Low-Complexity User Selection for Rate Maximization in MIMO Broadcast Channels with Downlink Beamforming

Research ArticleLow-Complexity User Selection for Rate Maximization inMIMO Broadcast Channels with Downlink Beamforming

Eduardo Castantildeeda12 Adatildeo Silva12 Ramiro Samano-Robles2 and Atiacutelio Gameiro12

1 Department of Electronics Telecommunications and Informatics Aveiro University Aveiro Portugal2 Instituto de Telecomunicacoes Campus Universitario de Santiago P-3810-193 Aveiro Portugal

Correspondence should be addressed to Eduardo Castaneda ecastanedaavitpt

Received 19 September 2013 Accepted 3 December 2013 Published 16 January 2014

Academic Editors L Jacob and Y Shi

Copyright copy 2014 Eduardo Castaneda et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present in this work a low-complexity algorithm to solve the sum rate maximization problem in multiuser MIMO broadcastchannels with downlink beamforming Our approach decouples the user selection problem from the resource allocation problemand its main goal is to create a set of quasiorthogonal users The proposed algorithm exploits physical metrics of the wirelesschannels that can be easily computed in such a way that a null space projection power can be approximated efficiently Based on thederived metrics we present a mathematical model that describes the dynamics of the user selection process which renders the userselection problem into an integer linear program Numerical results show that our approach is highly efficient to form groups ofquasiorthogonal users when compared to previously proposed algorithms in the literature Our user selection algorithm achievesa large portion of the optimum user selection sum rate (90) for a moderate number of active users

1 Introduction

Multiple-inputmultiple-output (MIMO) systems have a hugepotential to attain high throughput in wireless systems [1 2]MIMOsystems can be employed to exploit space-time codingand spatial multiplexing When channel state information(CSI) is known at the transmitter the overall system through-put can be increased by beamforming transmission In thewireless scenario of interest a transmitter encodes differentinformation for different receivers in a common signal whichis referred to the literature as a broadcast channel (BC) For aclassic deployment with one base station (BS) equipped with119873119905antennas and119870 single antenna users the overall through-

put for a MIMO system increases by a factor of min119873119905 119870

the capacity of a time-division-multiple-access (TDMA)scheduling system if the transmitted signals are uncorrelated[1] The TDMA system cannot exploit the multiple antennadeployment at the BS which leads to a waste of systemresources and a limited system performance The naturalsolution to this problem is to transmit simultaneously tomore than one user A strategy to accomplish this goal is toimplement a nonlinear coding scheme called dirty paper

(DPC) which is a multiplexing technique based on codingknown interference [3]TheDPC exploits the full CSIT (at thetransmitter) achieving the same capacity of an interferencefree MIMO BC system [2] and when the number of singleantenna users119870 is larger than119873

119905at the BS DPC can achieve

a linear capacity increase in119873119905

DPC is the optimal throughput maximization schemein a MIMO BC system However it requires huge compu-tation complexity and feedback information which rapidlyincreases with119873

119905 Two reduced-complexity suboptimal solu-

tions to the throughput maximization problem were pro-posed in [4] The first solution is the channel inversion zero-forcing beamforming (ZFBF) which is an orthogonal trans-mit spatialmultiplexing linear precoding schemewhosemainobjective is to nullify the mutual interference among usersaccording to perfect CSIT Despite its simplicity ZFBF hasbeen shown to achieve the same asymptotic sum capacity ofDPC when high multiuser diversity is ensured The secondsolution called zero-forcing dirty-paper (ZFDP) is an asymp-totically optimal beamforming scheme that combines a QRdecomposition of the channel matrix with DPC at the trans-mitter In this ranked known interference scheme the first

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 865905 13 pageshttpdxdoiorg1011552014865905

2 The Scientific World Journal

user is not affected by interference while the second user isonly affected by interference coming from the first user Thisprocedure is repeated for subsequent users

The throughputmaximization using ZFBF (eg [4ndash8]) orZFDP (eg [4 9 10]) can be further improved in scenarioswhere the number of single antenna users is larger thanthe number of antennas at the BS (119870 gt 119873

119905) The users

can be seen as an extra dimension of adaptation which isreferred to in the literature as multiuser diversity In orderto exploit such diversity it is necessary to select a set ofactive users whose channel characteristics result in a perfor-mance improvement (eg throughput) when they transmitsimultaneously in the same radio resourceThe user selection(scheduling) is a medium access control (MAC) processthat can use information from the adaptive physical-layer(PHY) design so that temporal dimension (scheduling) andspatial dimension (multiple antennas) can be fully exploitedThe scheduling is a real time process whose computationalcomplexity and implementation efficiency affect directly theperformance of upper-layers Moreover finding the set ofusers that optimizes a given global utility function is a highlycomplex combinatorial problem whose optimal solution isgiven by an exhaustive search and its associated search spacegrows geometrically with the number of users Since the com-putation of the optimal solution to the scheduling problem isprohibited for most practical systems for moderate119870 and119873

