Ergodic capacity analysis for DF strategies in cooperative FSO systems

Rubén Boluda-Ruiz; Antonio García-Zambrana; Beatriz Castillo-Vázquez; Carmen Castillo-Vázquez

doi:10.1364/OE.23.021565

1. Introduction

Free-space optical (FSO) systems have gained a significant popularity within the research community due to the fact that FSO links have a very high optical bandwidth available, allowing much higher data rates. These systems are used for high rate communication between two fixed points over distances up to several kilometers. This technology is not without drawbacks despite the major advantages of FSO technology such as unlicensed spectrum and an excellent security and, a wide range of applications such as metropolitan area network (MAN) extension as well as local area network (LAN)-to-LAN connectivity, among other applications [1]. On the one hand, atmospheric turbulence is one of the most important impairments and can affect the FSO link performance [2]. This phenomenon is the result of random variations in the refractive index along the optical path. This causes fluctuations in both the intensity and the phase of the received signal. On the other hand, an unsuitable alignment between transmitter and receiver can also degrade the performance, being mainly caused by building motion due to wind as well as differential heating and cooling [3]. In order to improve the performance in this turbulence FSO scenario, the adoption of cooperative communications has demonstrated that it is possible to achieve spatial diversity by creating diversity using the transceivers available at the other nodes of the network [4–15].

Much research over the last decade has focused on the study of the ergodic capacity for FSO communication links [16–23]. Ergodic capacity, also known as average channel capacity, defines the maximum data rate that can be sent over the channel with asymptotically small error probability, without any delay or complexity constraints [24]. It is commonly known that the atmospheric turbulence channels can be well described as slow fading or block fading channels and, hence, outage capacity becomes a more realistic measure of channel capacity than ergodic capacity in FSO systems. However, ergodic capacity can be perfectly applied to FSO links when the following aspects are taken into account. The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fast fading statistics. In spite of scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, this approach is valid because temporal correlation can in practice be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [16,25–28]. This assumption has to be considered like an ideal scenario where the latency introduced by the interleaver is not an inconvenience for the required application, being interpreted the results so obtained as upper bounds on the system performance. In addition to this, the obtained results in ergodic capacity are also applicable to slowly varying (block-fading) channels, i.e. FSO links, when the message is long enough to reveal long-term ergodic properties of the turbulence process [29]. This fact was taken into account in [30] in the context of FSO communication systems. Ergodic capacity analysis is attracting widespread interest in cooperative FSO systems. Few researchers have addressed the study of the ergodic capacity in the context of cooperative FSO communication systems [31–36]. In [31], a cooperative FSO system is analyzed based on users selection, where the best user is selected, being obtained an approximate expression for the ergodic capacity over log-normal and gamma-gamma (GG) fading channels. In [32], the end-to-end ergodic capacity of dual-hop FSO system employing amplify-and-forward (AF) relaying is evaluated over gamma-gamma fading channels with pointing errors, by approximating the probability density function (PDF) of the end-to-end signal to noise ratio (SNR), by the α−μ distribution. In [33], the analytical channel capacity of the decode-and-forward (DF) based dual-hop FSO system is studied under gamma-gamma fading channels with pointing errors. In [34,35], the capacity performance of dual-hop subcarrier intensity modulation (SIM)-based FSO system with DF and AF relay is evaluated, respectively, over gamma-gamma fading channels with pointing errors. The derived results are obtained in terms of special function known as generalized bivariate Meijers G-function (GBMGF). In [36] a dual-hop fixed and variable gain relay system over the asymmetric links composed of both Nakagami-m and unified gamma-gamma fading is studied, wherein the closed-form expression for the ergodic capacity is also presented in terms of the GBMGF.

The purpose of this paper is to study and comprehend the ergodic capacity in the context of cooperative FSO systems when the line of sight is available. Motivated by the fact that the performance of cooperative FSO systems has been widely analyzed in terms of the bit error-rate (BER) and outage probability, we focus on the study of the ergodic capacity for two different DF strategies over atmospheric turbulence channels with pointing errors. In this way, novel closed-form approximate ergodic capacity expressions are obtained for a 3-way FSO communication system using the well-known inequality between arithmetic and geometric mean of positive random variables (RV) in order to derive an approximate closed-form expression of the sum of gamma-gamma with pointing errors variates. The derived results are obtained in terms of Meijer’s G-function as well as GBMGF. In addition, we present based on this analysis novel asymptotic expressions at high SNR for the ergodic capacity of DF strategy as well as adaptive DF strategy. Here, the atmospheric turbulence is modeled by a gamma-gamma distribution of parameters α and β, which allows to study a wide range of turbulence conditions (moderate-to-strong) as well as the effect of the misalignment with zero boresight. It can be concluded that a greater and robust capacity strongly dependent on the relay location is achieved compared to a direct transmission without cooperative communication when line of sight is available. Unlike [33, 34], here the line of sight is taken into account in order to achieve a significant robustness under different turbulence conditions and more severe pointing errors regardless of the relay location. Moreover, it can be also concluded that the adaptive DF strategy is able to achieve a better performance in terms of the ergodic capacity than the DF strategy regardless of the relay location and pointing errors effect.

2. System and channel model

Following a generic DF strategy in order to study the ergodic capacity, we adopt a three-node cooperative system based on three separate FSO links, assuming laser sources intensity-modulated and ideal non-coherent (direct-detection) receivers, as shown in Fig. 1, wherein different relay-destination link distances are considered in order to perform a more realistic approach in the context of cooperative FSO systems. Ergodic capacity is analyzed for two different DF strategies, which have been explained in more detail in [7, 13, 14], but both of them are briefly here explained for convenience. The bit detect-and-forward (BDF) cooperative protocol works in two phases. In the first phase, the source node S sends its own data to the relay node R and the destination node D. In the second phase, the relay node R sends the received data from the source node S in the first phase to the destination node D. In this fashion, the relay node R detects each code bit to “0” or “1” and sends the bit with the new power to the destination node D regardless of these bits are detected correctly or incorrectly. The adaptive detect-and-forward (ADF) cooperative protocol is similar to the BDF cooperative protocol. ADF selects between direct transmission or BDF on the basis of the value of the fading gain. When the irradiance of the source-destination link (I_SD) is greater than the irradiance corresponding to the relay-destination link (I_RD), 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 R if the irradiance of the relay-destination (R-D) link is greater than the irradiance corresponding to the source-destination (S-D) link. It is taking into account that channel side information (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 [30, 37]. In this sense, the receiver always knows if the cooperative protocol is being used. 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 without using any cooperative strategy.

Fig. 1 Block diagram of the considered 3-way FSO communication system, where d_SD is the S-D link distance and (x_R, y_R) represents the location of the node R.

Download Full Size | PDF

The received electrical signal for each link is given by Y_m = XI_m + Z_m, where X is the binary transmitted signal, I_m is the equivalent real-value fading gain (irradiance) through the optical channel, and Z_m is additive white Gaussian noise (AWGN) with zero mean and variance σ² = N₀/2, i.e. Z_m ∼ N(0, N₀/2), independent of the on/off state of the received bit. Here, on-off keying (OOK) modulation scheme is used, X is either 0 or $2 P_{opt} \sqrt{T_{b} ξ}$ , where P_opt is the average transmitted optical power from each node, T_b is the bit period, and ξ represents the square of the increment due to the use of a pulse shape of high peak-to-average optical power ratio (PAOPR). The received instantaneous electrical SNR, γ, can be written as in [38], as

γ = \frac{1}{2} \frac{d_{E}^{2} I_{m}^{2}}{σ_{m}^{2}} = \frac{4 P_{opt}^{2} T_{b} ξ I_{m}^{2}}{N_{0}} = 4 γ_{0} ξ I_{m}^{2},

where d_E is the Euclidean distance and γ₀ represents the received electrical SNR in absence of turbulence when the classical rectangular pulse shape is adopted for OOK formats. The irradiance I_m is considered to be a product of three factors i.e.

