Available online at www.sciencedirect.com

H O S T E D B Y

Digital Communications and Networks (2015) 1, 171–180

http://dx.doi.org/12352-8648/& 2015 Carticle under the CC

☆This work wasCommunications anTurkey, April 2014.

nCorresponding auE-mail addresses

smao@ieee.org (S. M

journal homepage: www.elsevier.com/locate/dcan

Optical power allocation for adaptivetransmissions in wavelength-divisionmultiplexing free space optical networks$

Hui Zhoua,b, Shiwen Maoa,n, Prathima Agrawala

aDepartment of Electrical and Computer Engineering, Auburn University, Auburn, AL 36849-5201, USAbAmazon Web Service, Amazon Inc., 345 Boren Ave N, Seattle, WA 98109, USA

Received 17 August 2015; accepted 5 September 2015Available online 28 September 2015

KEYWORDSFree space optics;Radio over free spaceoptics;Wavelength-divisionmultiplexing

0.1016/j.dcan.201hongqing UniversiBY-NC-ND license

presented in partd Networking Co

thor.: hzz0016@tigermao), agrawpr@aub

AbstractAttracting increasing attention in recent years, the Free Space Optics (FSO) technology hasbeen recognized as a cost-effective wireless access technology for multi-Gigabit rate wirelessnetworks. Radio on Free Space Optics (RoFSO) provides a new approach to support variousbandwidth-intensive wireless services in an optical wireless link. In an RoFSO system usingwavelength-division multiplexing (WDM), it is possible to concurrently transmit multiple datastreams consisting of various wireless services at very high rate. In this paper, we investigatethe problem of optical power allocation under power budget and eye safety constraints foradaptive WDM transmission in RoFSO networks. We develop power allocation schemes foradaptive WDM transmissions to combat the effect of weather turbulence on RoFSO links.Simulation results show that WDM RoFSO can support high data rates even over long distance orunder bad weather conditions with an adequate system design.& 2015 Chongqing University of Posts and Telecommunications. Production and Hosting by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Recently, demand for multimedia service with high quality ofservice (QoS) requirements has been drastically increasing. To

5.09.001ty of Posts and Telecommunicatio(http://creativecommons.org/lic

at the 2014 IEEE Wirelessnference (WCNC), Istanbul,

ail.auburn.edu (H. Zhou),urn.edu (P. Agrawal).

cater to the increasing demand on capacity, optical fibers havebeen utilized for years to deliver high volume of data, oftenbetween central and remote universal stations. Due to therelatively high cost of deploying optical fibers, Free SpaceOptics (FSO) has been developed as a cost-effective alter-native wireless access technology for multi-Gigabit rate com-munication networks. FSO provides an excellent alternative tooptical fiber systems for last-mile access networks [1], rangingfrom local area network (LAN)-to-LAN connection for enter-prise/campus, high capacity military communications, todisaster recovery and emergency response, among others.

ns. Production and Hosting by Elsevier B.V. This is an open accessenses/by-nc-nd/4.0/).

H. Zhou et al.172

Attracting considerable attention from the researchcommunity, the FSO technology has also been developedfor carrying various wireless services, as known as Radio onFree Space Optics (RoFSO). FSO systems can operate onwavelengths in the 1520–1600 nm range, which makes thedevelopment of wavelength-division multiplexing (WDM)RoFSO systems feasible. Advanced Dense Wavelength Divi-sion Multiplexing (DWDM) RoFSO systems have been devel-oped to support the simultaneous transmission of multiplewireless signals [1]. Despite of its great potential ofsupporting data intensive communications for various appli-cations, a line-of-sight (LOS) path is required in any FSOsystem. Consequently, FSO is highly susceptible to theatmospheric environment due to the inhomogeneity of airtemperature and pressure, or flying objects [2]. To harvestthe high potential of RoFSO, fading-mitigation techniquesshould be employed to mitigate atmospheric turbulence-induced intensity fluctuations.

To this end, topology control [3,2,4,5], load-balancing[6], and spatial diversity techniques [7] have been studiedand proved to be effective in maintaining a good systemperformance. Adaptive transmissions have been recentlyintroduced into FSO systems and is emerging as a potentialsolution to mitigate the effect of atmospheric turbulence[8]. In FSO systems, channels are usually slow-fading andFSO transceivers have full-duplex capabilities. With negli-gible effect on data rates, a small portion of the bandwidthcan be used for feedback of channel state information (CSI).In some hybrid RF/FSO systems, the RF channel can be usedfor CSI feedback [9]. Thus reliable CSI could be available inFSO systems, which will be highly useful for designingadaptive transmission schemes.

