robust power allocation schemes for multibeamopportunistic transmission strategies underquality of...

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 26, NO. 6, AUGUST 2008 1025

Robust Power Allocation Schemes for MultibeamOpportunistic Transmission Strategies Under

Quality of Service ConstraintsNizar Zorba, Member, IEEE, and Ana I. Perez-Neira, Senior Member, IEEE,

Abstract—Scheduling in a Broadcast (BC) channel based onpartial Channel State Information at the Transmitter (CSIT) iscarried out in an opportunistic way, where several orthogonalbeams are randomly generated at the Base Station transmitter tosimultaneously deliver several users with their intended data. Thepaper presents a power allocation over the transmitting beams,where a minimum rate per user restriction is required for eachscheduled user, standing as a potential Quality of Service (QoS)indicator for the system behaviour. However, in practical wirelessscenarios the CSIT is imperfect due to non-accurate estimation,so that robust schemes are required to meet the system demands.Based on the allowed system outage in the QoS achievement,different robust power allocation schemes are proposed, whichare efficiently solved through convex optimization tools. Thepresented strategies are later compared via simulations for thedifferent scenarios and the system specifications.

Index Terms—Multiuser MIMO, opportunistic beamforming,QoS demands, robust designs, partial CSIT.

I. INTRODUCTION

ONE OF THE major transmission techniques within par-tial CSIT scenarios is Multibeam Opportunistic Beam-

forming (MOB), where several beams are generated at theBase Station (BS) to serve various users at the same time,obtaining an additional multiplexing gain [1]. Its partialchannel state information is in the form of Signal-to-Noise-Interference-Ratio (SNIR) that seems to be quite reasonable incommercial wireless systems. The MOB schemes are attrac-tive due to their high performance, while at the same time, lowcomplexity design. As the rate benefits of these schemes havebeen stated in several works in the literature [1][2], it turns tobe the time to analyze their QoS performance and suitabilityfor implementation in realistic commercial systems, where thecostumers demand some QoS requirements for their correctoperation.

A potential measure of the system QoS is through theminimum rate per user [3], so that each served user isguaranteed a minimum SNIR, allowing it to properly decodeits intended data with a predefined Packet Error Rate (PER).Regarding the minimum requirement per user, previous studies[4] have shown that the user satisfaction is insignificantly

Manuscript received July 8, 2007; accepted February 4, 2008. This workwas partially supported by the Catalan Government under grant SGR2005-00996; by the Spanish Government under project TEC2005-08122-C03; andby the European MEDEA+ under project FIT-330225-2007-2 (MIMOWA).This work was partially presented at the IEEE International Conference onCommunications (ICC’07), Glasgow, United Kingdom, June 2007.N. Zorba is with the Centre Tecnologic de Telecomunicacions de Catalunya

(CTTC), 08860-Castelldefels, Barcelona, Spain (e-mail: [email protected]).A.I. Perez-Neira is with Universitat Politecnica de Catalunya (UPC), 08034-

Barcelona, Spain (e-mail: [email protected]).Digital Object Identifier 10.1109/JSAC.2008.080818.

increased by a performance higher than its demands, whileon the other hand, if the provided resources fail to guaranteeits requirements, the satisfaction drastically decreases. Thus,an attractive transmission scheme is accomplished by meetingthe minimum rate requirement for each scheduled user, whileminimizing the total transmitted power.Some QoS studies have been presented [5] for the sin-

gle beam opportunistic beamforming [6], but no previousproposals (up to the authors’ knowledge) for the multibeamopportunistic beamforming are suggested in literature. Pro-viding QoS communications in the wireless environment is achallenging aspect due to the dynamic (and random) natureof the wireless channel, where users deal with a large amountof interference and channel fluctuations. This challenge iseven larger in the MOB schemes, mainly due to the crossinterference terms that this scheme originates in the multiuserscenario, as each user receives an interference componentfrom the multiple beams, while only partial CSIT is available.This paper presents the QoS fulfillment of the multibeamopportunistic beamforming as an optimization problem, thatcan be solved by Convex Optimization tools [7] to minimizethe total transmitted power.In practical wireless scenarios the SNIR information avail-

able at the transmitter is not perfect due to fast fading, limitedduration of the training step, quantization noise and/or non-accurate estimation during the training process. Therefore, thetransmitter has to allocate the available resources to satisfythe QoS requirements, while it knows that the availableSNIR information is imperfect. Based on the commercialapplication and the operator requirement, some failure in theQoS achievement can be allowed, for what is known as serviceoutage [8]. Motivated by this fact, the paper presents twodifferent policies to deal with the possibility of outage, wherea first philosophy, targeted to obtain the QoS restrictions overall the time (i.e. zero outage), is shown through two practicalrobust power allocation schemes that consider instantaneousCSIT measures in their operation, where the imperfectnessis assumed to belong to a bounded uncertainty region. Bothrobust power allocation strategies are intended to minimizethe total transmitted power, constrained to specific QoS regu-lations in terms of minimum rate requirements. Related to thesecond policy when some outage is allowed, the paper presentsa scheme to minimize the transmitted power when restrictedto a maximum probability of outage, where this scheme usesstatistical CSIT in its power allocation process.

Furthermore, with the target of achieving some level offairness among the served users, a maximization of the lowest

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1026 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 26, NO. 6, AUGUST 2008

QoS per user can be performed through the power allocationprocess, so that an alternative robust power allocation schemeis presented, standing as a realistic option for commercialimplementation, when the wireless operator asks for somefairness in the resource distribution among the users.As a summary, the contributions of this work are in the field

of robust-QoS multibeam schemes as follows

• A power allocation for MOB systems is provided, run-ning the system under QoS restrictions in terms of min-imum rate. Minimum transmitted power is guaranteed.

• Robust multibeam opportunistic schemes are presented,where the transmitter not only has to deal with partialCSIT, but it also has to take into consideration that thepartial CSIT is also imperfect.

• On the basis of the allowed outage, two different philoso-phies for the power allocation are proposed, even usingthe instantaneous or statistical CSIT for their operation.

