Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications

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

doi:10.1364/OE.23.002600

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 [3–9].

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 [10–21], 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 β_{SR_min} > β_SD + β_{R_minD} is satisfied, wherein β_{SR_min}, β_{R_minD} and β_SD 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 R_k for k = {1,2,...,N}, and one destination node D. Furthermore, different source-relay link distances within the range d′_SR to d_SR (d′_SR < d_SR) 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 (I_SD) is greater than the irradiance values corresponding to the relay-destination links (I_{R_kD} 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 R_k if the irradiance of the R_k-D link is greater than the irradiance corresponding to the S-D link and those corresponding to the R_i-D links for i ≠ k, 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 [16–18, 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.

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

Download Full Size | PDF

3. Channel model

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

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

where η is the detector responsivity, assumed hereinafter to be the unity, X ≜ x_m(t) represents the optical power supplied by the source, I_m ≜ i_m(t) the equivalent real-value fading gain (irradiance) through the optical channel between the laser and the receive aperture. Z_m ≜ z_m(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. It can usually be modeled to high accuracy as AWGN with zero mean and variance σ² = N₀/2, i.e. Z_m ∼ N(0, N₀/2), independent of the on/off state of the received bit. We use X, Y_m, I_m and Z_m to denote random variables and x_m(t), y_m(t), i_m(t) and z_m(t) their corresponding realizations. The irradiance 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^−Φ^d_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) [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 I_m 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

f_{I_{m}} (i) = a_{m} i^{b_{m} - 1} + c_{m} i^{b_{m}} + O (i^{b_{m} + 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

f_{I_{m}} (i) \approx a_{m} i^{b_{m} - 1} e^{i \frac{c_{m}}{a_{m}}} .

Different expressions for f_{I_m} (i), depending on the relation between the jitter variance and turbulence conditions [9], can be written as

f_{I_{m}} (i) \approx {\begin{array}{l} \frac{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 \frac{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} \\ \frac{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} \end{array}

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 (0.49 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}

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, 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 [32, eqn. (8.250)]. It must be noted that the parameter c_m is equal to 0 when the diversity order gain is independent of the pointing error, i.e. when the relation

φ_{m}^{2} > β_{m}

is satisfied.

In the following section, the fading coefficient I_m for the paths S-D, S-R_k and R_k-D is indicated by I_SD, I_{SR_k} and I_{R_kD}, 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 P_opt, 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 $d_{E} = 2 P_{opt} \sqrt{T_{b} ξ}$ , where the parameter T_b is the bit period and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high 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

Y_{0} = X I_{SD} + Z_{SD}, X \in {0, d_{E}}, Z_{SD} ~ N (0, N_{0} / 2) I_{R_{k} D} < I_{SD}

Y_{1} = \frac{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}

being R_k the relay node whose value of the fading gain corresponding to the R_k-D link is maximum, denoted as I_{R_kD}, i.e. I_{R_kD} = max{I_R₁D,I_R₂D,...,I_{R_ND}}. X^* represents the random variable corresponding to the information detected at the relay node R_k and, hence, X^* = X when the bit has been detected correctly, and X^* = d_E − X when the bit has been detected incorrectly at the relay node R_k. 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 I_SD > I_{R_kD} at the destination node D is given by

P_{b}^{SD - RD} (I_{SD}) = Q (\sqrt{d_{E}^{2} i^{2} / 2 N_{0}}) = Q (\sqrt{2 γ ξ i}),

where Q(·) is the Gaussian Q-function and

γ = P_{opt}^{2} T_{b} / N_{0}

represents the received electrical SNR in absence of turbulence. Hence, the average BER,

P_{b}^{SD - RD}

, can be obtained by averaging

P_{b}^{SD - RD} (I_{SD})

over the PDF as follows

P_{b}^{SD - RD} ≃ \int_{0}^{\infty} Q (\sqrt{2 γ ξ} i) \prod_{j = 1}^{N} F_{I_{R_{j} D}} (i) f_{I_{SD}} (i) d i .