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

, where L_m is the deterministic propagation loss,

I_{m}^{(a)}

is the attenuation due to atmospheric turbulence and

I_{m}^{(p)}

the attenuation due to geometric spread and pointing errors. L_m is determined by the exponential Beers-Lambert law as L_m = e^–Φd_m, where d_m 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) [39]. To consider a wide range of turbulence conditions (moderate to strong), the gamma-gamma turbulence model proposed in [2, 40] is assumed, whose PDF depends on the parameter α and β. Assuming plane wave propagation and negligible inner scale, α and β can be directly linked to physical parameters through the following expressions [40, 41]:

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},

where

σ_{R}^{2} = 1.23 C_{n}^{2} κ^{7 / 6} d^{11 / 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.

C_{n}^{2}

is the refractive index structure parameter, which is the most significant parameter that determines the turbulence strength. Clearly,

C_{n}^{2}

not only depends on the altitude, but also on the local conditions such as terrain type, geographic location, cloud cover, and time of day [3]. In addition,

C_{n}^{2}

is typically within the range 10⁻¹³−10⁻¹⁷ m^−2/3 [2]. Regarding the impact of pointing errors, we use the general model of misalignment fading given in [38], wherein the effect of beam width, detector size and jitter variance is considered. Assuming a Gaussian spatial intensity profile of beam waist radius, ω_z, on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, φ = ω_{z_eq}/2σ_s is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver,

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

,

v = \sqrt{π} r / \sqrt{2} ω_{z}

, A₀ = [erf(v)]² and erf(·) is the error function [42, eqn. (8.250)]. A closed-form expression of the combined PDF of I_m was derived in [43] as

f_{I_{m}} (i) = \frac{φ_{m}^{2} i^{- 1}}{Γ (α_{m}) Γ (β_{m})} G_{1, 3}^{3, 0} (\begin{array}{l} \frac{α_{m} β_{m} i}{A_{m} L_{m}} | & \begin{matrix} φ_{m}^{2} + 1 \\ φ_{m}^{2}, α_{m}, β_{m} \end{matrix} \end{array}),

where

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

is the Meijer’s G-function [42, eqn. (9.301)]. Moreover, a closed-form expression of the cumulative density function (CDF) was also obtained in [43] in terms of the Meijer’s G-function as

F_{I_{m}} (i) = \frac{φ_{m}^{2}}{Γ (α_{m}) Γ (β_{m})} G_{2, 4}^{3, 1} (\frac{α_{m} β_{m} i}{A_{m} L_{m}} | \begin{matrix} 1, φ_{m}^{2} + 1 \\ φ_{m}^{2}, α_{m}, β_{m}, 0 \end{matrix}) .

In the following section, the fading coefficient I_m for the paths S-D, source-relay (S-R) and R-D is indicated by I_SD, I_SR and I_RD, respectively. Here, we assume that all coefficients are statistically independent.

3. Ergodic capacity analysis

In this section, we analyze the ergodic capacity of the cooperative FSO system under study. Firstly, we study the ergodic capacity corresponding to the DF strategy following the BDF cooperative protocol analyzed in [7], depending on the fact that the bit from the relay S-R-D is detected correctly or incorrectly. Assuming a statistical channel model as follows

Y_{BDF} = \frac{1}{2} X I_{SD} + Z_{SD} + X^{*} I_{RD} + Z_{RD}, X \in {0, d_{E}}, Z_{SD}, Z_{RD} ~ N (0, N_{0} / 2)

where X^* represents the random variable corresponding to the information detected at node R and, hence, X^*= X when the bit has been detected correctly at node R and, X^* = d_E − X when the bit has been detected incorrectly. I_SD and I_RD are irradiances corresponding to the S-D and R-D links, respectively and, Z_SD and Z_RD are AWGN corresponding to the S-D and R-D links, respectively. The division by 2 in Eq. (5) is considered so as to maintain the average optical power in the air at a constant level of P_opt, being transmitted by each laser at source node an average optical power P_opt/2. Assuming channel side information at the receiver (CSIR), the ergodic capacity corresponding to the BDF cooperative protocol is given by

C_{BDF} = C_{0} \cdot (1 - P_{b}^{SR}) + C_{1} \cdot P_{b}^{SR} = C_{0} - (C_{0} - C_{1}) \cdot P_{b}^{SR},

where

P_{b}^{SR}

denotes the BER corresponding to the S-R link and, C₀ and C₁ are the ergodic capacity when the bit is correctly and incorrectly detected at node R, respectively. The resulting received electrical SNR when X^*= X can be defined as

γ_{BDF}^{0} = \frac{γ_{0} ξ}{2} {(I_{SD} + 2 I_{RD})}^{2}

and, when X^* = d_E − X, the resulting received electrical SNR can be defined as

γ_{BDF}^{1} = \frac{γ_{0} ξ}{2} {(I_{SD} - 2 I_{RD})}^{2}

. It must be noted that the ergodic capacity corresponding to BDF relaying scheme in Eq. (6) can be accurately approximated as follows

C_{BDF} \approx C_{0} = \frac{B}{2 ln (2)} \int_{0}^{\infty} \int_{0}^{\infty} ln (1 + \frac{γ_{0} ξ}{2} {(i_{1} + 2 i_{2})}^{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2},

where B is the channel bandwidth, ln(·) is the natural logarithm [42, eqn. (1.511)], and f_{I_m}(i) is the corresponding PDF given by Eq. (3). It should be noted that the factor 1/2 in Eq. (7) is because the source node S is assumed to operate in half-duplex mode. It must be noted that this approximation is completely valid as SNR increases, since the term

P_{b}^{SR}

tends to zero as SNR increases. In addition, the expression in Eq. (7) is not depend on the modulation schemes due to the fact that the term

P_{b}^{SR}

tends to zero fast when the simplest modulation scheme is implemented, i.e. OOK format. Moreover, this approximation has been numerically corroborated by Monte Carlo simulation and, it will be checked in following sections. Hence, the ergodic capacity of BDF cooperative protocol can be written as

C_{BDF} \approx \frac{B}{ln (4)} \int_{0}^{\infty} ln (1 + \frac{γ_{0} ξ}{2} i^{2}) f_{I_{T}} (i) d i,

where I_T = I_SD + 2I_RD. It should be noted that obtaining the corresponding PDF of I_T is remarkably tedious and not easily tractable due to the difficulty in finding its statistics. Hence, a lower bound (LB) for the sum I_T can be obtained by using the well-known inequality between arithmetic mean (AM) and geometric mean (GM) given by

AM \geq GM,

where AM = (I_SD + 2I_RD)/2 and

GM = \sqrt{I_{SD} \cdot 2 I_{RD}}

are the arithmetic and geometric means, respectively. The equality AM = GM holds when 𝔼[I_SD] = 2𝔼[I_RD], wherein 𝔼[·] denotes the expectation operator. Therefore, a lower bound for the sum I_T can be obtained as

I_{SD} + 2 I_{RD} \geq \sqrt{8 \cdot I_{SD} \cdot I_{RD}} = \sqrt{8 \cdot I_{T}^{LB}} .

It should be noted that

f_{I_{T}^{LB}} (i)

is mathematically more tractable than f_{I_T} (i) and can be efficiently applied to the analysis of FSO communication systems. From Eq. (10), it can be easily deduced that the mathematical expectation of both sides of inequality take different values and, hence, a correcting factor F should be added to the inequality in order to maintain the same value in both sides. Taking into account this fact, the correcting factor F is added to Eq. (10) in order to obtain a strict approximation on the ergodic capacity analysis. This correcting factor can be derived from Eq. (10) when both averages hold and, it can be seen in greater detail in appendix A. The closed-form expression of the correcting factor F can be seen in Eq. (44). Substituting Eq. (10) into Eq. (8) and, after performing some algebraic manipulations, the ergodic capacity for BDF relaying scheme here analyzed can be accurately approximated as follows

C_{BDF} \leq \frac{B}{ln (4)} \int_{0}^{\infty} ln (1 + 4 γ_{0} ξ Fi) f_{I_{T}^{LB}} (i) d i .

