′SR 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. 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 when different number of relays N = {1,2,3} are assumed. Additionally, we consider the performance analysis for the direct path link (non-cooperative link S-D) obtained in Eq. (12) to establish the baseline performance. The BER results corresponding to the ADF-SR cooperative protocol analyzed in [21] are also included in Fig. 4, but considering in the corresponding error-rate performance analysis the approximated PDF here considered in Eq. (3), as shown in the appendix. In this sense, we can see in Figs. 4(a) and 4(b) diversity gains of 2.36, 3.73 and 4.83 for a source-relay link distance of dSR = 0.6 km under moderate turbulence conditions corresponding to ADF-RD cooperative protocol, or diversity gains of 2.04, 3.09 and 3.43 for a source-relay link distance of dSR = 2.4 km under strong turbulence conditions corresponding to ADF-SR cooperative protocol, when the number of relays is 1, 2 and 3, respectively.

 figure: Fig. 3

Fig. 3 Maximum source-relay link distance when different weather conditions and different maximum relay-destination link distances are assumed, once the condition φ2 > β is satisfied for each link.

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 figure: Fig. 4

Fig. 4 BER performance for a source-destination link distance of dSD=3 km when different number of relays are assumed. Different relay placement are assumed together with values of normalized beam width and normalized jitter of (ωz/r,σs/r) =(7,1) and (ωz/r,σs/r)=(10,1).

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5.2. Impact of pointing errors

The effect of misalignment on the diversity order gain in this cooperative FSO system has been evaluated in Fig. 5 when more than one relay is considered. Hence, Gd is depicted for a source-destination link distance dSD = 3 km and dSR = d′SR when the number of relays is N = {2,3}. In this configuration, a normalized beam width of ωz/r = 7 and different values of normalized jitter σs/r = {1, 1.5, 2, 2.5, 3} are assumed in order to contrast the impact of pointing error when the relation φ2 > β is or not satisfied for each link under moderate turbulence conditions. It can be observed that diversity gains even greater than 5 are achieved when the number of relays is equal to 3 and (ωz/r,σs/r) = (7,1), not being affected by pointing error effects. However, the maximum values of diversity gain corresponding to this configuration are dramatically decreased as the normalized jitter increases, as a consequence that the relation φ2 > β is not satisfied. Moreover, it must be noted that the ADF-RD cooperative protocol does not present a better performance than ADF-SR cooperative protocol regardless of the relay locations when the pointing errors are more severe. When normalized jitter σs/r is greater than 1.5, the diversity order gain is completely determined by the ADF-SR cooperative protocol for all source-relay link distances. It can be concluded that the knowledge of CSI of the relay-destination links is not required in order to implement the relay selection in adaptive selective relaying when the pointing error effects are more severe for each link regardless of the relay locations. In this way, for values of normalized beam width and jitter of (ωz/r,σs/r) = (7,3), i.e. φ2 < 1, the diversity order gain is saturated for both cooperative protocols, as can be deduced from Eq. (28) and Eq. (29). Achieving diversity gains of 1 and 2 for ADF-RD and ADF-SR cooperative protocol respectively, regardless of the number of relays and relay placement. Hence, it is demonstrated that ADF-SR cooperative protocol presents a greater robustness when the relation φ2 < β is satisfied for each link. In this sense, the optimum beam width subject to φSR2>βSD+βRminD can be achieved using numerical optimization methods for different values of normalized jitter in order to obtain a better performance when ADF-RD cooperative protocol is being used, and pointing errors are more severe. Here, it is also corroborated as in [8] that the impact of the pointing errors is more severe as the link distance is smaller, and, hence, the atmospheric turbulence strength is lower. Numerical results for the optimum beam width are depicted in Fig. 6 for a source-destination link distance of dSD = 3 km and a source-relay link distance of dSR = 0.6 km together with different values of normalized jitter. From this figure, it can be deduced that the optimization method provides numerical results following a linear performance. This leads to easily obtain a first-degree polynomial as follows

ωz/roptimum4.013σs/r0.153.

 figure: Fig. 5

Fig. 5 Diversity order gain for a source-destination link distance of dSD=3 km when values of normalized beam width of ωz/r = 7 and normalized jitter of σs/r = {1, 1.5, 2, 2.5, 3} are considered. Moderate turbulence conditions are assumed.

