Multiple User Pair Scheduling in TWR Assisted FSO ... - OSA Publishing

290 J. OPT. COMMUN. NETW./VOL. 8, NO. 5/MAY 2016

Puri et al.

Multiple User Pair Scheduling in TWR Assisted FSO Systems Parul Puri, Parul Garg, and Mona Aggarwal

Abstract—In this paper, various multiuser scheduling schemes are applied to a two-way relay (TWR) assisted multiple user pair free-space optical (FSO) communication network, when the optical beam is subjected to path loss, misalignment errors, and atmospheric turbulence. The FSO network consists of a single, half-duplex, decodeand-forward, two-way optical relay that serves multiple user pairs by employing a scheduler. Three types of schedulers based on absolute signal-to-noise ratio (SNR), normalized SNR, and selective multiuser diversity are considered. Further, the atmospheric turbulence is modeled by a recently proposed statistical model, namely the Málaga M– distribution. For the presented system and channel models, closed-form and asymptotic expressions for outage probability and bit error rate are derived. The mathematical analysis is accompanied by Monte Carlo simulations and several numerical examples to illustrate the effect of the key system parameters. Index Terms—Free-space optics; Meijer’s G-function; M– distribution; Multiuser scheduling; Two-way relay.

I. INTRODUCTION

F

ree-space optics (FSO) has gained significant attention as it provides a solution for the ever-increasing demand for high-speed communication. Among its several advantages, the FSO technology offers a license-free bandwidth with a high transmission capacity. However, the performance of FSO systems is highly vulnerable to various impairments, such as path loss, misalignment errors, and atmospheric turbulence [1,2].

A popular technique used to overcome these limitations is the relay-assisted FSO communication. The relayassisted FSO systems employ intermediate relay(s) to establish a line-of-sight (LOS) link between two user nodes that may have substantial distance or obstructions between them. The FSO literature describes one-way relay (OWR) [3,4] and two-way relay (TWR) assisted FSO systems [5,6]. In the TWR assisted FSO system in [5], a single Manuscript received August 25, 2015; revised February 23, 2016; accepted March 14, 2016; published April 22, 2016 (Doc. ID 248400). P. Puri is with the Department of Electronics and Communication Engineering, Jaypee Institute of Information Technology, Noida 201301, India. P. Garg is with the Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, New Delhi 110078, India (e-mail: [email protected]). M. Aggarwal is with the Department of Electrical, Electronics and Communication Engineering, The Northcap University, Gurgaon 122017, India. http://dx.doi.org/10.1364/JOCN.8.000290

1943-0620/16/050290-12

TWR was employed to carry out data exchange between two users in two time phases. Furthermore, [6] described a parallel relayed FSO system, in which TWR selection was carried out to establish a two-time-phase bi-directional communication link between two users. Since the TWRFSO system required two time phases for complete data exchange between two users, it was considered spectrally more efficient in comparison to OWR assisted FSO systems, which require three or four time phases. In many practical applications, data exchange has to be carried out among multiple non-LOS user pairs. One approach is to employ dedicated relays for each user pair. However, this results in an inefficient utilization of the resources (relays), when all the user pairs do not have data for exchange. Furthermore, this approach requires a large number of relays as the number of user pairs increases. Therefore, in order to achieve resource optimization, the available relays must be shared. The radio-frequency literature describes two techniques for sharing the available resources [7,8], where a single TWR was shared by multiple user pairs. In the first technique [7], all the user pairs transmitted to a single TWR simultaneously; then the TWR broadcasted a combined signal to all the user pairs. However, this technique is not suitable for FSO systems since the FSO links are inherently point-to-point and the relay must use separate lasers for each user (without broadcasting). Further, using multiple lasers requires power-distribution strategies, which increases the complexity at the relay. The second technique [8] described a single TWR that served only one user pair at any given time. This was achieved by selecting a single user pair by using the multiuser diversity techniques. The motive was to maximize the system performance by allocating at any time the common TWR to the user pair that can best exploit it. Various multiuser diversity techniques based on absolute signal-to-noise ratio (SNR), normalized SNR, and selective multiuser diversity techniques were described in [9,10]. Recently, multiuser diversity techniques have attracted considerable interest in the FSO community [11–16]. These systems exploit the time-varying behavior of the turbulence-induced fading to achieve diversity gain. In [11,12], diversity techniques were applied to receive the optical signals in dual-branch FSO systems. Further, [13] described a point-to-multipoint FSO system based on multiuser diversity scheduling. In [14], a single-relay selection-based parallel relayed FSO system with one user pair was described. However, the literature lacks a

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multiuser (multipoint-to-multipoint) system in which users can exchange data with each other by utilizing the available resources via scheduling. Also, the multiuser diversity-based systems discussed so far did not consider pointing errors. Since the severity of pointing errors is more in high-rise buildings [2], where FSO transceivers are deployed, it is important that these impairments are included for analysis. It is well known that the energy of the laser beam scatters as it propagates through the atmospheric turbulent channel. The atmospheric turbulence can be modeled using various statistical models such as log-normal (weak turbulence) [1], K (strong turbulence), and Gamma–Gamma distribution (moderate-strong turbulence) [17]. Most of the statistical models assume that the propagation geometry of the beam has two components, namely the LOS component and the scattered component, which are assumed to be independent of each other. However, in practice there exists a coupled-to-LOS component, which is due to the high directivity and narrow beam width of the laser beams. The M–distribution, which was recently proposed in [18], considers all three components of the propagation geometry. Further, M–distribution generalizes the probability density functions (PDFs) of most of the optical channel models developed so far and is suitable for all weakmoderate-strong turbulence strengths. Motivated by this background, we address a TWRassisted multiple user pair FSO system that provides high spectral efficiency and effective resource utilization. The main contributions of the paper are 1) to model a bi-directional, decode-and-forward (DF), single-relay assisted multiple user pair FSO system that establishes communication between a selected user pair in two time phases; 2) to apply various multiuser scheduling schemes; 3) to analyze the considered system over an optical channel affected by path loss, pointing errors, and atmosphericturbulence-induced fading, which is modeled using the M– distribution; 4) to derive closed-form expressions for the outage probability and bit error rate (BER) of the system for the various scheduling schemes; and 5) to derive asymptotic expressions and diversity order in order to get a physical insight into the derived analytical results.

