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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 9, DECEMBER 2009 1709

Outage Analysis of the Hybrid Free-SpaceOptical and Radio-Frequency Channel

Nick Letzepis, Member, IEEE, Khoa D. Nguyen, Student Member, IEEE,Albert Guillen i Fabregas, Senior Member, IEEE, and William G. Cowley, Member, IEEE

Abstract�—We study the hybrid free-space optical (FSO) andradio-frequency (RF) channel from an information theoreticperspective. Since both links operate at vastly different carrierfrequencies, we model the hybrid channel as a pair of parallelchannels. Moreover, since the FSO channel signals at a higherrate than the RF channel, we incorporate this key feature in theparallel channel model. Both channels experience fading due toscintillation, which is slow compared to typical signalling rates.Under this framework, we study the fundamental limits of thehybrid channel. In particular, we analyse the outage probabilityin the large signal-to-noise ratio (SNR) regime, and obtain theoutage diversity or SNR exponent of the hybrid system. First weconsider the case when only the receiver has perfect channelstate information (CSIR case), and obtain the exponents forgeneral scintillation distributions. These exponents relate keysystem design parameters to the asymptotic outage performanceand illustrate the benets of using hybrid systems with respect toindependent FSO or RF links. We next consider the case whenperfect CSI is known at both the receiver and transmitter, andderive the optimal power allocation strategy that minimises theoutage probability subject to peak and average power constraints.The optimal solution involves non-convex optimisation, whichis intractable in practical systems. We therefore propose asuboptimal algorithm that achieves signicant power savings (onthe order of tens of dBs) over uniform power allocation. Weshow that the suboptimal algorithm has the same diversity asthe optimal power allocation strategy.

Index Terms�—Optical communication, radio-frequency com-munication, scintillation, outage probability, information theory,coded modulation, Singleton bound, power allocation.

I. INTRODUCTION

IN FREE-SPACE optical (FSO) communication an opticalcarrier is employed to convey information wirelessly. FSO

systems have the potential to provide ber-like data rateswith the advantages of quick deployment times, high securityand no frequency regulations. Unfortunately such links arehighly susceptible to atmospheric effects. Scintillation inducedby atmospheric turbulence causes random uctuations in thereceived irradiance of the optical laser beam [1]. Numerousstudies have shown that performance degradation caused byscintillation can be signicantly reduced through the use

Manuscript received 2 February 2009; revised 14 May 2009. This work wassupported by the Sir Ross and Sir Keith Smith Fund, Cisco Systems and theAustralian Research Council under RN0459498, DP0558861 and DP088160.N. Letzepis, K. D. Nguyen and W. G. Cowley are with Institute for

Telecommunications Research, University of South Australia, SPRI Build-ing - Mawson Lakes Blvd., Mawson Lakes SA 5095, Australia (e-mail:[email protected]).A. Guillen i Fabregas is with the Department of Engineering, University

of Cambridge, Cambridge CB2 1PZ, UK (e-mail: [email protected]).Digital Object Identier 10.1109/JSAC.2009.091220.

of multiple-lasers and multiple-apertures, creating the well-known multiple-input multiple-output (MIMO) channel (seee.g. [2], [3], [4], [5], [6], [7], [8], [9], [10]). However, itis the large attenuating effects of cloud and fog that posethe most formidable challenge. Extreme low-visibility fog cancause signal attenuation on the order of hundreds of decibelsper kilometre [11]. One method to improve the reliability inthese circumstances is to introduce a radio frequency (RF)link to create a hybrid FSO/RF communication system [12],[13], [14], [15], [16], [11]. When the FSO link is blocked bycloud or fog, the RF link maintains reliable communications,albeit at a reduced data rate. Typically a millimetre wavelengthcarrier is selected for the RF link to achieve data ratescomparable to that of the FSO link. At these wavelengths, theRF link is also subject to atmospheric effects, including rainand scintillation [17], [18], [19], [20], [21], but less affected byfog. The two channels are therefore complementary: the FSOsignal is severely attenuated by fog, whereas the RF signal isnot; and the RF signal is severely attenuated by rain, whereasthe FSO is not. Both, however, are affected by scintillation.Most works on the hybrid channel [12], [13], [14], [16],

[11] consider the RF and FSO links as separate channels, i.e.the channels do not aid each other to compensate signal leveluctuations. In these works, the main purpose of the RF linkis to act as a backup when the FSO link is down. In [15]a hybrid channel coding scheme is proposed that combinesboth FSO and RF channels and adapts the code rate to thechannel conditions. Multilevel coding schemes have also beenproposed in [22] for the hybrid channel.Lacking so far in the literature on hybrid FSO/RF channels

is the development of a suitable channel model and itstheoretical analysis to determine the fundamental limits ofcommunication. This is the central motivation for our paper.We propose a hybrid channel model based on the well knownparallel channel [23], and take into account the differencesin signalling rate, and the atmospheric fading effects presentin both the FSO and RF links. These fading effects are slowcompared to typical data rates, and as such, each channel isbased on a block-fading channel model. Previously in [2], [3],using a block-fading channel model, we examined the outageprobability of the multiple-input multiple-output (MIMO) FSOchannel under the assumption of pulse-position modulation(PPM) for several well-known scintillation distributions, i.e.lognormal, exponential, gamma-gamma, lognormal-Rice andI-K distributed scintillation [1]. In particular, we examinedthe SNR exponent (or outage diversity) [24], [25], which

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1710 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 9, DECEMBER 2009

AtmosphericChannelEncoder

FSO Mod

RF Mod

PD

RF Up

Decoder

RF Down

Laser Aper-ture

Fig. 1. Hybrid FSO/RF communication system.

describes the high SNR slope of the outage probability curvewhen plotted on a log-log scale. In this paper, we extend thisanalysis to include an RF link to create a hybrid FSO/RFchannel. The message to be transmitted is encoded intoparallel FSO and RF bit streams which are sent across theFSO and RF channels simultaneously. We examine the casewhen perfect CSI is known at the receiver only (CSIR case),then we consider the case when CSI is also known at thetransmitter (CSIT case), and power allocation is employedto reduce the outage probability subject to power constraints.When CSI is not available at the transmitter, we calculate theSNR exponents of the hybrid channels for general scintillationdistributions in each of the channels. On the other hand, whenCSI is available at the transmitter, we derive the optimalpower allocation algorithm subject to both peak and averagepower constraints. This optimal solution involves non-convexoptimisation, which has prohibitive complexity in practicalsystems. We therefore propose a suboptimal solution and provethat it has the same SNR exponent as optimal power allocation.The remainder of this paper is organised as follows. In

Section II we present our channel model and assumptions.Section III reviews the required information-theoretic prelim-inaries. Section IV presents our main results for the CSIR-only case while Section V discusses power allocation andSNR exponents for the CSIT case. Section VI draws nalconcluding remarks. Proofs of our results can be found in theappendices.