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As can be seen in Fig. 6, it is clearly observed that the approximate analytical expression remains in excellent agreement with numerical results. These conclusions related to the misalignment effect are contrasted in Fig. 7, wherein BER performance is depicted for a source-destination link distance dSD = 3 km and dSR = d′SR when the number of relay is N = {2, 3} under moderate turbulence conditions. The use of the expression in Eq. (32) is also shown in Fig. 7, where results assuming the optimum beam width corresponding to a normalized jitter of σs/r = 2 are also included for different number of relays. As before, the performance analysis for the direct path link is considered to establish the baseline performance. As also displayed in Fig 4, simulation results corroborate that approximate expressions here derived lead to simple bounds on the bit error probability that get tighter over a wide range of SNR as the turbulence strength increases.

 figure: Fig. 6

Fig. 6 Optimum normalized beam width versus normalized jitter for a source-destination link distance of dSD = 3 km under moderate turbulence conditions.

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 figure: Fig. 7

Fig. 7 BER performance when different number of relays N={2, 3} and source-destination link distance of dSD=3 km are assumed when a normalized beam width of ωz/r= 7 and normalized jitter of σs/r={1, 2, 3} are considered under moderate turbulence conditions, as well as the FSO scenario without pointing errors.

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Finally, we conclude the paper by evaluating that the impact of the pointing errors translates into a coding gain disadvantage. As in [21], it is here convenient to note that the impact of the pointing errors on coding gain is not related to the diversity order when a realistic FSO scenario is assumed, obtaining the same coding gain disadvantage of 28.7 decibels for a value of (ωz/r,σs/r) = (7,1) regardless of the cooperative protocol (ADF-SR or ADF-RD) and number of relays. This result can be justified from the different values for the links distance and the fact that βSDβR1DβR2DβR3D holds in this scenario. Thus, the impact of the pointing errors translates into a coding gain disadvantage, Dpe[dB], as obtained in [21, eqn. (28)], depending finally on the parameters corresponding to the source-destination link as follows

Dpe[dB](20/βSD)log10(φSD2/(A0βSD(φSD2βSD))).

6. Conclusions

In this paper, the impact of relay placement on diversity order in adaptive selective DF cooperative strategies is here investigated in the context of free-space optical (FSO) communications over atmospheric turbulence channels with pointing errors when line of sight is available. Novel closed-form approximate bit error-rate (BER) expressions are obtained for a cooperative FSO communication setup with N relays, assuming that these relays are located in an area similar to an annulus around source or destination node. An analytical expression is here found that determines the best selection criterion based on the knowledge of the CSI of source-relay or relay-destination links in order to significantly increase the diversity order corresponding to the cooperative strategy under study. It is concluded that the highest diversity order is achieved when the relation βSRmin > βSD + βRminD is satisfied, wherein βSRmin, βRminD and βSD are parameters corresponding to the atmospheric turbulence conditions of source-relay and relay-destination link with the greatest scintillation index, and source-destination link, respectively. From Eq. (31), we can conclude that the knowledge of the CSI for all links is not required in a cooperative FSO system, unlike the selection criterion established in [16], because depending on relay placement, the CSI corresponding to the relay within the outer ring around source node or around destination node is only required. By other hand, we have corroborated that the diversity order gain depends not only on the number of relays but also on the relay placement. Regarding the more severe impact of pointing errors, once the condition φ2 > β is not satisfied, it is concluded that ADF-SR cooperative protocol always obtains a better performance than ADF-RD cooperative protocol regardless of relay placement and number of relays. Taking into account the analysis about the optimal relay placement for serial and parallel relaying presented in [22], being evaluated the outage probability over log-normal fading channels wherein pointing error effects were not considered, similar conclusions in relation to the closeness to the source node can also be contrasted in this work when the number of relays is set to 2 and, hence, the relay nodes are at the same distance to the source node. Nonetheless, it is here corroborated that the highest diversity order gain is 4, as shown in Fig. 5, being achieved not only when relays are closer to the source node, as concluded in [22] in a weak turbulence context, but also when these are closer to the destination node. In fact, taking into account the effect of misalignment on the diversity order gain and, hence, the condition φ2 > β is not satisfied for all links, the diversity order gain achieved is greater when the relay nodes are closer to the destination node than to the source node. Simulation results are further provided to confirm the accuracy and usefulness of the derived results. Finally, it is verified that adaptive selective DF cooperative strategies are able to achieve a higher diversity order strongly dependent not only on the number of relays but also on the relay placement. From the relevant results here obtained, investigating the impact of adaptive selective amplify-and-forward (AF) cooperative strategies on diversity order as well as the incorporation of mirrors as relay nodes for different cooperative FSO systems are interesting topics for future research in order to complement the analysis in this work.