• A fair power allocation scheme is also exposed, standingas an attractive option for its implementation in commer-cial systems.

The remainder of this paper is organized as follows: whilesection II deals with the system model, in section III a reviewof the MOB procedure is discussed. Section IV presents theproposed robust power allocation philosophies followed bysection V with the numerical results and simulations. Thepaper finally draws the conclusions in section VI.

II. SYSTEM MODEL

We focus on the BC channel where N receivers, each oneof them equipped with a single receiving antenna, are beingserved by a transmitter at the BS provided with nt transmittingantennas. N is supposed to be greater than nt, thus matchingwith the most practical situation for the multiuser scenarios.A multiantenna channel h[1×nt] is considered between eachof the users and the BS where a quasi static block fadingmodel is assumed, which keeps constant through the coherencetime, and independently changes between consecutive timeintervals with independent and identically distributed (i.i.d.)complex Gaussian entries ∼ CN (0, 1). Let x(t) be the nt × 1transmitted vector, and denote yi(t) as the ith user receivedsignal given by

yi(t) = hi(t)x(t) + zi(t) (1)

where zi(t) is an additive complex i.i.d. noise componentwith zero mean and E{|zi|2} = σ2. The transmitter deliversservice to nt simultaneous users, so that a more compactsystem formulation is obtained by stacking the received signalsand the noise components for the set of nt selected users asy(t) = H(t)x(t) + z(t), with H(t) = [h1(t); ...;hnt(t)] asthe compound channel. Notice that the transmitted signal x(t)encloses the uncorrelated data symbols si(t) to each one ofthe selected users, where E{|si|2} = 1 is considered. For easeof notation, time index is dropped whenever possible.

III. MULTIBEAM OPPORTUNISTIC BEAMFORMING (MOB)

One of the main transmission techniques in multiuser sce-narios is the multibeam opportunistic beamforming [1], where

nt random orthonormal spatial beams ({bi}nt

i=1) are generatedat the BS to simultaneously serve more than one user. Thisrandom beams development guarantees the low complexityrequirement for the practical implementation of MOB, butit also does not need CSIT for its generation. Within theacquisition step, each one of the users sequentially calculatesthe SNIR that it receives from each beam, and feeds backthese values to the BS1. The BS scheduler chooses the userwith the largest SNIR value for each one of the beams, entersthe transmission stage and forwards every one of the selectedusers with its intended data, where no user can obtain morethan one beam at a time.Notice that the role of the partial CSIT in MOB, where

only SNIR is available, is in the selection process, as the usersare selected on the basis of the SNIR fed back values, whilethe spatial processing is randomly generated. This multibeamstrategy achieves high system sum rate by obtaining thesystem Multiuser gain [6] through the largest SNIR valuesearch. Moreover, the achievable multiplexing gain from themultiantenna availability is obtained by serving several usersat the same time, making the transmitted signal to enclose thedata symbols for the nt selected users

x =nt∑

m=1

xm =nt∑

m=1

bm p1/2m sm = BP1/2s (2)

with bm as the beam assigned to the mth user, and pm as theassigned power to that beam. The matrix B = [b1, ...,bnt ]is randomly generated following an orthonormal policy toproduce the lowest possible interference among the servedusers, whileP1/2 is a diagonal matrix with p

1/2m as its diagonal

entries. Notice that this formulation is an upgraded version of[1], as a power allocation P over the transmitted beams isincorporated, thanks to this formulation feedback load. Thesystem sum rate capacity of the MOB strategy can be written[1] as

SR � E

{nt∑

m=1

log(1 + max1≤i≤N

SNIRi,m)

}(3)

where the � stands for the small probability that a singleuser achieves the maximum SNIR value for two beams at thesame time, making the scheduler to select the user offering thesecond largest SNIR value for the second beam, thus a smallloss is presented. In the previous expression, it appears theSNIR term due to the interference that each beam generatesto its non-intended users, representing a major drawback ofthis system. The SNIR formulation for the ith user throughthe mth beam, with several transmitting orthogonal beams, is

SNIRi,m =pm |hibm|2

σ2 +nt∑

u�=m

pu |hibu|2=

pm |ci,m|2

σ2 +nt∑

u�=m

pu |di,u|2

(4)where hibm = ci,m and hibu�=m = di,u denote the equivalentchannel seen by the ith receiver with respect to each one of

1Notice that this scheme saves � 50% of the full CSIT feedback load.Other proposals in [1] or [9] apply thresholds and/or highest SNIR feedbackto further decrease the feedback, but feedback reduction objectives are out ofthe scope of this paper.


ZORBA and PEREZ-NEIRA: ROBUST POWER ALLOCATION SCHEMES FOR MULTIBEAM OPPORTUNISTIC TRANSMISSION STRATEGIES 1027

the nt generated beams2. Note that the BS scheduler receivesnt SNIR values from each one of the users, so that for eachone of the scheduled users, the BS can calculate the valuesof |ci,m|2 and |di,u|2 from direct matrix algebra calculationswith nt unknowns within nt equations.To select the nt users within the training step, a uniform

power allocation is applied over all the beams in equation(4), but to guarantee the QoS for the selected users, a powerallocation strategy is next presented.