TABLE I NOTATIONS Notation

Meaning

Path loss Hl Ha Atmospheric turbulence Pointing errors Hp H Channel state information R Responsivity of photodetector Quantum efficiency of photodetector ηe L Link length C2n Refractive index structure constant σ Attenuation coefficient α Large-scale scintillation parameter β Small-scale scintillation parameter ρ Scattering power coupled to the LOS component γ0 Average power of independent scatter component Average power of total scatter components 2b0 Ω Average power of LOS component Ω0 Average power from coherent contributions A M–distribution parameter σ 2R Rytov variance r Aperture radius we Equivalent beam-width radius Standard deviation of pointing error displacement σs ξ Pointing error parameter σ 2n Variance of noise power Instantaneous SNR of intermediate link ΓS;u Ωu Instantaneous SNR of uth end–end link Average electrical SNR γ¯ S;u Threshold SNR γ th f X · PDF of random variable X CDF of random variable X F X · Γ· Gamma function Γ·; · Complementary incomplete gamma function Modified Bessel function of second kind and order v K v · a1 ;…;ap Gm;n Meijer’s G-function p;q  xjb1 ;…;bq 

optical relaying network consisting of U number of user pairs that communicate with each other via a single relay. The U-pair relaying system is depicted in Fig. 1. The relay R consists of 2U antenna apertures, which are directed toward the corresponding user nodes, T A;u and T B;u , indexed by u ∈ f1; 2; …; Ug. An optical switch is used to select the direction of transmission by switching between various

The rest of the paper is organized as follows: Section II provides the statistical characteristics of the system and channel models under consideration. This is followed by Section III, which introduces various multiuser diversity techniques. Closed form and asymptotic expressions for system outage probability and BER are derived in Sections IV and V, respectively. Section VI provides the numerical results, and finally, we conclude in Section VII. The main notations used in this paper are summarized in Table I.

II. STATISTICAL CHARACTERISTICS A. System Model We consider a multiuser, bi-directional, half-duplex subcarrier intensity modulation with a direct-detection-based

Fig. 1. Multiple-user-pair TWR assisted FSO network.

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transmit apertures at the relay [14]. Equivalently, each user node is equipped with a single aperture antenna directed toward the corresponding relay. Before the beginning of communication process, users having data for transmission feedback their channel state information (CSI) to the relay. Assuming perfect CSI at the relay, the relay selects a single user pair and establishes communication in two time phases. During the first time phase, the nodes of the selected user pair u, i.e., T A;u and T B;u , transmit information-bearing signals, xA;u and xB;u , respectively, to the relay over their respective FSO links. The signals received at the relay are yA;u ;R  RR H A;u ;R xA;u  nR ; y

B;u ;R

 RR H

B;u ;R

x

B;u

 nR ;

(1)

where RR  ηe q∕hf is the responsivity of the photodetector at the relay. Here, ηe is the quantum efficiency of the photodetector, q is the electron’s charge, h is the Planck’s constant, and f is the optical frequency [1, Eq. (25)]. Further, nR is the noise at the relay and H S;u ;R is the channel state between S; u , where S ∈ fA; Bg, and the relay, R. The photodetector at the relay generates an electrical signal, which is then decoded. After decoding, the relay re-encodes the signals and converts the electrical signals to optical signals, which are forwarded to the users in the second time phase. Assuming yA;u ;R yB;u ;R  in Eq. (1) is decoded as xˆ A;u ˆxB;u , the signals received at T A;u and T B;u from the relay can be written as

(2)

where RS;u , S ∈ fA; Bg, is the responsivity of the photodetector at the terminal node, S; u , and nS;u is the noise at S; u . Finally, the data received from the relay is decoded at the respective destinations to recover the information sent from the opposite terminal.

β X

ak 2 −1

a k ha

k1

 s αβha K α−k 2 0 ; γ β  Ω0

(4)

where K v · is a modified Bessel function of the second 2 kind of order v, and γ 0  EjU G S j   2b0 1 − ρ represents the average power of the independent scatter component, where E· is the expectation operator. Here, 0 ≤ ρ ≤ 1 is the amount of scattering power 2 coupled to the LOS component and 2b0  EjU C Sj   2 EjU G j  represents the average power of the total S scatter components. Furthermore, Ω0 in Eq. (4) represents the average power from the coherent contribup tions and is given by Ω0  Ω  2ρb0  2 2b0 Ωρ cos ϕA − ϕB . Here, Ω  EjU L j2  represents the average power of the LOS component, and ϕA and ϕB are the deterministic phases of the U L and U C S components. α and β denote the large-scale and small-scale scintillation parameters, respectively, and are given as [17, Eqs. (18) and (19)] 1 e

12 7 6 0.49σ 2R ∕11.11σ R5 

β≅

; −1

1 12 5 6 0.51σ 2R ∕10.69σ R5 

e

; −1 (5)

where σ 2R  1.23C2n k7∕6 L11∕6 is the Rytov variance for a plane wave optical model. Here, k is the optical wave number, C2n is the refractive index structure constant, and L is the link distance. Further, A and ak in Eq. (4) are defined as [18, Eq. (25)] βα  α 2 2α2 γ0β ; α 0 0 γ 012 Γα γ β  Ω       β − 1 γ 0 β  Ω0 1−k2 Ω0 k−1 α k2 : ak ≜ β k − 1! γ0 k−1 A≜

B. Optical Channel Model The CSI is represented as H  H l × H a × H p , where H l is the path loss, H a is the atmospheric turbulence-induced fading, and H p is the misalignment fading (pointing errors). All the channels are assumed to be stationary, memoryless, and ergodic. Furthermore, since the FSO links are distance dependent, we have assumed independent but not necessarily identically distributed (i.n.i.d.) fading statistics [11]. 1) Path Loss H l : The path loss factor, H l , is expressed using the exponential Beer’s–Lambert Law as [19] H l L  exp−σL;

f H a ha   A

α≅

yA;u  RA;u H R;A;u  xˆ B;u  nA;u ; yB;u  RB;u H R;B;u  xˆ A;u  nB;u ;

2) Atmospheric Turbulence H a : The turbulence-induced fading is modeled by M–distribution. In this model, the geometry of the laser beam observed at the receiver consists of a LOS component U L , a coupled-to-LOS scatter component U C S , and an independent scatter component U G . Thus, H a follows an M–distribution S and is denoted by Mα; β; γ 0 ; ρ; Ω0  with the PDF given as [18, Eq. (24)]

(3)

where L is the link length between the terminal nodes and σ is the attenuation coefficient.