Appendix

We here review the analysis in [21] but considering the approximate PDF here used in Eq. (3), obtaining expressions with better accuracy than previously reported. The BER corresponding to the ADF-SR cooperative protocol is here rewritten as follows

PbSR=PbSDSR+k=1NPbBDFk0PbSRk0SR+k=1NPbBDFk1PbSRk1SR.
In this way, the closed-form solution for the BER corresponding to the source-destination link under the assumption that ISD > ISRk, i.e. ISRk = max{ISR1,...,ISRN }, PbSDSR, can be written as in Eq. (9), but substituting the corresponding coefficients aRDT, bRDT and cRDT by the corresponding coefficients aSRT, bSRT and cSRT, respectively. Therefore, corresponding expressions for coefficients aSRT, bSRT and cSRT are given by
aSRT=aSDi=1NaSRii=1NbSRi,bSRT=bSD+i=1NbSRi,cSRTaSRT=cSDaSD+i=1NcSRibSRiaSRi(bSRi+1).
Next, the corresponding solution for the BER in source-relay link under the assumption that ISD < ISRk as follows
PbSRk1SR(bSRk/bSD)2(bSD+i=1NbSRi)PbSDSR.
By other hand, the probability when the bit has been detected correctly at the relay node Rk, i.e. PbSRk0SR, can be written as
PbSRk00j=1jkNFISRj(i)FISD(i)fISRk(i)di.
It can be noted that the asymptotic behavior in PbSRk0SR is independent of the SNR γ, resulting in a positive value that is upper bounded by 1. Two cases can be considered depending on the fact that the bit from the relay S-Rk-D is detected correctly or incorrectly. Hence, PbBDFk0 can be written as in Eq. (19) under the assumption that ISRk > ISD, but substituting the corresponding coefficients aDFT, bDFT and cDFT by the corresponding coefficients aBDFT, bBDFT and cBDFT, respectively. Therefore, corresponding expressions for coefficients aBDFT, bBDFT and cBDFT are given by
aBDFT=aSDaRkΓ(bSD)Γ(bRkD)2bRkDΓ(bSD+bRkD),bBDFT=bSD+bRkD,cBDFTaBDFT=2aRkDbSDcSD+aSDbRkDcRkD2aSDaRkD(bSD+bRkD).
Finally, when the bit is detected incorrectly, the BER can be written as
PbBDFk1002i2fISD(i1)fIRkD(i2)di1di2.
Finally, taking into account the asymptotic behavior previously obtained, the expression in Eq. (34) can be simplified as follows
PbSR{PbBDFmin0PbSRmin0bRminD<i=1NbSRi(1+2(bSD+i=1NbSRi)bSDk=1NbSRkPbBDFk1)PbSDSRbRminD>i=1NbSRi