A. QoS in Multibeam Opportunistic Beamforming

The requirement of a certain system QoS is presented interms of a minimum rate value per user, where each scheduleduser needs for this minimum rate to detect and manage itsreceived signal. This is easily accomplished when a single useris scheduled at a time, so that through the power control, thedelivered rate is regulated to the user requirements, but whenseveral simultaneous users are scheduled through MOB, withcross interference terms among them as shown in (4), thenthis task is a challenging one.The problem can be formulated through a minimization

of the total transmitted power (Pt) over all possible powerloading matrices (P), while the minimum rate requirementsare presented by minimum SNIR per user restrictions (snirth

i )as follows

minP

Pt

s.t. SNIRi,m(P) − snirthi ≥ 0 ∀ i, m (5)

where although this formulation is widely used in literature[10][11], but this paper characterizes it for the multibeamopportunistic schemes with partial CSIT. Minimizing thetransmit power is of great importance as it reduces the inter-cells interference in the system, while achieving an efficientuse of the available resources. Note that the total transmittedpower for MOB is expressed as Pt = Tr(BPBH) = Tr(P)because of the orthonormal transmitter processing matrix B.The consideration of the SNIR expression in (4) makes theoptimization problem in (5) to particularize for the multibeamopportunistic case as

minP

Tr(P)

s.t. pm |ci,m|2−snirthi

[σ2+

nt∑u�=m

pu |di,u|2]≥0 ∀i, m (6)

that stands as a linear programming problem solved throughany convex optimization software, so that its complexity isextremely low and defined by the used software. Observe thatthe SNIR constraints must be satisfied for all the selectedusers, where the set of loading matrices P who satisfy theseconstraints are called the problem feasibility region SP , thatdepends on the equivalent channels (HB) of the selected users

2Along the paper, all the users are assumed to have the same averagechannel characteristics, and showing the same distribution for the maximumSNIR value, so that each user has the same probability to be selected. If thisis not the case (e.g. heterogeneous users’ distribution in the cell, with someusers far from the BS), then a channel normalization (e.g. division by thepath loss) can be accomplished for such a scenario.

and on the required SNIR values snirthi for each one of them

as

SP(HB, snirth

i

)=

{P ∈ P

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪pm|ci,m|2−snirthi

[σ2+

nt∑u�=m

pu|di,u|2]≥0 ∀i, m

}(7)

with P as the set of all diagonal and positive definite nt ×nt

matrices. The feasibility region is an important characteriza-tion for MOB, as not all the required sets of snirth

i can beachieved by the system due to the cross interference termsamong the served users. Later simulations will show therelevance of this feasibility region.

B. Fairness Approach

The practical implementation of any transmission scheme incommercial systems can require for some kind of fairness, inthe resource allocation process among the users in the system[12]. The ultimate target for fairness is achieving an equalrate among all the served users, so that all users share thesame reception quality. Furthermore, some applications can berestricted by a maximum transmission power Pmax that cannotbe exceeded due to inter-cells interference in the system.Therefore, this subsection presents a transmission scheme

that operates under a maximum transmitted power Tr(P) ≤Pmax, while the objective is to increase the lowest QoS amongthe serviced users, where snirth

min is defined as the minimumover all snirth

i . This fair power allocation scheme states asmax

Psnirth

min

s.t. SNIRi,m(P) − snirthmin ≥ 0 ∀ i, m

s.t. T r(P) ≤ Pmax (8)

where the SNIR expression in (4) can be included in thismaximization of the minimum (MaxMin) SNIR value, as

maxP

snirthmin

s.t. pm |ci,m|2−snirthmin

[σ2+

nt∑u�=m

pu |di,u|2]≥0 ∀i, m

s.t. T r(P) ≤ Pmax (9)

that also stands as a linear programming problem to be solvedthrough the convex optimization software, where previousformulation ensures that the minimum rate is equal for all theserviced users, whatever are their channel characteristics. Thisformulation has a special feature as it is not restricted throughany feasibility boundary, where independently of the channelquality for all the serviced users, a MaxMin over their snirth

i

is possible. Notice that the MaxMin will always provide asolution to the problem where, obviously, a minimum valueof snirth

min = 0 can be obtained.

IV. ROBUST POWER ALLOCATION SCHEMES

In practical wireless scenarios the SNIR information avail-able at the transmitter is not perfect due to quantization,limited duration of the training step, and/or non-accurateestimation during the training process. The design of a MOBscheme that not only requires for partial CSIT, but alsobeing robust to uncertainty in the already partial information



available to the transmitter, is a challenging aspect that has notbeen developed in the literature. To deal with the measuresuncertainty, the system can allow for some outage in theQoS satisfaction, so that a predefined value of QoS failureis enabled, providing an additional degree of freedom to thesystem. Actually this is an important design parameter in thepaper, as it determines the robust policy that is followed toachieve the system QoS requirements. Motivated by this fact,the paper presents robust approaches to solve the optimizationproblem in (6), through two different philosophies on thebasis of the allowed outage. All approaches minimize the totaltransmitted power while meeting the SNIR requirements inpartial and imperfect CSIT scenarios. A robust scheme forthe fairness approach is also presented.

A. Worst Case Philosophy

For some applications, the system can be asked to satisfythe minimum QoS restrictions for all the cases, so that nooutage is allowed in the service, even imperfect CSIT ispresent in the scenario. This is a hard restriction for thesystem, as a 100% SNIR satisfaction is imposed and, for theachievement of the QoS over all possible cases of uncertainty,a worst case design is required.

1) Worst Case Philosophy - Power Uncertainty: Noticethat in the MOB scheme the generation of the B matrix isindependent of the channel information, so that regardless ofthe CSIT quality, the B matrix cannot be robustly designed.Also from the opportunistic policy, the users are only selectedon the basis of the SNIR measures that they feed back to theBS for each one of the generated beams, so that robust userselection schemes can neither be realized if all the users sharethe same uncertainty region in their measurements, which isthe most frequent case in commercial systems as all usersemploy the same kind of receiver, and cope with the samelength of the training process. In scenarios where the users donot share the same uncertainty region, robust user selectionschemes probably could be obtained by upgrading the robustsingle user selection scheme presented in [13], but this remainsas an open area that lies out of the scope of this paper.The only tool that remains for the transmitter to deliver

a robust transmission scheme is the power loading over thegenerated beams to meet some predefined minimum raterequirements, so that once the users are selected, the powerloading has to take into consideration that the SNIR measures,reflecting the actual channel quality for each selected user,are imperfect. Therefore for applications where zero-outageis imposed, the power loading has to consider the worst casescenario to guarantee that the QoS requirements are alwaysmet, while the minimum transmitted power is realized.Back to equation (4) and considering that the measurements

are imperfect, the SNIR reformulates as

SNIRpi,m=

pm |ci,m|2

σ2+nt∑

u�=m

pu |di,u|2=

pm( |ci,m|2 + δi,m)

σ2+nt∑

u�=m

pu(|di,u|2 + δi,u)

(10)with |ci,m|2 and |di,u|2 as the measured equivalent channelpowers that can be obtained from the partial CSIT, where

these values are affected by some uncertainty. On the other

hand, |ci,m|2 and |di,u|2 represent the exact equivalent channelpowers that are unknown to the transmitter side due tothe uncertainty, with δi,m being the estimation error in themeasurement of the equivalent channel power. The value ofδi,m is assumed to be an unknown value bounded by anuncertainty region with maximum power ε, i.e. |δi,m|2 < ε.Observe that the estimation process is independent for eachtransmitted beam, so that δi,m �= δi,n for m �= n. Also noticethat the power detection process is usually performed throughan envelope detector so that the uncertainty in the powermeasure seems to be realistic in commercial systems.