II. CHANNEL MODEL AND ASSUMPTIONS

A block diagram of the communication system of interestis shown in Fig. 1. A binary data sequence is binary encodedinto parallel FSO and RF bit streams. The RF link modu-lates the encoded bits and up-converts the baseband signalto a millimetre wavelength RF carrier frequency. The FSOlink employs intensity modulation and direct detection, i.e.information is modulated using only the irradiance of a laserbeam. The RF and FSO signals are transmitted simultaneouslythrough an atmospheric channel. The received RF signal isthen downconverted to baseband and sent to the decoder.At the same time, the received irradiance is collected by anaperture, converted to an electrical signal via photodetectionand sent to the decoder. The received signals are jointlydecoded to recover the transmitted message.In this paper we dene a hybrid channel symbol, (x, x) !

Xnfso " Xrf , consisting of component FSO and RF symbols,which are transmitted in parallel with perfect synchronismand have the same symbol period Ts. The RF component

symbol, denoted by x, is drawn uniformly from an M -arycomplex signal set Xrf # C representing e.g. quadratureamplitude modulation (QAM) or phase shift keying (PSK).We assume the RF symbols are transmitted with unit averageenergy, i.e. E[|x|2] = 1. The number of bits per RF symbolis m ! log2 M bits. Since the FSO link employs a muchhigher carrier frequency than the RF link, we assume theFSO component of the hybrid channel symbol consists ofn symbols drawn uniformly from a constellation Xfso. Mostpractical FSO systems employ a pulsed type of modulation,e.g. on-off keying (OOK), pulse-position modulation (PPM) ormulti-PPM (MPPM). Hence we assume Xfso represents a pulsetype modulation scheme, i.e. the FSO component of the hybridsymbol consists of n symbols, which are further composed ofQ pulse intervals of duration Tp, where Ts = nQTp. Thesignal set Xfso # (0, 1)Q is a set of Q length binary vectors,where a binary 1 at index i indicates a pulse of duration Tp intime slot i. Hence the FSO symbols x ! Xn # (0, 1)nQ are1"nQ binary vectors. We assume each Tp second �‘on�’ pulseis normalised to have unit energy and denote the average FSOsymbol energy by ! = E

!"nQi=1 xi

#. Let q ! log2(|Xfso|),

hence the total bits per hybrid channel symbol is m+nq bits.Note that one could also consider pulse-amplitude modulated(PAM) symbols. In this case the same analysis and result thatfollow will apply, but for simplicity we assume on-off typepulses.Both FSO and RF channels are affected by scintillation [1],

[17], [18], [19], which is a slow fading process compared totypical data rates.1 We therefore model each component chan-nel by a block-fading channel model, whereby the componentchannels are divided into a nite number of blocks of symbols,and each block experiences an i.i.d. fading realisation. Thescintillation experienced by each component channel is alsoassumed to be independent.2 Typically, the RF scintillationhas a coherence time on the order of seconds [17], [18],[19], whereas the FSO scintillation is much faster, havinga coherence time on the order of tens of milliseconds [1].We therefore decompose the FSO and RF components ofthe codeword into A and B blocks of K and L symbolsrespectively, where A $ B. Given that the coherence timeof the RF scintillation is on the order of seconds, the mostrealistic scenario is B = 1. However, for generality we willassume B is an arbitrary positive integer. Note that the totalnumber of symbols in each FSO/RF component codeword isthe same, i.e. AK = BL. We assume that the number ofsymbols in each block tends to innity, but the ratio remainsa xed constant, i.e.

limK,L!"

L

K=

A

B. (1)

We assume both FSO/RF component channels are mod-elled by independent additive white Gaussian noise (AWGN)channels. Note that this assumption for the FSO channel maynot be accurate under certain conditions. The photodetec-

1We use the terms scintillation and fading interchangeably to refer touctuations in power of the received FSO/RF signals.2This will be true over short time intervals, but over longer time scales

meteorological variations will result in correlated channel fades.


LETZEPIS et al.: OUTAGE ANALYSIS OF THE HYBRID FREE-SPACE OPTICAL AND RADIO-FREQUENCY CHANNEL 1711

tor output electrical signal is proportional3 to the incidentoptical irradiance waveform (instantaneous optical power).Under ideal photodetection, thermal noise is negligible andthe received signal represents detected photons. The numberof detected photons in a given time interval is governed bya Poisson process whose rate is proportional to the averageirradiance over the time interval [26]. This is the well-knownPoisson channel for which a number of works on FSO systemsemploy (see e.g. [10], [7], [27]). As the average irradianceincreases due to increased background irradiation or high op-tical signal power, these statistics can be well approximated byGaussian statistics. If avalanche photodetection is employed,then random photo-multiplication effects must be considered.A comprehensive comparison of the various photodetectionmodels, Poisson, Webb, Gaussian (signal dependent and signalindependent variations) is given in [28]. For simplicity andanalytical convenience, we assume the combined shot andthermal noise can be considered as signal independent AWGN(an assumption also commonly used in the literature [26], [28],[9], [1], [29], [6], [4]). Under this assumption, the receivedsignals can written as,

ya[k] = pa"haxa[k] + za[k] (2)

yb[l] =$

pb"!hbxb[l] + zb[l], (3)

for l = 1, . . . , L, k = 1, . . . , K a = 1, . . . , A and b =1, . . . , B, where: ya[k] ! RnQ and yb[l] ! C are the noisyreceived symbols for the FSO and RF channels respectively;xa[k] ! Xn