Acknowledgments

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

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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, vol. 99 (SPIE press, 2001).
    [Crossref]
  3. E. J. Lee and V. W. S. Chan, “Part 1: optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun. 22(9), 1896–1906 (2004).
    [Crossref]
  4. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009).
    [Crossref]
  5. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Trans. Commun. 57(11), 3415–3424 (2009).
    [Crossref]
  6. E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
    [Crossref]
  7. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011).
    [Crossref] [PubMed]
  8. 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]
  9. A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
    [Crossref]
  10. M. Safari and M. Uysal, “Relay-assisted free-space optical communication,” IEEE Trans. Wireless Commun. 7(12), 5441–5449 (2008).
    [Crossref]
  11. M. Karimi and M. Nasiri-Kenari, “BER analysis of cooperative systems in free-space optical networks,” J. Light-waveTechnol. 27(24), 5639–5647 (2009).
    [Crossref]
  12. M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010).
    [Crossref]
  13. 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]
  14. C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013).
    [Crossref]
  15. 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]
  16. 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]
  17. M. R. Bhatnagar, “Average BER analysis of relay selection based decode-and-forward cooperative communication over Gamma-Gamma fading FSO links,” in Communications (ICC), 2013 IEEE International Conference on, pp. 3142–3147 (IEEE, 2013).
    [Crossref]
  18. 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]
  19. J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).
  20. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Improved BDF Relaying Scheme Using Time Diversity over Atmospheric Turbulence and Misalignment Fading Channels,” The Scientific World Journal 2014213834 (2014).
    [Crossref]
  21. R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
    [Crossref]
  22. M. A. Kashani, M. Safari, and M. Uysal, “Optimal relay placement and diversity analysis of relay-assisted free-space optical communication systems,” J. Opt. Commun. Netw. 5(1), 37–47 (2013).
    [Crossref]
  23. A. Jabeena, T. Jayabarathi, P. R. Gupta, G. Hazarika, and et al., “Cooperative Wireless Optical communication system using IWO based optimal relay placement,” in Advances in Electrical Engineering (ICAEE), 2014 International Conference on, pp. 1–6 (IEEE, 2014).
    [Crossref]
  24. 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]
  25. S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. A. Neifeld, “Information Theoretic Limits for Free-Space Optical Channels With and Without Memory,” J. Lightwave Technol. 26(19), 3376–3384 (2008).
    [Crossref]
  26. N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” Communications, IEEE Transactions on 57(12), 3682–3690 (2009).
    [Crossref]
  27. I. B. Djordjevic and G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17(20), 18250–18262 (2009).
    [Crossref] [PubMed]
  28. 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).
  29. 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]
  30. 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]
  31. 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]
  32. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007).
  33. Y. Dhungana and C. Tellambura, “New simple approximations for error probability and outage in fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
    [Crossref]
  34. 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]
  35. 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]
  36. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 2: Special Functions (CRC Press, 1986).

2014 (4)

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
[Crossref]

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]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Improved BDF Relaying Scheme Using Time Diversity over Atmospheric Turbulence and Misalignment Fading Channels,” The Scientific World Journal 2014213834 (2014).
[Crossref]

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
[Crossref]

2013 (4)

2012 (3)

2011 (1)

2010 (3)

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

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

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]

2009 (7)

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]

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” Communications, IEEE Transactions on 57(12), 3682–3690 (2009).
[Crossref]

I. B. Djordjevic and G. T. Djordjevic, “On the communication over strong atmospheric turbulence channels by adaptive modulation and coding,” Opt. Express 17(20), 18250–18262 (2009).
[Crossref] [PubMed]

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

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

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

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

2008 (2)

2007 (1)

2006 (1)

2004 (1)

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

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, vol. 99 (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]

Anguita, J.

Bayaki, E.