As SNIRpi,m depends on both the power loading and

the level of uncertainty in the received power from eachtransmitted beam, then by using the SNIR formulation in (10),the power allocation problem with QoS constraints under theimperfect CSIT stands as

minP

Tr(P)

s.t. pm( |ci,m|2 + δi,m)−

−snirthi

[σ2+

nt∑u�=m

pu(|di,u|2+δi,u)]≥ 0, ∀i, m, δ (11)

to indicate that even the measured SNIR is imperfect, thetransmitter has to allocate the power in such a way tosatisfy the QoS restrictions whatever is the uncertainty level,obviously, bounded by a maximum power ε. In contrast tothe developed robust approaches in [10][11] where a jointoptimization of both the transmitting beams and the powerallocation is performed, the opportunistic schemes with itspartial CSIT can only optimize the transmitted power asalready mentioned, so that each constraint in previous problemstands as a hyperplane which always defines a convex set[7, Chp.2]. Furthermore, as the previous minimization hasto be accomplished over all users and kinds of uncertaintywithin the maximum power ε, then the intersection over a setof hyperplanes is also a convex set [7, Chp.2], allowing theproblem to be easily solved through any convex optimizationsoftware.The system is restricted not only by the equivalent channel

realization and the snirthi requirements, but also by the level

of uncertainty in the power measurements, then a smallerfeasibility region SP

δ than the perfect CSIT case is obtainedas

SPδ

(HB, snirth

i , δ)=

{P∈P

⎪⎪⎪⎪⎪⎪⎪⎪pm( |ci,m|2+δi,m)−

−snirthi

[σ2+

nt∑u�=m

pu(|di,u|2+δi,u)]≥0 ∀i,m,δ

}(12)

where δ denotes all the uncertainty components in the SNIRvalue. Remind that with the zero outage restriction, the powerallocation has to be performed for all cases of uncertainty,so that a worst case calculation is needed to guarantee theQoS fulfillment in (11). If the user delivers an SNIR estimatedvalue lower than its actual received SNIR, then the transmitterwill allocate more power to that user to satisfy its QoS, thus



incurring in higher transmitted power and higher interferencecomponents, but it ensures the minimum rate to be satisfied.Even more interference is generated to the other users, but asthe transmitter knows the assigned power and interference toeach user, then the QoS is satisfied not only for the considereduser but to all scheduled users, at the expenses of highertransmitted power. On the other hand, if the user reports anSNIR value that is higher than its actual received SNIR, thenthe transmitter allocates less power than required and the QoSis not met. The worst case for QoS fulfillment is thus definedby the situation where the power estimation for each one of thetransmitted beams, results in an estimated SNIR value higherthan the actual SNIR value.

Inspecting the SNIR formulation in (10) and realizing thatthere exist nt sources of uncertainty, then a higher SNIR valueis obtained if the uncertainty in |ci,m|2 makes the numerator toincrease or if any uncertainty in |di,u|2 makes the denominatorto decrease. Obviously, the worst case scenario relates to bothsimultaneous conditions presenting the worst case SNIRwcp

as

SNIRwcpi,m =

pm( |ci,m|2 + ε )

σ2 +nt∑

u�=m

pu(|di,u|2 − ε )(13)

Therefore, the robust power allocation employs the imper-fect measures (remember that |ci,m| is unknown to the trans-mitter) and the maximum uncertainty value, to reformulate theproblem in (11) for the worst case scenario as

minP

Tr(P)

s.t. pm(|ci,m|2−ε )−snirthi

[σ2+

nt∑u�=m

pu(|di,u|2+ε )]≥ 0 ∀i,m

(14)where this problem is numerically solved by any convexoptimization software, so that the previous expression guar-antees the SNIR minimum snirth

i for all the scheduled users,whatever is the effect of the uncertainty in the power mea-surements for each transmitted beam, obviously, as long asthe uncertainty remains within a maximum power ε.This section has just presented a robust transmission

scheme when the imperfectness is assumed in the powermeasurement (e.g. due to the envelope detector or thefeedback quantization [14]). But another uncertainty sourcein the system is the incorrect channel estimation processduring the training step (e.g. due to a short training time[15]), so that next section is in charge of this second kind ofsystem uncertainty.

2) Worst Case Philosophy - Channel Uncertainty: Up tonow, the paper has considered the practical situation wherethe uncertainty is assumed to be in the power measurementwith respect to each one of the transmitted beams (e.g. throughan envelope detector), but another source of imperfectness isobtained if the uncertainty is assumed to be in an incorrect

channel estimation process. Back to equation (4) and consid-ering that the equivalent channel measurements are imperfect,the SNIR reformulates as