The PDF

f_{I_{T}^{LB}} (i)

can be derived in closed-form via inverse Mellin transform, which is an essential tool in studying the distribution of products and quotients of independent random variables. Due to irradiances I_SD and I_RD are independent random variables, the Mellin transform of their product is equal to the product of the Mellin transforms of I_SD and I_RD [44]. Hence, the closed-form solution for the PDF corresponding to

I_{T}^{LB}

can be expressed in terms of the Meijer G-function by employing the definition of the Mellin transform [42, eqn. (17.41)] as follows

f_{I_{T}^{LB}} (i) = \frac{φ_{SD}^{2} φ_{RD}^{2} i^{- 1} G_{2, 6}^{6, 0} (\begin{array}{l} \frac{α_{SD} β_{SD} α_{RD} β_{RD}}{A_{SD} L_{SD} A_{RD} L_{RD}} i | & \begin{matrix} φ_{SD}^{2} + 1, φ_{RD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, φ_{RD}^{2}, α_{RD}, β_{RD} \end{matrix} \end{array})}{Γ (α_{SD}) Γ (β_{SD}) Γ (α_{RD}) Γ (β_{RD})} .

The integral in Eq. (11) can be solved using [45, eqn. (8.4.6.5)] in order to express the natural logarithm in terms of the Meijer’s G-function as

ln (1 + z) = G_{2, 2}^{1, 2} ({z |}_{1, 0}^{1, 1})

and, then, using [45, eqn. (2.24.1.2)], we can obtain the approximate closed-form solution for the ergodic capacity corresponding to the BDF cooperative protocol, C_BDF, as follows

C_{BDF} \approx \frac{β φ_{SD}^{2} φ_{RD}^{2} G_{4, 8}^{8, 1} (\begin{array}{l} \frac{α_{SD} β_{SD} α_{RD} β_{RD}}{γ_{0} ξ 4 F A_{SD} L_{SD} A_{RD} L_{RD}} | & \begin{matrix} 0, 1, φ_{SD}^{2} + 1, φ_{RD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, φ_{RD}^{2}, α_{RD}, β_{RD}, 0, 0 \end{matrix} \end{array})}{ln (4) Γ (α_{SD}) Γ (β_{SD}) Γ (α_{RD}) Γ (β_{RD})} .

At this point, it should be mentioned that an asymptotic ergodic expression can be derived from Eq. (13) by using the corresponding series expansion of the Meijer’s G-function [46, eqn. (07.34.06.0006.01)]. However, this ergodic capacity analysis might not result in a closed-form expression and, hence, we cannot always obtain an asymptotic expression from its corresponding closed-form expression. Within this context, an asymptotic expression for the ergodic capacity corresponding to the BDF cooperative protocol at high SNR can be readily and accurately lower-bounded due to the fact that ln(1 + z) ≈ ln(z) when z → ∞ as follows

C_{BDF} ≐ \frac{B ln (4 γ_{0} ξ F)}{ln (4)} + \frac{B}{ln (4)} \int_{0}^{\infty} ln (i) f_{I_{T}^{LB}} (i) d i .

To evaluate the integral in Eq. (14), we can express the natural logarithm as a subtraction of two Meijer’s G-function by adding redundancy as

ln (z) = \frac{z ln (z)}{z - 1} - \frac{ln (z)}{z - 1} .

In addition, we can use the fact that the natural logarithm is related to the Meijer’s G-function by

\frac{ln (z)}{z - 1} = G_{2, 2}^{2, 2} ({z |}_{0, 0}^{0, 0})

[46, eqn. (01.04.26.0004.01)], therefore, the expression in Eq. (14) can be re-written as

C_{BDF} ≐ \frac{B ln (4 γ_{0} ξ F)}{ln (4)} + \frac{B}{ln (4)} (\int_{0}^{\infty} i G_{2, 2}^{2, 2} ({i |}_{0, 0}^{0, 0}) f_{I_{T}^{LB}} (i) d i - \int_{0}^{\infty} G_{2, 2}^{2, 2} ({i |}_{0, 0}^{0, 0}) f_{I_{T}^{LB}} (i) d i) .

Both integrals in Eq. (16) can be found in [45, eqn. (2.24.1.2)] and, hence, the corresponding asymptotic expression is given by

\begin{array}{l} C_{BDF}^{H} & ≐ \frac{B ln (4 γ_{0} ξ F)}{ln (4)} \\ + \frac{B φ_{SD}^{2} φ_{RD}^{2} G_{4, 8}^{8, 2} (\begin{array}{l} \frac{α_{SD} β_{SD} α_{RD} β_{RD}}{A_{SD} L_{SD} A_{RD} L_{RD}} | & \begin{matrix} 0, 0, φ_{SD}^{2} + 1, φ_{RD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, φ_{RD}^{2}, α_{RD}, β_{RD}, 0, 0 \end{matrix} \end{array})}{ln (4) Γ (α_{SD}) Γ (β_{SD}) Γ (α_{RD}) Γ (β_{RD})} \\ - \frac{B φ_{SD}^{2} φ_{RD}^{2} G_{4, 8}^{8, 2} (\begin{array}{l} \frac{α_{SD} β_{SD} α_{RD} β_{RD}}{A_{SD} L_{SD} A_{RD} L_{RD}} | & \begin{matrix} 1, 1, φ_{SD}^{2} + 1, φ_{RD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, φ_{RD}^{2}, α_{RD}, β_{RD}, 1, 1 \end{matrix} \end{array})}{ln (4) Γ (α_{SD}) Γ (β_{SD}) Γ (α_{RD}) Γ (β_{RD})} . \end{array}

As can be seen in Eq. (17), the subtraction of two Meijer’s G-function is independent of the SNR, γ, resulting in a positive value, which is dependent on the relay location. In order to simplify the expression in Eq. (17) and observe how the relay location impacts on the cooperative FSO system, the series asymptotic expansion corresponding to the Meijer’s G-function [46, eqn. (07.34.06.0002.01)] is used in Eq. (17). Therefore, the following asymptotic expression can be derived as follows

\begin{array}{l} C_{BDF}^{H} ≐ \frac{B ln (γ_{0} ξ 4 F)}{ln (4)} + \frac{B}{ln (4)} (- \frac{1}{φ_{SD}^{2}} - \frac{1}{φ_{RD}^{2}}) \\ + \frac{B}{ln (4)} (ψ (α_{SD}) + ψ (β_{SD}) + ψ (α_{RD}) + ψ (β_{RD}) + ln (\frac{A_{SD} L_{SD} A_{RD} L_{RD}}{α_{SD} β_{SD} α_{RD} β_{RD}})), \end{array}

wherein ψ(·) is the psi (digamma) function [42, eqn. (8.360.1)]. It can be corroborated that only two first terms have influence on the series expansion and, hence, the rest of the summation terms corresponding to this series expansion have been canceled as a result of the subtraction of two Meijer’s G-function. This mathematical procedure can be seen in greater detail in appendix C. It must be noted that the asymptotic expression in Eq. (18) is identical to the asymptotic expression obtained by applying the moments method, presented for the first time in [47, Eqs. (8) and (9)], wherein the asymptotic ergodic capacity can be obtained from the first derivate of the nth order moment of the sum of I_SD + 2I_RD. Next, we analyze the ergodic capacity corresponding to the adaptive DF strategy following the ADF cooperative protocol presented in [14]. The statistical channel model corresponding to the ADF cooperative protocol can be written as