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

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

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 Communications (ICC), 2013 IEEE International Conference on, pp. 3142–3147 (IEEE, 2013).
[Crossref]

Boluda-Ruiz, R.

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
[Crossref]

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
[Crossref]

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]

Borah, D. K.

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 2: Special Functions (CRC Press, 1986).

Castillo-Vazquez, B.

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
[Crossref]

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]

Castillo-Vazquez, C.

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
[Crossref]

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]

Castillo-Vázquez, B.

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Improved BDF Relaying Scheme Using Time Diversity over Atmospheric Turbulence and Misalignment Fading Channels,” The Scientific World Journal 2014213834 (2014).
[Crossref]

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

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011).
[Crossref] [PubMed]

Castillo-Vázquez, C.

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Improved BDF Relaying Scheme Using Time Diversity over Atmospheric Turbulence and Misalignment Fading Channels,” The Scientific World Journal 2014213834 (2014).
[Crossref]

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

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011).
[Crossref] [PubMed]

Chan, V. W. S.

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

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

Chatzidiamantis, N. D.

Chen, M.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Cheng, J.

Denic, S. Z.

Dhungana, Y.

Y. Dhungana and C. Tellambura, “New simple approximations for error probability and outage in fading,” IEEE Commun. Lett. 16(11), 1760–1763 (2012).
[Crossref]

Djordjevic, G. T.

Djordjevic, I.

Djordjevic, I. B.

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.

R. Boluda-Ruiz, A. Garcia-Zambrana, C. Castillo-Vazquez, and B. Castillo-Vazquez, “Adaptive selective relaying in cooperative free-space optical systems over atmospheric turbulence and misalignment fading channels,” Opt. Express 22(13), 16,629–16,644 (2014).
[Crossref]

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]

García-Zambrana, A.

A. García-Zambrana, R. Boluda-Ruiz, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Transmit alternate laser selection with time diversity for FSO communications,” Opt. Express 22(20), 23,861–23,874 (2014).
[Crossref]

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Improved BDF Relaying Scheme Using Time Diversity over Atmospheric Turbulence and Misalignment Fading Channels,” The Scientific World Journal 2014213834 (2014).
[Crossref]

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

A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Outage performance of MIMO FSO links over strong turbulence and misalignment fading channels,” Opt. Express 19(14), 13480–13496 (2011).
[Crossref] [PubMed]

Gradshteyn, I. S.

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

Guan, R.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Guillen i Fabregas, A.

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” Communications, IEEE Transactions on 57(12), 3682–3690 (2009).
[Crossref]

Gupta, P. R.

A. Jabeena, T. Jayabarathi, P. R. Gupta, G. Hazarika, and et al., “Cooperative Wireless Optical communication system using IWO based optimal relay placement,” in Advances in Electrical Engineering (ICAEE), 2014 International Conference on, pp. 1–6 (IEEE, 2014).
[Crossref]

Hazarika, G.

A. Jabeena, T. Jayabarathi, P. R. Gupta, G. Hazarika, and et al., “Cooperative Wireless Optical communication system using IWO based optimal relay placement,” in Advances in Electrical Engineering (ICAEE), 2014 International Conference on, pp. 1–6 (IEEE, 2014).
[Crossref]

Hopen, C.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE press, 2001).
[Crossref]

Hranilovic, S.

Hu, Q.-S.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

Huang, N.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

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]

Jabeena, A.

A. Jabeena, T. Jayabarathi, P. R. Gupta, G. Hazarika, and et al., “Cooperative Wireless Optical communication system using IWO based optimal relay placement,” in Advances in Electrical Engineering (ICAEE), 2014 International Conference on, pp. 1–6 (IEEE, 2014).
[Crossref]

Jayabarathi, T.

A. Jabeena, T. Jayabarathi, P. R. Gupta, G. Hazarika, and et al., “Cooperative Wireless Optical communication system using IWO based optimal relay placement,” in Advances in Electrical Engineering (ICAEE), 2014 International Conference on, pp. 1–6 (IEEE, 2014).
[Crossref]

Jia, L.