SNIRci,m=

pm |ci,m|2

σ2+nt∑

u�=m

pu |di,u|2=

pm |ci,m + γi,m|2

σ2+nt∑

u�=m

pu

∣∣∣di,u + γi,u

∣∣∣2(15)

with |ci,m|2 and |di,u|2 as the measured equivalent channelpowers that can be obtained from the partial CSIT, wherethese values are affected by some uncertainty. On the other

hand, |ci,m|2 and∣∣∣di,u

∣∣∣2 represent the exact equivalent channelpowers that are unknown to the transmitter side due to theuncertainty, with γi,m being the error in the equivalent channelestimation. The value of γi,m is assumed to be an unknownamount bounded by an uncertainty region with maximumpower ω. Note that with this formulation, the power allocationproblem with QoS constraints in an imperfect CSIT casestands as

minP

Tr(P)

s.t. pm|ci,m+γi,m|2−

−snirthi

[σ2+

nt∑u�=m

pu

∣∣∣di,u+γi,u

∣∣∣2]≥0 ∀i, m, γ (16)

that following the same procedure for the previous case, alsostands as a linear programming problem. Inspecting the SNIRformulation in (15) for the worst case development, noticethat for the numerator, the highest value is obtained if theuncertainty aligns with the equivalent channel ci,m to obtain

|ci,m + γi,m| = |ci,m| + |γi,m| = |ci,m| + √ω (17)

and using the same procedure for the denominator, the worstcase SNIR expression SNIRwcc stands as

SNIRwcci,m =

pm

( |ci,m| + √ω

)2

σ2 +nt∑

u�=m

pu

( ∣∣∣di,u

∣∣∣ −√ω

)2(18)

so that, similar to the power uncertainty case in expression(14), the power allocation process has to compensate for thisworst case value, making the power allocation problem inequation (16) to restate as

minP

Tr(P)

s.t. pm

( |ci,m|−√ω)2−

−snirthi

[σ2+

nt∑u�=m

pu

( |di,u|+√

ω)2

]≥0 ∀i, m (19)

where this problem is numerically solved by any convexoptimization software.



3) Worst Case Philosophy - Fairness Objective: Whendealing with a zero-outage robust design that takes intoconsiderations the fairness among the users, the presentedMaxMin design in (9) can be operated under the worst casescenario, thus providing a robust transmission scheme thatoffers fairness among the users. Dealing with the uncertaintyin the channel measures and using the SNIR worst caseexpression in (18), the formulation for the robust and fairpower allocation scheme is as follows

maxP

snirthmin

s.t. pm

(|ci,m|−√ω)2−snirth

min

[σ2+

nt∑u�=m

pu

(|di,u|+√

ω)2

]≥0

s.t. T r(P) ≤ Pmax (20)

where similar to the two previous expressions, it is also nu-merically solved through convex optimization software. Nextwe propose a completely different robust philosophy wherethe system allows for some outage in the QoS satisfaction.The SNIR statistical distribution is used for the power allo-cation process instead of the SNIR instantaneous measures,through a philosophy that meets different system objectivesand restrictions.

B. Statistical Philosophy - Probability of Outage

In the worst case scenario just presented, there exists a verysmall probability that the uncertainty makes the numerator toincrease at the same time as each one of the interferencecomponents to decrease, so that the formulations in (14)and (19) are very pessimistic, as they have to guarantee theminimum rate for absolutely all cases. Moreover, in somescenarios, to obtain the maximum uncertainty power, eitherε or ω, is a difficult task for the system developer.The wireless operators realize that some users can pro-

vide deficient channel conditions for communication, anddelivering service to such users can be very expensive interm of system resources, driving down the whole systemperformance; so that if these users are dropped, the operatorcan offer better service to all the remaining users in the system.Based on this practical point of view, operators are moreinterested in probability of outage measures [8] rather thanabsolute QoS fulfillment for some applications, making thecommercial systems to fix a target probability of outage ξout inthe users QoS. Furthermore, the uncertainty values in practicalsystems do not show very large values, so that the uncertaintyis not expected to be more than 20% of the original signalvalue, otherwise, the system is not commercially feasible.By considering a probability of outage in the users QoS

service, this section presents a robust power allocation schemethat contemplates the statistical distribution of the servingSNIR. Based on the opportunistic policy to deliver service tothe users, the serving SNIR value corresponds to the maximumSNIR over the active users in the system, so that the calcula-tion of the distribution of the serving SNIR enables a powerloading that is not performed over the instantaneous SNIRfed back measures, but over the distribution of the servingSNIR. This makes the power loading to be unaffected by theuncertainty in the instantaneous SNIR values, and restricts the

relevance of the SNIR feedback to the user selection processand not as a channel quality indicator.Considering an application that allows for outage, with

a common restriction to the minimum SNIR per selecteduser (snirth), its power loading formulation with a targetprobability of outage in the QoS states as follows

minP

Tr(P)

s.t. P rob

{SNIR(P) − snirth ≤ 0

}≤ ξout (21)

where the SNIR’s cumulative distribution function (cdf) for theselected users (i.e. maximum SNIR’s cdf) is needed for thecalculation of the required power in the previous formulation.As assumed in [1], consider that all the scheduled users presentthe same distribution for the maximum SNIR value, so that auniform power distribution p is applied over all the transmittedbeams. Then note that the numerator of the SNIR equation in(4) follows a χ2(2) distribution while the interference terms inthe denominator are modeled as χ2(2(nt − 1)), which allowsto obtain the probability distribution function (pdf) as (similarpdf calculation is developed in [1][3])

f(x) =e−xσ2/p

(1 + x)nt

(σ2

p(1 + x) + nt − 1

)(22)

and the cdf is formulated as

F (x) = 1 − e−xσ2/p

(1 + x)nt−1(23)

Since the serving SNIR is the maximization over all theusers’ SNIR values, then the maximum SNIR cdf is stated as

FF (x) = (F (x))N =[1 − e−xσ2/p

(1 + x)nt−1

]N

(24)

which enables the restriction of outage probability in (21) tobe represented by the cdf(snirth) as follows

minP

Tr(P)

s.t.

