Dual-Hop Optical Wireless Relaying Over Turbulence Channels With Pointing Error Impairments

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 9, MAY 1, 2014 1821

Dual-Hop Optical Wireless RelayingOver Turbulence Channels With Pointing

Error ImpairmentsMona Aggarwal, Student Member, IEEE, Parul Garg, Member, IEEE, and Parul Puri, Student Member, IEEE

Abstract—In this paper, we present analytical expressions forthe performance of dual-hop urban optical wireless communica-tion system under the combined influence of atmospheric turbu-lence induced-fading and misalignment-fading (pointing errors).The system employs an amplify and forward relay with the as-sumption that channel state information is available at transmit-ting as well as receiving terminals. The atmospheric turbulencechannels are modeled by independent but not necessarily iden-tically distributed (i.n.i.d) Gamma–Gamma fading statistics withpointing error impairments. Using a tight upper bound for theend-to-end signal to noise ratio of the system, we derive the novelclosed form expressions for the moment generating function andprobability density function all in terms of Meijer’s G-function.Based on the aforementioned statistical results, accurate closed-form expressions for average symbol error rate for M-ary phaseshift keying modulation schemes, outage probability and the er-godic capacity under four adaptive transmission policies namelyoptimal rate adaptation, optimal power and rate adaptation, chan-nel inversion with fixed rate and truncated channel inversion withfixed rate are derived in terms of Meijer’s G-function function.Finally, the proposed mathematical analysis is accompanied withvarious numerical examples to demonstrate the effect of differentsystem parameters on the system performance.

Index Terms—Average symbol error rate, ergodic capacity,Meijer’s G-function, pointing errors, urban optical wireless com-munication (UOWC) system.

I. INTRODUCTION

THE urban optical wireless communication (UOWC), alsoknown as outdoor free space optics (FSO) refers to the

line of sight (LoS) propagation of optical signals through theatmosphere. The UOWC system has gained significant researchinterest in recent times, however its performance is highly vul-nerable to adverse atmospheric conditions which occur mainlydue to turbulence and pointing errors [1]. The turbulence arisesdue to inhomogeneities in temperature and pressure of the at-mosphere which cause a change in refractive index and leads toamplitude and intensity fluctuations in the transmitted optical

Manuscript received December 17, 2013; revised February 14, 2014 andMarch 18, 2014; accepted March 20, 2014. Date of publication March 25,2014; date of current version April 23, 2014.

The authors are with the Division of Electronics and Communication En-gineering, Netaji Subhas Institute of Technology, New Delhi 110078, India(e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2014.2313859

signal. The pointing errors occur due to the sway of high-risebuildings as a result of dynamic wind loads, thermal expansionsand weak earthquakes, and cause misalignment between sourceand destination transceivers [1], [2].

In literature, most of the research is focused on UOWC sys-tems employing on–off keying (OOK) with intensity modulationand direct detection (IM/DD), but OOK IM/DD system requireadaptive thresholds for optimal performance and are subjectedto channel estimation errors. To overcome the challenges ofOOK IM/DD optical communication systems, the error per-formance of pulse-position modulation (PPM) scheme has beeninvestigated for a terrestrial FSO link impaired by both turbulentatmosphere and pointing errors in [3]. Subcarrier intensity mod-ulation (SIM) for UOWC systems proposed in [4] studied the biterror rate (BER) of differential phase shift keying (DPSK) andM-ary phase shift keying (MPSK) modulation schemes usingLog-normal distributed turbulence. The idea of SIM is borrowedfrom widely used orthogonal frequency division multiplexing(OFDM) system. In SIM UOWC systems, several data streamsare modulated onto different radio frequency (RF) subcarrierfrequencies and then the composite RF signal is used to mod-ulate the laser irradiance. It has been shown that subcarrierphase shift keying (PSK) OWC system outperforms an OOKsystem and permits the use of M-ary modulation schemes [5].In [6], authors have studied adaptive and nonadaptive subcar-rier PSK systems over the Log-normal and the Gamma–Gammaturbulence channels. Authors in [7]–[10] have presented perfor-mance analysis of hybrid RF and RF/FSO dual hop systemsusing fixed-gain or variable-gain relays. Recently in [11] theerror rate performance of SIM UOWC system has been stud-ied for MPSK, DPSK and non-coherent frequency shift keyingmodulation schemes. In this paper, we are applying the conceptof SIM for MPSK signalling scheme in relayed UOWC systemsto study the error performance under the combined influence ofturbulence and pointing errors .

In [12] fundamental limits on the outage probability andachievable rates for FSO channels have been derived. Peppaset al. in [13] investigated the upper bounds on the ergodic ca-pacity of dual-hop UOWC systems employing OOK IM/DDassuming channel state information (CSI) is known only at thereceiving terminals i.e. at the relay and the destination. However,such an assumption is not realistic in practical fixed wireless ac-cess networks where reliable feedback of channel estimates isfeasible. Moreover the available CSI at the transmitting ter-minals can be used to design adaptive channel transmissionschemes [14] for improving the system performance and better

0733-8724 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

1822 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 9, MAY 1, 2014

utilization of the channel. In adaptive transmission policies, CSIis known either or both at source and destination and the sourceadapts its information rate and/or transmitted power according tothe channel conditions using a feedback of the received signal tonoise ratio (SNR) from the destination. In literature, there existsfour such policies namely optimal rate adaptation (ORA), opti-mal power and rate adaptation (OPRA), channel inversion withfixed rate (CIFR) and truncated channel inversion with fixedrate (TIFR) [14]. To this end, a unified framework has beendeveloped in [15] to find the average capacity of point-to-pointFSO channels under different adaptation policies. However, tothe best of authors’ knowledge, the ergodic capacity of relayedSIM-assisted UOWC systems under different channel adapta-tion policies has not yet studied in the open technical literature.