In this paper, we propose optical power allocationschemes for an adaptive WDM RoFSO system in whichvariable wavelengths are adopted to mitigate the effectof weather turbulence [10]. Proposed optical power alloca-tion schemes optimally allocate transmit power to achievemaximum capacity and enhance the performance of theWDM RoFSO system. Nowadays, the bandwith wavelengthsbetween 1520 nm and 1600 nm have already been used inFSO systems. Furthermore, the emerging quantum cascadelaser (QCL) technology can offer great flexibility on adjust-ing an RoFSO transceiver to operate on the optimal transmitwavelengths [11]. Under a total power constraint, differentoptical powers can be allocated to the chosen wavelengthsin a WDM RoFSO system, to achieve further enhancedsystem performance.

We investigate the problem of optical power allocationunder the total power budget and eye safety powerconstraints for adaptive WDM transmission in RoFSO sys-tems. To achieve capacity gain, we first analyze a conven-tional FSO system and develop a simple water-filling basedalgorithm to derive the optimal power allocation for thechosen wavelengths. For WDM RoFSO systems, a near-optimal RoFSO power allocation algorithm is developedbased on the reformulation–linearization technique (RLT)[12], which can provide a linear programming (LP) relaxa-tion of the complex problem. A computationally efficientscheme is also developed based on an approximation of thechannel model. Finally, we investigate the diversity gain inthe WDM RoFSO system. The performance of the proposedschemes is evaluated with simulations, and is demonstrated

to be highly effective for achieving high system capacityunder various scenarios.

The remainder of this paper is organized as follows. Therelated work is discussed in Section 2. The system model ispresented in Section 3, while the three power allocationschemes are developed in Section 4 to fully utilize DWDMRoFSO systems. Simulation results are analyzed in Section 5.Section 6 concludes this paper.

2. Related work

Attracting significant interest both in academia and indus-try, the FSO technology has been recognized as a promisingsolution for high capacity, long distance communications.The performance of FSO networks is highly depend on theavailability and reliability of the LOS path since an FSOtransmitter is highly directional. Weather turbulencestrongly affects FSO communication links. Weather effectson the connectivity of FSO networks were studied in [13].The influence of turbulence-accentuated interchannelcrosstalk on WDM FSO system performance has been studiedin [14].

Many fading-mitigation techniques have recently beenemployed to maintain a good FSO system performance. In[15], a multipath fading resistant FSO communicationsystem architecture was introduced to combat adverseweather conditions. Spatial diversity techniques, exten-sively studied in conventional RF communication systems[16], can also be applied in FSO systems to improve systemperformance. A multiple-input multiple-output (MIMO) FSOsystem can achieve significant diversity gain in the presenceof atmospheric fading by deploying multiple transmit orreceiver apertures [7,17]. A cooperative diversity technique[18–20] is also a cost-effective alternative to maintainsystem performance. In a recent work [21], a one-relaycooperative diversity scheme was proposed for combatingturbulence-induced fading, while cooperative diversity wasanalyzed for non-coherent FSO communications.

Adaptive transmission technology has been introducedinto FSO systems to mitigate weather turbulence. Djordje-vic in [22] applied the conventional wireless adaptivemodulation and coding method in an FSO system and furtherstudied adaptive low-density-parity-check (LDPC) codedmodulation to compensate performance degradation whenturbulence is strong. Karimi and Uysal in [8] designedtransmission algorithms with consideration of the numberof bits carried per chip time (BpC) in an FSO link, in whichintensity modulation/direct detection (IM/DD) with M-arypulse position modulation (M-PPM) was employed. Severalother adaptive schemes have also been proposed andstudied. In [11], the authors proposed using variablewavelength to combat the effects of atmospheric interfer-ence. Varying wavelength becomes feasible as the QCLtechnology becomes more mature. In [23], the authorsproposed an adaptive transmission scheme to satisfy therequirements of various wireless services. A WDM powerallocation method considering Optical Modulation Index(OMI) was proposed in their adaptive RoFSO system design.The authors in [24] studied the potential of the MIMOchannel for combating link fading. However, the perfor-mance of the FSO MIMO system is constrained by the

Table 1 Table of notation.