(6)

3) Misalignment Error H p : Building sway causes misalignment between the received optical beam and detector plane, which results in pointing errors. The PDF of H p is given as [19, Eq. (11)] f H p hp  

ξ2 2

Aξ0

2

hpξ −1 ;

0 ≤ hp ≤ A0 ;

(7)

where A0 is the fraction of the collected power at r  0, where r is the aperture radius, and ξ  we ∕2σ s , where we is the equivalent beam-width radius and σ s is the standard deviation of the pointing error displacement at the receiver.

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Using Eqs. (1), (4), and (7), we obtain the PDF of the combined channel statistics as [20, Eq. (23)]

f H h 

  −αk   β  ξ2  1 2 ξ2 A X αβ αβh 3;0  ; ak 0 G 1;3 2h k1 γ β  Ω0 γ 0 β  Ω0 A0 H l  ξ2 ; α; k

   βA;u ξ2A;u AA;u X κ1  κ10 6;0 f Ωu γ  κ G γ 4γ k1 8 2;6 γ A;u  κ11    βB;u ξ2 AB;u X κ4  κ 12 κ 9 G6;0 γ  B;u 2;6 4γ γ B;u  κ 13 j1

(8)

−

a1 ;…;ap where Gm;n p;q xj b1 ;…;bq  is the Meijer’s G-function [21, Eq. (8.2.1.1)].

  βB;u     κ7 κ1  κ10 X κ4  κ5 6;1 κ 8 G6;0 γ κ G γ 2;6 γ k1 γ A;u  κ11 j1 9 3;7 γ B;u  κ6 βA;u X

  βA;u     βB;u κ7 X κ4  κ 12 X κ1  κ2 6;0 6;1 ; κ G γ κ G γ − γ j1 9 2;6 γ B;u  κ 13 k1 8 3;7 γ A;u  κ3

C. Instantaneous Signal-to-Noise Ratio

(12)

The instantaneous SNR of the intermediate link between terminal node S; u, where S ∈ fA; Bg, u ∈ f1; 2; …:; Ug, and the relay, R, is given as ΓS;u 

ηe H S;u;R 2 , σ 2n;R

where σ 2n;R is the κ1 

variance of the noise power at the relay. The PDF of ΓS;u is obtained by applying transformation of random variables (RVs) in Eq. (8) and is given as [22, Eq. (19)]

f ΓS;u γ S;u  

 −αS;u k βS;u ξ2S;u AS;u X 2 α β ak 0 S;u S;u 0 4γ S;u k1 γ βS;u Ω   r 2 αS;u βS;u ξ2S;u γ S;u  ξS;u 1 ×G3;0 ; 1;3 γ 0 βS;u Ω0 ξ2S;u 1 γ S;u  ξ2S;u ;αS;u ;k (9)

where γ S;u is the average electrical SNR at node S; u and is defined as γ S;u 

where

 2 2 ξS;u ηe EH S;u;R 2 η2e A20;S;u H 2l;S;u  : (10) σ 2n;R σ 2n;R 1  ξ2S;u

α2A;u β2A;u ξ4A;u

; 16γ 0 βA;u  Ω0 2 ξ2A;u  12

κ2  1;

ξ2A;u  1 ξ2A;u  2 ; ; 2 2

ξ2A;u ξ2A;u  1 αA;u αA;u  1 k k  1 ; ; ; 0; ; ; ; 2 2 2 2 2 2 α2B;u β2B;u ξ4B;u ξ2B;u  1 ξ2B;u  2 κ4  ; κ  1; ; ; 5 2 2 16γ 0 βB;u  Ω0 2 ξ2B;u  12 κ3 

ξ2B;u ξ2B;u  1 αB;u αB;u  1 j j  1 ; ; ; 0; ; ; ; 2 2 2 2 2 2  −αA;u k ξ2 AA;u ξ2B;u AB;u 2 α β 2αA;u k−2 κ7  A;u ; ; κ8  ak 0 A;u A;u 0 16 π γ βA;u  Ω  −αB;u j ξ2  1 ξ2A;u  2 2 α β 2αB;u j−2 ; κ10  A;u ; ; κ9  aj 0 B;u B;u 0 2 2 π γ βB;u  Ω κ6 

ξ2A;u ξ2A;u  1 αA;u αA;u  1 k k  1 ; ; ; ; ; ; 2 2 2 2 2 2 ξ2  1 ξ2B;u  2 κ12  B;u ; ; 2 2 ξ2 ξ2  1 αB;u αB;u  1 j j  1 κ13  B;u ; B;u ; ; : ; ; 2 2 2 2 2 2

κ11 

D. Two-Way Relay Channel Proof: A detailed proof is provided in Appendix A. Using the statistics of the SNR of the intermediate FSO link defined in Eq. (9), we derive an exact expression for the cumulative distribution function (CDF) and PDF of the SNR of the TWR-FSO channel. Theorem 1. The CDF, F Ωu γ, and PDF, f Ωu γ, of the SNR of a TWR-FSO link, given as Ωu  minΓA;u ; ΓB;u , for any u ∈ U is given as F Ωu γ 

   βA;u ξ2A;u AA;u X κ1  κ2 κ 8 G6;1 γ 3;7 4 γ A;u  κ3 k1    βB;u ξ2 AB;u X κ4  κ 5 κ 9 G6;1 γ  B;u 3;7 4 γ B;u  κ 6 j1

− κ7

βA;u X k1

κ8 G6;1 3;7



  βB;u    κ1  κ2 X κ 4  κ5 ; γ κ9 G6;1 γ 3;7 γ A;u κ3 γ B;u  κ6 j1

(11)

III. MULTIUSER SCHEDULING In this section, we describe multiuser scheduling techniques and their respective u th user pair selection criteria. It is assumed the users feedback their CSI via a reliable feedback channel and user pair selection is made after every two time phases, i.e., time required for one complete data transfer. Further, all the participating user pairs have data for transmission at every time instant.