Motivated by above mentioned facts, we investigate the per-formance of subcarrier intensity modulated dual-hop UOWCsystem assuming that the CSI is available at both transmittingand the receiving terminals. The main contributions of this pa-per are to study the end-to-end statistical characteristics of thesystem using a tight upper bound on end-to-end SNR and derivenovel closed form expressions for its moment generating func-tion (MGF) and probability density function (pdf). Then usingthe MGF based approach, we analyze the error performance ofthe system in terms of average symbol error rate (SER) for SIMbased MPSK modulation schemes. Further using the pdf, we findthe novel closed form expressions for the outage probability andthe ergodic capacity of the system under four channel adaptationpolicies namely ORA, OPRA, CIFR and TIFR. The atmosphericchannels are modeled by independent but not necessarily identi-cally distributed (i.n.i.d) Gamma–Gamma fading statistics withpointing error impairments. Moreover, all the closed form ex-pressions are obtained in terms of Meijer’s G-function whichcan be written in terms of well-known hypergeometric functionand is easily available in software packages like Mathematicaand Mupad.

The rest of the paper is organized as follows: In Section II, weintroduce the system model and derive the statistics of an upperbound to the end-to-end SNR of the system. In Section III, wederive novel accurate expression of outage probability, averageSER for MPSK modulation schemes and ergodic capacity ofthe system under different channel adaptation policies. In Sec-tion IV, numerical results are discussed and illustrated throughplots. Finally we conclude the paper in Section V.

II. STATISTICAL CHARACTERISTICS OF END-TO-END SNR

A. System Model

We consider a dual-hop UOWC System employing SIM asshown in Fig. 1 where a source (s) is communicating with desti-nation (d) with the help of a relay (r). All the three transceiversare located on the high-rise buildings to enable LoS communi-cation. We assume that direct communication between sourceand destination is not feasible due to practical reasons whichmay be either non-LoS condition or the larger distance betweenthe two. The system employs a full-duplex AF relaying proto-col with the assumption that the source s is transmitting a signalwith average power normalized to unity. Further we assume

Fig. 1. Dual-hop system model.

that CSI is available at source, relay and the destination. Theequivalent instantaneous SNR at the destination γeq [16] can bewritten as

γeq =h2

1h22g

2

h22g

2N0,1 + N0,2(1)

where hi = ηIi is the turbulence induced light intensity of theith hop for i ∈ (1, 2), (I) denotes the channel irradiance, η isthe effective photo-current conversion ratio and N0,i is the powerspectral density of additive white Gaussian noise (AWGN) atthe ith hop. Further, g denotes the gain of the relay [17] given asg2 = 1

h21

so that the relay simply amplifies the incoming signalwith the inverse of the channel of first hop, regardless of averagenoise power of the first hop. Defining the instantaneous SNR ofs-r and r-d links as γ1 and γ2 respectively, the end to end SNRγeq , becomes the normalized harmonic mean of instantaneousSNRs given as

γeq =γ1γ2

γ1 + γ2=

(2∑

i=1

1γi

)−1

(2)

where γi = h2i /N0,i for i ∈ (1, 2). We consider an aggregated

channel model which takes into account the combined effectsof atmospheric turbulence-induced fading and misalignment-fading (pointing errors). All the channels are assumed to be sta-tionary, memoryless and ergodic with i.n.i.d. Gamma–Gammafading statistics. Therefore the pdf of instantaneous SNR γi fori ∈ (1, 2) [15, Eq. (20)]–[8] is given as

fγi(γi) =

ξi2

2γiΓ(αi)Γ(βi)G3,0

1,3

(αiβi

√γi

Ωi

∣∣∣∣∣ ξi2 + 1

ξi2 , αi, βi

)(3)

where Gm,np,q (x| a1 ,...,ap

b1 ,...,bq) is Meijer’s G-function [18,

Eq. (8.2.1.1)], Ω denotes the average electrical SNR and ξ =we/2σs with we being the ratio of the equivalent beam radius atthe receiver and σs being the standard deviation of the pointingerror displacement at the receiver. The scintillation parametersαi and βi (αi > 0, βi > 0) correspond to atmospheric turbu-lence strength of ith hop for i ∈ (1, 2) given by [15], Eq. (4),Eq. (5)].