Symbol Definition

hl FSO link attenuationhf Atmospheric turbulenceATX Aperture area of the transmitterARX Aperture area of the receiverα Atmospheric attenuation coefficientV Visibility in kilometersΛ Wavelengthq A parameter related to visibilityCNR Carrier to noise ratioOMI Optical modulation indexRIN Relative intensity noiseK Boltzman's constantT Temperaturem Photodiode gaine Electrical chargeF Excess noise factorGf Photodiode output conductanceh Channel state of a FSO linkB BandwidthN Number of available wavelength bandsM Number of used wavelength bandshi Channel gain for channel iPi Optical power allocated to channel i

P Peak power bound for transmitted pulse

Pmax Power budgetλ; λi Lagrange multipliers

173Optical power allocation for adaptive transmissions in wavelength-division multiplexing free space optical networks

thermal noise limited receivers and thus Avalanche photo-diodes (APDs) [25] were studied and commonly used in FSOsystems.

In this paper, we propose power allocation schemes foradaptive WDM transmissions to achieve capacity gain ordiversity gain. WDM has been employed in FSO transmissionsystems and has been shown to be capable of supportingvery high data rate transmission in [26]. RoFSO technologymakes it possible to transmit multiple RF signals using WDM.RoFSO provides a promising alternative to optical fibersystems. In [27], the authors designed and evaluated anRoFSO system as an universal platform for the integration ofoptical fiber and FSO networks. In [28], optical fading in FSOChannels was statistically analyzed, while [1] provides acomprehensive study of RoFSO and the satisfactory resultsconfirmed that the effect of scintillation on RoFSO perfor-mance can be estimated by an analytical model.

In the adaptive WDM RoFSO system we studied, a variablenumber of wavelengths are adopted to mitigate weatherturbulence and optical transmit power is optimally allo-cated to achieve maximum capacity gain. Although turbu-lence may not change significantly with wavelength in someweather conditions, considering the huge bandwidth thatthe DWDM RoFSO system can support, it is still non-trivial tostudy the problem of utilizing wavelength properly. More-over, since QCL technology can offer great flexibility onadjusting wavelengths, more wavelengths can be utilized asFSO systems advance. Alternatively, the multiple wave-lengths used in the WDM RoFSO system can be utilized toachieve robustness of the system.

3. System and channel model

In this section, we will introduce the channel model andsystem models. We summarize the notation used in thispaper in Table 1.

3.1. Channel model

FSO transceivers are highly directional, but FSO links areprone to degradation due to weather turbulence. Weconsider both effects of path loss and turbulence-inducedfading over FSO links [18]. The optical channel state h ismodeled as a product of two factors

h¼ hl � hf ; ð1Þwhere hl denotes the attenuation and hf represents theatmospheric turbulence. Attenuation hl is a function ofoptical wavelength Λ and link distance d, as

hl ¼ATXARX � e�αd

ðΛ � dÞ2; ð2Þ

where ATX and ARX are the aperture areas of transmitter andreceiver, respectively. The atmospheric attenuation coeffi-cient α is given by

α¼ 3:91V

� �� Λ

55

� ��q

; ð3Þ

where V is the visibility in kilometers and q is a parameterrelated to the visibility as [8]

q¼ 0:585V1=3; Vr6 km

1:3; 6 km rVr50 km:

(ð4Þ

For fading hf, we assume atmospheric turbulence can bemodeled as a log-normal distribution [29]. The log-normalmodel is a widely used fading model, especially under weak-to-moderate turbulence conditions.

The channel model for an FSO link can be written as

y ¼ h � Pt � xþn; ð5Þwhere x and y are transmitted and received signal respec-tively; Pt is the power of the transmitted pulse; and n is theadditive Gaussian noise.

3.2. System model

RoFSO is a new technology that provides high data rate andreliable transmission. The RoFSO system using WDM allowssimultaneous transmission of multiple data streams consist-ing of various wireless and wireline services at very highrates. An advanced DWDM RoFSO system is illustrated inFig. 1 [1].

We assume a WDM FSO system that is capable of operat-ing in the wavelength band from 1520 nm to 1600 nm. Apartfrom the data-transmitting antenna, we assume that anatmospheric influence measurement antenna or weathermeasurement device is equipped at the FSO BS. Thus, wecan estimate atmospheric loss for channels using different

WD

M

OE

/EO

FSO

Tra

nsce

iver

WD

M

OE

/EO

FSO

Tra

nsce

iver

Wire

less

/Wire

line

appl

icat

ions

Wire

less

/Wire

line

appl

icat

ions

Fig. 1 Illustration of an advanced DWDM RoFSO system.

H. Zhou et al.174

wavelengths. Alternatively, CSI can be obtained by using asmall portion of the bandwidth to provide channel informa-tion without tangibly affecting the data rate. In some hybridRF/FSO systems, the RF channel can be used for CSI feed-back. Since atmospheric turbulence is a major degradingfactor in FSO systems, we propose power allocation schemesfor the adaptive RoFSO system, in which both wavelengthand transmit power will be adaptively allocated accordingto channel conditions.