A. Absolute SNR-Based Scheduling In absolute SNR-based scheduling, the user pair with the maximum instantaneous SNR is chosen in each time slot. Hence, u is based on the condition [9, Eq. (7)]

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(13)

A. Absolute SNR-Based Scheduling Using the criterion defined in Eq. (13), system outage probability for absolute (ABS) SNR-based scheduling is given as [9, Eq. (7)]

where Ωu  minΓA;u ; ΓB;u  and u  1; 2; …:U.

B. Normalized SNR-Based Scheduling In normalized SNR-based scheduling, the user pair with the maximum value of normalized SNR is chosen in each time phase. Hence, u is based on the condition [9, Eq. (32)]   Ωu ; u∈U Ωu

u  arg max

(14)

where Ωu is the average value of Ωu for the uth user pair in a given time phase. This type of scheduling helps to achieve fairness among user pairs based on the long-term condition of the channel, unlike the absolute SNR-based scheduling scheme in which user pairs that are nearest to the relay are most likely to be selected.

Pout u ;ABS  maxfF Ωu γ th g  u∈U

U Y

F Ωu γ th ;

where F Ωu · is the CDF of SNR Ωu given in Eq. (11). Substituting Eq. (11) into Eq. (17), the closed-form expression for system outage probability with absolute SNRbased scheduling is obtained as Pout u ;ABS

   κ 1 γ th  κ2  4 γ A;u  κ3 u1 k1    βB;u κ ξ2B;u AB;u X 6;1 κ 4 γ th  5 κ9 G3;7   4 γ B;u κ 6 j1   βB;u     βA;u X κ κ1 γ th  κ2 X 6;1 κ 4 γ th  5 : κ8 G6;1 κ G −κ7 9 3;7 3;7   γ γ A;u κ 3 B;u κ 6 j1 k1 βA;u U  ξ2 A Y X A;u A;u

κ8 G6;1 3;7

C. Selective Multiuser Diversity Scheduling

(18)

Selective multiuser diversity (SMUD) scheduling is designed to reduce the CSI feedback sent by users to the relay at every time slot and to reduce the relay’s processing load. In this scheme, the number of feedback users Pt at any time t depends on γ T as [10, Section (4.2)] Pt  cardfτ; such that ΓS;u t ≥ γ T g;

(15)

where card is the cardinal operator and γ T is the predetermined threshold. Depending on system requirements, the threshold may be selected. When none of the user’s SNR is above γ T and we obtain Pt  0, the relay randomly selects a user pair for transmission. However, when Pt > 0, the relay selects a user pair with the maximum instantaneous SNR. Hence, the selection criterion for u is given as

u 

8 < arg randfΩu g;

Pt  0;

: arg maxfΩu g;

Pt > 0;

u∈U

u∈U

(17)

u1

(16)

where rand is the random operator. As evident, the number of feedback users is reduced in this type of scheduling scheme as compared to absolute and normalized SNRbased scheduling, thereby decreasing the feedback load.

Asymptotic Analysis: To get physical insight into the results, we perform the asymptotic analysis at high SNRs, assuming independent and identically distributed (i.i.d.) fading channels between user pairs. Rewriting Eq. (A2), we obtain F Ωu γ  2F ΓS;u γ − F 2ΓS;u γ  1 − 1 − F ΓS;u γ2 :

(19)

Substituting Eq. (19) into Eq. (17), we obtain Pout u ;ABS 

U Y

1 − 1 − F ΓS;u γ th 2 :

(20)

u1

In order to obtain the asymptotic expression for outage, we require the asymptotic expression of the CDF. The asymptotic expression for the CDF of the SNR is evaluated in Appendix B. Substituting the asymptotic expression of F ΓS;u γ from Eq. (B2) into Eq. (20), the asymptotic expression for system outage probability for absolute SNR-based scheduling is obtained as

Pout u ;ABS

U  Y



 ξ2S;u  ξ2S;u AS;u γ th 2 1− 1− ≈ ψ ξ2 S;u 4 γ S;u u1 α   S;u X  k 2 βS;u 2 2 γ γ ; ψ αS;u th  ψ k th γ S;u γ S;u k1

(21)

where ψ ξ2 , ψ αS;u , and ψ k are as in Eq. (B3). S;u

IV. OUTAGE PROBABILITY ANALYSIS B. Normalized SNR-Based Scheduling In this section, we derive closed-form and asymptotic expressions for the outage of a multiuser TWR-FSO system for different scheduling schemes described in the previous section.

Using the criterion defined in Eq. (14), the system outage probability for normalized (NORM) SNR-based scheduling is given as

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Pout u ;NORM  F Ωu ;NORM γ th ;

(22)

where F Ωu ;NORM · is the CDF of the SNR of the u th user pair post scheduling under normalized SNR-based scheduling, and is given as [9, Eq. (70)]   Ω Pr Ωu ≤ γ and u max ; Ωu u1 Z Ωu U U Y X Ω u f u x F m xdx 

F Ωu ;NORM γ 

0

m1;m≠u

 U U  1X Ω  Fm u ; U u1 Ωu

(23)

where f u · and F m · are the PDF and CDF of the normalized SNR, Ωu ∕Ωu . Now, using Eqs. (11), (22), and (23), we obtain    βA;u U  ξ2 A X  κ2 1X A;u A;u 6;1 κ 1 κ8 G3;7 γ th  U u1 4 Ω κ3 u k1    β 2 B;u ξ AB;u X κ4  κ5 κ 9 G6;1 γ th   B;u 3;7 4 Ω κ6 u j1    U    β β A;u u X  κ5 κ1  κ2 X 6;1 κ 4  κ8 G6;1 γ κ G γ : −κ7 th  9 3;7 th  3;7 Ω Ω κ κ6 u u 3 j1 k1