Y_{BDF} = \frac{1}{2} X I_{SD} + X^{*} I_{RD} + Z_{SD} + Z_{RD}, I_{RD} > I_{SD}

Y_{DT} = X I_{SD} + Z_{SD}, I_{RD} < I_{SD}

Assuming channel side information at the receiver, the ergodic capacity corresponding to the ADF cooperative protocol is given by

\begin{array}{l} C_{ADF} & \approx \frac{B}{2 ln (2)} \int_{0}^{\infty} \int_{0}^{i_{2}} ln (1 + \frac{γ_{0} ξ}{2} {(i_{1} + 2 i_{2})}^{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2} \\ + \frac{B}{2 ln (2)} \int_{0}^{\infty} ln (1 + 4 γ_{0} ξ i^{2}) F_{I_{RD}} (i) f_{I_{SD}} (i) d i . \end{array}

Similar to BDF cooperative protocol, the well-known inequality between arithmetic mean and geometric mean is also here used in order to obtain the corresponding asymptotic expression for the ADF cooperative protocol. Hence, the expression in Eq. (20) is re-written as follows

\begin{array}{l} C_{ADF} & \approx \frac{B}{ln (4)} \int_{0}^{\infty} \int_{0}^{i_{2}} ln (1 + 4 γ_{0} ξ F^{'} i_{1} i_{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2} \\ + \frac{B}{ln (4)} \int_{0}^{\infty} ln (1 + 4 γ_{0} ξ i^{2}) F_{I_{RD}} (i) f_{I_{SD}} (i) d i = C_{BDF}^{'} + C_{SD}, \end{array}

where F′ is another correcting factor, which can be derived in a similar way to F and, it can be seen in more detail in appendix B. The closed-form expression of the correcting factor F′ can be seen in Eq. (51). Knowing the fact that the irradiance values are statistically independent, the probability corresponding to I_SD > I_RD is computed by using F_{I_RD}(I_SD), being F_{I_m}(i) the CDF of the random variable I_m, which is given by F_{I_m}(i) = Prob(I_m ≤ i). Unfortunately, an approximate closed-form solution for the integral corresponding to C′_BDF is not available and, hence, a numerical integration must be used. However, the term C_SD in Eq. (21) can be expressed in closed-form in terms of GBMGF by using [46, eqn. (07.34.21.0081.01)]. The closed-form expression corresponding to C_SD has been omitted from this paper due to the fact that it cannot be possible to find the solution of the first integral in Eq. (21). As in Eq. (14), we can obtain an asymptotic expression for the ergodic capacity corresponding to the ADF cooperative protocol at high SNR,

C_{ADF}^{H}

, after performing some algebraic manipulations in Eq. (21) as follows

\begin{array}{l} C_{ADF}^{H} ≐ \frac{B ln (4 γ_{0} ξ F^{'})}{ln (4)} \int_{0}^{\infty} F_{I_{SD}} (i) f_{I_{RD}} (i) d i + \frac{B}{ln (4)} \int_{0}^{\infty} \int_{0}^{i_{2}} ln (i_{1} i_{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2} \\ + \frac{B ln (4 γ_{0} ξ)}{ln (4)} \int_{0}^{\infty} F_{I_{RD}} (i) f_{I_{SD}} (i) d i + \frac{B}{ln (2)} \int_{0}^{\infty} ln (i) F_{I_{RD}} (i) f_{I_{SD}} (i) d i = C_{BDF}^{H^{'}} + C_{SD}^{H} . \end{array}

It is noteworthy to mention that the average probability when the source node S selects cooperative transmission instead of direct transmission can be defined as follows

P ≐ \int_{0}^{\infty} F_{I_{SD}} (i) f_{I_{RD}} (i) d i .

It can be also noted that the asymptotic behavior corresponding to P is independent of the SNR γ, resulting in a positive value that is upper bounded by 1. To evaluate Eq. (23), we can use [45, eqn. (2.24.1.2)] in order to transform the integral expression in terms of the Meijer’s G-function. Hence, a closed-form solution is obtained as can be seen in

P = \frac{φ_{RD}^{2} φ_{SD}^{2} G_{5, 4}^{4, 3} (\begin{array}{l} \frac{α_{RD} β_{RD} A_{SD} L_{SD}}{α_{SD} β_{SD} A_{RD} L_{RD}} | & \begin{matrix} 1 - φ_{SD}^{2}, 1 - α_{SD}, 1 - β_{SD}, 1, φ_{RD}^{2} + 1 \\ α_{RD}^{2}, α_{RD}, β_{RD}, 0 - φ_{SD}^{2} \end{matrix} \end{array})}{Γ (α_{RD}) Γ (α_{SD}) Γ (β_{RD}) Γ (β_{SD})} .

An asymptotic expression for the ergodic capacity corresponding to ADF cooperative protocol

C_{ADF}^{H}

is given by

C_{ADF}^{H} ≐ \frac{B (ln (4 γ_{0} ξ) + P ln (F^{'}) + I_{1} + 2 I_{2})}{ln (4)},

wherein

I_{1} = \int_{0}^{\infty} \int_{0}^{i_{2}} ln (i_{1} i_{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2},

I_{2} = \int_{0}^{\infty} ln (i) F_{I_{RD}} (i) f_{I_{SD}} (i) d i .

Unfortunately, the corresponding closed-form solution for the integral I₁ cannot be determined and as such numerical integration should be used. However, the integral I₂ can be expressed in a closed-form solution. Note that the integrand in I₂ involves the product of the three independent Meijer G-functions which can be expressed in terms of GBMGF by using [46, eqn. (07.34.21.0081.01)]. Hence, the expression of

C_{SD}^{H}

can be seen in

\begin{array}{l} I_{2} & = \frac{φ_{SD}^{2} φ_{RD}^{2}}{Γ (α_{SD}) Γ (β_{SD}) Γ (α_{RD}) Γ (β_{RD})} \\ \times G_{2, 2 : 2, 4 : 1, 3}^{2, 2 : 3, 1 : 3, 0} (\begin{matrix} \begin{matrix} 1, 1 \\ 1, 1 \end{matrix} & | \begin{matrix} 1, φ_{RD}^{2} + 1 \\ φ_{RD}^{2}, α_{RD}, β_{RD} \end{matrix} | & \begin{matrix} φ_{SD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD} \end{matrix} & | \frac{α_{RD} β_{RD}}{A_{RD} L_{RD}}, \frac{α_{SD} β_{SD}}{A_{SD} L_{SD}} \end{matrix}) . \end{array}

To the best of the authors’ knowledge, the GBMGF is not available in standard mathematical packages such as Mathematica, Maple or Matlab. However, the GBMGF in Eq. (27) can be computed in a efficient manner by using the two-fold Mellin-Barnes representation of the Meijer-G function. Similar to [48, Table II] and [49, appendix], the GBMGF was implemented using a Mathematica implementation for the numerical evaluation of Eq. (27). Both the integral I₁ and I₂ present the same asymptotic behavior, which is independent of the SNR γ. Finally, we study the ergodic capacity corresponding to the direct transmission (DT) without cooperative communication in order to establish the baseline performance. This closed-form expression was obtained in [17, 20] and, it is here reproduced for convenience. Assuming channel side information at the receiver for an average power constraint P_opt, the ergodic capacity, C_DT, can be defined as

C_{DT} = \frac{B}{2 ln (2)} \int_{0}^{\infty} ln (1 + 4 γ_{0} ξ i^{2}) f_{I_{SD}} (i) d i .