An important parameter to evaluate the performance ofRF-FSO is the carrier to noise ratio (CNR). The CNR of anRoFSO system using an APD photo detector is given as [1]

CNR¼ 0:5ðOMI �mPrÞ2RIN � P2

r þ2emð2þFÞPrþ4KT � Gf; ð6Þ

where OMI is the optical modulation index, m is thephotodiode gain, T is the temperature, K is Boltzman'sconstant, e is the electrical charge and Gf is the photodiodeoutput conductance. In the denominator, 4 KTGf is thermalnoise; F is the excess noise factor; m is the photodiode gain;e is the electrical charge; 2emð2þFÞPr is the optical shortnoise; and RINP2

r is the relative intensity noise from Laserdiode (LD). The numerator represents the received signalpower. Pr ¼ rPpd, where r is the photodiode responsivity andPpd is the received power at the detector and is given asthe product of transmit power and channel gain. Withoutloss of generality, we assume all tones are modulated withthe same OMI that will not introduce intermodulationdistortion.

4. Adaptive WDM transmission

To mitigate the effect of weather turbulence, we adaptwavelength and adjust power allocation to achieve bettersystem performance. First, we consider an RoFSO systemusing wavelength Λ in the range of [1520,1600] nm. Theavailable wavelengths are divided into N parts, which arenon-overlapping with adequate spacing. We assume the FSOchannel is slow varying [8]. For each channel with wave-length Λi, we can estimate its channel state hi or obtain thechannel state through a feedback channel.

In different weather conditions, it is desirable to choosethe wavelengths that have the best channel conditions. Weassume that the WDM FSO system will use at most Mdifferent wavelengths. Among the N available wavelengths,we will first choose M wavelengths that have the greatestchannel gains. The next step is to allocate transmit powersto different wavelengths such that the system capacity ismaximized.

4.1. Conventional FSO system

We first consider conventional modeling of wireless channelsunder white Gaussian noise. For the sake of simplicity, wedefine channel gain over noise for channel i by abusing ofnotation as

hi ¼jhl � hf j2

N0: ð7Þ

According to the estimated channel gains, we choose Mwavelengths that can offer the greatest channel gains fromthe N available wavelengths. Here M is a constant jointlydetermined by the channel conditions and the capacityneed of the system. It can be set to N, for example, if allthe wavelengths are to be used.

Let Pi be the optical power allocated to channel i. Due tofixed power budget Pmax for each FSO base station, we havethe following total power constraint:

XMi ¼ 1

PirPmax : ð8Þ

In FSO systems, eye safety should always be taken intoconsideration in the system design. Thus we have theadditional power constraint

0rPirP for all i: ð9Þwhere P is the peak power bound for the transmit powers.Thus, we formulate the following capacity maximizationproblem:

maxXMi ¼ 1

logð1þPi � hiÞ ð10Þ

s:t:XMi ¼ 1

PirPmax ð11Þ

0rPirP ; for all i: ð12ÞBy applying Karush–Kuhn–Tucker (KKT) theorem, we can findthat optimal power allocation satisfies

Pi ¼min P ;1λ� 1hi

� �þ� �XMi ¼ 1

Pi ¼ Pmax ;

8>>>><>>>>:

ð13Þ

where λ is the Lagrange multiplier. The inverse of λ is oftenregarded as the water level.

The algorithm to solve the capacity maximization pro-blem is to first sort the channels according to their channelgains. We then find the number of channels n, which are

log(1+γ )

γ0

Tangent lines

Segment connecting the end points

γi

Fig. 2 Four-point polyhedral outer approximation forlogð1þγÞ.

175Optical power allocation for adaptive transmissions in wavelength-division multiplexing free space optical networks

allocated with a nonzero power, as in Steps 2–3. The watervolume Hn that is required to fill n channels can becalculated as

Xni ¼ 1

i � 1hiþ1

� 1hi

� �;

and Hn should not be greater than Pmax. Then we cancalculate the water level and allocate power to eachselected channel according to (13). This procedure is basedon the assumption that a feasible power should also satisfythe eye safety power constraint. If the allocated powerð1=λ�1=hiÞ is greater than P, we need to adjust the waterlevel accordingly. The detailed water-filling algorithm ispresented in Algorithm 1.

Algorithm 1. The water-filling algorithm.

4.2. DWDM RoFSO system

Next, we develop the model for the RoFSO channels asdescribed in Section 3, which is more suitable for RoFSOusing APD photo-detectors. CNR defined in Section 3.2 willbe an important parameter to evaluate the RoFSOperformance.