AGGARWAL et al.: DUAL-HOP OPTICAL WIRELESS RELAYING OVER TURBULENCE CHANNELS WITH POINTING ERROR IMPAIRMENTS 1823

B. MGF and Pdf of End-to-End SNR

The MGF of γeq , Mγe q (s) can be written as [19, Th. (1)]

Mγe q (s) =

−∫ ∞

0J0(2

√su)

2∑i=1

[∂

∂uMγ̃ i

(u)] 2∏

k = 1k �= i

Mγ̃k(u)d(u) (4)

where Mγ̃ i(s) is the MGF of the reciprocal distribution of the

fading γi of ith hop with i ∈ (1, 2) as given in [20, Th. (1)]

Mγ̃ i(s) = −

∫ ∞

0J0(2

√su)

[∂

∂uMγi

(u)]

d(u), �{s} ≥ 0

(5)where J0(.) is the zeroth-order Bessel function of firstkind and Mγi

(s) is the MGF of γi given as Mγi(s) =∫ ∞

0 exp(−sγi)fγi(γi)dγi . Now substituting fγi

(γi) from (3)and using [18, Eq. (2.24.3)], Mγi

(s) is evaluated as

Mγi(s) =

2αi +βi −1ξ2i

4πΓ(αi)Γ(βi)G6,1

3,6

(α2

i β2i

16Ωis

∣∣∣∣∣ 1, ϕi

�i

)(6)

where �i = ξi2

2 , ξi2 +12 , αi

2 , αi +12 , βi

2 , βi +12 and ϕi = ξi

2 +12 ,

ξi2 +22 for i ∈ (1, 2). Further substituting (6) in (5) and with

the help of [18, Eq. (8.2.2), Eq. (2.24.1.1)] and Jv (2√

x) =G1,0

0,2(x| −v/2,−v/2 ) [18, Eq. (8.4.19.1)], Mγ̃ i

(s) is obtained as

Mγ̃ i(s) =

2αi +βi −1ξ2i

4πΓ(αi)Γ(βi)G7,1

3,8

(α2

i β2i s

16Ωi

∣∣∣∣∣ 0, ϕi

0, �i , 0

). (7)

Substituting (7) in (4) and using [18, Eq. (8.2.2)], we obtainMγe q (s) as in (8)

Mγe q (s) =2∏

i=1

(2αi +βi −1ξ2

i

4πΓ(αi)Γ(βi)

)∫ ∞

0

1u

J0(2√

su)

×[G7,1

3,8

(α2

1β21 u

16Ω1

∣∣∣∣∣ 0, ϕ1

0, �1 , 0

)G7,1

3,8

(α2

2β22 u

16Ω2

∣∣∣∣∣ 0, ϕ2

1, �2 , 0

)

+ G7,13,8

(α2

2β22 u

16Ω2

∣∣∣∣∣ 0, ϕ2

0, �2 , 0

)G7,1

3,8

(α2

1β21 u

16Ω1

∣∣∣∣∣ 0, ϕ1

1, �1 , 0

)]d(u).

(8)

The closed form solution of (8) does not exist. Therefore tomake the analysis mathematically tractable, we proceed withthe upper bound on the end-to-end SNR using the inequalitybetween geometric and harmonic means for random variablesas given in [17]. An upper bound for the end-to-end SNR γb , ofdual-hop system with CSI-assisted relays can be re-written as

γeq ≤ γb =12

2∏i=1

γ1/2i . (9)

Using the same procedure as in [17], the MGF of end-to-endSNR γb , as defined in (9), Mγb

(s) can be written as

Mγb(s) =

2∏i=1

[ξi

2

Γ(αi)Γ(βi)

]G6,1

3,6

(2s

2∏i=1

[αiβi√

Ωi

] ∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)

(10)where ψ1 = α1 , β1 , α2 , β2 , ψ2 = ξ1

2 , ξ22 and ψ3 = ξ1

2 +1, ξ2

2 + 1.The pdf of end-to-end SNR, fγb

(γ) is defined as fγb(γ) =

L−1 [Mγb] where L−1(.) is the inverse Laplace trans-

form operator. Substituting Mγb(γ) from(10) and using [21,

Eq. (3.40.1.1)], we obtain the closed form expression of pdf ofend-to-end SNR given as

fγb(γ) =

1γ

2∏i=1

[ξi

2

Γ(αi)Γ(βi)

]G6,0

2,6

(2γ

2∏i=1

[αiβi√

Ωi

]∣∣∣∣∣ ψ3

ψ1 , ψ2

).

(11)

III. PERFORMANCE ANALYSIS

A. Outage Probability

The outage probability Pout , is defined as the probabilitythat the instantaneous SNR falls below a specified thresholdγ0 , given as Pout = Pr(γb < γ0) =

∫ γ0

0 fγb(γ)dγ. Substituting

fγb(γ) from (11) and using [18, Eq. (1.16.2)], Pout is derived to

Pout =2∏

i=1

[ξi

2

Γ(αi)Γ(βi)

]G6,1

3,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2 , 0

)

(12)where Ξ1 = α1 β1√

Ω1and Ξ2 = α2 β2√

Ω2. It can be easily shown that,

in case of system affected by atmospheric turbulence only i.e.(ξi → ∞), Pout converges to

Pout =2∏

i=1

[1

Γ(αi)Γ(βi)

]G4,1

1,5

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1ψ1 , 0

). (13)

The accurate closed form expressions for outage probabilityof considered system with and without pointing error impair-ments are given in (12) and (13).