The integral in Eq. (28) can be solved using [45, eqn. (8.4.6.5)] in order to express the natural logarithm in terms of the Meijer’s G-function as in Eq. (11), and afterwards using [45, eqn. (2.24.1.1)]. Hence, the closed-form solution for the ergodic capacity corresponding to the direct transmission can be seen in

\begin{array}{l} C_{DT} = \frac{B φ_{SD}^{2} 2^{α_{SD} + β_{SD} - 4}}{π ln (2) Γ (α_{SD}) Γ (β_{SD})} \\ \times G_{8, 4}^{1, 8} (\begin{array}{l} \frac{64 A_{SD}^{2} L_{SD}^{2} γ_{0} ξ}{α_{SD}^{2} β_{SD}^{2}} | & \begin{matrix} 1, 1, \frac{1 - α_{SD}}{2}, \frac{2 - α_{SD}}{2}, \frac{1 - β_{SD}}{2}, \frac{2 - β_{SD}}{2}, \frac{1 - φ_{SD}^{2}}{2}, \frac{2 - φ_{SD}^{2}}{2} \\ 1, 0, - \frac{φ_{SD}^{2}}{2}, \frac{1 - φ_{SD}^{2}}{2} \end{matrix} \end{array}) . \end{array}

Similar to Eq. (14), an asymptotic expression for the ergodic capacity corresponding to the direct transmission at high SNR can be readily and accurately lower-bounded as follows

C_{DT} ≐ \frac{B ln (4 γ_{0} ξ)}{ln (4)} + \frac{B}{ln (2)} \int_{0}^{\infty} i G_{2, 2}^{2, 2} ({i |}_{0, 0}^{0, 0}) f_{I_{SD}} (i) d i - \frac{B}{ln (2)} \int_{0}^{\infty} G_{2, 2}^{2, 2} ({i |}_{0, 0}^{0, 0}) f_{I_{SD}} (i) d i .

Both integrals in Eq. (30) can be found in [45, eqn. (2.24.1.2)] and, hence, the corresponding asymptotic expression is given by

\begin{array}{l} C_{DT}^{H} & ≐ \frac{B ln (4 γ_{0} ξ)}{ln (4)} + \frac{B φ_{SD}^{2} G_{3, 5}^{5, 2} (\begin{array}{l} \frac{α_{SD} β_{SD}}{A_{SD} L_{SD}} | & \begin{matrix} 0, 0, φ_{SD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, 0, 0 \end{matrix} \end{array})}{ln (2) Γ (α_{SD}) Γ (β_{SD})} \\ - \frac{B φ_{SD}^{2} G_{3, 5}^{5, 2} (\begin{array}{l} \frac{α_{SD} β_{SD}}{A_{SD} L_{SD}} | & \begin{matrix} 1, 1, φ_{SD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, 1, 1 \end{matrix} \end{array})}{ln (2) Γ (α_{SD}) Γ (β_{SD})} . \end{array}

As in Eq. (18), it can be applied the same series asymptotic expansion in order to simplify the asymptotic expression in Eq. (31). Hence, the asymptotic expression for the ergodic capacity corresponding to the direct transmission can be also expressed as

C_{DT}^{H} ≐ \frac{B ln (4 γ_{0} ξ)}{ln (4)} + \frac{B}{ln (2)} (ψ (α_{SD}) + ψ (β_{SD}) - \frac{1}{φ_{SD}^{2}} - ln (\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})) .

As in Eq. (18), it can be also corroborated that only two first terms have influence on the series expansion and, hence, the rest of the summation terms corresponding to this series expansion have been canceled as a result of the subtraction of two Meijer’s G-function. This mathematical procedure can be seen in greater detail in appendix C. The asymptotic expression in Eq. (32) is identical to the asymptotic expression obtained in [22, eqn. (24)] by applying the moments method. From this ergodic capacity analysis, it can be deduced that the main aspect to consider in order to optimize the ergodic capacity is the relay location as well as channel parameters. Both aspects play an important role on ergodic capacity analysis for cooperative FSO systems. The results corresponding to this ergodic capacity analysis are depicted in Fig. 2 when different relay locations y_R={0.5 km, 2 km} are considered for a source-destination link distance of d_SD = 3 km. Different weather conditions are here adopted: haze visibility of 4 km with

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

and clear visibility of 16 km with

C_{n}^{2} = 8 \times 10^{- 14} m^{- 2 / 3}

, corresponding to moderate and strong turbulence conditions, respectively. Here, the parameters α and β are calculated from Eq. (2), and a value of λ = 1550 nm is assumed. Pointing errors are here present assuming a value of normalized beam width of ω_z/r = 5 and values of normalized jitter of σ_s/r = {1, 3} for each link. A remarkable improvement in performance can be observed at high SNR in both DF strategies, when comparing to the direct transmission without cooperative communication. Monte Carlo simulations results are furthermore included as a reference (Eq. (6) for BDF cooperative protocol, Eq. (20) for ADF cooperative protocol and Eq. (28) for direct transmission), confirming the accuracy of the proposed approximation, and usefulness of the derived results. In addition, this figure shows the high accuracy of the asymptotic results based on the logarithm approximation given in Eqs. (18), (25) and (32) at high SNR.

Fig. 2 Ergodic capacity at high SNR for a source-destination link distance of d_SD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (5, 3) are assumed.

Download Full Size | PDF

This ergodic capacity analysis can be extended in order to obtain a point where the asymptotic ergodic capacity at high SNR intersects with the γ₀-axis. This point can be understood as a SNR threshold, i.e. $γ_{BDF}^{th}$ , in which the ergodic capacity corresponding to the BDF cooperative protocol is significantly increased. From Eq. (18) the corresponding expression of $γ_{BDF}^{th}$ in terms of the channel parameters can be derived as

\begin{array}{l} γ_{BDF}^{th} [d B] = - \frac{10 ln (4 F ξ)}{ln (10)} + \frac{10}{ln (10)} (\frac{1}{φ_{SD}^{2}} + \frac{1}{φ_{RD}^{2}}) \\ + \frac{10}{ln (10)} (ln (\frac{α_{SD} β_{SD} α_{RD} β_{RD}}{A_{SD} L_{SD} A_{RD} L_{RD}}) - ψ (α_{SD}) - ψ (β_{SD}) - ψ (α_{RD}) - ψ (β_{RD})) . \end{array}

Similar to Eq. (33), from Eq. (25) we can obtain the corresponding SNR threshold corresponding to the ADF cooperative protocol,

γ_{ADF}^{th}

, as follows

γ_{ADF}^{th} [d B] = - \frac{10}{ln (10)} (I_{1} + 2 I_{2} + P ln (F^{'}) + ln (4 ξ))) .

Similar to Eqs. (33) and (34), from Eq. (32) we can obtain the corresponding SNR threshold corresponding to the direct transmission,

γ_{DT}^{th}

, as follows

γ_{DT}^{th} [d B] = - \frac{10 ln (4 ξ)}{ln (10)} + \frac{20}{ln (10)} (ln (\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}}) + \frac{1}{φ_{SD}^{2}} - ψ (α_{SD}) - ψ (β_{SD})) .

Finally, it can be easily deduced from asymptotic analysis at high SNR that the shift of the ergodic capacity versus SNR is here more relevant than the slope of the curve in SNR compared to other performance metrics such as BER and outage probability. This shift can be interpreted as an improvement on ergodic capacity in order to maintain the same performance in terms of capacity with less SNR. From Eqs. (33) and (35), we can obtain this improvement or gain corresponding to the BDF cooperative protocol, i.e. G_BDF[dB], as follows

G_{BDF} [d B] = γ_{DT}^{th} [d B] - γ_{BDF}^{th} [d B] .

At the same time, from Eqs. (34) and (35), we can also obtain the improvement or gain corresponding to the ADF cooperative protocol, i.e. G_ADF[dB], as follows

G_{ADF} [d B] = γ_{DT}^{th} [d B] - γ_{ADF}^{th} [d B] .