4.2.1. Reformulation and relaxation based approachTo simplify notation, we denote the constant 0:5ðmOMIÞ2 bya, the relative intensity noise level RIN by b, the opticalshort noise 2emð2þFÞ by c, and the thermal noise 4 KTGf byd. Thus, our adaptive power allocation problem becomes

maxXMi ¼ 1

log 1þ aiðPihiÞ2biðPihiÞ2þciPihiþdi

!ð14Þ

s:t:XMi ¼ 1

PirPmax ð15Þ

0rPirP for all i: ð16Þ

If we denote CNR by

γi ¼aiðPihiÞ2

biðPihiÞ2þciPihiþdi; ð17Þ

the optimization problem, termed Problem OPT-RoFSO, canbe rewritten as

maxXMi ¼ 1

logð1þγiÞ ð18Þ

s:tXMi ¼ 1

PirPmax ð19Þ

0rPirP for all i ð20Þ

γi ¼aiðPihiÞ2

biðPihiÞ2þciPihiþdifor all i: ð21Þ

It is challenging to solve this problem due to its complexityand nonlinear nonconvex properties. In the following, weadopt the RLT technique to obtain an LP relaxation ofProblem OPT-RoFSO and derive a feasible near-optimalsolution [12].

The RLT relaxation is as follows. Letting

ci ¼ logð1þγiÞ;the objective function

PMi ¼ 1 ci will now be linear and new

constraints ci ¼ logð1þγiÞ are introduced. We first linearizethe logarithmic terms in the new constraints using thepolyhedral outer approximation as follows.

From (21), it can be seen that γi is a monotone increasingfunction of Pi. Letting Pi be 0 and P, we obtain the lowerand upper bounds of γi, respectively. We denote the upperbound of γi by γ i, while the lower bound is 0. We use thefour-point approximation and obtain the following newlinear constraints:

ciZγiγ i� log 1þγ i

� cir log 1þγki

� þγi�γki1þγki

;

8>>><>>>:

ð22Þ

where

γki ¼13� kγ i for k¼ 0; 1; 2; 3:

H. Zhou et al.176

The first equation in (22) is for the segment connecting thetwo end points of the logarithm function and the secondequation in (22) is for the tangent lines at the four points onthe logarithm function respectively. A four-point approx-imation for logð1þγÞ is illustrated in Fig. 2. The correspond-ing convex envelope is formed by four tangent lines and achord connecting the two end points.

Problem OPT-RoFSO now becomes a polynomial program-ming problem. We next introduce substitution variables andthe corresponding RLT bound-factor product constraints toremove the quadratic terms and to obtain an LP relaxation.Specifically, constraint (21) contains quadratic terms. Wecan rewrite (21) as

biðPihiÞ2γiþcihiPiγiþdiγi�aiðPihiÞ2 ¼ 0: ð23ÞTo remove quadratic terms, we define substitution variables

ui ¼ P2i for all i

vi ¼ γiPi for all i

wi ¼ γiui for all i

βi ¼ civi for all i:

8>>>><>>>>:

ð24Þ

Thus constraint (21) becomes

bih2i wiþhiβiþdiγi�aih

2i ui ¼ 0: ð25Þ

Since Pi is bounded, it can easily show that uiA ð0; P2Þ. Forvariables vi, with 0rγirγ i and 0rPirP, we can obtainthe following RLT bound-factor product constraints:

ðγi�0ÞðPi�0ÞZ0

ðγ i�γiÞðPi�0ÞZ0

ðγi�0ÞðP�PiÞZ0

ðγ i�γiÞðP�PiÞZ0:

8>>>><>>>>:

ð26Þ

Substituting vi ¼ γiPi, we obtain the following four linearconstraints for variable vi:

viZ0

γ iPi�viZ0

Pγi�viZ0

Pγ i�γ iPi�PγiþviZ0:

8>>>><>>>>:

ð27Þ

For variable wi, with 0rγirγ i and 0ruirP2, we

obtain the following four linear constraints in the samemanner:

wiZ0

γ iui�wiZ0

P2γi�wiZ0

P2γ i�γ iui�P

2γiþwiZ0:

8>>>>><>>>>>:

ð28Þ

We deal with the variables βi in the same manner, with0rcirci and 0rvirγ iP, and obtain the following fourlinear constraints for variable βi.

βiZ0

civi�βiZ0

γ iPci�βiZ0

ciγ iP�γ iPci�civiþβiZ0;

8>>>><>>>>:

ð29Þ

where ci ¼ logð1þγ iÞ.