[1 − e−snirthσ2/p

(1 + snirth)nt−1

]N

≤ ξout (25)

where the relationship between the transmitted power and boththe ξout and snirth restrictions is established, to provide theminimum required power to satisfy the QoS restrictions as

p ≥ snirthσ2

ln((1 − N

√ξout)(1 + snirth)nt−1

)−1 (26)

As already stated, this formulation is not affected by theSNIR uncertainty in the power allocation procedure, as itis based on the SNIR distribution in (22) that is availableat the BS, by just knowing that the scheduler is runningthe MOB scheme. Nevertheless, the SNIR measures are stillused for the user selection process, but this has a negligibleeffect in changing the distribution of the serving SNIR ifthe uncertainty region is common for all the users. If anerror happens in the users’ selection process, the N effectis decreased (the uncertainty can make the selected user tobe the best one within N − 1, N − 2, ... users), but notice



that the multiuser effect is only present in the exponent ofthe expression (25), so that this effect is minor as long asN > nt and the uncertainty values are not larger than 20% ofthe original signal. These two assumptions are very realistic inevery commercial system, which is actually the target scenarioof this power allocation philosophy. Next section will showthis result by simulations.

Comparing the previous result to the statistical formulationspresented in [16][17], this paper considers a more challengingscenario, as the users are opportunistically selected withseveral users being serviced at the same time, but also a closedform expression for the power allocation is provided.An interesting property of this scheme when compared to

the worst case philosophy is that the wireless operator does notcare about the measures uncertainty, so that the knowledge ofthe uncertainty region ε and ω values (as in equations (13) and(18)) is not required, which reduces the system setups priorto transmission, and enables the system to work in scenarioswith unbounded uncertainty levels. A drawback of this schemestands in a lower benefit from the multiuser gain, as the worstcase design in (14) makes the power loading on the basis ofthe instantaneous SNIR values, while for the probability ofoutage design in (21) the power allocation is based on theserving SNIR statistical distribution, so that it does not fullyexploit the instantaneous SNIR measures. Nevertheless, themultiuser gain is still presented in terms of the user selectionprocess, and as reflected by the exponent of expression (24),also in the distribution of the serving SNIR.As previously commented, there exist several uncertainty

sources in the communication process, where a short trainingtime has a bounded effect on the estimation error, as theconnection is known between the uncertainty and the lengthof the training time [15]. The use of quantization to makethe feedback process also generates a restricted uncertainty inaccordance with the length of the quantization step [14]. Thepower detector used at the receivers has also a constrainederror margin due to the used electronics. On the other hand,the transmission through the feedback channel can generateunbounded uncertainty. Based on the commercial systemcharacteristics (training time, selected quantization, ...) thenthe weight of each uncertainty source can be determined, sothat a better selection among the two proposed philosophiesalong this paper can be performed.A remark to avoid misleading conclusions to the reader

about feasibility and outage: notice that the worst caseschemes aim to guarantee 100% of the QoS requirements(i.e. outage is not enabled), but this objective may not bealways satisfied (i.e. falls outside the feasibility region). Thisprocess is completely different from a scheme that predefinesan outage in its performance where 100% of the QoS is neverprovided (i.e. outage is allowed). Both philosophies operatethrough different procedures, so that as already stated, they areproposed for different applications and system requirements.

V. SIMULATIONS AND RESULTS

The performance of the proposed schemes is presented byMonte Carlo simulations, where the objective is to guaranteethe QoS restrictions in terms of minimum rate per served user,

Fig. 1. Performance of the robust1 scheme compared to the non-robustscheme in an imperfect CSIT scenario, with a minimum rate of 500 kbps.

regardless of the system sum rate. We consider a wirelessscenario with nt = 4 transmitting antennas in a cell with N =10 active users, each one equipped with a single receivingantenna. The transmitter runs the MOB technique where a totalof 4 orthogonal beams are set up. A bandwidth Bw = 1MHzis assumed together with a noise variance σ2 = 1. For ease inthe notation, the robust worst case with power uncertainty isdenoted as robust1 scheme, while the robust worst case withchannel uncertainty is presented as robust2 scheme.In a first approach the system outage is set to zero, and the

behaviour of both the worst case scenario robust1 scheme in(14) and the non-robust scheme in (6) are compared when thelevel of uncertainty increases, to show the benefits of the pro-posed robust scheme when applied to a more realistic scenariowhere the CSIT is imperfect. Fig. 1 exhibits the performanceof both schemes with a common QoS rate restriction of 500kbps over all the users and a varying level of uncertainty ε. Itexposes the failure in the QoS fulfillment for the non-robuststrategy as the uncertainty power increases, where due to thecross interference terms, a failure in the QoS requirement fora specific user usually drags all the other users to fail in theirQoS fulfillment.

In Fig. 2 the performance of the robust1 proposed schemeis presented, where an environment with a variable requiredrate is shown. It plots the performance for two numbers ofactive users in the cell N = 10, 30 where an uncertaintyvalue ε = 0.1 is considered. Observe that as the requiredrate increases, then more power is needed to satisfy suchrequirements, certainly, as the rate requirements are within thefeasibility region. Also note that a rate higher than 765kbpsfalls outside the feasibility region of the considered systemwith N = 10 and ε = 0.1 parameters, and how an increasein the number of active users improves the probability to findusers with better channel characteristics, so that the fulfillmentof the QoS minimum rate requirements is less power consum-ing, and consequently remaining within the feasibility regionfor N = 30. Certainly, this figure also exhibits the multiusergain of the opportunistic schemes in terms of lower requiredtransmitting power. The non-robust scheme is not included



Fig. 2. Performance of the robust1 scheme with an uncertainty level=0.1

Fig. 3. robust2 scheme against the robust ZFB in a scenario with minimumrate of 140 kbps and uncertainty=0.1

in the comparison as it does not guarantee the QoS for anuncertainty value of ε = 0.1. Along the simulations section, afixed number of 4 beams is generated (equal to the number ofavailable nt antennas), but transmitting with a smaller numberof beams would certainly decrease the interference amongthe served users, with the resultant increase in the systemfeasibility region. But the impact of the different scenariovariables on the system performance is the same, regardlessof the number of beams and/or antennas, so that the presentedresults stand for any number of transmitting beams.In order to compare the performance of the robust2 scheme

to other non-complex practical transmission schemes withuncertainty in the channel measures, the Zero-Forcing Beam-forming (ZFB) is a transmitter processing that offers highperformance but at the expenses of full CSIT requirements,where [14] studied the behaviour of ZFB under QoS restric-tions within an imperfect full CSIT scenario. Opportunisticschemes have an implicit user selection strategy for theirdirect application over cellular systems with high number ofactive users, while on the other hand, ZFB requires for anattached user selection strategy to choose the best nt users to