For the better understanding of the impact of the three-node cooperative FSO system under study, gains G_BDF[dB] in Eq. (36) and G_ADF[dB] in Eq. (37) as a function of the horizontal displacement of the relay node R, x_R, are depicted in Fig. 3 for a source-destination link distance of d_SD = 3 km when different relay locations y_R = {0.5 km, 1 km, 1.5 km, 2 km} are assumed. Eqs. (33), (34) and (35) are included in Fig. 3 as well as Fig. 4 to display gains G_BDF[dB] and G_ADF[dB]. In order to show the effect of the pointing errors on the ergodic capacity, different values of normalized beam width of ω_z/r = {5, 7} and normalized jitter of σ_s/r = {1, 3} are considered. It should be noted that in Fig. 3 as well as in Fig. 4, both BDF and ADF present the same gain under moderate turbulence conditions as well as strong turbulence conditions when values of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r,σ_s/r) = (7, 1) are considered. This behavior is related to the fact that the relation φ² > β is satisfied and, hence, pointing errors do not affect the obtained gain. In addition to this, when pointing errors are more severe, i.e. when values of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 3) and (ω_z/r, σ_s/r) = (7, 3) are considered, the relation φ² > β is not satisfied and, hence, the gain is totally dependent on pointing errors. Moreover, it can be observed that cooperative communications can be used in order to achieve a greater robustness against pointing errors since the obtained performance is much better with respect to the direct transmission when more severe pointing errors. It should be also noted that ergodic capacity is severely degraded as normalized jitter increases. As expected, the ergodic capacity is strongly dependent on the relay location regardless of cooperative protocol adopted. It must be highlighted that the ergodic capacity shows one maximum value when the horizontal displacement of the relay node R is equal to the source-destination link distance regardless of the atmospheric turbulence and pointing errors. This maximum gain is related to the minimum relay-destination link distance due to the relation x_R = d_SD holds and, hence, the received SNR at the destination node is maximum. On the one hand, it can be seen that ADF cooperative protocol always presents a better and robust performance than BDF cooperative protocol due to the fact that ADF is based on the selection of the optical path with a greater value of irradiance. On the other hand, the BDF cooperative protocol is only able to achieve a better performance than direct transmission for specific relay locations. In other words, BDF cooperative protocol does not improve the ergodic capacity when the relay-destination link distance is greater than source-destination link distance.

Fig. 3 Gain, G[dB], for a source-destination link distance of d_SD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (5, 3) are assumed.

Download Full Size | PDF

Fig. 4 Gain, G[dB], for a source-destination link distance of d_SD = 3 km when different weather condition and values of of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (7, 1) and (ω_z/r, σ_s/r) = (7, 3) are assumed.

Download Full Size | PDF

Finally, it can be seen in Fig. 2 that gains of 4.83 dB and 7.26 dB corresponding to ADF, in contrast to gains of 2.96 dB and 4.05 dB corresponding to BDF for moderate turbulence when values of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (5, 3) are considered, respectively. Analogously, it can be also seen in Fig. 2 that gains of 5.1 dB and 7.38 dB corresponding to ADF, in contrast to gains of 3.08 dB and 4.62 dB corresponding to BDF for strong turbulence when values of normalized beam width and normalized jitter of (ω_z/r, σ_s/r) = (5, 1) and (ω_z/r, σ_s/r) = (5, 3) are considered, respectively.

4. Conclusions

In this paper, the ergodic capacity of two different DF strategies is analyzed over gamma-gamma fading channels with pointing errors when line of sight is available. Novel closed-form approximate expressions are obtained as well as asymptotic expressions for the ergodic capacity using the well-known inequality between arithmetic and geometric mean of positive random variables in order to derive an approximate closed-form expression of the distribution of the sum of gamma-gamma with pointing errors variates. Obtained results have confirmed the accuracy of the proposed approximation based on the logarithm function as well as the validity of the moments method, which is being used in many recently reported works. From this asymptotic ergodic capacity analysis is easily deduced that the shift of the ergodic capacity versus SNR is here more relevant than the slope of the curve in SNR compared to other performance metric such as BER and outage probability. This shift is interpreted as an improvement on ergodic capacity in order to maintain the same performance in terms of capacity with less SNR, which is dependent on the relay location. It is demonstrated that cooperative communications are able to achieve not only a better performance in terms of the error rate performance as well as outage probability than direct transmission [7, 13, 14], but also in terms of the channel capacity in the context of FSO systems without much increase in hardware. In addition, it can be concluded that the ADF cooperative protocol is able to achieve a greater ergodic capacity than the BDF cooperative protocol regardless of the relay location and pointing errors effect due to the fact that ADF is based on the selection of the optical path with a greater value of fading gain and, hence, an increase of hardware complexity and cost is required. Unlike [33, 34], here the line of sight is taken into account in order to achieve a significant robustness under different turbulence conditions and more severe pointing errors regardless of the relay location. Finally, it is verified that DF strategies are able to achieve a higher ergodic capacity when the line of sight is available, being strongly dependent on the relay location without much increase in hardware. From the relevant results here derived, researching the impact on ergodic capacity when the number of relays is increased as well as employing space-time trellis code to increase the ergodic capacity in cooperative FSO systems are interesting topics for future research in order to complement the study in this paper.

Appendix A

In this appendix the correcting factor F is obtained. This correcting factor can be derived from Eq. (10) when both averages hold and, hence, it can be expressed as follows

𝔼 [I_{T}] = \sqrt{8 F} \cdot 𝔼 [\sqrt{I_{T}^{LB}}] .

After performing some straightforward manipulations in Eq. (38), we can express the correcting factor, F, as follows

F = \frac{𝔼 {[I_{T}]}^{2}}{8 \cdot 𝔼 {[\sqrt{I_{T}^{LB}}]}^{2}} .

firstly, before evaluating the parameter F, we obtain the mean of I_T as 𝔼[I_T ] = 𝔼[I_SD] + 2𝔼[I_RD] since the variates I_SD and I_RD are statistically independent. Therefore, the mean of a generic random variable I_m is given by

𝔼 [I_{m}] = \int_{0}^{\infty} i f_{I_{m}} (i) d i .

The integral in Eq. (40) can be evaluated with the help of [45, eqn. (2.24.2.1)] and, hence, 𝔼[I_m] can be expressed as follows

𝔼 [I_{m}] = \frac{A_{m} L_{m} φ_{m}^{2}}{1 + φ_{m}^{2}} .

Secondly, we obtain the expectation of positive square root of

I_{T}^{LB}

as

𝔼 [\sqrt{I_{T}^{LB}}] = \int_{0}^{\infty} i^{1 / 2} f_{I_{T}^{LB}} (i) d i .

In order to evaluate the integral in Eq. (42), we can use [45, eqn. (2.24.2.1)] as in Eq. (41) and, hence,

𝔼 [\sqrt{I_{T}^{LB}}]

is given by

\begin{array}{l} 𝔼 [\sqrt{I_{T}^{LB}}] = \sqrt{\frac{A_{SD} A_{RD} L_{SD} L_{RD}}{α_{SD} β_{SD} α_{RD} β_{RD}}} \\ \times \frac{4 φ_{SD}^{2} Γ (α_{SD} + 1 / 2) Γ (β_{SD} + 1 / 2) φ_{RD}^{2} Γ (α_{RD} + 1 / 2) Γ (β_{RD} + 1 / 2)}{(1 + 2 φ_{SD}^{2}) Γ (α_{SD}) Γ (β_{SD}) (1 + 2 φ_{RD}^{2}) Γ (α_{RD}) Γ (β_{RD})} . \end{array}

Finally, the correcting factor F is easily derived from Eqs. (41) and (43) as follows

\begin{array}{l} F & = {(\frac{A_{SD} L_{SD} φ_{SD}^{2}}{1 + φ_{SD}^{2}} + \frac{2 A_{RD} L_{RD} φ_{RD}^{2}}{1 + φ_{RD}^{2}})}^{2} \frac{α_{SD} β_{SD} {(1 + 2 φ_{SD}^{2})}^{2} Γ {(α_{SD})}^{2} Γ {(β_{SD})}^{2}}{128 A_{SD} L_{SD} φ_{SD}^{4} Γ {(α_{SD} + 1 / 2)}^{2} Γ {(β_{SD} + 1 / 2)}^{2}} \\ \times \frac{α_{RD} β_{RD} {(1 + 2 φ_{RD}^{2})}^{2} Γ {(α_{RD})}^{2} Γ {(β_{RD})}^{2}}{A_{RD} L_{RD} φ_{RD}^{4} Γ {(α_{RD} + 1 / 2)}^{2} Γ {(β_{RD} + 1 / 2)}^{2}} . \end{array}

Appendix B

In this appendix the correcting factor F′ is obtained. This correcting factor can be derived from Eq. (10) when both averages hold under the assumption that I_RD > I_SD and, hence, it can be expressed as in Eq. (38). After performing some straightforward manipulations in Eq. (38), we can express the correcting factor, F′, as follows

F^{'} = \frac{𝔼 {[I_{SD} + 2 I_{RD}]}^{2}}{8 \cdot 𝔼 {[\sqrt{I_{SD} I_{RD}}]}^{2}}, I_{RD} > I_{SD} .