fso and xb[l] ! Xrf denote the transmitted symbols;za[k] ! RnQ is a i.i.d. vector of zero mean unit variancereal Gaussian noise, and zb[l] ! C is unit variance complexGaussian noise (CN (0, 1)); ha > 0 and hb > 0 are inde-pendent random power uctuations due to scintillation, eachi.i.d. drawn from distributions fH and fH respectively, withnormalisation E[ha] = E[hb] = 1; pa and pb denotes the powerof block a and b for the FSO and RF channels respectively.The ! parameter in (3) ensures both FSO and RF symbolshave the same energy. The parameters 0 < ", " < 1 in (2)and (3) model differences in the relative strengths of the twoparallel channels, e.g. it reects long-term fading effects dueto rain, fog or cloud as well as other parameters such asaperture/antenna gains and propagation loss. When " > ",the FSO channel is much stronger than the RF, e.g. modellingthe effects of severe rain attenuation. On the other hand, if" < ", then the RF channel is stronger than the FSO channel,e.g. modelling the effects of severe fog/cloud attenuation.Although in practise " and " are randomly varying with time(and are also most likely correlated random variables), weassume they remain unchanged over many codeword timeintervals and therefore are xed constants. As we shall seelater, these parameters will not affect the asymptotic outageanalysis that is to follow.In this paper, we consider two CSI scenarios. First we will

assume only the receiver has perfect CSI and the transmitterallocates power uniformly across all blocks (CSIR case). Thenwe will consider the case where perfect CSI is also known

3The proportionality constant is referred to as the responsivity (measuredin Watts per Ampere), and is dependent on the optical wavelength andphotodetector material.

at the transmitter (CSIT case). The transmitter then performspower allocation to reduce the outage probability subject topower constraints. The transmit power for both channels isultimately drawn from the same power resource. As such, weassume the long-term average power consumed by the hybridsystem is constrained according to

E [%p&] + E [%p&] ' Pav, (4)

where %p& = 1A

"Aa=1 pa and %p& = 1

B

"Bb=1 pb. Practical

communication systems have limitations on the peak powerthat can be transmitted. Since the FSO and RF links willrealistically have different peak power limitations, we assumeeach is subjected to its own individual peak (or short-term)power constraint, i.e.

%p& ' P fsopeak and %p& ' P rf

peak, (5)

where P fsopeak = #fsoPav and P rf

peak = #rfPav for xed peak-to-average power ratios #fso and #rf .It is important to note the difference in signal scaling

between the FSO and RF AWGN channels, (2) and (3) respec-tively. In standard RF channels, the amplitude of the receivedelectrical signal is proportional to the square-root of the powerof the transmitted electromagnetic signal [30]. For the FSOchannel, the amplitude of the received electrical signal isdirectly proportional to the transmitted optical power, due tothe photodetection process [26]. This FSO model is slightlydifferent to [2], where the AWGN FSO channel model waswritten in terms of the received electrical power (proportionalto the square of the optical power). In this paper, we write theFSO AWGN channel (2) and power constraints (4) and (5)with respect to the transmitted optical and RF power. This isnecessary to ensure consistency between the power constraintsover the FSO and RF channels. As we shall see later, thisscaling will signicantly affect the design of power allocationstrategies when CSI is known at the transmitter.

III. INFORMATION THEORETIC PRELIMINARIES

The channels described by (2) and (3) under the quasi-staticassumption are not information stable [31] and therefore, theirindividual channel capacities in the strict Shannon sense arezero. For these channels, it is more appropriate to analysethe information outage probability, which lower bounds thecodeword error probability of any coding scheme [32], [33].Dene the vector channel realisations h = (h1, . . . , hA)and h = (h1, . . . , hB), and vector power allocations p =(p1, . . . , pA) and p = (p1, . . . , pB). The total instantaneousmutual information in bits per channel use is therefore [23]

Itot(p, p, h, h) =

n

A

A%

a=1

IawgnXfso

(h2a"

2p2a) +

1B

B%

b=1

IawgnXrf

(hb"!pb), (6)

where IawgnX (u) ! (0, log2 |X |) denotes the input-output

mutual information of the AWGN channel with input con-stellation X and SNR u. Note that the achievable rate (6)implicitly assumes joint encoding and decoding across FSOand RF channels.



TABLE ISNR EXPONENTS FOR SOME TYPICAL SCINTILLATION DISTRIBUTIONS.

Distribution PDF f(h) SNR Exponent

Exponential exp("h) d(1) = 1

Lognormal 1h!

!2"

e"(log h"µ)2/(2!2) d(2) = 14 log(1+!2)

Gamma-gamma 2(ab)a+b2

!(a)!(b) ha+b2 "1 Ka"b(2

#abh) d(1) = min(a, b)

The information outage probability of the hybrid system isgiven by

Pout(Pav, R) ! Pr&

Itot(p, p, h, h) < R'

, (7)

where R is the target rate of the system in bits per hybridchannel use. We dene the overall binary code rate of thehybrid system as Rc ! R/(m + nq), 0 < Rc < 1. Inaddition, we dene 0 < $ < 1 as the ratio of FSO bits tototal transmitted bits, i.e. $ ! nq

m+nq .In this paper we study the SNR exponent of the system,

dened as

d(k) ! limPav!"

( log Pout(Pav, R)(log Pav)k

, (8)

where k = 1, 2. Note that by including the integer k, (8)is more general than the SNR exponent normally dened inRF systems [24], [25]. This modied denition is required toallow for scintillation cases where the resulting outage proba-bility curve will not converge to a constant slope when plottedon a log-log scale, but does when plotted on a log( log2

scale (most notably under weak turbulence conditions wherethe scintillation is log-normal distributed [2]).The SNR exponent will depend on the distribution of the

fading coefcients. Rather than assuming a specic distribu-tion, we characterise the fading via the component channel�’ssingle block transmission SNR exponent, dened as

d(i)fso ! lim

u!"(

log Pr{Iawgnfso (h2u2) < Rfso}

(log u)i(9)

d(j)rf ! lim

u!"( log Pr{Iawgn

rf (hu) < Rrf}(log u)j

, (10)

for given component channel rate constraints Rfso and Rrf ,where i, j ! {1, 2}. In Table I we list some single blocktransmission SNR exponents (derived in [2]) for some typicalscintillation distributions [1].4 Note that the exponents derivedin [2] dened the SNR exponents in terms of the receivedelectrical SNR. In this paper, we dene the FSO exponent interms of the transmitted optical power. This results in a factorof 2 in the exponents listed in Table I compared to those givenin [2].