B. Symbol Error Rate

In this section, we use the MGF based approach to obtainthe average SER performance of dual-hop subcarrier intensitymodulated UOWC system employing a CSI-assisted AF relay.The average SER [22] for MPSK modulation schemes is givenas

P(e)MPSK(γb) =

1π

∫ (M −1 )πM

0Mγb

(sin2(π/M)

sin2(φ)

)dφ. (14)

Substituting Mγbfrom (10) in (14), we get

P(e)MPSK(γb) =

A1

π

∫ ΘM

0G6,1

3,6

(2 sin2(φ)

sin2(π/M)Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)dφ

(15)where ΘM = π (M −1)

M and A1 =∏2

i=1[ξi

2

Γ(αi )Γ(βi )]. The closed

form solution for equation (15) exists only for M = 2 i.e. for


binary phase shift keying (BPSK) modulation scheme. Alter-natively for M > 2, the average SER can be written as [23,Eq. (10)]

P(e)MPSK(γb)

≈ A1

[(ΘM

2π− 1

6

)G6,1

3,6

(2

sin2( πM )

Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)

+14G6,1

3,6

(3

2 sin2( πM )

Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)

+(

ΘM

2π− 1

4

)×G6,1

3,6

(2 sin2 ΘM

sin2( πM )

Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)]. (16)

The average SER given in (16) is a tight approximation of(15) as illustrated through numerical plots. Further for a specialcase of M = 2 i.e. BPSK modulation scheme which generallyholds true for UOWC systems, (15) can be rewritten as

P(e)BPSK(γb) =

A1

π

∫ π2

0G6,1

3,6

(2 sin2(φ)Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2

)dφ.

(17)With a change of variable sin2(φ) = x and integrating with

the help of [18, Eq. (2.24.2), Eq (8.2.2)] and after some math-ematical manipulations, the exact closed form expression ofP

(e)BPSK(γb) is derived to

P(e)BPSK(γb) =

A1

2√

πG6,2

4,7

(2Ξ1Ξ2

∣∣∣∣∣ 1/2, 1, ψ3

ψ1 , ψ2 , 0

)(18)

In case of negligible pointing errors i.e. (ξi → ∞), the averageBER for M = 2 simplifies to

P(e)BPSK(γb) =

12√

π

2∏i=1

[1

Γ(αi)Γ(βi)

]G4,2

2,5

(2Ξ1Ξ2

∣∣∣∣∣ 1/2, 1ψ1 , 0

)

(19)Similarly using (16), the average SER of MPSK modulation

schemes for M > 2 simplifies to

P(e)MPSK(γb)

≈ A2

[(ΘM

2π− 1

6

)G4,1

1,4

(2Ξ1Ξ2

sin2( πM )

∣∣∣∣∣ 1ψ1

)

+14G4,1

1,4

(3Ξ1Ξ2

2 sin2( πM )

∣∣∣∣∣ 1ψ1

)

+(

ΘM

2π− 1

4

)×G4,1

1,4

(2 sin2 ΘM

sin2( πM )

Ξ1Ξ2

∣∣∣∣∣ 1ψ1

)](20)

where A2 =∏2

i=1[1

Γ(αi )Γ(βi )]. Thus (19) and (20) give accu-

rate closed form expressions for average SER of BPSK andMPSK modulation schemes for the system considering negligi-ble pointing error impairments.

C. Ergodic Capacity

The ergodic capacity of fading channel is defined as the max-imum achievable transmission rate under which errors are re-coverable. Various channel adaptive transmission policies areavailable in literature [14] depending upon the knowledge ofthe CSI at transmitter and/or receiver, transmitter power andinformation rate and result in the existance, or not, of an out-age probability. In this section we use the pdf based approachto derive the closed form expressions for ergodic capacity ofthe system under four channel adaptive transmission schemeswhich are ORA, OPRA, CIFR and TIFR.

1) Optimal Rate Adaptation: The ORA transmission policyrequires that transmitted power is constant and CSI is availableonly at the receiver side. The channel capacity (in bits per secondper hertz) with ORA policy Cora in terms pdf of end-to-endSNR [14] is given as

Cora =1

ln(2)

∫ ∞

0ln(1 + γ)fγb

(γ)dγ. (21)

Substituting (11) into (21), Cora may be expressed as

Cora =A1

ln(2)

∫ ∞

0

1γ

ln(1 + γ)G6,02,6

(2γΞ1Ξ2

∣∣∣∣∣ ψ3

ψ1 , ψ2

)dγ.

(22)Now expressing logarithmic integrand in terms of Meijer’sG-function using ln(1 + x) = G1,2

2,2(x | 1,11,0 ), and integrating

(22) using [18, eq (2.24.1.1)], the closed form expression ofCora is derived to

Cora =A1

ln(2)G8,1

4,8

(2Ξ1Ξ2

∣∣∣∣∣ 0, 1, ψ3

ψ1 , ψ2 , 0, 0

)dγ. (23)

In case of negligible pointing errors (ξi → ∞), Cora reducesto

Cora =A2

ln(2)G6,1

2,6

(2Ξ1Ξ2

∣∣∣∣∣ 0, 1ψ1 , 0, 0

)dγ. (24)

2) Optimal Power and Rate Adaptation: With an averagetransmitted power constraint, the OPRA transmission policyrequires that CSI is available both at the transmitter and thereceiver side so that the transmitter can adapt both its transmitpower and rate according to the variations of the fading channels,such that high power levels and rates are allocated when the end-to-end SNR γb is high and vice versa. The ergodic capacity (inbits per second per hertz) with OPRA policy Copra , in terms ofpdf of end-to-end SNR [14] is given as