Knowing the fact that the irradiance values are statistically independent, the probability corresponding to I_SD > I_{R_kD} is computed by using

\prod_{j = 1}^{N} F_{I_{R_{j} D}} (I_{SD})

, being F_{I_m} (i) the cumulative density function (CDF) of the random variable I_m, which is given by F_{I_m} (i) = Prob(I_m ≤ i). 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

F_{I_{m}} (i) \approx \frac{a_{m}}{b_{m}} i^{b_{m}} e^{i \frac{c_{m} b_{m}}{a_{m} (b_{m} + 1)}},

where the value of the parameters a_m, b_m and c_m 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) = 2 Q (\sqrt{2} x)

[32, eqn. (6.287)] and the fact that

\int_{0}^{\infty} erfc (c x) x^{α - 1} e^{- p x} d x

can be found in [36, eqn. (2.8.5.2)]. Therefore, a approximate closed-form solution for the BER,

P_{b}^{SD - RD}

, can be expressed after some algebraic manipulations as follows

\begin{array}{l} P_{b}^{SD - RD} ≃ \frac{a_{RD}^{T} Γ ((b_{RD}^{T} + 1) / 2) {(γ ξ)}^{- b_{RD}^{T} / 2}}{2 b_{RD}^{T} \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{RD}^{T}}{2}, \frac{b_{RD}^{T} + 1}{2}; \frac{b_{RD}^{T} + 2}{2}, \frac{1}{2}; \frac{c_{RD}^{T^{2}}}{4 a_{RD}^{T^{2}} γ ξ}) \\ + \frac{c_{RD}^{T} Γ ((b_{RD}^{T} + 2) / 2) {(γ ξ)}^{- (b_{RD}^{T} + 1) / 2}}{2 (b_{RD}^{T} + 1) \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{RD}^{T} + 1}{2}, \frac{b_{RD}^{T} + 2}{2}; \frac{b_{RD}^{T} + 3}{2}, \frac{3}{2}; \frac{c_{RD}^{T^{2}}}{4 α_{RD}^{T^{2}} γ ξ}), \end{array}

where ₂F₂(a₁,a₂;b₁,b₂;z) is the generalized hypergeometric function whose coefficients p and q are equal to 2, i.e.

{}_{p}{F_{q}} (a_{1}, \dots, a_{p}; b_{1}, \dots, b_{q}; z) = \sum_{k = 0}^{\infty} \frac{\prod_{j = 1}^{p} {(a_{j})}_{k} z^{k}}{\prod_{j = 1}^{q} {(b_{j})}_{k} k!}

[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

a_{RD}^{T}

,

b_{RD}^{T}

and

c_{RD}^{T}

are given by

a_{RD}^{T} = \frac{a_{SD} \prod_{i = 1}^{N} a R_{i} D}{\prod_{i = 1}^{N} b_{R_{i} D}}, b_{RD}^{T} = b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D}, \frac{c_{RD}^{T}}{a_{RD}^{T}} = \frac{c_{SD}}{a_{SD}} + \sum_{i = 1}^{N} \frac{c_{R_{i} D} b_{R_{i} D}}{a_{R_{i} D} (b_{R_{i} D} + 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),

P_{b}^{DL}

, can be determined as follows

P_{b}^{DL} = \int_{0}^{\infty} Q (\sqrt{2 γ ξ} i) f_{I_{SD}} (i) d i .

Evaluating this integral as in Eq. (7), we can obtain the approximate closed-form solution for the BER as follows

\begin{array}{l} P_{b}^{DL} ≃ \frac{a_{SD} Γ ((b_{SD} + 1) / 2) {(γ ξ)}^{- b_{SD} / 2}}{2 b_{SD} \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{SD}}{2}, \frac{b_{SD} + 1}{2}; \frac{b_{SD} + 2}{2}, \frac{1}{2}; \frac{c_{SD}^{2}}{4 a_{SD}^{2} γ ξ}) \\ + \frac{c_{SD} Γ ((b_{SD} + 2) / 2)}{{(γ ξ)}^{(b_{SD} + 1) / 2} (b_{SD} + 1) \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{SD} + 1}{2}, \frac{b_{SD} + 2}{2}; \frac{b_{SD} + 3}{2}, \frac{3}{2}; \frac{c_{SD}^{2}}{4 α_{SD}^{2} γ ξ}) . \end{array}