IV. ASYMPTOTIC OUTAGE ANALYSIS: CSIR CASE

First let us assume that perfect CSI is known only atthe receiver (CSIR case). The transmitter allocates poweruniformly across all blocks, i.e. p1 = . . . = pA = p1 =. . . = pB = p = Pav.4Note that if MIMO FSO with transmit repetition and equal gain combining

is employed, then the exponents listed in Table I are simply multipliedby NtNr , where Nt and Nr denote the number of lasers and aperturesrespectively [2].

Theorem 4.1: Dene component channel SNR exponentsd(i)fso and d(j)

rf as in (9) and (10) respectively. Suppose ", " > 0and i = j = k. Then the SNR exponent is given by

d(k) = infK(!,Rc)

&d(k)fso%1 + d(k)

rf %2

', (11)

where

K($, Rc) !&%1,%2 ! Z : $

%1

A+ (1 ( $)

%2

B> 1 ( Rc,

0 ' %1 ' A, 0 ' %2 ' B'. (12)

Proof: See Appendix A.From Theorem 4.1, we see that the overall SNR exponentdepends on Rc, $, A, B and the individual SNR exponents dfso

and drf in a non-trivial way. Although in general, the solutionto (11) can be straightforwardly determined numerically, itis difcult to obtain insight as to how the various systemparameters inuence the overall SNR exponent. However, forthe most basic and interesting scenario, A = B = 1, thesolution to (11) reduces to a simple intuitive form.Corollary 4.1: Suppose A = B = 1. The solution to (11)

is divided into two cases as follows.1) If $ ' 1

2 , then

d(k) =

()*

)+

d(k)fso + d(k)

rf 0 < Rc ' $

d(k)rf $ < Rc ' 1 ( $

min(d(k)fso , d(k)

rf ) 1 ( $ < Rc < 1.

(13)

2) If $ $ 12 , then

d(k) =

()*

)+

d(k)fso + d(k)

rf 0 < Rc ' 1 ( $

d(k)fso 1 ( $ < Rc ' $

min(d(k)fso , d(k)

rf ) $ < Rc < 1.

(14)

Proof: See Appendix B.From Corollary 4.1 we can see directly how the hybrid systemparameters will affect the asymptotic slope of the outageprobability curve for the single block transmission case. Inparticular, we see that the asymptotic slope is affected by theSNR exponent of the individual component channels (whichis in turn dependent on the scintillation distribution), thesize of the FSO/RF signal set constellation, and the overallbinary code rate of the system. In most practical systems,$ $ 1

2 , i.e. in a hybrid symbol period, the number oftransmitted FSO bits will be greater than the number of RFtransmitted bits. From (14), we see that the highest diversityis achieved if the binary code rate Rc is set to be less than1($ = m/(m+nq), i.e. the total information rate is less thanthe maximum information rate of the stand-alone RF channel.If 1 ( $ < Rc ' $, the exponent is the same as a single FSOlink, i.e. the additional coding over an RF link will not improvethe asymptotic slope of the outage probability curve. For highbinary code rates, $ < Rc < 1, the asymptotic performance isdominated by the worst of the two exponents. Note that coderates above $ are not achievable with a stand-alone FSO link.Although we have concentrated on the single block case,

given the short coherence time of the optical channel comparedto the RF channel, the cases of A = 2, 3 and B = 1 are alsoof practical interest and are readily evaluated from Theorem4.1. In particular, these provide signicant SNR exponent



0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

Rc

d(k

)

A = 1, B = 1A = 3, B = 1A = 3, B = 0

Fig. 2. Hybrid channel SNR exponent (11) with d(k)fso = 2, d

(k)rf = 1 and

! = 0.8: A = 1, B = 1 system (dashed line); A = 3, B = 1 system (solidline); and a stand-alone FSO link with three blocks, i.e. A = 3, B = 0(dot-dashed line).

improvements for lower rates, compared to the single blockcase. This is illustrated in Fig. 2, which plots the hybridchannel SNR exponent with, d(k)

fso = 2, d(k)rf = 1 and $ = 0.8

for a: A = 1, B = 1 system (dashed line); A = 3, B = 1system (solid line); and a stand-alone FSO link with threeblocks5, i.e. A = 3, B = 0 (dot-dashed line). By coding overmore FSO blocks, vast improvements in the SNR exponent canbe seen, particularly as the code rate decreases. In addition,we see that the A = 3, B = 1 exponent exceeds that of thestand-alone FSO system by d(k)

rf for most code rates.Theorem 4.2: Dene component channel SNR exponents

d(i)fso and d(j)

rf as in (9) and (10) respectively. Suppose i > jthen the SNR exponent is

d(i) = d(i)fso

,1 +

-A

$($ ( Rc)

./0 < Rc ' $ (15)

d(j) = d(j)rf

,1 +

-B

1 ( $(1 ( Rc))

./$ < R < 1. (16)

Otherwise, if i < j then the SNR exponent is

d(j) = d(j)rf

,1 +

-B

1 ( $(1 ( $ ( Rc)

./0 < Rc ' 1 ( $

(17)

d(i) = d(i)fso

,1 +

-A

$(1 ( Rc)

./1 ( $ < Rc < 1.

(18)

Proof: See Appendix C.Theorem 4.2 shows how the overall performance of the hybridchannel will be affected when one of the component channelshas an asymptotic outage probability that decays with SNRmuch faster than the other. In particular, we see that the overallSNR exponent will be dominated by the worst of the twocomponent channel SNR exponents unless the binary coderate is below a certain threshold dependent on the ratio of FSO

5Note that for the stand-alone FSO system with A blocks, the exponent isgiven by d(k) = d(k)

fso (1 + $A(1"Rc/!)%) for 0 < Rc < !, since we havedened Rc with respect to the hybrid system.

bits to total transmitted bits ($). Note that these exponents arerelated to how outage-approaching codes should be designed[34].