119905

it is necessary to find efficient suboptimal scheduling schemesthat can provide a good trade-off between performance andcomplexity

2 Related Works and Contributions

A considerable amount of work focused on the asymptoticsum rate of MIMO BC systems with user selection has beendone over the last ten years (eg [5 6 8 9]) and severalpublished works presented efficient suboptimal algorithmsthat attempt to overcome the prohibitively high complexityof exhaustively searching users Most of the works thatsuboptimally solve the problem of sum rate maximizationinmultiusermultiple-antenna systems implement cross-layerdesigns where the scheduling decisions are made based oninstantaneous CSI or link-level metrics

Since the aforementioned problem can be tackled indifferent ways we propose a classification of the algorithmsthat can be found in the literature based on the methodologyfollowed to solve the mixed convex and combinatorial prob-lem of throughput maximization in multiuser MIMO BCsystems We use this classification to make a clear distinctionbetween the metrics used by each class and to fairly comparethe performance achieved by algorithms of different classes

We say that a class-A algorithm is the one that performsa joint user selection and power allocation optimization Anew user 119896 is added to the set of selected users S only if fora given utility function 119880 the aggregation of 119896 to S increasesthe value of the utility function that is 119880(S) lt 119880(S + 119896)This kind of greedy algorithms [10ndash15] are highly effective for

throughput maximization However they still employ a highcomputational power since the selection process requires theevaluation of the global utility function (this requires a water-filling power allocation evaluation and the computation of theShannon capacity) for each unselected user in every iterationof the algorithm

The algorithm class-B operates in two phases In the firstphase a set of users is selected based on specific channel char-acteristics and in the second phase the algorithm evaluatesthe global utility function for the previously defined set [5 68 9 16] This means that the user selection and the resourceallocation (powers and beamforming weights) problems arecarried out independently and the throughput maximizationheavily depends on the channel characteristics of the selectedusers Furthermore the cardinality of the set of selected usersis fixed in the first phase and it might be modified duringthe second phase when the global utility function is evalu-ated For instance if water-filling based power allocation isperformed to evaluate the global utility function this mightresult in zero power allocation for some selected users dueto the channel characteristics of the selected users the powerconstraints and the SNR regime In [5] the authors designeda greedy algorithm that performs a semiorthogonal userselection (SUS) in order tomaximize the total sum rate imple-menting ZFBF In this class-B algorithm the new selected usermaximizes the component of the channel that is orthogonalto the subspace spanned by the channels of the previouslyselected users The evaluation of that orthogonal componentrequires the multiplication of the unselected channel vectorsby a matrix that describes the subspace defined by channelsof the selected users The authors of [5] showed that theaverage sum rate of ZFBF combined with their proposed userselection technique achieves asymptotically the average sumrate of DPC when the number of users is infinite (119870 rarr infin)Tu and Blum [9] proposed a class-B greedy algorithm forthroughput maximization and ZFDP The metric for userselection is based on the channel component projected ontothe null space of the space spanned by the previously selecteduser channels This metric is used to estimate the powerdegradation that a new user will experience if it interactswith the orthogonal subspace spanned by the other selectedusers A statistical analysis of this methodology was done in[10] where it was shown that the greedy user selection basedon channel component projection is a suboptimal yet highlyefficient way to form groups of quasiorthogonal users thatsuboptimally maximize the sum rateThemain drawbacks ofthis approach are the following one is the computation of anull space projectormatrix unsing the channels of all selectedusers and two is the multiplication of such projector matrixby the channels of all unselected users in order to identify thebest unselected user A similar approach to [9] was presentedin [8] for throughputmaximizationwith ZFBFThedifferencebetween these two approaches lies in the fact that the latterperforms singular value decomposition (SVD) in order toevaluate null space of the selected user channels The userselection of [8] requires for each iteration the multiplicationof the matrix that defines the null space of the selectedchannels by all nonselected channels

The Scientific World Journal 3

21 Contributions Both classes of algorithms require exten-sive use of matrix operations to perform the user selectionClass-A algorithms use matrix inversion in order to performpower allocation per each possible set of selected users andclass-B algorithms require the computation of either theprojector or the orthogonal projectormatrix [17] per iterationand amatrix inversion for the final power allocation based onwater-filling

In this work we design a low-complexity suboptimalgreedy class-B algorithm for throughput maximization thatmakes scheduling decisions based on simple physical metricsof the channels that is information extracted from thechannel norms and the orthogonality between channels Wepropose ametric that approximates the one used in [8 9] withthe advantage that we only require multiplication of scalarsdefined by the correlation coefficient between any twochannels We quantitatively compare the MIMO BC systemperformance in terms of the throughput (measured by theaverage sum rate) achieved by the proposed algorithm andseveral state-of-the-art algorithms (classes A and B)