Pout u ;NORM 

(24) It may be noted when the average value of Ωu Ωu  is the same for all users, i.e., the i.i.d. case, normalized SNRbased scheduling is equivalent to absolute SNR-based scheduling. Asymptotic Analysis: Substituting the asymptotic expression of F ΓS;u γ from Eq. (B2) into Eq. (23) and using Eq. (22), the asymptotic expression for system outage probability for normalized SNR-based scheduling is obtained as Pout u ;NORM

  ξ2S;u  U  ξ2S;u AS;u 1X γ th 2 1− 1− ≈ ψ ξ2 S;u U u1 4 γ S;u  αS;u X  k 2 U βS;u 2 2 γ γ  ψ αS;u th  ψ k th : γ S;u γ S;u k1

Pout u ;SMUD

System outage probability for SMUD scheduling is given as [10, Eqs. (6) and (7)] Pout u ;SMUD  F Ωu ;SMUD γ th ;  PrΩu ≤ γ th ; Ω < γ T ; Pt  0;  PrΩu ≤ γ th ; Ω ≥ γ T ; Pt > 0;

U X

u1

C. Selective Multiuser Diversity Scheduling

where F Ωu ;SMUD · is the CDF of the SNR of the u th user pair post scheduling under SMUD-based scheduling. In Eq. (26), when Pt  0, it is observed that the system is in outage when the SNR of all user pairs is below γ T and the randomly picked user pair SNR is below γ th. Generalizing the expression in [10, Eq. (6)] for the i.n.i.d. case, PrΩu ≤ γ; Ω < γ T  is obtained as  F Ωu ;SMUD γ  Pr Ωu ≤ γ and 

U Y

 Ωu < γ T ;

u1;u≠u

U U Y 1X F γ F γ: U u1 j1;j≠u Ωj T Ωu

(27)

When γ  γ T , this expression is equal to the CDF for the ABS SNR-based scheduling given in Eq. (17). When Pt > 0, the system is in outage when the SNR of τ user pairs given by Eq. (15) is above γ T and the user pair with maximum SNR is below γ th. Generalizing the expression in [10, Eq. (7)] for the i.n.i.d. case, PrΩu ≤ γ; Ω ≥ γ T  is obtained as

F Ωu ;SMUD γ 

 u U  U X Y U Y F Ωi γ − F Ωi γ T  F Ωj γ: u u0

i1

ju1

(28)

(25)

Hence, by substituting Eqs. (27) and (28) into (26), the closed-form expression for system outage probability for SMUD scheduling is obtained as

8 U U > X Y > > 1 > F Ωj γ T F Ωu γ th ; > > U Y > > F Ωi γ th  − F Ωi γ T  F Ωj γ th ; > : u u0 i1

(26)

γ T ≥ γ th ; (29) γ T < γ th :

ju1

Asymptotic Analysis: Substituting the asymptotic expression of the CDF from Eq. (B2) into Eq. (27), we obtain the asymptotic expression for the multiuser TWR-FSO system outage probability with SMUD scheduling for γ T ≥ γ th in Eq. (30). Similarly, by substituting Eq. (B2) into Eq. (28), one can obtain the asymptotic expression of outage for SMUD scheduling when γ T < γ th :    ξ2S;j  αS;j X  k 2  βS;u 2 A U U X Y ξ 1 γT 2 γT 2 γ 2 S;j S;j out Pu ;SMUD ≈ 1− 1−  ψ αS;j  ψk T ψ ξ2 S;j U u1 j1;j≠u 4 γ S;j γ S;j γ S;j k1    ξ2S;u  αS;u X  k 2  βS;u ξ2S;u AS;u γ th 2 γ th 2 γ th 2 × 1− 1− ;  ψ αS;u  ψk ψ ξ2 S;u 4 γ S;u γ S;u γ S;u k1

γ T ≥ γ th :

(30)

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V. ERROR RATE ANALYSIS In this section, we derive analytical and asymptotic expressions for the average BER for different scheduling schemes described in Section II. The average BER based on the CDF is defined as [23, Eq. (21)] Z Pe  −

∞

P0e εjγF Ωu γdγ;

0

(31)

where P0e εjγ is the first-order derivative of the conditional error probability (CEP) for the given SNR. For coherent and non-coherent binary modulation schemes, a unified expression for CEP over the additive white Gaussian noise channel is given as [23, Eq. (25)] Pe εjγ  Γb;aγ 2Γb , where Γ·; · is the complementary incomplete gamma function defined in [24, Eq. (8.350.2)] and the values of parameters a and b corresponding to different modulation schemes are as summarized in Table II. Differentiating Pe εjγ and substituting into Eq. (31) we obtain Pe as [11, Eq. (12)] Pe 

ab 2Γb

Z

∞

0

e−aγ γ b−1 F Ωu γdγ:

(32)

Now, by substituting the CDFs of various scheduling schemes in Eq. (32), the respective expressions for BER are evaluated as follows.

and RN is the error term. The error becomes negligible for higher values of N. The roots and corresponding weights for various values of n and order N are available in the literature in the form of tables [26,27]. Asymptotic Analysis: In order to get physical insight into the results, we derive asymptotic expressions for the BER assuming coherent-BPSK modulation and i.i.d. fading statistics among all users. For PSK modulation, the error probability in terms of the CDF can be written as [28, Eq. (4)]   2  X Pe  EX F Ωu ; (35) 2 where X ∼ N0; 1 is a standardR normal-distributed ran∞ gxf xdx, Eq. (35) dom variable. Using EgX  −∞ X can be rewritten as 1 Pe  p 2π

b

Pe;ABS

a  2Γb

Z

∞

0

e−aγ γ b−1

U Y

F Ωu γdγ:

F Ω ;ABS γ 

 2u   U  X U X 2u u0

u

l0

N 1 X ≈ w f z   RN ; 2Γb i1 i i

(34)

Q where f zi   U u1 F Ωu zi ∕a, zi is the ith root of the Nth order Laguerre polynomial, wi is the corresponding weight, TABLE II BINARY MODULATION SCHEMES