Copra =1

ln(2)

∫ ∞

γ0

ln(

γ

γ0

)fγb

(γ)dγ (25)

where γ0 is the optimal cutoff SNR below which data transmis-sion is suspended and must satisfy the following equality [14,Eq. (6)] ∫ ∞

γ0

(1γ0

− 1γ

)fγ (γ)dγ = 1. (26)


Substituting (11) into (25), the Copra can be expressed as

Copra =A1

ln(2)

∫ ∞

γ0

γ−1 ln(

γ

γ0

)G6,0

2,6

(2γΞ1Ξ2

∣∣∣∣∣ ψ3

ψ1 , ψ2

).

(27)By expressing logarithmic function in terms of Meijer’s

G-function using ln xH(x − 1) = G2,20,2(x| 1,1

0,0 ) [18, Eq.(8.4.6.2)] and integrating using [18], eq (2.24.1.1)], the fol-lowing expression of Copra may be obtained

Copra =A1

ln(2)G8,0

4,8

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1, 1, ψ3

ψ1 , ψ2 , 0, 0

)(28)

In case of negligible pointing errors i.e. (ξi → ∞), Copra sim-plifies to

Copra =A2

ln(2)G6,0

2,6

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1, 1ψ1 , 0, 0

)(29)

To obtain Copra , we need to evaluate the optimal cutoff SNRγ0 as defined in (26). Substituting (11) in (26), γ0 is obtained as

A1

∫ ∞

γ0

(1γ0

− 1γ

)1γ

G6,02,6

(2γΞ1Ξ2

∣∣∣∣∣ ψ3

ψ1 , ψ2

)dγ = 1.

(30)Further using [18, Eq. (8.4.2)] and applying [18, Eq. (2.24.1)],

(30) can be re-written as

γ02

A1= G7,0

3,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ ψ3 , 1ψ1 , ψ2 , 0

)

− G7,03,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ ψ3 , 2ψ1 , ψ2 , 1

). (31)

The optimal cutoff threshold SNR is obtained as a solutionof (31) using numerical methods. Since no data is sent whenγb < γ0 , the OPRA policy suffers with outage probability Poutgiven by (12).

3) Channel Inversion with Fixed Rate: The CIFR adapta-tion policy also known as zero outage capacity requires thattransmitter exploits the CSI to maintain a constant SNR at thereceiver i.e. it inverts the channel fading. The ergodic capacity(in bits per second per hertz) for CIFR policy Ccifr , in terms ofpdf of end-to-end SNR [14] can be expressed as

Ccifr = log2

[1 +

1∫ ∞0

1γ fγb

(γ)dγ

]. (32)

Substituting (11) in (32), the integral in (32) can be expressedas

K(γ) =∫ ∞

0

1γ

fγb(γ)dγ =

∫ ∞

0

A1

γ2 G6,02,6

(2γΞ1Ξ2

∣∣∣∣∣ ψ3

ψ1 , ψ2

).

(33)Now using [18, Eq. (2.24.2)], properties of gamma function

and after some mathematical manipulations, expression ofK(γ)is derived to

K(γ) = 22∏

i=1

[ξ2i αiβi

(ξ2i − 1)(αi − 1)(βi − 1)(

√Ωi)

]. (34)

Substituting (34) in (32), Ccifr is obtained as

Ccifr = log2

(1 +

12

2∏i=1

[(ξ2

i − 1)(αi − 1)(βi − 1)(√

Ωi)ξ2i αiβi

]).

(35)For the case of negligible pointing errors i.e. (ξi → ∞), Ccifr

reduces to

Ccifr = log2

(1 +

12

2∏i=1

[(αi − 1)(βi − 1)(

√Ωi)

αiβi

]). (36)

The accurate closed form expressions of ergodic capacity ofconsidered system with and without pointing error impairmentsunder CIFR policy are given in (35) and (36).

4) Truncated Channel Inversion with Fixed Rate: The CIFRpolicy suffers with a large capacity penalty since a large amountof transmitted power is required to compensate for the deepchannel fading. A better approach is to use truncated CIFRpolicy named as TIFR that inverts the channel fading abovea fixed cut-off fade depth γ0 . The TIFR policy improves thechannel capacity but at the expense of outage probability. Theergodic capacity (in bits per second per hertz) for TIFR [14]scheme Ctifr , is obtained by maximizing over all possible γ0and can be expressed in terms of pdf of end-to-end SNR as

maxγ0

Ctifr = log2

[1 +

1∫ ∞γ0

1γ fγb

(γ)dγ

](1 − Pout(γ0))

(37)where Pout is the outage probability given by (12). Substituting(11) in (37), the integral in (37) can be expressed as

N (γ0) =∫ ∞

γ0

1γ

fγb(γ)dγ =

∫ ∞

γ0

A1

γ2 G6,02,6

(2γΞ1Ξ2

∣∣∣∣∣ ψ3

ψ1 , ψ2

).