The nature of the quasiorthogonal user grouping yieldsthemaximization of the sum projection power of the selectedusers The optimum sum projection power can be approx-imated as the optimization of a global objective functionwhich is given by the sum of individual weighted convexfunctions For this problem the constraints are given by affinefunctions and the weights are given by binary variablesTherefore we show that it is possible to render the sum pro-jection power problem into a convex integer program whichcan be efficiently solved using available numerical packagesIn contrast to previous works (eg [13]) that only providea description of the user selection problem as an integerprogram (due to the high complexity of the problem for-mulation) we provide a complete mathematical model forthe integer constrained program based on the derived metricwhose solution asymptotically approximates the optimumone for moderate values of119870

Numerical results show that our proposed algorithmscan achieve a large portion of the optimum sum rate with alow-computational complexity price and high performancefor both precoding schemes ZFBF and ZFDP Moreoverthe proposed algorithms outperform state-of-the-art class-Balgorithms for low values of 119870 and achieve asymptoticallyoptimal behavior for large values of 119870

22 Organization The remainder of the paper is organized asfollows In Section 3 we present the system model Section 4describes the throughput maximization and the user selec-tion problems and the optimization metric that is studiedalong the paper Section 5 presents the design of a greedyalgorithm that performs quasiorthogonal user selection and ageneralmathematicalmodel that represents the user selectionproblem as an integer programming problem Section 6shows numerical examples for the assessment of the proposedalgorithms using different performance metrics The mainconclusions are drawn in Section 7

Some notational conventions are as follows Matrices andvectors are set in boldface ⟨sdot⟩ (sdot)119879 (sdot)119867 | sdot | sdot

119865 and Esdot

denote the inner product transpose hermitian transpose setcardinality Frobenius norm and the expectation operationrespectively Sp(A) denotes the subspace spanned by therows of matrix A rank(A) is the rank of matrix A and(119909)+ represents max119909 0 diag(x) denotes a diagonal matrix

whose main diagonal is x [A]119894119895is the element 119886

119894119895of matrix A

and I is the identity matrix of compatible size

3 System Model

Consider a single-cell with a single base station equippedwith119873119905antennas and 119870 single antenna active users competing

for resources We assume perfect CSI at the base station andthe channel coefficients are modeled as independent randomvariables with a zero-mean circularly symmetric complexGaussian distribution (Rayleigh fading) The signal receivedby the 119894th user is given by

y119894= h119894x + 119899119894 (1)

where x isin C119873119905times1 is the transmitted signal vector from thebase station antennas and h

119894isin C1times119873119905 is the channel vector

to the user 119894 Each user treats the signals intended for otherusers as interference and 119899

119894simCN(0 1205902

119899) is the additive zero-

mean white Gaussian noise with variance 1205902119899 The entries

of the block fading channel H = [h1198671 h119867

119870]119867 and n =

[1198991 119899

119870]119879 are normalized so that they have unitary vari-

ance and the transmitter has an average power constraintEx119867x le 119875 Since the noise has unit variance 119875 representsthe total transmit signal-to-noise-ratio (SNR)

For linear spatial processing at the transmitter the beam-forming matrix can be defined as W = [w

1w2 w

119870]

the symbol vector as s = [1199041 1199042 119904

119870]119879 and P =

diag(1199011 119901

119870) is the power loading so that the transmitted

signal is given by x = sum119870

119896=1radic119901119896w119896119904119896 The signal-to-inter-

ference-plus-noise ratio (SINR) of the 119894th user is

SINR119894=

119901119894

1003816100381610038161003816h119894w11989410038161003816100381610038162

sum119895 = 119894

119901119895

10038161003816100381610038161003816h119894w119895

10038161003816100381610038161003816

2

+ 1205902119899

(2)

Assuming 119873119905ge 119870 the sum rate maximization problem

using beamforming (BF) can be formulated as

119877BF

= maxWP

119870

sum119896=1

log2(1 + SINR

119896)

subject to WP2119865le 119875

(3)

31 Zero-Forcing Beamforming In ZFBF the channel matrixH at the transmitter is processed so that orthogonal channelsbetween the transmitter and the receiver are created defininga set of parallel subchannels Assuming 119870 active users thenfor the case where 119870 le 119873

119905and rank(119867) le 119873

119905 the ZF beam-

formingmatrix is given by theMoore-Penrose pseudoinverseofH [17 18] as

W = Hdagger = H119867(HH119867)minus1

(4)

4 The Scientific World Journal

The throughput when ZFBF is applied to (3) is given by[4]

119877ZFBF

(H) =

119870

sum119894=1

(log (120583119887119894))+

(5)

where 119887119894= [(HH119867)minus1]