Similar to the correcting factor F, firstly, we obtain the expectation of 𝔼[I_SD + 2I_RD] under the assumption that I_RD > I_SD as

\begin{array}{l} E [I_{SD} + 2 I_{RD}] & = \int_{0}^{\infty} \int_{0}^{i_{2}} (i_{1} + 2 i_{2}) f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2} = A + 2 B \\ = \int_{0}^{\infty} [\int_{0}^{i_{2}} i_{1} f_{I_{SD}} (i_{1}) d i_{1}] f_{I_{RD}} (i_{2}) d i_{2} + 2 \int_{0}^{\infty} i_{2} F_{I_{SD}} (i_{2}) F_{I_{RD}} (i_{2}) d i_{2} . \end{array}

In order to evaluate the double integral in A, we have to evaluate the inner one first. In this way, we can obtain the closed-form expression corresponding to A with the help of [45, eqn. 1.16.2.1] and [45, eqn. 2.24.1.2] for the inner integral and the outer integral, respectively. Hence, the integral A can be written as

A = \frac{φ_{RD}^{2} φ_{SD}^{2} A_{SD} L_{SD} G_{5, 5}^{4, 3} (\begin{array}{l} \frac{α_{RD} β_{RD} A_{SD} L_{SD}}{α_{SD} β_{SD} A_{RD} L_{RD}} | & \begin{matrix} - φ_{SD}^{2}, - α_{SD}, - β_{SD}, 1, φ_{RD}^{2} + 1 \\ α_{RD}^{2}, α_{RD}, β_{RD}, 0, - φ_{SD}^{2} - 1 \end{matrix} \end{array})}{α_{SD} β_{SD} Γ (α_{RD}) Γ (α_{SD}) Γ (β_{RD}) Γ (β_{SD})} .

Similar to A, we can obtain the corresponding closed-form expression for the integral B with the help of [45, eqn. 2.24.1.2] as

B = \frac{2 φ_{RD}^{2} φ_{SD}^{2} A_{SD} L_{SD} G_{5, 5}^{4, 3} (\begin{array}{l} \frac{α_{RD} β_{RD} A_{SD} L_{SD}}{α_{SD} β_{SD} A_{RD} L_{RD}} | & \begin{matrix} - φ_{SD}^{2}, - α_{SD}, - β_{SD}, 0, φ_{RD}^{2} + 1 \\ α_{RD}^{2}, α_{RD}, β_{RD}, - 1, - φ_{SD}^{2} - 1 \end{matrix} \end{array})}{α_{SD} β_{SD} Γ (α_{RD}) Γ (α_{SD}) Γ (β_{RD}) Γ (β_{SD})} .

Secondly, we obtain the expectation of positive square root of the product of I_SDI_RD under the assumption that I_RD > I_SD as follows

\begin{array}{l} 𝔼 [\sqrt{I_{SD} I_{RD}}] & = \int_{0}^{\infty} \int_{0}^{i_{2}} \sqrt{i_{1} i_{2}} f_{I_{SD}} (i_{1}) f_{I_{RD}} (i_{2}) d i_{1} d i_{2} = C \\ = \int_{0}^{\infty} \sqrt{i_{2}} [\int_{0}^{i_{2}} \sqrt{i_{1}} f_{I_{SD}} (i_{1}) d i_{1}] f_{I_{RD}} (i_{2}) d i_{2} . \end{array}

In the same way as A, we can obtain the expectation of positive square root of the product of I_SD and I_RD as

C = \frac{φ_{RD}^{2} φ_{SD}^{2} A_{SD} L_{SD} G_{5, 5}^{4, 3} (\begin{array}{l} \frac{α_{RD} β_{RD} A_{SD} L_{SD}}{α_{SD} β_{SD} A_{RD} L_{RD}} | & \begin{matrix} - φ_{SD}^{2}, - α_{SD}, - β_{SD}, \frac{1}{2}, φ_{RD}^{2} + 1 \\ α_{RD}^{2}, α_{RD}, β_{RD}, - \frac{1}{2}, - φ_{SD}^{2} - 1 \end{matrix} \end{array})}{α_{SD} β_{SD} Γ (α_{RD}) Γ (α_{SD}) Γ (β_{RD}) Γ (β_{SD})} .

Finally, the correcting factor F′ is easily derived from Eqs. (47), (48) and (50) as follows

F^{'} = \frac{{(A + 2 B)}^{2}}{8 \cdot C^{2}} .

Appendix C

In this appendix, we explain in more detail how to obtain the expression in Eq. (32). It should be noted that adding the series asymptotic expansion corresponding to the subtraction of the Meijer’s G-function in Eq. (17) to this appendix corresponding to the BDF cooperative protocol is remarkably tedious due to the fact that both series asymptotic expansion are too long. Nevertheless, for the sake of simplicity we have added both series asymptotic expression corresponding to direct transmission since this mathematical procedure is identical to BDF cooperative protocol and, hence, it can be better understood. Therefore, the subtraction of two Meijer’s G-function in Eq. (31) is given by

\begin{array}{l} S = \frac{B φ_{SD}^{2} (M_{1} - M_{2})}{ln (2) Γ (α_{SD}) Γ (β_{SD})} = \frac{B φ_{SD}^{2}}{ln (2) Γ (α_{SD}) Γ (β_{SD})} \\ \times (G_{3, 5}^{5, 2} (\begin{array}{l} \frac{α_{SD} β_{SD}}{A_{SD} L_{SD}} | & \begin{matrix} 0, 0, φ_{SD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, 0, 0 \end{matrix} \end{array}) - G_{3, 5}^{5, 2} (\begin{array}{l} \frac{α_{SD} β_{SD}}{A_{SD} L_{SD}} | & \begin{matrix} 1, 1, φ_{SD}^{2} + 1 \\ φ_{SD}^{2}, α_{SD}, β_{SD}, 1, 1 \end{matrix} \end{array})) . \end{array}

Now, we can express M₂ function by using the series asymptotic expansion corresponding to the Meijer’s G-function [46, eqn. (07.34.06.0002.01)] as follows

\begin{array}{l} M_{2} = {(\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})}^{φ_{SD}^{2}} \frac{π^{4} csc {[π φ_{SD}^{2}]}^{2} csc [π (α_{SD} - φ_{SD}^{2})] csc [π (β_{SD} - φ_{SD}^{2})]}{Γ (1 - α_{SD} + φ_{SD}^{2}) Γ (1 - β_{SD} - φ_{SD}^{2})} \\ + {(\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})}^{α_{SD}} \sum_{k = 0}^{\infty} {(\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})}^{k} \frac{π^{3} csc {[π α_{SD}]}^{2} csc [π (α_{SD} - β_{SD})]}{k! (k + α_{SD} - φ_{SD}^{2}) Γ (1 + k + α_{SD} - β_{SD})} \\ - {(\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})}^{β_{SD}} \sum_{k = 0}^{\infty} {(\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})}^{k} \frac{π^{3} csc {[π β_{SD}]}^{2} csc [π (α_{SD} - β_{SD})]}{k! (k + β_{SD} - φ_{SD}^{2}) Γ (1 + k + β_{SD} - α_{SD})} \\ + \sum_{k = 1}^{\infty} \frac{Γ (α_{SD} - k) Γ (β_{SD} - k)}{φ_{SD}^{2} - k} (ψ (α_{SD} - k) + ψ (β_{SD} - k) + \frac{1}{k - φ_{SD}^{2}} - ln (\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})) . \end{array}

In Eq. (53), it has been used that the cosecant function is related to the gamma function by

csc (π z) = \frac{Γ (1 - z) Γ (z)}{π}

[46, eqn. (01.10.26.0025.01)]. Analogously, we can obtain the series asymptotic expansion corresponding to M₁, which can be easily expressed in terms of M₂ as follows

M_{1} = M_{2} + \frac{Γ (α_{SD}) Γ (β_{SD})}{φ_{SD}^{2}} (ψ (α_{SD}) + ψ (β_{SD}) - \frac{1}{φ_{SD}^{2}} - ln (\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})) .