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 I_{R_kD} > I_SD. Hence, two cases can be considered depending on the fact that the bit from the relay S-R_k-D is detected correctly or incorrectly. In this way, considering that the bit has been detected correctly at relay the node R_k, the statistical channel model corresponding to this case can be expressed as

Y_{1} = \frac{1}{2} X (I_{SD} + 2 I_{R_{k} D}) + Z_{SD} + Z_{R_{k} D}, X^{*} = X

Assuming CSIRT, the conditional BER at the destination node D when the bit is detected correctly (

{DF}_{k}^{0}

) is given by

P_{b}^{{DF}_{k}^{0}} (I_{SD}, I_{R_{k} D}) = Q (\sqrt{\frac{γ ξ}{4}} (i_{1} + 2 i_{2})),

Hence, the average BER,

P_{b}^{{DF}_{k}^{0}}

, can be obtained by averaging over the PDFs as follows

P_{b}^{{DF}_{k}^{0}} = \int_{0}^{\infty} \int_{0}^{\infty} Q (\sqrt{\frac{γ ξ}{4}} (i_{i} + 2 i_{2})) f_{I_{SD}} (i_{1}) F_{I_{SD}} (i_{2}) \prod_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{N} F_{I_{R_{j} D}} (i_{2}) f_{I_{R_{k} D}} (i_{2}) d i_{1} d i_{2} .

Before evaluating the integral in Eq. (15) we define

I_{DF}^{T}

, which represents the sum of variates

I_{DF}^{T} = I_{SD} + 2 I_{R_{k} D}

, under the assumption that I_{R_kD} > I_SD, as determined by the terms F_{I_SD} (i₂) and F_{I_{R_jD}} (i₂) for j ≠ k. Knowing that I_SD and I_{R_kD} are independent statistically, the resulting PDF of their sum

I_{DF}^{T}

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 I_{R_kD} = max{I_R₁_D,I_R₂_D,...,I_{R_N}_D} under the assumption that I_{R_kD} > I_SD, i.e.

f_{I_{{RD}_{max}}} (i) = F_{I_{SD}} (i) \prod_{j \neq k}^{N} F_{I_{R_{j} D}} (i) f_{I_{R_{k} D}} (i)

and, then, its corresponding approximate expression,

f_{I_{{RD}_{max}}} (i) \approx a_{{RD}_{max}} i^{b_{{RD}_{max}} - 1} e^{i (c_{{RD}_{max}} / a_{{RD}_{max}})}

, can be written as

f_{I_{{RD}_{max}}} (i) \approx \frac{a_{SD} b_{R_{k} D} \prod_{j = 1}^{N} a_{R_{j} D}}{b_{SD} \prod_{j = 1}^{N} b_{R_{j} D}} i^{b_{SD} + \sum_{j = 1}^{N} b R_{j} D^{- 1}} e^{i (\frac{c R_{k} D}{a R_{k} D} + \frac{c_{SD} b_{SD}}{a_{SD} (b_{SD} + 1)} + \sum_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{N} \frac{c_{R_{j} D^{b} R_{j} D}}{a_{R_{j} D} (b_{R_{j} D^{+ 1}})})} .

, of combined variates can be easily derived as

f_{I_{DF}^{T}} \approx \frac{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} (\frac{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}})}}) .

From this expression, the average BER,

P_{b}^{{DF}_{k}^{0}}

, can be determined as follows

P_{b}^{{DF}_{k}^{0}} = \int_{0}^{\infty} Q (\sqrt{\frac{γ ξ}{4}} i) f_{I_{DF}^{T}} (i) d i .