(38)Re-writing (38) using xvH(x − 1) = G0,1

1,1(x| v+1v ) [18,

Eq. (8.4.2.2)] and integrating using [18, Eq. (2.24.1)], the ex-pression of N (γ0) is derived to

N (γ0) = A11γ0

G7,03,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ ψ3 , 2ψ1 , ψ2 , 1

). (39)

Substituting (39) and (12) into (37), the closed form expressionof Ctifr is derived to

Ctifr = log2

⎡⎣1 +

γ0

A1

[G7,0

3,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ ψ3 , 2ψ1 , ψ2 , 1

)]−1⎤⎦

×[1 − A1G

6,13,7

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1, ψ3

ψ1 , ψ2 , 0

)]. (40)

In case of negligible pointing errors i.e. (ξi → ∞), Ctif r

reduces to

Ctifr = log2

⎡⎣1 +

γ0

A2

[G5,0

1,5

(2γ0Ξ1Ξ2

∣∣∣∣∣ 2ψ1 , 1

)]−1⎤⎦

×[1 − A2G

4,11,5

(2γ0Ξ1Ξ2

∣∣∣∣∣ 1ψ1 , 0

)]. (41)


Fig. 2. Average BER of BPSK modulation scheme versus average SNR perhop Ω(dB) of the system in moderate turbulence (α = 5.42, β = 3.8) and strongturbulence (α = 4, β = 1.71).

It is worth noting that both OPRA and TIFR schemes sufferwith outage probability when the end-to-end SNR is below thecut-off SNR, whereas the ORA and CIFR schemes do not sufferany outage event. Also the optimal cutoff SNR of TIFR schemeis different than the cutoff SNR of OPRA scheme and can beobtained by solving the equation ∂C t i f r

∂γ0= 0, given as

∂

∂γ0

[log2

(1 +

1N (γ0)

)(1 − Pout(γ0))

]= 0 (42)

Now Substituting N (γ0) from (39) and Pout from (12) respec-tively, the cut-off SNR is numerically obtained.

IV. NUMERICAL RESULTS

In this section we present numerical results for the aver-age SER, outage probability and ergodic capacity of consid-ered system. Without loss of generality, we assume turbulenceparameters for s − r link and r − d link are α1 = α2 = α,β1 = β2 = β with ξ1 = ξ2 = ξ. The value of C2

n is 3 × 10−14 or1 × 10−13 for moderate and strong turbulence regimes with av-erage SNR per hop Ω1 = Ω2 = Ω. The wavelength is assumedto be λ = 1550 nm and r = 1 cm is the radius of receiver aper-ture [13], [15].

Fig. 2 illustrates a plot of average BER of BPSK modulationscheme as a function of average SNR per hop Ω (dB) for var-ious turbulence conditions with distance of s-r link (dsr ) anddistance of r-d link (drd) as dsr = drd = 1000 m. The turbu-lence parameters for moderate turbulence (α = 5.42, β = 3.8)and for strong turbulence (α = 4, β = 1.71) are calculated us-ing [15, Eq. (4), Eq. (5)]. The exact average BER bound using(18) and approximated average BER bound using (16) has beencompared and it can be seen that there exists a close matchbetween the two. Therefore (16) has been used to find averageSER for other values of M e.g. QPSK and 16-PSK modulationschemes. The dashed curves for the exact BER performance,using (8), are also included for comparison purposes. It hasbeen observed that the difference between the exact value of theBER and the obtained bound is tighter at low values of average

Fig. 3. Average SER versus average SNR per hop Ω(dB) of dual-hop UOWCsystem for QPSK and 16-PSK modulation schemes for various values of we /rwith α = 4, β = 1.65 and σs /r = 2.

Fig. 4. Ergodic Capacity under OPRA policy versus average SNR per hopΩ(dB) showing impact of pointing errors with parameters given as α = 4, β =1.65 and we /r = 12 for various values of σs /r.

SNR per hop Ω (dB). However, the bound loses tightness asaverage SNR per hop increases. It is noted that the obtainedresults are in agreement with [17], [24]. Also, it is evident thataverage BER increases with increase in turbulence strength andadversely affects the system performance.

Fig. 3 shows a plot of average SER of QPSK and 16-PSKmodulation schemes as a function of average SNR per hopΩ(dB) demonstrating the effect of both turbulence and point-ing errors on the system performance. We consider moderateturbulence strength with dsr = drd = 2000 m for a fixed valueof σs/r = 2 and various values of we/r [13]. It can be seenthat for both QPSK and 16-PSK, average SER decreases withdecrease in pointing errors (i.e. we/r increases) and is leastwith no pointing errors. It can also be observed that for giventurbulence strength and pointing errors, average SER for QPSKis less as compared to 16-PSK modulation scheme.

In Fig. 4, ergodic capacity per unit bandwidth of the sys-tem under OPRA transmission scheme is plotted as a functionof average SNR per hop Ω(dB) using (28). The curve is plot-ted assuming moderate turbulence strength (α = 4, β = 1.65)with a fixed value of we/r = 12 and various values of σs/r.


TABLE IVALUES OF OPTIMUM CUTOFF THRESHOLD SNR γ0 FOR DIFFERENT

TURBULENCE PARAMETERS AND POINTING ERRORS

Fig. 5. Outage Probability for OPRA policy versus average SNR per hopΩ(dB) for dsr = drd = 1000 m with moderate turbulence α = 5.42, β = 3.8and strong turbulence α = 4.0, β = 1.71 for different values of ξ.