Fig. 4. robust2 scheme against the robust ZFB in a scenario with minimumrate of 140 kbps and N=12 active users.

award them service. A high performance and low complexityselection scheme suggested by [18] is implemented to realize acomparison between the performance of both the opportunisticand ZFB schemes in a wireless cell with minimum rate of140 kbps and ω = 0.1. A small rate value is employed toremain within the feasibility region for both schemes in theconsidered scenario, as a larger rate cannot be satisfied withthe available number of users and the scenario uncertaintylevel. Fig. 3 exposes the performance of both schemes for avariable number of active users, and how the robust2 schemerequires for lower power than the robust ZFB scheme to meetthe QoS constraints for a number of users lower than 10. Alsoobserve that the robust2 scheme has a feasibility region largerthan the one corresponding to ZFB, as the system restrictionscan be met in scenarios with smaller number of users.While in Fig. 3 we fix the uncertainty and show the systems

performance as the number of users changes, now in Fig. 4we fix the number of users and see both schemes behaviourfor un increasing level of uncertainty. Notice how the robustZFB seems to be better than the robust2 scheme for largeruncertainty values, but as the uncertainty continues to increase,then the ZFB approaches its feasibility region which explainsthe sudden increase in its required power. This shows thatthe robust2 strategy is an attractive scheme under QoS ofminimum rate constraints, that with a 50% feedback loadsaving it outperforms a full CSIT scheme for certain scenarioconditions.

Now concerned with the fair power allocation strategy, itsperformance is studied under imperfect CSIT, to show thebenefits of the proposed scheme for a fair distribution ofresources among the users, where a value for Pmax = 1is considered. A useful tool to measure the fairness of theresource allocation is through the relation ”mean/variance” ofthe Jain index [19] that states as follows

IF =(∑nt

i Ri,m)2

nt ·∑nt

i R2i,m

(27)

with Ri,m as the obtained rate for each one of the selectedusers in its assigned beam. Fig. 5 shows the fairness perfor-mance of the presented scheme as compared to the standard



Fig. 5. Performance of the fair scheme in imperfect CSIT scenarios.

Fig. 6. Performance of the robust statistical power allocation scheme, witha minimum rate of 500 kbps and N=10 users.

approach in (3) that maximizes the system sum rate, wherea value for IF = 1 reflects a totally fair system while an IF= 0 shows a totally unfair resource distribution. Observe bothschemes behaviour as the number of users rises, where the fairscheme achieves a very high fairness index over all number ofusers, while for the standard approach in (3), its fairness indexincreases with the number of users. This dependence on thenumber of users is due to the higher probability of finding a setof users, with similar high channel characteristics, thus makingthe resources distribution to be more fair among them. Theperformance of the proposed scheme is an important aspect forcommercial implementation of the MOB scheme, as fairnessis needed within the costumers for some applications.Finally, Fig. 6 studies the performance of the statistical

scheme for an increasing probability of outage ξout value,where a required minimum rate of 500 kbps is used. Theresults show how a higher allowed ξout value gives morefreedom to the scheduler in the power allocation procedure,so that the worst channel users can be discarded with theconsequent saving in the power budget, as plotted in the figure.

Fig. 7. The robust statistical power allocation performance for a variableuncertainty value. A minimum rate of 500 kbps is considered.

The statistical design is shown to be a robust power alloca-tion strategy as the users’ SNIR values are not employed forthe power allocation process, but these values are still usedfor the users selection process. Fig. 7 shows the uncertaintyeffect on the whole system performance, where as previouslystated, the effect of the imperfect measures is negligible onthe serving SNIR distribution, where for a practical uncertaintyvalue of ε = 0.1 (as the channel and noise powers are normal-ized, then it represents a 10% error in the measures), the QoSfailure is only 0.2% more than the predefined outage valueof ξout = 6%. A comparison between the worst case designrobust2 and the statistical design is shown in Fig. 8. As thestatistical design is controlled by the outage ξout, two differentoutage values are shown. On the other hand, as the worst casedesign is highly affected by the maximum uncertainty ω value,two uncertainties are also considered. Both philosophies do notshare the same restrictions nor performance, but they deliverthe operator the freedom to choose among them for the bestmatching with its specifications. The performance with perfectpartial CSIT in (6) is also shown in the figure, standing as theperformance upper bound when no outage nor uncertaintiesare present.

VI. CONCLUSIONS

The paper presents the performance of the MOB schemewhen the system is restricted by QoS in terms of a minimumrate per user, which is a very practical approach for theimplementation of any opportunistic scheme.As a step towards more realistic transmission schemes, the

paper proposes power allocation philosophies that are robust tothe uncertainty in the CSIT information, where a system thatoperates with partial and imperfect CSIT is a very attractiveoption in commercial wireless systems. Based on the systemallowed outage in the QoS achievement, a first philosophythat considers the worst case scenario for the power loadingprocedure, is proposed when hard QoS restrictions (i.e. zero-outage constraint) are required by the system administrator forall served users.A second philosophy that provides outage measures in

the resource allocation is presented, where the serving SNIR



Fig. 8. Comparison of power allocation philosophies: worst case robust2against the outage scheme.

distribution is considered for the power loading. The perfor-mance of all schemes is shown through simulations where thebenefits of each scheme are highlighted. Both philosophiesdo not share the same objectives and requirements, where theoutage scheme provides lower QoS fulfillment than the worstcase scheme while the outage scheme works in all scenariosregardless of the uncertainty level and the feasibility region.We propose both strategies, to provide the wireless operatorwith the freedom to choose the scheme that better matcheswith its available resources and demanded restrictions.

For the worst case design, two alternatives are providedupon the source of uncertainty in the partial CSIT, so thatrobust schemes with power uncertainty as well as, channeluncertainty are provided. Moreover, with the objective offairness among the users, another power allocation schemeis provided to guarantee the same minimum rate for all theserved users.