J.-Y. Wang, J.-B. Wang, M. Chen, Q.-S. Hu, N. Huang, R. Guan, and L. Jia, “Free-space optical communications using all-optical relays over weak turbulence channels with pointing errors,” in Wireless Communications & Signal Processing (WCSP), 2013 International Conference on, pp. 1–6 (IEEE, 2013).

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-waveTechnol. 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.

Lee, E. J.

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

Letzepis, N.

N. Letzepis and A. Guillen i Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” Communications, IEEE Transactions on 57(12), 3682–3690 (2009).
[Crossref]

Mallik, R. K.

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

Marichev, O. I.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 2: Special Functions (CRC Press, 1986).

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-waveTechnol. 27(24), 5639–5647 (2009).
[Crossref]

Neifeld, M. A.

Phillips, R.

L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, vol. 99 (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]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series Volume 2: Special Functions (CRC Press, 1986).

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.

Sandalidis, H. G.

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

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]

Schober, R.

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]

E. Bayaki and R. Schober, “On space-time coding for free-space optical systems,” IEEE Trans. Commun. 58(1), 58–62 (2010).
[Crossref]

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

Tellambura, C.

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Figures (7)

Fig. 1
Fig. 1 Diagram showing the cooperative FSO communications system, where dSD is the source-destination link distance, Rk are the relay nodes for k = {1,2,...,N}, and (dSRk,θRk) represents the relay placement using polar coordinates.
Fig. 2
Fig. 2 Diversity order gain for a source-destination link distance of dSD=3 km when different weather conditions are assumed. Different relay locations are assumed together with values of normalized beam width and normalized jitter of (ωz/r,σs/r) = (7,1).
Fig. 3
Fig. 3 Maximum source-relay link distance when different weather conditions and different maximum relay-destination link distances are assumed, once the condition φ2 > β is satisfied for each link.
Fig. 4
Fig. 4 BER performance for a source-destination link distance of dSD=3 km when different number of relays are assumed. Different relay placement are assumed together with values of normalized beam width and normalized jitter of (ωz/r,σs/r) =(7,1) and (ωz/r,σs/r)=(10,1).
Fig. 5
Fig. 5 Diversity order gain for a source-destination link distance of dSD=3 km when values of normalized beam width of ωz/r = 7 and normalized jitter of σs/r = {1, 1.5, 2, 2.5, 3} are considered. Moderate turbulence conditions are assumed.
Fig. 6
Fig. 6 Optimum normalized beam width versus normalized jitter for a source-destination link distance of dSD = 3 km under moderate turbulence conditions.
Fig. 7
Fig. 7 BER performance when different number of relays N={2, 3} and source-destination link distance of dSD=3 km are assumed when a normalized beam width of ωz/r= 7 and normalized jitter of σs/r={1, 2, 3} are considered under moderate turbulence conditions, as well as the FSO scenario without pointing errors.

Equations (41)