To demonstrate the implications of our asymptotic results,we conducted a number of Monte Carlo simulations. Thisinvolved computing the mutual information curves IXfso(u)and IXrf (u) for a given constellation. These curves are thenused as look-up-tables to determine the total mutual informa-tion (6) for a given set of vector fading realisations h andh, which are generated randomly according to one of thedistributions listed in Table I. The simulation results are shownin Fig. 3. In this example we chose A = B = 1, n = 4 (hence! = 4), 2PPM for the FSO channel and 16QAM for the RFchannel. Therefore m = nq = 4 bits, and a maximum of 8bits per hybrid channel symbol can be transmitted ($ = 0.5).For simplicity we have chosen " = 1 ( ". The dot-dashedand dashed curves illustrate the case when only the FSOand RF channels are available, i.e. when " = 1 and " = 0respectively, and must support R = 3 bits per symbol. Thesolid and dot marked curves show the hybrid channel outageperformance to support a Rc = 3/8 and Rc = 6/8 binarycode rate respectively when " = 0.5. In Fig. 3(a), bothFSO and RF channels experience exponential scintillation,corresponding to very strong turbulence conditions. In thiscase, d(1)

fso = d(1)rf = 1 and hence Corollary 4.1 applies. As

expected the dashed and dot-dashed curves have the sameslope since they both have the same SNR exponent. WhenRc = 3/8 < $, we see the slope of the outage curve becomessteeper when both channels are available (solid curve), infact the SNR exponent is now d(1) = 2. However, whenRc = 6/8 > $ (dot-marked curve), from (14) the exponent isd(1) = 1, and hence the curve has the same slope as the FSOand RF only cases previously described. Fig. 3(b) shows thecase when the FSO channel now experiences gamma-gammascintillation with a = 2 and b = 3, therefore d(1)

fso = 2.When Rc = 3/8 the system benets from both channels,the overall exponent increases to d(1) = 3. However, aspredicted by our asymptotic results, if Rc = 6/8 the overallexponent is dominated by the RF component and the exponentis d(1) = 1. Fig 3(c) shows the case when the FSO and RFchannels experience exponential and lognormal scintillation(with µ = ( log(2)/2 and &2 = log(2)) respectively. FromTable I, d(1)

fso = 1 and d(2)rf = 1

4"2 . Hence Theorem 4.2applies in this case. In particular, for the Rc = 3/8 system,from (17), the hybrid channel exponent is dominated by theRF channel exponent and hence the hybrid outage curve(solid) has the same asymptotic slope as the RF outage curve(dashed). If the required rate is increased to Rc = 6/8, theexponent is now given by (18), hence the exponent of theFSO channel dominates overall hybrid channel performance.This behaviour can be seen in Fig. 3(c), where the dot markedcurve (corresponding to the Rc = 6/8 hybrid case) has thesame slope as the FSO channel (dot-dashed curve). Fig. 3(d)illustrates the case when both channels experience lognormalscintillation corresponding to weak turbulence conditions. Inthis case, d(2)

fso = d(2)rf = 1

4"2 , and we see that when Rc = 3/8the hybrid channel benets from both component channelexponents.



0 10 20 30 40 5010 6

10 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

Rc = 3/8, ! = 0.5Rc = 3/8, ! = 0.0Rc = 3/8, ! = 1.0Rc = 6/8, ! = 0.5

(a) Both Exponential.

0 10 20 30 40 5010 6

10 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

Rc = 3/8, ! = 0.5Rc = 3/8, ! = 0.0Rc = 3/8, ! = 1.0Rc = 6/8, ! = 0.5

(b) FSO Gamma-gamma, RF Exponential.

0 10 20 30 40 5010 6

10 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

Rc = 3/8, ! = 0.5Rc = 3/8, ! = 0.0Rc = 3/8, ! = 1.0Rc = 6/8, ! = 0.5

(c) FSO Exponential, RF Lognormal.

0 5 10 15 20 25 3010 6

10 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

Rc = 3/8, ! = 0.5Rc = 3/8, ! = 0.0Rc = 3/8, ! = 1.0Rc = 6/8, ! = 0.5

(d) Both Lognormal.

Fig. 3. Outage performance of the hybrid FSO/RF channel with A = B = 1, n = 4 (" = 4), 2PPM FSO and 16QAM RF: solid, dashed and dot-dashedcurves correspond to hybrid channel with Rc = 3/8 for # = 0.5, 0, 1 respectively (where # = 1 " #). The dot-marked curves show the hybrid outageperformance with Rc = 6/8 and # = 0.5.

V. ASYMPTOTIC OUTAGE ANALYSIS: CSIT CASE

In this section we assume both the transmitter and receiverhave perfect knowledge of the FSO and RF fading coefcients.Due to the slow nature of the scintillation processes, this is arealistic assumption, which, as we shall see, induces signicantpower gains. In this case the transmitter adapts the power(subject to power constraints) to compensate for channeluctuations to signicantly reduce the outage probability.

A. Power Allocation Strategies

To nd the optimal power allocation strategy, we requirethe solution to the following minimisation problem.

())*

))+

Minimise: Pr&

Itot(p, p, h, h) < R'

Subject to: E[%p&] + E[%p&] ' Pav,

%p& ' P fsopeak, %p& ' P rf

peak.

(19)

Theorem 5.1: The solution to problem (19) is given by

(!#, !#) =

0(!, !) %!& + %!& ' s#

(0,0) otherwise,(20)

where (!, !) is the solution to the following minimisationproblem

()))*

)))+

minimise %p& + %p&subject to Itot(p, p, h, h) $ R

%p& ' P fsopeak, %p& ' P rf

peak

p, p ) 0.