Finally, it is demonstrated that the result of the subtraction of two Meijer’s G-function in Eq. (31) is given by

S = \frac{B}{ln (2)} (ψ (α_{SD}) + ψ (β_{SD}) - \frac{1}{φ_{SD}^{2}} - ln (\frac{α_{SD} β_{SD}}{A_{SD} L_{SD}})) .

Acknowledgments

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

References and links

1. M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Communications Surveys Tutorials 16(4), 2231–2258 (2014). [CrossRef]

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

3. A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances, vol. 2 (Springer Science & Business Media, 2010).

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

5. 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]

6. M. Karimi and M. Nasiri-Kenari, “Outage analysis of relay-assisted free-space optical communications,” IET Communications 4(12), 1423–1432 (2010). [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. C. Abou-Rjeily, “Performance Analysis of Selective Relaying in Cooperative Free-Space Optical Systems,” J. Lightwave Technol. 31(18), 2965–2973 (2013). [CrossRef]

9. 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]

10. 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]

11. 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]

12. 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).

13. 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), 16629–16644 (2014). [CrossRef] [PubMed]

14. R. Boluda-Ruiz, A. García-Zambrana, B. Castillo-Vázquez, and C. Castillo-Vázquez, “Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications,” Opt. Express 23(3), 2600–2617 (2015). [CrossRef] [PubMed]

15. C. del Castillo-Vazquez, R. Boluda-Ruiz, B. del Castillo-Vazquez, and A. Garcia-Zambrana, “Outage performance of DF relay-assisted FSO communications using time-diversity,” IEEE Photon. Technol. Lett. 27(11), 1149–1152 (2015).

16. H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios, and G. S. Tombras, “Average Capacity of Optical Wireless Communication Systems Over Atmospheric Turbulence Channels,” J. Lightwave Technol. 27(8), 974–979 (2009). [CrossRef]

17. C. Liu, Y. Yao, Y. Sun, and X. Zhao, “Analysis of average capacity for free-space optical links with pointing errors over gamma-gamma turbulence channels,” Chinese Opt. Lett. 8(6), 537–540 (2010). [CrossRef]

18. J. M. Garrido-Balsells, A. Jurado-Navas, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “On the capacity of ℳ-distributed atmospheric optical channels,” Opt. Lett. 38(20), 3984–3987 (2013). [CrossRef] [PubMed]

19. M. Z. Hassan, M. J. Hossain, and J. Cheng, “Ergodic capacity comparison of optical wireless communications using adaptive transmissions,” Opt. Express 21(17), 20,346–20,362 (2013). [CrossRef]

20. M. Petkovic and G. Dordevic, “Effects of pointing errors on average capacity of FSO links over gamma-gamma turbulence channel,” in Telecommunication in Modern Satellite, Cable and Broadcasting Services (TELSIKS), 2013 11th International Conference on, vol. 02, pp. 481–484 (2013). [CrossRef]

21. F. Benkhelifa, Z. Rezki, and M. Alouini, “Low SNR Capacity of FSO Links over Gamma-Gamma Atmospheric Turbulence Channels,” IEEE Commun. Lett. 17(6), 1264–1267 (2013). [CrossRef]

22. I. Ansari, F. Yilmaz, and M. Alouini, “A unified performance of free-space optical links over Gamma-Gamma turbulence channels with pointing errors,” submitted to IEEE Trans. Communications, technical report available at http://hdl.handle.net/10754/305353 (2015).

23. J. Zhang, L. Dai, Y. Han, Y. Zhang, and Z. Wang, “On the Ergodic Capacity of MIMO Free-Space Optical Systems over Turbulence Channels,” IEEE J. Sel. Areas Commun. (to be published) (2015). [CrossRef]

24. M. K. Simon and M.-S. Alouini, Digital Communications Over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005).

25. J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005). [CrossRef]

26. M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006). [CrossRef]

27. H. Sandalidis and T. Tsiftsis, “Outage probability and ergodic capacity of free-space optical links over strong turbulence,” Electron. Lett. 44(1), 46–47 (2008). [CrossRef]

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

29. E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Trans. Inf. Theory 44(6), 2619–2692 (1998). [CrossRef]

30. 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]

31. L. Yang, X. Gao, and M.-S. Alouini, “Performance analysis of free-space optical communication systems with multiuser diversity over stmospheric turbulence channels,” IEEE Photonics J. 6(2), 7901217 (2014). [CrossRef]

32. K. P. Peppas, A. N. Stassinakis, H. E. Nistazakis, and G. S. Tombras, “Capacity analysis of dual amplify-and-forward relayed free-space optical communication systems over turbulence channels with pointing errors,” J. Opt. Commun. Netw. 5(9), 1032–1042 (2013). [CrossRef]

33. S. Anees and M. R. Bhatnagar, “On the capacity of decode-and-forward dual-hop free space optical communication systems,” in Wireless Communications and Networking Conference (WCNC), 2014 IEEE, pp. 18–23 (IEEE, 2014). [CrossRef]

34. M. Aggarwal, P. Garg, and P. Puri, “Ergodic capacity of SIM based DF relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(10), 1104–1107 (2015). [CrossRef]

35. M. Aggarwal, P. Garg, and P. Puri, “Exact capacity of amplify-and-forward relayed optical wireless communication systems,” IEEE Photon. Technol. Lett. 27(8), 903–906 (2015). [CrossRef]

36. E. Zedini, I. S. Ansari, and M.-S. Alouini, “Performance analysis of mixed Nakagami-and Gamma–Gamma Dual-Hop FSO Transmission Systems,” IEEE Photonics J. 7(1), 1–20 (2015).

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

38. 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]

39. 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).

40. 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]

41. 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]

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

43. 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]

44. J. Galambos and I. Simonelli, Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions (CRC Press, 2004).

45. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and series Volume 3: More Special Functions, vol. 3 (Gordon and Breach Science Publishers, 1999).

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

47. F. Yilmaz and M.-S. Alouini, “Novel asymptotic results on the high-order statistics of the channel capacity over generalized fading channels,” in Signal Processing Advances in Wireless Communications (SPAWC), 2012 IEEE 13th International Workshop on, pp. 389–393 (IEEE, 2012). [CrossRef]

48. I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M.-S. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K,” IEEE Trans. Commun. 59(10), 2654–2658 (2011). [CrossRef]

49. K. P. Peppas, “A new formula for the average bit error probability of dual-hop amplify-and-forward relaying systems over generalized shadowed fading channels,” IEEE Wireless Commun. Lett. 1(2), 85–88 (2012). [CrossRef]

Ergodic capacity analysis for DF strategies in cooperative FSO systems

Abstract

1. Introduction

2. System and channel model

3. Ergodic capacity analysis

4. Conclusions

Appendix A

Appendix B

Appendix C

Acknowledgments

References and links

Cited By

Figures (4)

Equations (58)

Optics Express