We observe that for a given turbulence strength, the ergodiccapacity under OPRA transmission schemes increases with de-crease in pointing errors (i.e. σs/r increases) and is maximumfor no pointing errors case. When the end-to-end SNR falls be-low the optimal cut-off threshold SNR given by (31), no datais transmitted and the OPRA transmission scheme suffers anoutage probability. Table I shows the values of optimum cut-off threshold SNR γ0 for dsr = drd = 1000 m with moderateand strong turbulence with different values of ξ. These resultsclearly demonstrate the effect of turbulence strength, pointingerrors and average SNR per hop on the cut-off threshold SNRlevel. The corresponding outage probability evaluated for theoptimal cut-off SNR of OPRA transmission scheme is plottedin Fig. 5. It can be seen that with increase in turbulence strengthand/or pointing errors, the outage performance deteriorates andPout increases.

Fig. 6 shows the plot of Ctifr using (40), versus the thresholdcut-off SNR γ0 , for different values of average SNR per hop Ωwith dsr = drd = 2000 m for moderate turbulence (α = 4, β =1.65) and pointing errors given as we/r = 12, σs/r = 2. Thefigure shows that for a given turbulence strength and pointingerrors, there exists a cut-off SNR for which the capacity per unitbandwidth is maximized and this cut-off SNR level increaseswith increase in the value of average SNR Ω(dB).

Fig. 7 shows a comparison plot for ergodic capacities un-der different channel adaptation schemes namely OPRA, ORA,CIFR and TIFR using closed form expressions given in equa-tions (28), (23), (35) and (40) after maximizing over all γ0 .

Fig. 6. Ergodic Capacity under TIFR policy versus Cut-off SNR γ0 (dB) ofthe system for various values of average SNR Ω(dB) with turbulence parametersas α = 4 and β = 1.71 and pointing errors given as we /r = 12, σs /r = 2.

Fig. 7. Comparison of ergodic capacities under ORA, OPRA, CIFR and TIFRpolicies versus average SNR per hop Ω(dB) with moderate atmospheric turbu-lence given as α = 4 and β = 1.71 and pointing errors we /r = 12, σs /r = 2.

The atmospheric turbulence parameters are chosen (α = 4, β =1.71) for dsr = drd = 2000 m with pointing errors we/r = 12,σs/r = 2. We observe that for a given value of average SNR Ω,the OPRA transmission scheme achieves the highest capacityand CIFR transmission scheme achieves the lowest capacity ascompared to other schemes with Copra ≥ Cora ≥ Ctifr ≥ Ccifr .It can also be observed that capacity under CIFR transmissionscheme suffers with the largest capacity penalty as compared toother schemes.

V. CONCLUSION

In this paper, we investigated the end-to-end performance ofdual-hop subcarrier intensity modulated UOWC system withCSI-assisted AF relay operating over Gamma–Gamma turbu-lence channels with pointing errors. Using a tight upper boundon end-to-end SNR, we studied the statistical characteristics ofthe system in terms of MGF and pdf of end-to-end SNR. Fur-ther utilizing theses results, we obtained the novel closed formexpressions for average SER of M-ary modulation schemes,outage probability and ergodic capacity under different channel


adaptation schemes. At the end of paper, extensive numericalresults clearly demonstrate the effect of atmospheric turbulenceand pointing error impairments on the system performance.

REFERENCES

[1] D. Kedar and S. Arnon, “Urban optical wireless communication net-works: The main challenges and possible solutions,” IEEE Commun. Mag.,vol. 42, no. 5, pp. S2–S7, May 2004.

[2] S. Arnon, “Effects of atmospheric turbulence and building sway on opticalwireless-communication systems,” Opt. Lett., vol. 28, pp. 129–131, Jan.2003.

[3] W. Gappmair, “Novel results on pulse-position modulation performancefor terrestrial free-space optical links impaired by turbulent atmosphereand pointing errors,” IET Commun., vol. 6, no. 10, pp. 1300–1305, 2012.

[4] W. Huang, J. Takayanagi, T. Sakanaka, and M. Nakagawa, “Atmosphericoptical communication system using subcarrier PSK modulation,” IEICETrans. Commun., vol. E76-B, pp. 1169–1177, Sep. 1993.

[5] J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcar-rier PSK intensity modulation through atmospheric turbulence channels,”IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007.

[6] N. D. Chatzidiamantis, A. S. Lioumpas, G. K. Karagiannidis, andS. Arnon, “Adaptive subcarrier PSK intensity modulation in free spaceoptical systems,” IEEE Trans. Commun., vol. 59, no. 5, pp. 1368–1377,May 2011.

[7] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “Impact of pointing errors onthe performance of mixed RF/FSO dual-hop transmission systems,” IEEEWireless Commun. Lett., vol. 2, no. 3, pp. 351–354, Jun. 2013.

[8] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “On the performance of hybridRF and RF/FSO fixed gain dual-hop transmission systems,” in Proc. SaudiInt. Elect. Commun. Photon. Conf., Apr. 27–30, 2013, pp. 1–6.

[9] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “On the performance of hybridRF and RF/FSO dual-hop transmission systems,” presented at the 2ndInter. Work. Opt. Wir. Commun., Newcastle Upon Tyne, U.K., Oct. 2013.