(21)

In (20), s# is a threshold determined by

s# = sup&s : E(h,h)$R(s) [%!& + %!&] ' Pav

', (22)

where

R(s) !&

(h, h) ! RA+B : %!& + %!& ' s'

. (23)

Proof: The proof follows similar arguments describedin [35], [36]. Essentially, (21) gives the minimum set of power



allocations that satisfy the peak power constraints such thattotal mutual information is greater than the rate requirement.The inclusion of threshold s# ensures that the long-term powerconstraint is also satised. The larger s# is, the smaller theoutage probability will be. Since the solution (!, !) from (21)gives the minimum sum power to satisfy the rate constraint,then this solution also gives the maximum s#, which in turnminimises the outage probability.Unfortunately, (21) does not lend itself to a simple solution,

since in general IawgnX (p2) is not a concave function in p for all

p > 0. Thus (21) is a non-convex optimisation problem [37].Instead of solving (21), we propose a suboptimal algorithmthat, as we shall see, exhibits the same asymptotic behaviouras the optimal solution. In this direction, rst consider thefollowing lemmas. The proofs follow straightforwardly viathe Karush-Kahn-Tucker conditions [37] and using the rela-tionship between mutual information and the minimum mean-square error (MMSE) for Gaussian channels [38], as donein [39].Lemma 5.1: Dene the minimisation problem,

()*

)+

minimise %p2& + %p2&subject to Itot(p, p, h, h) $ R

p, p ) 0,

(24)

where %p2& ! 1A

"Aa=1 p2

a. The solution to (24) is

p#a =

1

!Xfso

,h2

a"2,1

n'

/and p#b = "Xrf

2"hb!,'

3,

(25)where !X (u, t) ! 1

ummse%1X

4min

5mmseX (0), t

u

67,

"X (u, t) is the solution x to the equation mmseX (xu) = 2xtu ,

mmseX (p) denotes the MMSE of a Gaussian channelwith discrete input constellation X , mmse%1

X (u) is theinverse MMSE function, and ' is chosen such thatItot(p, p, h, h) = R.Lemma 5.2: Dene the minimisation problem

())))*

))))+

minimise %p2& + %p2&subject to Itot(p, p, h, h) $ R

8%p2& ' P fso

peak,$%p2& ' P rf

peak

p, p ) 0.

(26)

Let p# and p# be the solution to (24) in Lemma 5.1, and !and ! be the solution to (26). The solution to (26) is separatedinto four cases depending on p# and p#.1) If p# and p# satisfy the constraints in (26). Then! = p#

and ! = p#.2) If

8%(p#)2& ' P fso

peak and8%(p#)2& > P rf

peak. Then

(a =

1

!Xfso

,h2

a"2,1

n'1

/(27)

(b = "Xrf

2hb!",'2

3, (28)

where '2 is chosen such that$%!2& = P rf

peak and '1

is chosen such that

n

A

A%

a=1

IawgnXfso

("2h2a(

2a) = R ( 1

B

B%

b=1

IawgnXrf

((bhb"!).

If8%!2& > P fso

peak then the solution to (26) is infeasible.3) If

8%(p#)2& > P fso

peak and8%(p#)2& ' P rf

peak. Then thesolution to (26) is the same as the previous case, withthe roles of rf and fso interchanged.

4) If8%(p#)2& > P fso

peak and8%(p#)2& > P rf

peak, then thesolution to (26) is infeasible.

Comparing (21) with (26), we see that (26) is minimisingthe sum of the mean-square power of the FSO and RFchannels, subject to individual short-term root mean-square(RMS) power constraints. By applying Jensen�’s inequality [23]to these constraints, we see that a solution to (26) will alsosatisfy the constraints in (21) and hence can be considered asuboptimal solution to (21). Therefore to nd a suboptimalsolution to the original minimisation problem (19) we use thesolutions in Lemma 5.2 for (!, !) instead of solving (21).

B. Asymptotic AnalysisThe asymptotic outage performance of optimal power al-

location for discrete-input block-fading AWGN channels wasanalysed by Nguyen et al. in [35], [40]. In particular, from [35,Prop. 3], if the peak-to-average power ratios #fso and #rf arenite, then the SNR exponent will be the same as the CSIRcase given in Theorems 4.1 and 4.2. When there are no peak-to-average power constraints then the SNR exponent of theoptimal power allocation strategy is [40, Th. 2]

d(1)csit =

(*

+* d(1)

csir > 1d(1)csir

1%d(1)csir

d(1)csir < 1,

(29)

where d(1)csir is the SNR exponent for the CSIR case. We

now prove that the SNR exponent of our suboptimal powerallocation strategy in Section V-A is the same as (29).Theorem 5.2: Suppose #fso,#rf + *. Then the SNR

exponent of the suboptimal power allocation scheme describedby (19) with (26) is given by (29).

Proof: A sketch of proof proceeds as follows. Let(p#, p#) solve (24). Now consider the following power al-location rule

(!†, !†) =

0(p#, p#)

8%(p#)2& + %(p#)2& ' s†

(0,0) otherwise,(30)

where s† is a threshold such thatE(h,h)$R(s)

9$%p2& + %p2&

:= Pav and

R(s) !;

(h, h) ! RA+B :$%p2& + %p2& ' s

<. (31)

Using similar arguments as [40] it can be proved that thehybrid system with power allocation rule (30) has diver-sity (29). This diversity lower bounds the diversity achievedby the suboptimal algorithm (19) with (26), since (30) satisesmore stringent power constraints. Hence we conclude thesuboptimal power allocation algorithm also has diversity (29).

The implications of (29) are described as follows. Whend(1)csit = *, then the outage probability curve will be verticalat a certain threshold of average power, i.e. the hybrid system



is able to maintain a constant level of instantaneous input-output mutual information. The threshold at which this occursis referred to as the delay-limited capacity of the system [41].Note that if d(1)

csir = 1 in (29) then d(1)csit = *, however, the

outage curve will not go vertical, nor will it converge to aconstant slope when plotted on a log-log scale [35]. Whenthe peak-to-average power ratios are nite, the peak powerconstraints introduce an error oor with a slope equal to theCSIR case. The height of the error oor is dependent on #fso

and #rf [35].To demonstrate the benets of power allocation and its

asymptotic behaviour, we simulated the suboptimal hybridpower allocation strategy for a A = B = 1, n = 4, 2PPMFSO and 16QAM RF system with " = " = 0.5 and peak-to-average power ratios #fso = #rf = #. Exponential distributedfading was applied to both channels. Fig. 4(a) shows the hybridoutage performance with our suboptimal power allocationstrategy compared to uniform power allocation (cross markedcurve) when Rc = 3/8. Since, from Corollary 4.1, d(1)

csir = 2,then from (29), the SNR exponent is d(1)

csit = *, i.e. whenthere are no peak power constraints, the curve will go verticalat a certain average power threshold. This can be seen inFig. 4(a) (thick solid curve), for Pav > 8 dB outages arecompletely removed. We see that there is a power saving ofmore than 20 dB compared to uniform power allocation toachieve 10%5 outage probability. When peak power constraintsare introduced, as expected, we see that an error oor isintroduced with the same slope as the CSIR case. The oorshifts down in probability as the peak-to-average power ratioincreases. Fig. 4(b) shows the case when Rc = 6/8. Sinced(1)csir = 1, when there are no peak power constraints, theoutage curve will no longer go vertical (thick solid curve).As expected we see an error oor is introduced when thepeak-to-average power ratio is nite. Nonetheless, we stillsee signicant power savings compared to uniform powerallocation.