119894119894minus1 is the effective channel gain of the

119894th user and its allocated power is

119901119894= (120583119887119894minus 1)+

(6)

and the water level 120583 is chosen to satisfy

sum119894isinΩ

(120583 minus1

119887119894

)

+

= 119875 (7)

32 Zero-Forcing Dirty Paper Beamforming Suboptimalthroughput maximization in Gaussian BC channels has beenproposed in several works [4 9 10] based on the QR-typedecomposition [18] of the channel matrix H = LQ obtainedby applying Gram-Schmidt orthogonalization to the rows ofHL is a lower triangularmatrix andQhas orthonormal rowsThe beamforming matrix given byW = Q119867 generates a set ofinterference channels

119910119894= 119897119894119894radic119901119894119904119894+ sum119895lt119894

119897119894119895radic119901119895119904119895+ 119899119894 119894 = 1 119896 (8)

while no information is sent to users 119896 + 1 119870 In orderto eliminate the interference component 119868

119894= sum119895lt119894

119897119894119895radic119901119895119904119895of

the 119894th user the signals radic119901119894119904119894for 119894 = 1 119896 are obtained

by successive dirty-paper encoding where 119868119894is noncausally

known This precoding scheme was proposed in [4] and theauthors showed that the precoding matrix forces to zero theinterference caused by users 119895 gt 119894 on each user 119894 therefore thisscheme is called zero-forcing dirty-paper (ZFDP) codingThethroughput achieved in (3) under the ZFDP scheme is givenby [4]

119877ZFDP

(H) =

119870

sum119894=1

(log (120583119889119894))+

(9)

where 119889119894= |119897119894119894|2 and 120583 is the solution to the water-filling

equation

sum119894isinΩ

(120583 minus1

119889119894

)

+

= 119875 (10)

which defines the 119894th power as 119901119894= (120583119889

119894minus 1)+

4 The User Selection Problem

Let Ω = 1 119870 be the set of all competing users where119870 is larger than the number of available antennas at the basestation that is |Ω| = 119870 gt 119873

119905 Under this condition user

selection is required and the joint sum rate maximization (3)and user selection problem can be defined as

R = maxSsubΩ|S|=119873

119905

119877(type)

(H (S)) (11)

where S sub Ω H(S) is a row-reduced channel matrixcontaining only the channel vectors of the selected users andtype denotes the precoder that is used either ZFBF or ZFDPObserving that in (11) the set of selected users is constrainedto have maximum cardinality full spatial multiplexing issought For the high SNR regime and ZFBF using water-filling based power allocation it is possible to achieve a finalsubset with cardinality119873

119905as long as the given SNR is above a

critical value [4]The optimum solution to (11) requires an exhaustive

search over a search space of size ( 119870119873119905) and for large values

of 119870 its computation has prohibitive complexity Thereforelow-complexity suboptimal algorithms have been proposedin the literature in order to maximize the throughput solving(11) in two phases (class-B approach) first by finding a set Sof quasiorthogonal users (combinatorial search) and secondby allocating resources to such a set (convex optimization)[5 6 8]

41 Metric of Orthogonality In the literature of user selectionfor MIMO systems [5 9 10] one of the most commonapproaches to form the set of selected users S is to find iter-atively the user that locally maximizes the sum power pro-jection This means that given S = 0 the optimum new userformΩ achieves the largest amount of projection power onceits channel is projected onto the subspace spanned by thepreviously selected users Sp(H(S))

This procedure is optimum when only 1 element from Ω

must be selected to be added toS In the case of |S| lt 119873119905the

aggregation of a new user is required to meet the constraintof (11) and the aforementioned procedure results in a subop-timal maximization of the total sum of projection powers

Let QS be the orthogonal complement projector matrixof Sp(H(S)) defined as [17]

QS = I119873119905

minus PS = I119873119905

minusH(S)119867(H (S)H(S)

119867)minus1

H (S)

(12)

where PS is the orthogonal projector matrix of Sp(H(S))In [9] Tu and Blum proposed a greedy algorithm originallydesigned to be applied to ZFDP coding scheme which selects119873119905out of119870 rows of the channel matrixH Such user selection

methodology is based on an iterative null space projection(NSP) and it achieves the best suboptimal solution to theproblem (11) for a class-B algorithm regardless of the codingscheme which will be elaborated upon in the followingsections In [9] given S = 0 the new selected user is the onethat maximizes the following metric

119903S119894 = h119894QSh119867

119894= h119894h119867119894minus h119894PSh119867

119894 (13)

where the term h119894PSh119867119894 represents the power loss due to

the imperfect orthogonality between h119894and Sp(H(S)) In

other words the metric 119903S119894 measures the amount of powerpreserved by user 119894 when h

119894is projected onto the null space

ofH(S) The same idea of [9] has been applied by Wang andYeh [8] for ZFBF calculating the null space ofH(S) via SVD