Now the original problem is relaxed to an LP problemwith the additional constraints and variables as follows:

maxXMi ¼ 1

ci ð30Þ

s:t:XMi ¼ 1

PirPmax ð31Þ

0rPirP for all i ð32Þ

0ruirP2

for all i ð33Þ

bih2i wiþhiβiþdiγi�aih

2i ui ¼ 0;

for all i ð34Þ

New linear constraints ð22Þ forall i ð35Þ

RLT bound� factor constraints ð27Þ;ð28Þ and ð29Þ for all i: ð36Þ

The relaxed problem can be solved in polynomial time withan LP solver. Note that during the procedure of reformula-tion and linearization, we preserve the original powerconstraints of Problem OPT-RoFSO, i.e., (19) and (20).Hence the optimal transmit power allocation policyobtained for the LP relaxation is also feasible to the originalproblem OPT-RoFSO. The feasibility of the LP solution issummarized in the following proposition:

Proposition 1. The optimal transmit power allocationpolicy to the LP relaxation of Problem OPT-RoFSO is afeasible solution to the original problem.

4.2.2. A faster near-optimal algorithmDue to the complexity of the RLT-based method, it may notbe suitable when the channels vary quickly. We nextdevelop a more computationally cost-effective scheme inthe following. If we ignore the relative intensity noise andoptical short noise, CNR can be approximated by0:5P2

r ðmOMIÞ2=4KTGf. To simplify notation, we denote con-stant value 0:5ðmOMIÞ2=4KTGf by a. We obtain the followingoptimization problem:

maxXMi ¼ 1

log 1þaiðPihiÞ2 �

ð37Þ

s:t:XMi ¼ 1

PirPmax ð38Þ

0rPirP for all i: ð39Þ

According to Karush–Kuhn–Tucker (KKT) theorem, ifP⋆ ¼ ½P⋆

1 ; P⋆2 ;⋯; P⋆

M� is a local maximizer for the aboveoptimization problem, there exists λAR and λiAR, for

Table 2 Simulation parameters.

Symbol Value Definition

Dt 15 mm Tx. aperture diameterDr 0.1 m Rx. aperture diameterB 1 GHz Bandwidthd 1 km DistancePmax 0.5 W Power budget

P 0.1 W Peak power constraint

OMI 17.5% Optical modulation indexm 5 Photodiode gainRIN �150 dB/Hz Relatively intensity noise

0.3 0.4 0.5 0.6 0.7 0.8

100

120

140

160

180

200

220

240

260

280

300

Power Budget (W)

Cap

acity

(Gbp

s)

RLT AlgorithmWater FillingRoFSO Power Allocation

Fig. 3 System capacity versus the power budget Pmax.

177Optical power allocation for adaptive transmissions in wavelength-division multiplexing free space optical networks

iAf1; 2;⋯;Mg, such that

∂½logð1þaiðP⋆i hiÞ2Þ�

∂Pi�λ�λi ¼ 0 for all i

λXMi ¼ 1

P⋆i �Pmax

!¼ 0

λiðP⋆i �PÞ ¼ 0 for all i

λZ0;

λiZ0 for all i:

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð40Þ

According to (40), if λi40, then P⋆i ¼ P; and if λi ¼ 0, we

can solve (40) to have

P⋆i ¼ 1

λþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

λ2� 1

aih2i

s: ð41Þ

Thus we find that the optimal power allocation satisfies

Pi ¼ min P ;1λþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

λ2� 1

aih2i

s( ); if λ2λaih

2i

Pi ¼ 0; otherwisePMi¼1 Pi ¼ Pmax:

8>>>><>>>>:

ð42Þ

Regarding the inverse of λ as some kind of “water level,”we find that the greater the channel gain, the larger thedeviation of the power allocated to this channel from thewater level. Usually we cannot directly solve from (42) forthe optimal power allocation. An iterative algorithm isneeded to obtain an appropriate λ and solve this optimiza-tion problem. The detailed algorithm is presented inAlgorithm 2.

Algorithm 2. RoFSO power allocation algorithm.

5. Performance evaluation

In this section, we evaluate the performance of theproposed algorithms with MATLAB simulations. We calculatechannel gains as shown in Section 3.1 and CNR are calcu-lated according to (6) for the evaluated schemes. Thesimulation parameters are the same from prior work andare listed in Table 2 [1,23].

We investigate the three algorithms introduced in theprevious section for power allocation in the WDM RoFSOsystem. Wavelengths in the band 1520–1580 nm are assumed

in our WDM RoFSO simulations. We adopt a 5 nm spacingbetween adjacent wavelengths used in the simulations.