VI. CONCLUSIONSWe proposed a simple hybrid FSO/RF channel model based

on parallel block fading channels. This hybrid model takesinto account differences in signalling rates and fading effectstypically experienced by the component channels involved.Under this framework, we examined the information theo-retic limits of the hybrid channel. In particular, we studiedits asymptotic high SNR outage performance by analysingthe outage diversity or SNR exponents. When CSI is onlyavailable at the receiver, in the general case, the exponentis not available in closed form. Instead, we derived simpleexpressions from which it can be computed numerically. Forthe special case when transmission consists of single FSOand RF blocks, we derived the SNR exponent in closed formin terms of each component channel�’s SNR exponent, theratio component channel bits to total bits, and the overallbinary code rate of the system. The highest outage diversityis achieved if the hybrid binary code rate is set less than theminimum of the two component channel�’s maximum binarycode rates. At the other extreme, if the hybrid binary code rateis set larger than the maximum of the two component chan-nel�’s maximum binary code rates, then the SNR exponent is

dominated by the worst of the two component channel�’s SNRexponents. When CSI is also available at the transmitter, wederived the optimal power allocation scheme that minimisesthe outage probability subject to peak and average powerconstraints. Due to the power scaling of the FSO channel, thisrequires the solution to a non-convex optimisation problem,which is intractable in practical systems. We proposed asuboptimal power allocation strategy, which is much simplerto implement and has the same SNR exponent as the optimalpower allocation. Our results indicate that signicant powersavings (on the order of tens of dBs) are achievable using thesuboptimal algorithm compared to uniform power allocation.Future extensions to the work presented in this paper include:asymptotic analysis with a joint distribution on parameters "and "; independent non-identically distributed fading; adaptivecoding and modulation; practical outage-approaching codedesign; and lower complexity power adaptation strategies.

ACKNOWLEDGEMENTWe thank Professor Robert Schober and the anonymous

reviewers for their helpful comments and suggestions that havesignicantly improved this work.

APPENDIX APROOF OF THEOREM 4.1

Under uniform power allocation, p1 = . . . = pA =p1 = . . . = pB = p = Pav. Dene the normalised fadingcoefcients

)a ! ( log(ha)log p

, *b ! ( log(hb)log p

. (32)

Assume the FSO and RF component channels have SNRexponents d(k)

fso and d(k)rf as dened in (9) and (10) respectively.

This implies6

f#()).= exp

2(d(k)

fso )(log p)k3

f$(*).= exp

2(d(k)

rf *(log p)k3

,

and hence

f(", #) .= exp

=((log p)k

=d(k)fso

A%

a=1

)a + d(k)rf

B%

b=1

*b

>>.

(33)In the large SNR limit, the total instantaneous mutual infor-mation becomes

limp!"

Itot(p, p, h, h) =

nq

A

A%

a=1

(1 ( 11{)a > 1}) +m

B

B%

b=1

(1 ( 11{*b > 1}), (34)

where 11{·} denotes the indicator function. Let

O !0

", # :nq

A

A%

a=1

(1 ( 11 {)a > 1})

+m

B

B%

b=1

(1 ( 11 {*b > 1}) < R

?(35)

6Note that g(z).= exp("d log z) indicates that limz#$ " log g(z)

log z = d.



0 5 10 15 20 25 3010 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

CSIR" = 5" = 10" = 15" = !

(a) Rc = 3/8.

0 5 10 15 20 25 3010 5

10 4

10 3

10 2

10 1

100

Pav (dB)

Pou

t(P

av,R

)

CSIR" = 5" = 15" = 25" = !

(b) Rc = 6/8.

Fig. 4. Outage performance of the hybrid FSO/RF channel with CSIT (solid) and uniform power allocation (dashed). System parameters included: # = # = 0.5,A = B = 1, n = 4, 2PPM FSO and 16QAM RF with peak and average power constraints, and peak-to-average power ratios $fso = $rf = $ in decibels.Exponential distributed fading on both channels.

be the outage set. Hence the outage probability is

Pout(p, R) =@

Of(", #)d"d#. (36)

From the SNR exponent denition (8), using (33) and Varad-han�’s lemma [42] we have

d(k) = infO

=d(k)fso

A%

a=1

)a + d(k)rf

B%

b=1

*b

>. (37)

Let 0 ' %1 ' A and 0 ' %2 ' B be the number elementsof " and # set to one respectively. Dividing both sides of theinequality in (35) by m + nq we obtain the set K($, Rc) asdened in (12), and the inmum (37) is now (11), as given inthe theorem.

APPENDIX BPROOF OF COROLLARY 4.1

Proof: From Theorem 4.1 we must nd the inmum (11)over the set

K($, Rc) ! {%1,%2 ! (0, 1) : $%1 + (1 ( $)%2 > 1 ( Rc} .(38)

Possible solutions to %1 and %2 include: (a) %1 = %2 = 1,d(k) = d(k)

fso + d(k)rf , 0 < Rc < 1; (b) %1 = 1, %2 = 0,

d(k) = d(k)fso , 1 ( $ < Rc < 1; (c) %1 = 0, %2 = 1, d(k) =

d(k)rf , $ < Rc < 1. Now suppose $ ' 1

2 , then $ ' 1 ( $,hence: for 0 < Rc ' $ only solution (a) is possible, therefored(k) = d(k)

fso + d(k)rf ; for $ < Rc ' 1 ( $ solutions (a) and

(c) are valid, hence taking the inmum results in d(k) = d(k)rf ;

for 1 ( $ < Rc < 1, solutions (a), (b) and (c) are valid, andhence the resulting inmum is d(k) = min(d(k)

fso , d(k)rf ). The

case when $ $ 12 follows similar arguments.