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y m ( t ) = η i m ( t ) x m ( t ) + z m ( t ) ,
f I m ( i ) a m i b m 1 e i c m a m .
f I m ( i ) { 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 e i α m β m ( φ m 2 β m ) ( A 0 L m ) 1 ( α m β m 1 ) ( β m β m φ m 2 + 1 ) , φ m 2 > β m 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
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
Y 0 = X I SD + Z SD , X { 0 , d E } , Z SD ~ N ( 0 , N 0 / 2 ) I R k D < I SD
Y 1 = 1 2 X I SD + X * I R k D + Z SD + Z R k D , Z R k D ~ N ( 0 , N 0 / 2 ) I R k D > I SD
P b SD RD ( I SD ) = Q ( d E 2 i 2 / 2 N 0 ) = Q ( 2 γ ξ i ) ,
P b SD RD 0 Q ( 2 γ ξ i ) j = 1 N F I R j D ( i ) f I SD ( i ) d i .
F I m ( i ) a m b m i b m e i c m b m a m ( b m + 1 ) ,
P b SD RD a RD T Γ ( ( b RD T + 1 ) / 2 ) ( γ ξ ) b RD T / 2 2 b RD T π F 2 2 ( b RD T 2 , b RD T + 1 2 ; b RD T + 2 2 , 1 2 ; c RD T 2 4 a RD T 2 γ ξ ) + c RD T Γ ( ( b RD T + 2 ) / 2 ) ( γ ξ ) ( b RD T + 1 ) / 2 2 ( b RD T + 1 ) π F 2 2 ( b RD T + 1 2 , b RD T + 2 2 ; b RD T + 3 2 , 3 2 ; c RD T 2 4 α RD T 2 γ ξ ) ,
a RD T = a SD i = 1 N a R i D i = 1 N b R i D , b RD T = b SD + i = 1 N b R i D , c RD T a RD T = c SD a SD + i = 1 N c R i D b R i D a R i D ( b R i D + 1 ) .
P b DL = 0 Q ( 2 γ ξ i ) f I SD ( i ) d i .
P b DL a SD Γ ( ( b SD + 1 ) / 2 ) ( γ ξ ) b SD / 2 2 b SD π F 2 2 ( b SD 2 , b SD + 1 2 ; b SD + 2 2 , 1 2 ; c SD 2 4 a SD 2 γ ξ ) + c SD Γ ( ( b SD + 2 ) / 2 ) ( γ ξ ) ( b SD + 1 ) / 2 ( b SD + 1 ) π F 2 2 ( b SD + 1 2 , b SD + 2 2 ; b SD + 3 2 , 3 2 ; c SD 2 4 α SD 2 γ ξ ) .
Y 1 = 1 2 X ( I SD + 2 I R k D ) + Z SD + Z R k D , X * = X
P b DF k 0 ( I SD , I R k D ) = Q ( γ ξ 4 ( i 1 + 2 i 2 ) ) ,
P b DF k 0 = 0 0 Q ( γ ξ 4 ( i i + 2 i 2 ) ) f I SD ( i 1 ) F I SD ( i 2 ) j = 1 j k N F I R j D ( i 2 ) f I R k D ( i 2 ) d i 1 d i 2 .
f I RD max ( i ) a SD b R k D j = 1 N a R j D b SD j = 1 N b R j D i b SD + j = 1 N b R j D 1 e i ( c R k D a R k D + c SD b SD a SD ( b SD + 1 ) + j = 1 j k N c R j D b R j D a R j D ( b R j D + 1 ) ) .
f I DF T a SD a RD max Γ ( b SD ) Γ ( b RD max ) 2 b RD max Γ ( b SD + b RD max ) i b SD + b RD max 1 e i ( 2 a RD max b SD c SD + a SD b RD max c RD max 2 a SD a RD max ( b SD + b RD max ) ) .
P b DF k 0 = 0 Q ( γ ξ 4 i ) f I DF T ( i ) d i .
P b DF k 0 a DF T Γ ( ( b DF T + 1 ) / 2 ) ( γ ξ ) b DF T / 2 2 ( 2 3 b DF T ) / 2 b DF T π F 2 2 ( b DF T 2 , b DF T + 1 2 ; b DF T + 2 2 , 1 2 ; 2 c DF T 2 a DF T 2 γ ξ ) + c DF T Γ ( ( b DF T + 2 ) / 2 ) ( γ ξ ) ( b D F T + 1 ) / 2 2 ( 3 b DF T + 1 ) / 2 ( b DF T + 1 ) π F 2 2 ( b DF T + 1 2 , b DF T + 2 2 ; b DF T + 3 2 , 3 2 ; 2 c DF T 2 a DF T 2 γ ξ ) .