The Scientific World Journal 5

hk

QShkSp(H( ))perp119982

Sp(H( ))119982

(a)

hk120579ij

120579kj

120579ki

hi

hj

Sp(H( ))119982

(b)

Figure 1 (a) The orthogonal component of vector h119896to Sp(H(S))

(b) Physical components of the interaction of two selected users 119894

and 119895 with third unselected user 119896

This concept is represented in Figure 1(a) where the chan-nel h119896of the 119896th unselected user is projected onto the null

space Sp(H(S))perp using (12)

Several user selection algorithms (eg [5 6 11 16 19 20])attempt to create groups of quasiorthogonal users based onthe information provided by the coefficient of correlation 120578

119894119895

which for two users 119894 and 119895 is defined as [17 21]

120578119894119895= cos (120579

119894119895) =

⟨h119894 h119895⟩

1003817100381710038171003817h119894100381710038171003817100381710038171003817100381710038171003817h119895

10038171003817100381710038171003817

0 le 120579119894119895le 120587 (14)

where the coefficient 0 le |120578119894119895| le 1 geometrically represents

the cosine of the angle between the two channel vectors [17]In [22] the authors presented an algorithm that selects thebest 2 users out of 119870 The first user 119894 isin S is given by theuser with the highest channel norm as in [5 6 8 9] and thesecond user 119895 isin Ω is the one that maximizes the producth1198952(1 minus 1205782

119894119895) = h

1198952sin2(120579

119894119895) In the particular case of [22]

when |S| = 1 h119895QSh119867119895 = h

1198952sin2(120579

119894119895) that is scaling the

squared norm by the squared sine of the angle between user119894 and 119895 is equivalent to projecting h

119895onto the null space of

h119894[17] When zero-forcing-based precoding is used the term

sin2(120579119894119895) can be viewed as a projection power loss factor [21]

In the following section we derive a metric to approximatethe projection of a given h

119894 forall119894 notin S onto Sp(H(S))

perp for thegeneral case where |S| gt 1

5 Power Projection Based User Selection

In this section we propose a cross-layer design that subopti-mally solves the sum rate maximization problemThis designonly considers the physical layer model and we ignore theapplication level delay effects and assume that all users haveinfinite information to transmit when they are scheduledThegeneralization of the user selection problem is modeled as aninteger convex program and we analyze the suboptimality ofthe selection metrics

51 Iterative Power Projection (IPP) Algorithm Based on thefact that (13) has a fundamental connection to the coefficientsof correlation we design an algorithm that attempts to find

a quasiorthogonal set of users S using exclusively the infor-mation provided by the channel norms and the orthogonalitybetween any two user channels given by (14) Figure 1(b)exemplifies the required information used to find the set Sand for two selected users 119894 and 119895 the figure shows the physicalcomponents that affect the interactionwith a third unselecteduser 119896

In order to start the users selection process we assumethat the base station knows the coefficients of correlation forall users in Ω = 1 119870 which requires (1198702 minus 119870)2 com-putations of (14) since 120578

119894119895= 120578119895119894and the computation of the

coefficients (inner product and vector norm operations) canbe done within time O(119870) For the sake of notation let 984858

119894119895=

1 minus 1205782119894119895 984858119894119895= 1 minus |120578

119894119895| and define the following geometric and

arithmetic means for the elements 984858 associated with user 119894 isinΩ as

119872119892(119894)

= ( prod119895 = 119894119895isinΩ

984858119894119895)

1(|Ω|minus1)

le (1

|Ω| minus 1) sum119895 = 119894119895isinΩ

984858119894119895 (15)

where 119872119892(119894)

is a lower bound of the arithmetic mean of theprojection power loss factors of user 119894 We select the first useras the one that preserves the highest amount of average poweronce it is projected onto all other users such that

119894lowast= argmax

119894isinΩ

1003817100381710038171003817h11989410038171003817100381710038172

119872119892(119894)

(16)

and the sets of selected and unselected users are updatedS = 119894lowast and Ω = Ω minus 119894lowast By selecting the first userusing (16) the goal is assigning priority weights to the channelnorms that is users with large channel norms are penalized iftheir associated correlation coefficients have a large varianceFurthermore the geometric mean 119872

119892(119894)minimizes the bias

created by the terms 984858 with very large or small values whichwould be neglected if the arithmetic mean of the projectionpower loss factors were considered in (16)

The following user to be selected must maximize twocriteria at the same time On the one hand it must maximizeits own projected power which is affected by the coefficients984858 of the already selected users in S The effective projectedpower of the user 119894 isin Ω is given by

120595119894=1003817100381710038171003817h119894

10038171003817100381710038172

prod119895isinS

984858119894119895 (17)

On the other hand the users in S have already achievedan effective projected power that is defined as

120601119895=10038171003817100381710038171003817h119895

10038171003817100381710038171003817

2

prod119896 = 119895119896isinS

984858119895119896 119895 isin S (18)