Evaluating this integral as in Eq. (7), we can obtain the approximate closed-form solution for the BER,

P_{b}^{{DF}_{k}^{0}}

as follows

\begin{array}{l} P_{b}^{{DF}_{k}^{0}} ≃ \frac{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} \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{DF}^{T}}{2}, \frac{b_{DF}^{T} + 1}{2}; \frac{b_{DF}^{T} + 2}{2}, \frac{1}{2}; \frac{2 c_{DF}^{T^{2}}}{a_{DF}^{T^{2}} γ ξ}) \\ + \frac{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) \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{DF}^{T} + 1}{2}, \frac{b_{DF}^{T} + 2}{2}; \frac{b_{DF}^{T} + 3}{2}, \frac{3}{2}; \frac{2 c_{DF}^{T^{2}}}{a_{DF}^{T^{2}} γ ξ}) . \end{array}

Alternatively, considering now that the bit has been detected incorrectly at the relay node R_k, the statistical channel model for this case can be expressed as

Y_{1} = \frac{1}{2} X (I_{SD} - 2 I_{R_{k} D}) + d_{E} \cdot I_{R_{k} D} + Z_{SD} + Z_{R_{k} D}, X^{*} = d_{E} - X

Assuming CSIRT and given that the statistics corresponding to the term d_E · I_{R_kD} become irrelevant to the detection process, the conditional BER at the destination node D when the bit is detected incorrectly (

{DF}_{k}^{1}

) is given by

P_{b}^{{DF}_{k}^{1}} (I_{SD}, I_{R_{k} D}) = Q (\sqrt{\frac{γ ξ}{4}} (i_{1} - 2 i_{2})) .

Hence, the average BER,

P_{b}^{{DF}_{k}^{1}}

, can be obtained by averaging over the PDFs as follows

P_{b}^{{DF}_{k}^{1}} = \int_{0}^{\infty} \int_{0}^{\infty} Q (\sqrt{\frac{γ ξ}{4}} (i_{1} - 2 i_{2})) f_{I_{SD}} (i_{1}) F_{I_{SD}} (i_{2}) \prod_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{N} F_{I_{R_{j} D}} (i_{2}) f_{I_{R_{k} D}} (i_{2}) d i_{1} d i_{2} .

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

P_{b}^{{DF}_{k}^{1}} \dot{=} \int_{0}^{\infty} \int_{0}^{2 i_{2}} f_{I_{SD}} (i_{1}) F_{I_{SD}} (i_{2}) \prod_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{N} F_{I_{R_{j} D}} (i_{2}) f_{I_{R_{k} D}} (i_{2}) d i_{1} d i_{2} .

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-R_k link, as follows

\begin{array}{l} P_{b}^{{SR}_{k} - RD} ≃ \frac{Γ ((b_{{SR}_{k}} + 1) / 2) {(γ ξ)}^{- b_{{SR}_{k}} / 2}}{{(a_{{SR}_{k}})}^{- 1} 2^{(1 - b_{{SR}_{k}})} b_{{SR}_{k}} \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{{SR}_{k}}}{2}, \frac{b_{{SR}_{k}} + 1}{2}; \frac{b_{{SR}_{k}} + 2}{2}, \frac{1}{2}; \frac{c_{{SR}_{k}}^{2}}{a_{{SR}_{k}}^{2} γ ξ}) \\ + \frac{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) \sqrt{π}} {}_{2}{F_{2}} (\frac{b_{{SR}_{k}} + 1}{2}, \frac{b_{{SR}_{k}} + 2}{2}; \frac{b_{{SR}_{k}} + 3}{2}, \frac{3}{2}; \frac{c_{{SR}_{k}}^{2}}{a_{{SR}_{k}}^{2} γ ξ}), \end{array}

wherein the division by 2 has been considered to maintain the average optical power in the air at a constant level of P_opt, being transmitted by each laser an average optical power P_opt/2 in the first phase when only relay R_k is selected by the source node. The BER corresponding to the ADF-RD cooperative protocol here proposed is given by

P_{b}^{RD} = P_{b}^{SD - RD} + \sum_{k = 1}^{N} P_{b}^{{DF}_{k}^{0}} \cdot (1 - P_{b}^{{SR}_{k} - RD}) + \sum_{k = 1}^{N} P_{b}^{{DF}_{k}^{1}} \cdot P_{b}^{{SR}_{k} - RD} .