In Fig. 3, we compare the three algorithms introduced inSection 4 and examine the impact of the power budget onthe total system capacity. We increase Pmax from 0.5 to1 with step-size 0.1 and plot the total capacity. As can beseen in Fig. 3, the RoFSO power allocation algorithm out-performs the other two algorithms with considerable gains.Since the relative intensity noise and the optical short noiseare very small in our simulations, the RoFSO power alloca-tion algorithm will achieve the best near-optimal solution.The RLT algorithm, although consumes much running time,can only produce an optimal power allocation for therelaxed LP problem. The power allocation solution obtainedfrom RLT is feasible but achieves the worst performance interms of capacity gain due to relaxation. We also find thatthe total capacity increases along with the power budget forboth the water-filling algorithm and RoFSO power allocationalgorithm, albeit not obviously for the latter. After thepower budget becomes even larger, there is much less spacefor capacity increment due to the eye safety powerconstraint.

Next, we examine the effect of weather conditions on thesystem capacity in Fig. 4. The distance between FSO BS's isset to 500 m. We change the weather condition from clearto foggy by changing the atmospheric attention coefficient.The coefficient is 0.48 db/km for clear weather, 2.8 db/km

Clear Hazy Foggy

50

100

150

200

250

300

Weather Conditions

Cap

acity

(Gbp

s)RLT AlgorithmWater−fillingRoFSO Power Allocation

Fig. 4 System capacity versus weather condition.

0.5 1 1.5 2 2.5 3

50

100

150

200

250

300

Distance(Km)

Cap

acity

(Gbp

s)

RLT AlgorithmWater FillingRoFSO Power Allocation

Fig. 5 System capacity versus distance.

15 25 35

100

200

300

400

500

600

700

No. of Subcarriers in clear weather

Cap

acity

(Gbp

s)

RLT AlgorithmWater FillingRoFSO Power Allocation

Fig. 6 System capacity versus the number of subcarriers inclear weather.

15 25 35

100

200

300

400

500

600

700

No. of Subcarriers in hazy weather

Cap

acity

(Gbp

s)

RLT AlgorithmWater FillingRoFSO Power Allocation

Fig. 7 System capacity versus the number of subcarriers inhazy weather.

H. Zhou et al.178

for hazy weather, and 15 db/km for foggy weather [30]. Ascan be seen from Fig. 4, the system capacity decreases asweather gets worse. We also find that the simple water-filling algorithm which is developed for conventional RFsystems is not suitable for the RoFSO system when weathercondition is severe. The capacity achieved by the simplewater-filling algorithm decrease the most as the weatherbecomes worse. When the weather is foggy, the capacityachieved by the water-filling algorithm is about 10 Gbps.

We also plot the total capacity vs. the distance betweenthe FSO BS's in Fig. 5. As expected, the total capacitydecreases as the distance is increased. When the distancebetween the FSO transceivers is relatively small, thedifference between the capacities achieved by three algo-rithms is also small. But as we increase the distance from500 m to 1 km or even greater, the capacities achieved bysimple water filling algorithm and RLT algorithm dropdramatically. The capacity obtained by using the RoFSOpower allocation algorithm decreases the least and is alwaysgreater than capacities produced by the other two schemes.

Finally, we examine the impact of the number ofsubcarriers used in an adaptive WDM RoFSO system on thesystem capacity. In the wavelength bands starting from1520 nm, we adopt a 1 nm spacing between adjacent

wavelengths. The simulation results in clear weather arepresented in Fig. 6. As we can see in Fig. 6, a systemcapacity increase if we adopt more subcarriers in theadaptive WDM RoFSO system. When there are 15 channelsin the system, the difference between the achieved capa-city of the three schemes is not much great. However, as thenumber of subcarriers increases, the advantage of theRoFSO power allocation scheme becomes greater.

We also run our simulations in hazy and foggy weatherand presented in Figs. 7 and 8, respectively. In Fig. 7, theRoFSO power allocation algorithm achieves greater capacitywith more subcarriers. System capacities increased with thenumber of subcarriers for all three algorithms. However,due to the simplified objective function used in the devel-opment of the water-filling algorithm, the water-fillingalgorithm does not achieve as much capacity as the RLTalgorithm when the number of sub-carriers is 35. The RoFSOpower allocation algorithm makes better use of availablesubcarriers to enhance system capacity and thus providethe greatest capacity. When weather conditions becomeworse, the system capacity become much small and the

15 25 35

50

100

150

200

250

No. of Subcarriers in foggy weather

Cap

acity

(Gbp

s)

RLT AlgorithmWater FillingRoFSO Power Allocation

Fig. 8 System capacity versus the number of subcarriers infoggy weather.