For a new user candidate 119894 isin Ω its aggregation to the setS implies a reduction of the total sum of projected powersof the selected users (sum

119895isinS 120601119895) by the factors 984858 associated

with the new selected user Using the arithmetic and geomet-ric means lower bounds of the average projected power of

6 The Scientific World Journal

the selected users in (18) can be defined for the 119894th unselecteduser as follows

prod119895isinS

120601119895984858119894119895le (

1

|S|sum119895isinS

120601119895984858119894119895)

|S|

le (1

|S|sum119895isinS

120601119895)

|S|

(19)

The total effective projection power 119894of the unselected

user 119894 takes into account both the average projection powerover the elements inS computed for the lower bound in (19)and the projection power of user 119894 isin Ω (17) Consider

119894= (prod119895isinS

120601119895984858119894119895)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gainforall119895isinS

(1003817100381710038171003817h119894

10038171003817100381710038172

prod119895isinS

984858119894119895)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟gainforall119894isinΩ

= (prod119895isinS

120601119895)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟constantforall119894isinΩ

(prod119895isinS

984858119894119895)(

1003817100381710038171003817h11989410038171003817100381710038172

prod119895isinS

984858119894119895)

(20)

By taking the square of the product of the terms 984858119894119895 both

effects are considered the impact of the selected users overuser 119894 and the power degradation that the users inSwill haveif user 119894 is selected

Since the effective projected power of the selected usersremains constant for all users in Ω the metric in (20) can benormalized as follows

120593119894=1003817100381710038171003817h119894

10038171003817100381710038172

prod119895isinS

9848582

119894119895 (21)

Given S the next selected user is found using the metricdefined in (21) as

119894lowast= arg max

119894isinΩ

120593119894 (22)

where the selection of the locally optimum 120593(119899) in a giveniteration 119899 is conditioned on the choice of 120593(1) 120593(119899 minus 1)

As119870 rarr infin the number of total operations to solve prob-lem (11) becomes computationally costly and a more efficientupdate of the setΩ can be performed By selecting a new userusing (22) each iteration requires the comparison of |Ω|

elements in order to select the user whose projection poweris maximum Considering that the cardinality of the final setmust be 119873

119905 without modifying Ω this algorithm would

require a total of 119871 = 119873119905(119870 minus (119873

119905minus 1)2) comparison

operations For our case the projection power evaluations forthe metric used in (21) will use all coefficients 984858 associatedwith the elements of S The algorithms proposed in [6 8 9]also require 119871 comparison operations versus the elements ofS However the computational complexity is quite differentsince each comparison requires a matrix multiplicationwhilst the metric used in (21) is a multiplication of realpositive numbers

In [5 20 22] after a new user 119894 is added to S the set ofunselected userΩ is reduced by keeping the users whose cor-relation factors are above a threshold 120572th that isΩ(119899) = 119895 isin

Ω(119899 minus 1) 120578119894119895lt 120572th where 119899 stands for the iteration number

and 119894 is the selected user of iteration 119899 minus 1 This subselection

within the algorithm has the drawback that the value of theparameter 120572th is fixed which might result in a drastic reduc-tion of the size of Ω and the degradation of the multiuserdiversity According to [5] there exists an optimum valueof the threshold 120572th for each value of 119870 and 119873

119905 but the

mathematical relationship between these terms is not givenin a closed form The statistical dependence of the averagethroughput due to 120572th has been established only for the casewhere the cardinality of the set of selected users is constrainedto be 2 that is |S| = 2 in [21]

We propose a dynamic reduction of the setΩ consideringtwo factors to discard users at each iteration The first crite-rion is related to the statistics of the projection powers regard-ing the users that have been selected The second criterionweights the first criterion based on the number of active usersand the number of antennas 119873

119905 Let us define the arithmetic

mean of the projected powers given the new selected user 119894lowastas

119872119886(119894lowast)=

1

|Ω|sum119895isinΩ

10038171003817100381710038171003817h119895

10038171003817100381710038171003817

2

984858119894lowast119895 (23)

Notice that the power projection computation is per-formed considering only the power projection loss factorsassociated with 119894

lowast and each term of the sum in (23) is themultiplication of two real numbers The metric defined in(23) is used to discard users whose projection powers arebelow the arithmetic mean which results in a reduction of thenumber of comparisons for the next iteration Neverthelesswhen the number of total users is low (119870 asymp 119873

119905) the number

of users in Ω should not be reduced drastically in order topreserve enoughmultiuser diversity and to achieve full spatialmultiplexing We define a weight factor based on the numberof antennas119873

119905and the size of the sets S andΩ as follows

119908(119873119905SΩ) = 1 minus (

119873119905minus |S|

|Ω|)

1(119873119905minus|S|)

(24)