[10] I. S. Ansari, F. Yilmaz, and M.-S. Alouini, “On the performance of mixedRF/FSO variable gain dual-hop transmission systems with pointing er-rors,” in Proc. IEEE 78th Veh. Tech. Conf., Las Vegas, CA, USA, Sep.2013.

[11] X. Song, F. Yang, and J. Cheng, “Subcarrier intensity modulated opticalwireless communications in atmospheric turbulence with pointing errors,”IEEE/OSA J. Opt. Commun. Net., vol. 5, no. 4, pp. 349–358, Apr. 2013.

[12] A. A. Farid and S. Hranilovic, “Outage capacity optimization for freespaceoptical links with pointing errors,” IEEE J. Lightw. Tech., vol. 25,pp. 1702–1710, Jul. 2007.

[13] K. P. Peppas, A. N. Stassinakis, H. E. Nistazakis, and G. S. Tombras, “Ca-pacity analysis of dual amplify-and-forward relayed free-space opticalcommunication systems over turbulence channels with pointing errors,”IEEE/OSA J. Opt. Commun. Net., vol. 5, no. 9, pp. 1032–1042, Sep. 2013.

[14] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with chan-nel side information,” IEEE Trans. Inf. Theory, vol. 43, no. 11, pp. 1986–1992, Nov. 1997.

[15] W. Gappmair, “Further results on the capacity of free-space optical chan-nels in turbulent atmosphere,” IET Commun., vol. 5, no. 9, pp. 1262–1267,2011.

[16] M. Hasna and M. S. Alouini, “End-to-end performance of transmissionsystems with relays over Rayleigh-fading channel,” IEEE Trans. Wirel.Commun., vol. 2, no. 6, pp. 1126–1131, Jun. 2003.

[17] G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds of multihoprelayed communications in Nakagami-m fading,” IEEE Trans. Commun.,vol. 54, no. 1, pp. 18–22, Jan. 2006.

[18] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and SeriesVolume 3: More Special Functions, 1st ed. New York, NY, USA: Gordonand Breach Science Publishers, 1986.

[19] F. Yilmaz, O. Kucur, and M.-S. Alouini, “A novel framework on exactaverage symbol error probabilities of multihop transmission over amplify-and-forward relay fading channels,” in Proc. 7th Inter. Symp. Wir. Com-mun. Sys., York, U.K., Nov. 2010, pp. 546–550.

[20] F. Yilmaz, O. Kucur, and M.-S. Alouini, “Exact capacity analysis ofmultihop transmission over amplify-and-forward relay fading channels,”in Proc. IEEE Inter. Symp. Personal Indoor Mobile Radio Commun.,Istanbul, Turkey, Sep. 2010, pp. 2293–2298.

[21] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and SeriesVolume 4: Direct Laplace Transforms, 1st ed. Boca Raton, FL, USA:CRC, 1992.

[22] M. K. Simon and M. S. Alouini, Digital Communication over FadingChannels. New York, NY, USA: John Wiley and Sons, 2000.

[23] M. McKay, A. Zanella, I. Collings, and M. Chiani, “Error probabilityand SINR analysis of optimum combining in Rician fading,” IEEE Trans.Commun., vol. 57, no. 3, pp. 676–687, Mar. 2009.

[24] C. K. Datsikas, K. P. Peppas, N. C. Sagias, and G. S. Tombras, “Serial free-space optical relaying communications over Gamma-Gamma atmosphericturbulence channels,” IEEE/OSA J. Opt. Commun. Net., vol. 2, no. 8,pp. 576–586, Aug. 2010.

Mona Aggarwal received the B.Tech. and M.Tech. degrees in electron-ics and communication engineering from Punjab Technical University, Ja-landhar, India, and Guru Gobind Singh Indraprastha University, Delhi,India, in 2003 and 2011 respectively. She is currently working as a Teach-ing cum Research Fellow at Netaji Subhas Institute of Technology, New Delhi,India. Her current research interests include optical wireless communicationand cooperative communication.

Parul Garg received the B.Sc.(Engg.) and M.Sc.(Engg.) degrees fromAligarh Muslim University, Aligarh, India, in 1990 and 1994, respectively,all in electronics engineering and the Ph.D. degree in electrical engineeringfrom the Indian Institute of Technology, Delhi, India, in 2005.

From May 1996 to July 2000, she worked as a Faculty Member at the Insti-tute of Engineering and Technology, Lucknow, India. Since July 2000, she hasbeen working as a Faculty Member at the Netaji Subhas Institute of Technology,New Delhi, India.

Her research interests include different aspects of wireless communicationswith emphasis on channel estimation, diversity techniques, cooperative com-munication, and free space optics.

Parul Puri received the B.Tech. and M.Tech. degrees in electronics and com-munication engineering from National Institute of Technology, Hamirpur, India,and Jaypee Institute of Information Technology, Noida, India, in 2008 and 2012,respectively. She is currently working as a Teaching cum Research Fellow atNetaji Subhas Institute of Technology, New Delhi, India. Her current researchinterests include wireless communication and free space optics.

Dual-Hop Optical Wireless Relaying Over Turbulence Channels With Pointing Error Impairments

Documents

Transcript of Dual-Hop Optical Wireless Relaying Over Turbulence Channels With Pointing Error Impairments