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 c_m = 0, it is straightforward to show that the average BER behaves asymptotically as (G_cγξ) ^−G_d due to ₂F₂(a₁,a₂;b₁,b₂;0) = 1, where G_d and G_c 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

P_{b}^{RD} \dot{=} {\begin{array}{l} P_{b}^{{DF}_{min}^{1}} \cdot P_{b}^{{SR}_{min} - RD}, & b_{{SR}_{min}} < b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D} \\ P_{b}^{SD - RD}, & b_{{SR}_{min}} > b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D} \end{array}

being b_{SR_min}=min{b_SR₁,...,b_{SR_N} } and

b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D}

the diversity order gain corresponding to

P_{b}^{{SR}_{min} - RD}

and

P_{b}^{SD - RD}

, 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,

G_{d}^{RD}

, relative to the non-cooperative link S-D of

G_{d}^{RD} = min (b_{{SR}_{min}}, b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D}) / b_{SD} .

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 $G_{d}^{RD}$ 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

G_{d}^{SR} = 1 + min (b_{R_{min} D}, \sum_{i = 1}^{N} b_{{SR}_{i}}) / b_{SD} .

being b_{R_minD}=min{b_R₁D,...,b_{R_ND}}. The diversity order gain dependent on relay placement is defined as the maximum between

G_{d}^{RD}

and

G_{d}^{SR}

, and it can be written as

G_{d} = {\begin{array}{l} G_{d}^{RD}, & b_{{SR}_{min}} > b_{SD} + b_{R_{min} D} \\ G_{d}^{SR}, & b_{{SR}_{min}} < b_{SD} + b_{R_{min} D} \end{array}

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 b_{SR_min} and b_SD + b_{R_minD}, 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 G_d in Eq. (30) as a function of the radius of a circumference, d_SR, whose center is the source node, is depicted in Fig. 2 for a source-destination link distance d_SD=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 $C_{n}^{2} = 1.7 \times 10^{- 4} m^{- 2 / 3}$ and clear visibility of 16 km with $C_{n}^{2} = 8 \times 10^{- 14} m^{- 2 / 3}$ , corresponding to moderate and strong atmospheric turbulence conditions, respectively. In these figures, the performance analysis is evaluated for d_SR = d′_SR, d_SR − d′_SR = 0.3 km and d_SR − d′_SR = 0.6 km. Hereinafter, $φ_{{SR}_{k}}^{2} ≜ φ_{SR}^{2}$ and $φ_{R_{k} D}^{2} ≜ φ_{RD}^{2}$ are assumed for k = {1,2,...,N}. Moreover, the relay locations assumed in this paper are (θ_R₁,θ_R₃) = (π/12,π/18) and (θ_R₁,θ_R₃) = (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 $b_{{SR}_{min}} = b_{SD} + \sum_{i = 1}^{N} b_{R_{i} D}$ holds, while the other closer to the destination node, when the relation $b_{R_{min} D} = \sum_{i = 1}^{N} b_{{SR}_{i}}$ 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 b_{SR_min} = b_SD + b_{R_minD} 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 b_{SR_min} > b_SD + b_{R_minD} holds. By other hand, the ADFSR cooperative protocol presents a better performance when the relation b_{SR_min} <b_SD + b_{R_minD} holds, corroborating the results obtained in [21]. It should be noted that the parameters b_{SR_min} and b_{R_minD} do not necessarily have to be related to the same relay node. According to Fig. 2 when d_SR − d′_SR = 0.3 km or d_SR − d′_SR = 0.6 km, only one relay is around of the circumference of radius d_SR (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.