The objective of 119908(119873119905SΩ) is to scale 119872

119886(119894lowast)in iteration 119899

taking into account the degrees of freedom available at thebase station (rank(H(S))) and the current size ofΩ Given thenew selected user 119894lowast and weighting (23) by (24) the modifiedset of users that will compete to be scheduled in the nextiteration 119899 + 1 is defined as

Ω (119899 + 1) = 119895 isin Ω (119899) 10038171003817100381710038171003817h119895

10038171003817100381710038171003817

2

984858119894lowast119895ge 119908(119873119905SΩ)119872119886(119894lowast) (25)

The procedure to generate the quasiorthogonal set of userthat solves problem (11) is described in Algorithm 1

52 User Selection as an Integer Linear Program (ILP) Theoptimization performed in Algorithm 1 can be described as agreedy search over a tree structure [23] where the treersquos rootis given by the element of Ω that preservers a higher averageprojected power (16) Similar approaches are implemented in[5 6 8 9] considering the user with the maximum channelnorm as the root of tree The greedy Algorithm 1 makes asequence of decisions in order to optimize the metric in (22)However this local optimization might not lead to a global

The Scientific World Journal 7

(1)Ω = 1 119870 S = 0 119899 = 0

(2)while |S| lt 119873119905do

(3) if 119899 = 0 then(4) Compute 119894lowast by (16)(5) else(6) Compute 119894lowast by (22)(7) end if(8) 119899 = 119899 + 1 S(119899) = S(119899 minus 1) cup 119894lowastΩ(119899) = Ω(119899 minus 1) minus 119894lowast

(9) Update Ω(119899) by (25)(10) end while(11)Power Loading Principle water-filling

Algorithm 1 Iterative power projection (IPP)

optimal solution Moreover since the first user is foundby (16) the correlation of such a user with the futureselected users is neglected when S is initialized A generalmathematical model of the interaction of all elements in Sthat exploits themetrics used in (16) and (22) can be designedDue to the structure of (16) and (22) which maximizes thesquared channel norm weighted by the product (interaction)of the correlation coefficients we canmodel a relaxed versionof the user selection problem (11) as an integer programmingproblem

Let us define the interaction of the user 119894 isin Ωwith the restof the users as a function 119891

119894considering the structure of (21)

as

119891119894=1003817100381710038171003817h119894

10038171003817100381710038172

prod119895 = 119894

9848582

119894119895 forall119894 119895 isin Ω (26)

and by applying a change of variables the function 119891119894

=

log(119891119894) is given by

119891119894= 119886119894+ sum119895 = 119894

119887119894119895 (27)

where 119886119894= 2 log(h

119894) and 119887

119894119895= 2 log(984858

119894119895) Our objective

is to maximize the total sum of the projected powers whichis a function of two factors the orthogonality between theselected channels and the amount of remaining power aftera projection Therefore (11) can be thought of as the maxi-mization of sum

119894119891119894with the constraint that |S| = 119873

119905 In order

to introduce such constraint we define the following binaryvariable 119910

119894as

119910119894=

1 if user 119894 is selected0 otherwise

(28)

In the same way we can define a set of binary variables 119909119894119895

that relate to the common coefficient 984858119894119895of two users as

119909119894119895=

1 if both users 119894 and 119895 are selected0 otherwise

(29)

Themathematicalmodel for the user selection problembasedexclusively on the channel norms and correlation coefficientsis given by

maximize sum119894

119886119894119910119894+ 2sum119894

sum119895=119894+1

119887119894119895119909119894119895

subject to sum119894

119910119894= 119873119905

119910119894+ 119910119895le 1 + 119909

119894119895 forall119894 119895

119909119894119895le 119910119894 forall119894 119895

119909119894119895le 119910119895 forall119894 119895

119910119894isin 0 1 forall119894

119909119894119895isin 0 1 forall119894 119895

variables 119910119894 119909119894119895

(30)

where (30) is a binary programming problem that general-izes the objective function optimized by Algorithm 1 Theadvantage of this formulation is that the order in which theusers are selected has no impact on the orthogonality of theelements ofH(S) that is the negative effects of selecting localoptimum users in each iteration are canceled The solutionto the user selection problem is given by the binary variables119910119894and power allocation based on water-filling is performed

over the set of selected users according to the employedprecoding scheme Observe that a conversion from 119891

119894to 119891119894

is not required because the relevant information to formthe set S is given by the variables 119910

119894that have achieved a

value of one Since the objective function is convex and theconstraints are given by affine functions this problem can besolved by the pseudodual simplex method [24] for integerprograms or by using standard optimization packages [2526] Moreover problem (30) always has a feasible solutionbecause the only constraint that might lead to infeasibility isthe equality constraint that is always met due to the fact that119870 ge 119873

119905 Problem (30) is a relaxed version of (11) and it finds

a suboptimal solution to the user selection problem owingto the nature of the coefficients 119887