Y 1 = 1 2 X ( I SD 2 I R k D ) + d E I R k D + Z SD + Z R k D , X * = d E X
P b DF k 1 ( I SD , I R k D ) = Q ( γ ξ 4 ( i 1 2 i 2 ) ) .
P b DF k 1 = 0 0 Q ( γ ξ 4 ( i 1 2 i 2 ) ) f I SD ( i 1 ) F I SD ( i 2 ) j = 1 j k N F I R j D ( i 2 ) f I R k D ( i 2 ) d i 1 d i 2 .
P b DF k 1 = ˙ 0 0 2 i 2 f I SD ( i 1 ) F I SD ( i 2 ) j = 1 j k N F I R j D ( i 2 ) f I R k D ( i 2 ) d i 1 d i 2 .
P b SR k RD Γ ( ( b SR k + 1 ) / 2 ) ( γ ξ ) b SR k / 2 ( a SR k ) 1 2 ( 1 b SR k ) b SR k π F 2 2 ( b SR k 2 , b SR k + 1 2 ; b SR k + 2 2 , 1 2 ; c SR k 2 a SR k 2 γ ξ ) + Γ ( ( b SR k + 2 ) / 2 ) ( γ ξ ) ( b SR k + 1 ) / 2 ( c SR k ) 1 2 b SR k ( b SR k + 1 ) π F 2 2 ( b SR k + 1 2 , b SR k + 2 2 ; b SR k + 3 2 , 3 2 ; c SR k 2 a SR k 2 γ ξ ) ,
P b RD = P b SD RD + k = 1 N P b DF k 0 ( 1 P b SR k RD ) + k = 1 N P b DF k 1 P b SR k RD .
P b RD = ˙ { P b DF min 1 P b SR min RD , b SR min < b SD + i = 1 N b R i D P b SD RD , b SR min > b SD + i = 1 N b R i D
G d RD = min ( b SR min , b SD + i = 1 N b R i D ) / b SD .
G d SR = 1 + min ( b R min D , i = 1 N b SR i ) / b SD .
G d = { G d RD , b SR min > b SD + b R min D G d SR , b SR min < b SD + b R min D
d SR max ( km ) = ( 1 1.23 C n 2 κ 7 / 6 ) 6 / 11 ( ln 6 / 11 ( 1 + 1 β SD + β R min D ) ( 0.44 0.69 ln 6 / 5 ( 1 + 1 β SD + β R min D ) ) 5 / 11 ) × 10 3 ,
ω z / r optimum 4.013 σ s / r 0.153 .
D pe [ dB ] ( 20 / β SD ) log 10 ( φ SD 2 / ( A 0 β SD ( φ SD 2 β SD ) ) ) .
P b SR = P b SD SR + k = 1 N P b BDF k 0 P b SR k 0 SR + k = 1 N P b BDF k 1 P b SR k 1 SR .
a SR T = a SD i = 1 N a SR i i = 1 N b SR i , b SR T = b SD + i = 1 N b SR i , c SR T a SR T = c SD a SD + i = 1 N c SR i b SR i a SR i ( b SR i + 1 ) .
P b SR k 1 SR ( b SR k / b SD ) 2 ( b SD + i = 1 N b SR i ) P b SD SR .
P b SR k 0 0 j = 1 j k N F I SR j ( i ) F I SD ( i ) f I SR k ( i ) d i .
a BDF T = a SD a R k Γ ( b SD ) Γ ( b R k D ) 2 b R k D Γ ( b SD + b R k D ) , b BDF T = b SD + b R k D , c BDF T a BDF T = 2 a R k D b SD c SD + a SD b R k D c R k D 2 a SD a R k D ( b SD + b R k D ) .
P b BDF k 1 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 SR { P b BDF min 0 P b SR min 0 b R min D < i = 1 N b SR i ( 1 + 2 ( b SD + i = 1 N b SR i ) b SD k = 1 N b SR k P b BDF k 1 ) P b SD SR b R min D > i = 1 N b SR i

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