REFERENCES

[1] M. Sharif and B. Hassibi, “On the Capacity of MIMO Broadcast Channelwith Partial Side Information,” IEEE Trans. Inform. Theory, vol. 51, no.2, February 2005.

[2] N. Zorba and A.I. Perez-Neira, “Opportunistic Grassmannian Beamform-ing for Multiuser and Multiantenna Downlink Communications”, IEEETrans. Wireless Commun., vol.7, no.3, March 2008.

[3] N. Zorba and A.I. Perez-Neira, “Robust Multibeam OpportunisticSchemes Under Quality of Service Constraints,” IEEE-ICC, Glasgow-UK, 2007.

[4] N. Enderle and X. Lagrange, “User Satisfaction Models and SchedulingAlgorithms for Packet-switched Services in UMTS,” IEEE VTC Spring,New Orleans-USA, 2003.

[5] P. Svedman, S.K. Wilson, and B. Ottersten, “A QoS-aware ProportionalFair Scheduler for Opportunistic OFDM,” IEEE VTC, Los Angeles-USA,2004.

[6] P. Viswanath, D.N.C. Tse, and R. Laroia, “Opportunistic BeamformingUsing Dumb Antennas,” IEEE Trans. Inform. Theory, vol. 48, no. 6, June2002.

[7] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge Univer-sity Press, 2004.

[8] N. Ahmed and R.G. Baraniuk, “Throughput Measures for Delay-Constrained Communications in Fading Channels,” Allerton Conference,Monticello-USA, 2003.

[9] D. Gesbert and M.S. Alouini, “How Much Feedback is Multi-userDiversity Really Worth?” IEEE ICC, Paris-France, 2004.

[10] M. Bengtsson and B. Ottersten, “Optimal Downlink Beamforming UsingSemidefinite Optimization,” Allerton Conference, Monticello-USA, 1999.

[11] M. Schubert and H. Boche, “Solution of the Multiuser DownlinkBeamforming Problem with Individual SINR Constraints,” IEEE Trans.Veh. Technol., vol. 53, no. 1, January 2004.

[12] D. Bartolome and A.I. Perez-Neira, “Mean vs. Variance Trade-offAnalysis in Multi-antenna Multi-user Channels,” IEEE Trans. WirelessCommun., vol. 5, no. 3, March 2006.

[13] A. Vakili, M. Sharif, and B. Hassibi, “The Effect of Channel EstimationError on The Throughput of Broadcast Channels”, IEEE ICC, Istanbul-Turkey, 2006.

[14] M. Payaro, A. Pascual-Iserte, and M.A. Lagunas, “Robust Power Alloca-tion Designs for Multiuser and Multiantenna Downlink CommunicationSystems through Convex Optimization”, IEEE J. Sel. Areas Commun.,vol. 25, no. 7, September 2007.

[15] B. Hassibi and B.M. Hochwald, “How Much Training is Needed inMultiple-Antenna Wireless Links?,” IEEE Trans. Inform. Theory, vol.49, no. 4, April 2003.

[16] T. Heikkinen and A. Prekopa, “Optimal Power Control in a WirelessNetwork Using a Model with Stochastic Link Coefficients”, NavalResearch Logistics, vol. 52, no. 2, March 2005.

[17] S. Boyd, S. Kandukuri, “Optimal Power Control in Interference-limitedFading Wireless Channels with Outage Probability Specifications,” IEEETrans. Wireless Commun., vol. 2, no. 1, January 2002.

[18] T. Yoo and A. Goldsmith, “On the Optimality of Multi-Antenna Broad-cast Scheduling Using Zero-Forcing Beamforming”, IEEE J. Sel. AreasCommun., vol. 24, no. 3, March 2006.

[19] R. Jain, D. Chiu, and W. Hawe, “A Quantitative Measure of Fairness andDiscrimination for Resource Allocation in Shared Computer Systems,”DEC Research Rep. TR-301, vol. 2, no. 5, September 1984.

Nizar Zorba (Jordan, 1980) holds a BSc. in Electri-cal Engineering by JUST University (2002, Jordan),a MPhil in Mobile Communications by University ofZaragoza (2002-2004, Spain), a MPhil in Economicsby University of Zaragoza (2003-2005, Spain), aMPhil in Signal Processing for Communications byUniversitat Politecnica de Catalunya-UPC (2004-2006, Spain), and a PhD in Signal Processing forCommunications by UPC (2004-2007, Spain). Heis now working at CTTC (Centre Tecnologic deTelecomunicacions de Catalunya) as an associate

researcher, and since arriving at CTTC he participated in several R&D projectsfor European MEDEA+, FP7 and ESA.

Ana I. Perez-Neira graduated in telecommunicationengineering in 1991 and received the Ph.D. degreein 1995 from the Technical University of Catalonia(UPC), Barcelona, Spain. In 1991, she joined theDepartment of Signal Theory and Communicationof the UPC, where she carried on research activitiesin the field of higher order statistics and statisticalarray processing. In 1992, she became Lecturer, in1996, Associate Professor. Since 2006 she is fullprofessor for Signal Theory and Communication atUPC. From 2000 to 2003 she was member of the

Board of Directors of the Telecommunications School of Barcelona, ETSETB,and since 2002 she is research associate at CTTC (Centre Tecnologicde Telecomunicacions de Catalunya - Castelldefels). She teaches and co-ordinates graduate and undergraduate courses in statistical signal processing,array & MIMO processing, analog and digital communications, mathematicalmethods for communications and nonlinear signal processing. She is theeditor of 3 special issues for Eurasip SP and the author of: 3 book chapters,3 patents, 24 journal papers and more than 150 conference papers (17invited) in the area of statistical signal processing and fuzzy processing,with applications to mobile/satellite communication systems, physical andaccess layers. She has participated in the organization of 2 conferences (ESAconference’96, SAM’04) and she has coordinated national public and privatefounded projects (INTAS, RACE, ACTS, IST, Eureka and ESA Europeanprojects): 15 European competitive projects, 9 National competitive projectsand 4 industry non-competitive projects.


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