APPENDIX CPROOF OF THEOREM 4.2

The proof follows the same arguments as the proof of The-orem 4.1. First we dene the normalised fading coefcients

)a and *b as in (32). From (9) and (10) we have

f(", #) .= exp

=((log p)id(i)

fso

A%

a=1

)a ( (log p)jd(j)rf

B%

b=1

*b

>.

(39)

From the SNR exponent denition (8), using (33) and Varad-han�’s lemma [42] we have

d(i) = infK(!,Rc)

2d(k)fso%1 + (log p)j%id(j)

rf %2

3, (40)

d(j) = infK(!,Rc)

2(log p)i%jd(k)

fso%1 + d(j)rf %2

3, (41)

where the outage set K is dened as in (12). Now, if i > j,then from (41) we need to set %1 as small as possible, whilstsatisfying (12). If we set %1 = 0 then we require %2 = 1 +, B

1%! (1(Rc)-. Since %2 ' B, then we can only set %1 = 0 ifRc > $. In this case the exponent is d(j) = d(j)

rf (1+ , B1%! (1(

Rc)-. If 0 < Rc ' $ then we must choose %1 > 0. In thiscase we set %2 = B and hence to satisfy (12) we require%1 = 1 + ,A

! ($ ( Rc)-. Since i > j the SNR exponent ofthe FSO channel dominates, and the overall SNR exponentbecomes d(i) = 1 + ,A

! ($(Rc)- as given in the statement ofthe theorem. The case when i < j follows the same arguments.

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Nick Letzepis (S �’03 �– M �’06) received the B.E.degree in electrical and electronic engineering fromFlinders University, Adelaide, South Australia in1998, and the Ph.D. degree from the Institute forTelecommunications Research (ITR) at the Univer-sity of South Australia, Adelaide, in 2006.He worked with DSpace Pty. Ltd. from 1998 to

2002 as a systems/software engineer, specialisingin modem and codec design for satellite commu-nications. He is currently a Research Fellow inthe Coding and Information Theory Group at the

ITR. His research interests include information theory and coding for mul-tiuser/multiantenna communications, random matrix theory, digital modula-tion and coding for satellite communications, iterative decoding, and free-space optical communications.

Khoa D. Nguyen (SM �’06) received the B.E. inElectrical Engineering at the University of Mel-bourne, Australia in 2005. Since 2006, he has beenpursuing the Ph.D. degree in electrical engineeringat the Institute for Telecommunications Research,University of South Australia, Australia. He was asummer research scholar at the Australian NationalUniversity in 2004 and held a visiting appointment atUniversity of Cambridge, United Kingdom in 2007.His research interests are in the areas of informa-tion theory and error control codes. He has mainly

worked on adaptive techniques and theoretical limits of communications withpractical constraints.



Albert Guillen i Fabregas (S �’01 �– M �’05 �–SM�’09) was born in Barcelona, Catalunya, Spain,in 1974. He received the Telecommunication En-gineering Degree and the Electronics EngineeringDegree from Universitat Politecnica de Catalunya,Barcelona, Catalunya, Spain, and the Politecnico diTorino, Torino, Italy, respectively, in 1999, and thePh.D. in Communication Systems from Ecole Poly-technique Federale de Lausanne (EPFL), Lausanne,Switzerland, in 2004.

From August 1998 to March 1999, he conducted his Final Research Projectat the New Jersey Institute of Technology, Newark, NJ, USA. He was withTelecom Italia Laboratories, Italy, from November 1999 to June 2000 andwith the European Space Agency (ESA), Noordwijk, The Netherlands, fromSeptember 2000 to May 2001. During his doctoral studies, from 2001 to2004, he was a Research and Teaching Assistant at Institut Eurcom, Sophia-Antipolis, France. From June 2003 to July 2004, he was a Visiting Scholarat EPFL. From September 2004 to November 2006, he was a ResearchFellow at the Institute for Telecommunications Research, University of SouthAustralia, Mawson Lakes, Australia. Since 2007, he has been a Lecturerin the Department of Engineering, University of Cambridge, Cambridge,U.K., where he is also a Fellow of Trinity Hall. He has held visitingappointments at Ecole Nationale Superieure des Telecommunications, Paris,France, Universitat Pompeu Fabra, Barcelona, Spain, and the University ofSouth Australia, Australia. His research interests are in communication theory,information theory, coding theory, digital modulation, and signal processingtechniques with wireless applications.Dr. Guillen i Fabregas is currently an Editor of the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS. He received a pre-doctoral Research

Fellowship of the Spanish Ministry of Education to join ESA. He received theYoung Authors Award of the 2004 European Signal Processing ConferenceEUSIPCO 2004, Vienna, Austria and the 2004 Nokia Best Doctoral ThesisAward in Mobile Internet and 3rd Generation Mobile Solutions from theSpanish Institution of Telecommunications Engineers. He is also a member ofthe ARC Communications Research Network (ACoRN) and a Junior Memberof the Isaac Newton Institute for Mathematical Sciences.

Bill Cowley (M �’78) received his B.Sc. and B.E.degrees from the University of Adelaide in 1974and 75. After working initially for the Post Mas-ter Generals Department, he joined the DefenceScience and Technology Organisation in 1979. In1985 he completed his Ph.D. with Adelaide Uni-versity and then he moved to the Digital Commu-nications Group at the South Australian Instituteof Technology. This organisation later became theInstitute for Telecommunications Research (ITR) atthe University of South Australia. During the last

20 years Bill Cowley has worked mainly in modem signal processing. Hehas recently been leader of the Satellite Communications Program in theAustralian Cooperative Research Centre for Satellite Systems (CRCSS). Heis currently Professor of Communications Signal Processing and Leader ofthe Applied Signal Processing Research Group in ITR. His technical interestsinclude modem synchronisation, iterative decoding, satellite communications,speech processing and real-time signal processing implementation methods.


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