FOR

Modulation Scheme Coherent binary frequency shift keying Coherent binary phase shift keying Non-coherent binary frequency shift keying Differential binary phase shift keying

b

a

0.5 0.5 1 1

0.5 1 0.5 1

(37)

l

−1lu F ΓS;u γl ;

(38)

where F ΓS;u γ is the asymptotic CDF given by Eq. (B2). Using Eq. (B2), F ΓS;u γl is obtained as F ΓS;u γl 

ξ2S;u AS;u 4

l X l  X l−r l ψ 2 l−r−t r t0 ξS;u r0

rβ S;u X

 χv

vr

(33)

−x n

(36)

Using binomial expansion, Eq. (37) is given as

× ψ αS;u 

u1

 2  2 x x exp − dx: 2 2

F Ω ;ABS γ  1 − 1 − F ΓS;u γ2 U :



The integrals of the form 0 e x f xdx in Eq. (33) can be easily computed using the Gauss–Laguerre quadrature technique [25, Eq. (25.4.45); 26,27]. Hence, Eq. (33) evaluates to

BER PARAMETERS

0

F Ωu

t

R∞

Pe;ABS

∞

Further, Eq. (20) for the i.i.d. case is written as

A. Absolute SNR-Based Scheduling The BER for an absolute SNR-based scheduling scheme is obtained by substituting Eq. (17) into Eq. (32) as

Z

coefficient of where

PβS;uχ v is

γ the k r 2 of . k1 ψ k γ u

l−r−tξ2S;u tαS;u v

γ

2

γ S;u

γ v γu

2

;

(39)

in the expansion

Substituting Eq. (38) into Eq. (36), the asymptotic error evaluates to Pe;ABS ≈

  U U X 2u  2u  X −1lu ξ2S;u AS;u l p 2 π 4 l u0 u l0   l l−r X l X × ψ ξ2 l−r−t ψ αS;u t S;u r0 r t0   l−r−tξ2S;u tαS;u v1 χ Γ rβ v S;u 2 X × : l−r−tξ2 tα v vr

γ¯ S;u

S;u 2

(40)

S;u

Diversity Order: At high SNRs, it is observed that Eq. (40) is dominated by the least power of γ S;u. Hence, the diversity order, Gd , is obtained by taking the summation of the dominating term for all users as [15] ξ 2

P S;u αS;u βS;u Gd  U u1 min 2 ; 2 ; 2 .

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Diversity Order: We can (48) that ξEq.

the diver2 P see from S;u αS;u β S;u sity order for SMUD is U min ; ; u1 2 2 2 .

B. Normalized SNR-Based Scheduling The BER for a normalized SNR-based scheduling scheme is obtained by substituting Eq. (23) into Eq. (32) as  U Z∞ U  X ab Ωu F ψu Pe;NORM  e−aγ γ b−1 dγ: (41) 2UΓb 0 Ωu u1 Using the Gauss–Laguerre quadrature technique, Eq. (41) evaluates to Pe;NORM ≈

N X 1 w f z   RN ; 2UΓb i1 i i

(42)

where  U U  X z F ψu i : a u1

f zi  

Asymptotic Analysis: The asymptotic error with i.i.d. statistics for normalized SNR-based scheduling is the same as absolute SNR-based scheduling and is given by Eq. (40).

C. SMUD Scheduling The BER for a SMUD scheduling scheme is obtained by substituting Eqs. (27) and (28) into Eq. (32) as Z∞ ab Pe;SMUD  e−aγ γ b−1 F Ωu ;SMUD γdγ: (43) 2Γb 0 Using the Gauss–Laguerre quadrature technique, Eq. (43) evaluates to Pe;SMUD ≈

N 1 X w f z   RN ; 2Γb i1 i i

(44)

where f zi  is given as 0

1 U

VI. NUMERICAL ANALYSIS In this section, we investigate the outage and error performance of various scheduling schemes for a five-user-pair TWR-FSO system. The system parameters have been taken from [20]. In all numerical examples, the link length between any transmitting node T A;u or T B;u and the relay is 1 km, λ  785 nm, and H l;S;u  0.9033; C2n is 7.2 × 10−15 m−2∕3 at night, 1.2 × 10−14 m−2∕3 at sunrise, and 2.8 × 10−14 m−2∕3 near midday. For these values, σ 2R evaluates to 0.32, 0.52, and 1.2, respectively [18, Eq. (27)]. Further, using [18, Eq. (27)] and [18, Eq. (48)], the M–distribution parameters α; β; γ 0 ; ρ; Ω0  are evaluated such that β is a natural number. Additionally, the transmitted power is normalized, i.e., Ω  2b0  1, by keeping Ω  0.5 and b0  0.25. The transmitted optical power is 11.5 dBm, the radius of the receiver aperture r  0.1 m, and the normalized beam width we ∕r  10. Figure 2 is a plot of the analytical, asymptotic, and simulated system outage probabilities with respect to the average electrical SNR, when γ th  4 dB, γ T  2 dB, ξ  0.833 and 1.66, and σ 2R  1.2 (midday). For σ 2R  1.2, we obtain Mα; β; γ 0 ; ρ; Ω0   M8.1; 4; 0.45; 0.1; 0.55. Monte Carlo simulation results for 106 samples have been included. As can be seen, the analytical curves have a close match with the simulation curves, thereby demonstrating the correctness of the derived formulas. Further, in terms of the performance of various scheduling schemes, it is observed that the absolute SNR-based scheduling outperforms normalized SNR-based scheduling. However, the NORM-SNR scheme provides scheduling fairness among all user pairs such that each user pair has an equal probability 1∕U to

U U X Y

F Ωj γ T F Ωu zi ∕a; γ T ≥ γ zi ∕a ; B u1 j1;j≠u B f zi   B U   u U Y @X U Y F Ωi zi ∕a − F Ωi γ T  F Ωj zi ∕a; γ T < γ zi ∕a : u u0 i1 ju1 Asymptotic Analysis: For i.i.d. statistics, Eq. (27) can be written as F Ωu ;SMUD γ  F Ωu γ T U−1 F Ωu γ:

(45)

(46)

Using Eq. (19), Eq. (46) can be re-written as F Ωu ;SMUD γ  1 − 1 − F ΓS;u γ T 2 U−1 × 2F ΓS;u γ − F 2ΓS;u γ:

(47)

Substituting Eqs. (47) and (B2) into Eq. (36), the asymptotic expression for error probability using the SMUD scheme is obtained in Eq. (48): "  ξ2S;j  αS;j X  k 2 #U−1   βS;u ξ2S;j AS;j γT 2 γT 2 γ 2 Pe;SMUD ≈ 1 − 1 −  ψ αS;j  ψk T ψ ξ2 S;j 4 γ¯ S;j γ¯ S;j γ ¯ S;j k1 2        2  βS;u X 6ξ2S;u AS;u 1  ξ2S;u −ξS;u 1  αS;u −αS;u 1  k −k2 2 2 6 γ¯ S;u × 4 p ψ kΓ ψ ξ2 Γ γ¯ S;u  ψ αS;u Γ γ¯ S;u  S;u 2 4 π 2 2 k1

−

ξ2S;u AS;u 2 p 32 π

2 2X 2−r X r0

r

t0

ψ ξ2 2−r−t ψ αS;u t S;u

rβ S;u χ X vr

2−r−tξ2 tα v1 3 S;u S;u Γ v 7 2 7: 5 2−r−tξ2 tαS;u v S;u 2 γ¯ S;u

(48)

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among user pairs are T S;1  2 km, T S;2  2.2 km, T S;3  1.8 km, T S;4  2.4 km, and T S;5  1.9 km, where T S;u is the distance between T A;u and T B;u , u ∈ f1; 2; …; 5g, assuming the relay is equidistant to T A;u and T B;u . As can be observed, the outage performance improves as the number of users increases. This is due to the fact that, as the number of user pairs increases, the effective SNR also increases. Further, when the users are i.i.d., i.e., Ωu  Ω for all user pairs, the outage performance of NORM-SNR scheduling is equivalent to ABS-SNR scheduling. However, for the practical scenario when users are i.n.i.d., ABS-SNR outperforms NORM-SNR. Also, it is observed for both i.i.d. and i.n.i.d. cases that SMUD-based scheduling gives the best performance.

Fig. 2. Outage probability versus average electrical SNR plot for the five-user-pair TWR-FSO network for various scheduling schemes when Mα; β; γ 0 ; ρ; Ω0   M8.1; 4; 0.45; 0.1; 0.55 and ξ  0.833 and 1.66.

be scheduled. However, in both these schemes, all the U user pairs are required to send their CSI to the relay, while only one pair is selected for transmission. Hence, an additional processing load (APL) depending on factor ≃2U − 1 is incurred. The SMUD-based scheduling can reduce the APL factor to ≃2τ − 1 and also achieve a better performance than the ABS-SNR and NORM-SNR schemes when γ T ≤ γ th . Additionally, the plot compares the outage performance for different values of ξ, which is the ratio between the equivalent beam-width radius and the pointing error displacement standard deviation (jitter) at the receiver. As is evident, poor performance is achieved as ξ decreases, i.e., as the jitter increases. Figure 3 is a plot of outage probability with respect to number of user pairs, when γ th  4 dB, γ T  2 dB, ξ  1.66, and σ 2R  0.52 (at sunrise). The lengths assumed

Fig. 3. Outage probability versus number of user pairs plot for the multiuser TWR-FSO network for i.i.d. and i.n.i.d. user pairs when ξ  1.66.

Figure 4 is a plot of system outage probability with respect to the average electrical SNR for the SMUD-based scheduling scheme for three cases, i.e., γ T < γ th ; γ T  γ th , and γ T > γ th . The system parameters considered are γ th  5 dB, ξ  1.66, and σ 2R  0.32 (at night). For σ 2R  0.32, we obtain M10; 5; 0; 1; 1. It may be noted that this set of parameters when ρ  1 and γ  0 corresponds to Gamma–Gamma distribution [18, Table I)]. As can be inferred, the best performance for SMUD scheduling is achieved when γ T < γ th . Further, when γ T  γ th , SMUD scheduling is equivalent to ABS-SNR scheduling. The third case, when γ T > γ th , has significance when Pt  0 and the randomly chosen user’s SNR is above γ th . Hence, γ T is one of the key parameters for optimum system design. Figure 5 is a plot of the BER with respect to the average electrical SNR for a three-user-pair TWR-FSO system with BPSK modulation, when ξ  0.833 and σ 2R  1.2. The values of Gauss–Laguerre quadrature roots and weights in the analytical expressions for the BER given in Eqs. (34), (42), and (44) are taken from [26] for n  0; N  15 and [27, Table II)] for n  − 12 ; N  15. As can be observed, the analytical curves have a close match with the simulation curves. Also, it is observed that the SMUD scheme outperforms both ABS-SNR- and NORM-SNRbased scheduling schemes. Further, the convergence of

Fig. 4. Outage probability versus average electrical SNR plot for the five-user-pair TWR-FSO network with SMUD-based scheduling when Mα; β; γ 0 ; ρ; Ω0   M10; 5; 0; 1; 1.

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VII. CONCLUSION In this paper, we modeled a two-way, single, DF relayassisted multiuser FSO network. The performance of various scheduling algorithms based on absolute SNR, normalized SNR, and SMUD were studied. Further, closedform expressions of the outage probability and BER for the considered system for different scheduling schemes in a turbulence regime modeled using M–distribution and also affected by pointing errors and path loss were derived. One round of communication between a selected user pair consumed two time phases, thereby overcoming the spectral loss incurred in half-duplex communication. Further, a comparative analysis of system outage probability, APL, and BER of various scheduling schemes demonstrated the superiority of SMUD-based scheduling over absoluteSNR- and normalized-SNR-based scheduling schemes. Fig. 5. BER versus average electrical SNR plot for the three-userpair TWR-FSO network for the BPSK modulation scheme when Mα; β; γ 0 ; ρ; Ω0   M8.1; 4; 0.45; 0.1; 0.55.

asymptotic results at high SNRs proves a tight approximation of the asymptotic results. Also, it can be seen from Fig. 5 that the BER is 5.17 × 10−3 at a SNR of 30 dB and 4.37 × 10−4 at a SNR of 40 dB, for absolute SNR-based scheduling. The diversity order obtained from these values is Gd ≈ 1.07, which is approximately equal to the theoretical value of Gd ≈ 1.04. Figure 6 is a plot of the analytical and simulated BERs with respect to the average electrical SNR for a TWR-FSO network using absolute SNR-based scheduling and employing various modulation schemes, including coherent BPSK and BFSK, non-coherent BFSK, and differential BPSK. The system parameters are the same as in Fig. 5. It is observed that coherent BPSK has the best performance. Also, the performance improvement with the increase in the number of user pairs brings out the effectiveness of using multiuser systems.