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

Download Full Size | PDF

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 d_SR = d′_SR. This maximum source-relay link distance, d_{SR_max}, 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 β_{SR_min} = β_SD + β_{R_minD} 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 β_{SR_min} by its corresponding expression in Eq. (4b), we can obtain the closed-form expression for d_{SR_max} through some algebraic manipulations as follows

d_{{SR}_{max}} (km) = {(\frac{1}{1.23 C_{n}^{2} κ^{7 / 6}})}^{6 / 11} (\frac{{ln}^{6 / 11} (1 + \frac{1}{β_{SD} + β_{R_{min} D}})}{{(0.44 - 0.69 {ln}^{6 / 5} (1 + \frac{1}{β_{SD} + β_{R_{min} D}}))}^{5 / 11}}) \times 10^{- 3},

where β_SD and β_{R_minD} 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,

C_{n}^{2}

, 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 β_SD → 1 and β_{R_minD} → 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 d_SD=3 km, together with values of normalized beam width and normalized jitter of (ω_z/r,σ_s/r)=(7,1) and (ω_z/r,σ_s/r)=(10,1). For the sake of simplicity, hereinafter it is assumed that d_SR = 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 d_SR = 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 d_SR = 2.4 km under strong turbulence conditions corresponding to ADF-SR cooperative protocol, when the number of relays is 1, 2 and 3, respectively.

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.

Download Full Size | PDF

Fig. 4 BER performance for a source-destination link distance of d_SD=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 (ω_z/r,σ_s/r) =(7,1) and (ω_z/r,σ_s/r)=(10,1).

Download Full Size | PDF

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, G_d is depicted for a source-destination link distance d_SD = 3 km and d_SR = d′_SR when the number of relays is N = {2,3}. In this configuration, a normalized beam width of ω_z/r = 7 and different values of normalized jitter σ_s/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 (ω_z/r,σ_s/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 σ_s/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 (ω_z/r,σ_s/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 $φ_{SR}^{2} > β_{SD} + β_{R_{min} D}$ 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 d_SD = 3 km and a source-relay link distance of d_SR = 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

ω_{z} / r_{optimum} \approx 4.013 σ_{s} / r - 0.153 .

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

Download Full Size | PDF

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 d_SD = 3 km and d_SR = 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 σ_s/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.

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

Download Full Size | PDF

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

Download Full Size | PDF

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 (ω_z/r,σ_s/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 β_SD ≈ β_R₁D ≈ β_R₂D ≈ β_R₃D holds in this scenario. Thus, the impact of the pointing errors translates into a coding gain disadvantage, D_pe[dB], as obtained in [21, eqn. (28)], depending finally on the parameters corresponding to the source-destination link as follows

D_{pe} [dB] \approx (20 / β_{SD}) {log}_{10} (φ_{SD}^{2} / (A_{0}^{β_{SD}} (φ_{SD}^{2} - β_{SD}))) .

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 β_{SR_min} > β_SD + β_{R_minD} is satisfied, wherein β_{SR_min}, β_{R_minD} and β_SD 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

P_{b}^{SR} = P_{b}^{SD - SR} + \sum_{k = 1}^{N} P_{b}^{{BDF}_{k}^{0}} \cdot P_{b}^{{SR}_{k}^{0} - SR} + \sum_{k = 1}^{N} P_{b}^{{BDF}_{k}^{1}} \cdot P_{b}^{{SR}_{k}^{1} - SR} .

In this way, the closed-form solution for the BER corresponding to the source-destination link under the assumption that I_SD > I_{SR_k}, i.e. I_{SR_k} = max{I_SR₁,...,I_{SR_N} },