179Optical power allocation for adaptive transmissions in wavelength-division multiplexing free space optical networks

results are shown in Fig. 8. When the weather is foggy, thecapacity achieved by the water-filling algorithm is verysmall. But the RoFSO power allocation algorithm and theRLT algorithm can maintain system performance in foggyweather. Also when more subcarriers are utilized in thesystem, more system capacity can be achieved by usingthese two algorithms.

To conclude, the system performance in terms of capacityby using the RoFSO power allocation algorithm is the best inall situations studied here. These simulation results indicatethat with proper system design, the RoFSO system cansupport high data rates even over long distance and underbad weather conditions.

6. Conclusions

In this paper, we investigated the problem of optical powerallocation under a power budget constraint and eye safetypower constraint for adaptive WDM transmission to mitigatethe effect of weather turbulence, and solution algorithmsare developed. For convectional transmission systems, asimple water-filling algorithm can be adopted to allocatepower to different wavelengths; for WDM RoFSO systems, anRoFSO power allocation algorithm was demonstrated toachieve the greatest system capacity. It is capable ofsupporting high data rates even over long distance or underbad weather conditions. Utilizing multiple wavelengths inthe WDM FSO system for diversity gain was also investigatedin the paper.

Acknowledgements

This work was supported in part by the U.S. NationalScience Foundation (NSF) under Grant CNS-1320664, andthrough the Wireless Engineering Research and EducationCenter (WEREC) at Auburn University. Any opinions, find-ings, and conclusions or recommendations expressed in thismaterial are those of the author(s) and do not necessarilyreflect the views of the NSF.

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Hui Zhou received the Ph.D. in Electricaland Computer Engineering from AuburnUniversity, Auburn, AL in 2014. She receivedthe M.S. degree in 2009, and the B.S.degree in 2007 from Huazhong Universityof Science and Technology, both in Electro-nic and Information Engineering. Currently,she is a Software Development Engineer atAmazon Web Services, Amazon, Inc. Shewas a Senior DSP Software Engineer with

Zhongxing Telecommunication Equipment Corporation, Shanghai,China from 2009 to 2011. Her research interests include femtocellnetworks and free space networks. She is a co-recipient of the IEEEInternational Conference on Communications (ICC) 2013 BestPaper Award.

Shiwen Mao received his Ph.D. in electrical

and computer engineering from PolytechnicUniversity, Brooklyn, NY in 2004. He was theMcWane Associate Professor in the Depart-ment of Electrical and Computer Engineer-ing from 2012 to 2015, and is the SamuelGinn Endowed Professor and the Director ofthe Wireless Engineering Research and Edu-cation Center (WEREC) since 2015 at AuburnUniversity, Auburn, AL, USA.

His research interests include wireless networks and multimediacommunications. He is a Distinguished Lecturer of the IEEEVehicular Technology Society in the Class of 2014, and the ViceChair-Letters and Member Communications of IEEE CommunicationsSociety Multimedia Communications Technical Committee. He is onthe Editorial Board of IEEE Transactions on Multimedia, IEEEInternet of Things Journal, IEEE Communications Surveys andTutorials, and IEEE Multimedia, among others. He serves as SteeringCommittee Member for IEEE ICME and AdhocNets, Area TPC Chair ofIEEE INFOCOM 2016 and Technical Program Vice Chair for Informa-tion Systems of IEEE INFOCOM 2015, symposium co-chairs for manyconferences, including IEEE ICC, IEEE GLOBECOM, ICCCN, et al. Hereceived the 2013 IEEE ComSoc MMTC Outstanding LeadershipAward and the NSF CAREER Award in 2010. He is a co-recipient ofThe IEEE WCNC 2015 Best Paper Award, The IEEE ICC 2013 BestPaper Award, and The 2004 IEEE Communications Society LeonardG. Abraham Prize in the Field of Communications Systems.

Prathima Agrawal is an Emeritus professorof electrical and computer engineering,Auburn University, Auburn, Alabama. Priorto her present positions, she worked inTelcordia Technologies (formerly Bellcore),Morristown, New Jersey, and at AT&T/Lucent Bell Laboratories, Murray Hill, NewJersey. She created and served as head ofthe Networked Computing Research Depart-ment in Murray Hill. She is widely published

and holds 51 US patents. She is a Fellow of IEEE. She received the BEand ME degrees in electrical communication engineering from theIndian Institute of Science, Bangalore, India, and the PhD degree inelectrical engineering from the University of Southern Californiain 1977.