APPENDIX A: EXACT CDF

AND

PDF

The CDF of a TWR channel is the probability that the SNR of any of the intermediate links of the uth user pair, i.e., T A;u ; R and T B;u ; R, is less than γ. Hence, the CDF F Ωu γ of Ωu, where Ωu  minΓA;u ; ΓB;u , for any u ∈ U using [14, Eq. (7)] is given as F Ωu γ  PrΩu ≤ γ  PrminΓA;u ; ΓB;u  ≤ γ:

(A1)

Since ΓA;u and ΓB;u are independent, Eq. (A1) can be rewritten as F Ωu γ  1 − PrΓA;u > γ PrΓB;u > γ;  F ΓA;u γ  F ΓB;u γ − F ΓA;u γF ΓB;u γ:

(A2)

In order to calculate F ΓA;u γ and F ΓB;u γ in Eq. (A2), we integrate Eq. (9) using [21, Eq. (8.2.2.19)] and [29, Eq. (26)], and obtain

F ΓS;u γ 

 −αS;u k βS;u ξ2S;u AS;u X 2 α β ak 0 S;u S;u 0 4 γ βS;u  Ω k1 0 1  ξ2S;u 1 ξ2S;u 2 2αS;u k−1 6;1 @ κ 0 γ 1; 2 ; 2 A; (A3) × G3;7 2 2 2π γ ξS;u ξS;u 1 αS;u αS;u 1 k k1 S;u

where S ∈ fA; Bg and κ0  16γ 0 β

2

;

2

;

2

;

;2;

2

α2S;u β2S;u ξ4S;u 0 2 2 2 S;u Ω  ξS;u 1

2

;0

.

Substituting Eq. (A3) into Eq. (A2), the CDF of the TWRFSO link is obtained in Eq. (11).

Fig. 6. BER versus average electrical SNR plot for the TWR-FSO network for various modulation schemes under absolute-SNRbased scheduling when Mα; β; γ 0 ; ρ; Ω0   M8.1; 4; 0.45; 0.1; 0.55.

Further, the PDF of Ωu is obtained by differentiating Eq. (A2) and is given as f Ωu γ  f ΓA;u γ  f ΓB;u γ− f ΓA;u γF ΓB;u γ − F ΓA;u γf ΓB;u γ. Substituting Eqs. (9) and (A3) and using [21, Eq. (8.2.2.19)], the PDF of Ωu is as given in Eq. (12).

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APPENDIX B: ASYMPTOTIC CDF Expanding the Meijer’s G-function in Eq. (A3) using [29, Eq. (18)] for the conditions z  0, q > p, and limz→0 p F q a; b; z  1, the expression for the asymptotic CDF at high SNRs is obtained as

F ΓS;u γ 

Q3

βS;u 3 X ξ2S;u AS;u X j1;j≠t Γbj − bt  ak 2 4 ΓξS;u  1 − bt bt ∕2 t1 k1



×

αS;u βS;u γ 0 βS;u  Ω0

b −αS;u k  t

2

ξ2u 2 ξu  1

b  t

γ

bt 2

γ S;u

;

(B1)

where b1  ξ2u , b2  αu , and b3  k. Further, Eq. (B1) can be rewritten as  ξ2S;u  ξ2S;u AS;u 2 γ F ΓS;u γ  ψ ξ2 S;u 4 γ S;u  αS;u X  k βS;u 2 γ γ 2 ;  ψ αS;u  ψk γ S;u γ S;u k1

βS;u X k1

×

 ak Q3

αS;u βS;u γ 0 βS;u  Ω0

b −αS;u k  t

2

j1;j≠t Γbj − bt  : 2 ΓξS;u  1 − bt bt ∕2

ξ2S;u

[11] 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 composite fading channels,” IEEE Trans. Commun., vol. 59, no. 10, pp. 2654–2658, Oct. 2011. [12] H. Moradi, H. H. Refai, and P. G. LoPresti, “Switch-and-stay and switch-and-examine dual diversity for high-speed free-space optics links,” IET Optoelectron., vol. 6, no. 1, pp. 34–42, Feb. 2012. [13] J. Abouei and K. N. Plataniotis, “Multiuser diversity scheduling in free-space optical communications,” J. Lightwave Technol., vol. 30, no. 9, pp. 1351–1358, May 2012.

(B2)

[14] 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., vol. 5, no. 1, pp. 92–103, Jan. 2013. [15] L. Yang, M. O. Hasna, and X. Gao, “Asymptotic BER analysis of FSO with multiple receive apertures over M-distributed turbulence channels with pointing errors,” Opt. Express, vol. 22, no. 15, pp. 18238–18245, July 2014.

(B3)

[16] S. Zhalehpour and M. Uysal, “Performance of multiuser scheduling in free space optical systems over atmospheric turbulence channels,” IET Optoelectron., vol. 9, no. 5, pp. 275–281, Oct. 2015. [17] M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance PDF of a laser beam propagating through turbulent media,” Opt. Eng., vol. 40, no. 8, pp. 1554–1562, Feb. 2001.

where

ψ bt 

[9] L. Yang and M.-S. Alouni, “Performance analysis of multiuser selection diversity,” IEEE Trans. Veh. Technol., vol. 55, no. 6, pp. 1848–1861, Nov. 2006. [10] D. Gesbert and M.-S. Alouni, “Selective multi-user diversity,” in Proc. 3rd IEEE Int. Symp. on Signal Processing and Information Technology, Dec. 14–17, 2003, pp. 162–165.

bt

ξ2S;u  1

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