P_{b}^{SD - SR}

, can be written as in Eq. (9), but substituting the corresponding coefficients

a_{RD}^{T}

,

b_{RD}^{T}

and

c_{RD}^{T}

by the corresponding coefficients

a_{SR}^{T}

,

b_{SR}^{T}

and

c_{SR}^{T}

, respectively. Therefore, corresponding expressions for coefficients

a_{SR}^{T}

,

b_{SR}^{T}

and

c_{SR}^{T}

are given by

a_{SR}^{T} = \frac{a_{SD} \prod_{i = 1}^{N} a_{{SR}_{i}}}{\prod_{i = 1}^{N} b_{{SR}_{i}}}, b_{SR}^{T} = b_{SD} + \sum_{i = 1}^{N} b_{{SR}_{i}}, \frac{c_{SR}^{T}}{a_{SR}^{T}} = \frac{c_{SD}}{a_{SD}} + \sum_{i = 1}^{N} \frac{c_{{SR}_{i}} b_{{SR}_{i}}}{a_{{SR}_{i}} (b_{{SR}_{i}} + 1)} .

Next, the corresponding solution for the BER in source-relay link under the assumption that I_SD < I_{SR_k} as follows

P_{b}^{{SR}_{k}^{1} - SR} ≃ (b_{{SR}_{k}} / b_{SD}) 2^{(b_{SD} + \sum_{i = 1}^{N} b_{{SR}_{i}})} \cdot P_{b}^{SD - SR} .

By other hand, the probability when the bit has been detected correctly at the relay node R_k, i.e.

P_{b}^{{SR}_{k}^{0} - SR}

, can be written as

P_{b}^{{SR}_{k}^{0}} ≐ \int_{0}^{\infty} \prod_{\begin{array}{l} j = 1 \\ j \neq k \end{array}}^{N} F_{I_{{SR}_{j}}} (i) F_{I_{SD}} (i) f_{I_{{SR}_{k}}} (i) d i .

It can be noted that the asymptotic behavior in

P_{b}^{{SR}_{k}^{0} - SR}

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-R_k-D is detected correctly or incorrectly. Hence,

P_{b}^{{BDF}_{k}^{0}}

can be written as in Eq. (19) under the assumption that I_{SR_k} > I_SD, but substituting the corresponding coefficients

a_{DF}^{T}

,

b_{DF}^{T}

and

c_{DF}^{T}

by the corresponding coefficients

a_{BDF}^{T}

,

b_{BDF}^{T}

and

c_{BDF}^{T}

, respectively. Therefore, corresponding expressions for coefficients

a_{BDF}^{T}

,

b_{BDF}^{T}

and

c_{BDF}^{T}

are given by

\begin{array}{l} a_{BDF}^{T} = \frac{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}, \\ \frac{c_{BDF}^{T}}{a_{BDF}^{T}} = \frac{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})} . \end{array}

Finally, when the bit is detected incorrectly, the BER can be written as

P_{b}^{{BDF}_{k}^{1}} ≐ \int_{0}^{\infty} \int_{0}^{2 i_{2}} f_{I_{SD}} (i_{1}) f_{I_{R_{k} D}} (i_{2}) d i_{1} d i_{2} .

Finally, taking into account the asymptotic behavior previously obtained, the expression in Eq. (34) can be simplified as follows

P_{b}^{SR} ≐ {\begin{array}{l} P_{b}^{{BDF}_{min}^{0}} \cdot P_{b}^{{SR}_{min}^{0}} & b_{R_{min} D} < \sum_{i = 1}^{N} b_{{SR}_{i}} \\ (1 + \frac{2 (b_{SD} + \sum_{i = 1}^{N} b_{{SR}_{i}})}{b_{SD}} \sum_{k = 1}^{N} b_{{SR}_{k}} P_{b}^{{BDF}_{k}^{1}}) \cdot P_{b}^{SD - SR} & b_{R_{min} D} > \sum_{i = 1}^{N} b_{{SR}_{i}} \end{array}

Acknowledgments

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

References and links

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

2. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications, 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, 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).

Impact of relay placement on diversity order in adaptive selective DF relay-assisted FSO communications

Abstract

1. Introduction

2. System model

3. Channel model

4. Error-rate performance analysis

5. Numerical results and discussions

5.1. Diversity order gain analysis

5.2. Impact of pointing errors

6. Conclusions

Appendix

Acknowledgments

References and links

Cited By

Figures (7)

Equations (41)

Optics Express