119894119895which is analyzed in the

following subsection

8 The Scientific World Journal

53 Suboptimality of the User Selection Process The projec-tion power found by (13) has a direct relationship with thecorrelation coefficients 120578 of the users in S and the channelvector h of the candidate user in Ω The normalized powerloss of such user once it is projected onto PS is called thecoefficient of determination and is given by [17]

1198772

Sh =hPSh119867

hh119867 (31)

where1198772Sh measures howmuch the vector h can be predicted(correlated) from the selected vectors of H(S) Notice thatfrom (13) and (31) the projection of h onto the null space ofSp(H(S)) is equivalent to 1 minus 1198772Sh which can be evaluatedfrom the correlation coefficients 120578 as follows [17]

1 minus 1198772

Sh

= (1 minus 1205782

h120587(1)) (1 minus 1205782

h120587(2)|120587(1)) sdot sdot sdot (1 minus 1205782

h120587(119896)|120587(1)sdotsdotsdot120587(119896minus1))

(32)

where 120587(119894) is the 119894th ordered element of H(S) and120578h120587(119896)|120587(1)sdotsdotsdot120587(119896minus1) is the partial correlation between the can-didate vector h and the ordered channel vector h

120587(119896)isin

H(S) associated with 120587(119896) eliminating the effects due to120587(1) 120587(2) 120587(119896minus1)The exact computation of the last 119896minus1partial correlation coefficients in (32) requires the implemen-tation of recursive algorithms whose analysis and efficientimplementation are a subject of future research It can beobserved that the product that scales the squared channelnorm of user 119894 in (21) contains all the information of thecorrelation coefficients of elements ofS which resembles theproduct (32) However (21) considers redundant informationof how all elements in H(S) interact with h which resultsin a suboptimal evaluation of (32) Notice that as 119870 growsthe probability that basis of Sp(H(S)) can describe a newcandidate userrsquos channel h decreases Therefore the gapbetween the correlation and the partial correlation factorsreduces as well This characteristic is used in [6] to prove thatfor 119870 rarr infin the performance of an SVD-based schedulingalgorithm that generates a quasiorthogonal set of users byapproximating (31) achieves asymptotical optimal user selec-tion performance

The optimum metric for user selection varies accordingto the precoding scheme that is implemented For the caseof ZFDP the fact that (21) considers redundant informationwhen all terms 984858 are multiplied can be compensated by theelimination of the noncausally known interference In thecase of ZFBF the orthogonality among selected channels playsa more important role in terms of throughput maximizationIn order to compensate the lack of knowledge of the partialcorrelation coefficients in (32) we consider larger values ofthe power loss factors that is the procedure for user selectionis the one described inAlgorithm 1with the difference that forthe ZFBF scheme we use 984858

119894119895instead of 984858

119894119895 Due to the fact that

984858119894119895

le 984858119894119895(with equality when the channels are uncorrelated)

the projection power loss factor increases its value and in thisway the poor orthogonality between channels has a higherimpact when the squared channel norms are scaled in (21)

6 Numerical Results

We compare the proposed user selection algorithm with sev-eral state-of-the-art algorithms namely the semiorthogonaluser selection (SUS) proposed in [5] with threshold parame-ter120572th and the null space projection based approach (NSP) [89] The upper bound of the sum rate is given by the expectedvalue of the solution of (11) found by an exhaustive searchIn order to highlight the contribution of multiuser diversitywe compare performance with respect to two simplistic userselection approaches one based on the maximum channelgain (MCG) criterion (selecting the 119873

119905users with higher

channels norms) and a second approach performing roundrobin user scheduling (RRS) policy We also compare theperformance of the proposed Algorithm 1 (IPP) with twogreedy class-A algorithms one proposed by Dimic andSidiropoulos [10] and the other proposed by Karachontzitisand Toumpakaris [11] The solution of the integer linearprogram (ILP) optimization in (30) is presented and used asan upper bound of the performance of Algorithm 1 (IPP) andcompared to the optimum solution of (11) The simulationsconsider perfect CSIT fading channels are generated follow-ing a complex Gaussian distribution with unit variance andthe average sum rate is given in [bpsHz] Since we evaluatesystem performance via Shannon capacity by means of (5)and (9) the results are independent of the specific imple-mentation on the coding and modulation schemes whichprovides us with a general design insight

61 Throughput (119877) versus Number of Active Users (119870) InFigures 2 and 3 we compare the throughput performance ofdifferent user selection strategies and Algorithm 1 regardingthe number of competing users119870 The performance of ZFBFis highly susceptible to the characteristics of the set of selectedusersS IPP algorithm performs the user selection exploitingthe information of the terms 984858 Since 984858

119894119895le 984858119894119895 the conse-

quence is a more drastic reduction in the power projectionin (21) due to the value of the correlation coefficient 120578

119894119895

Figure 2 shows that IPP achieves a considerable portion ofthe average sum rate of the optimum selection in the casewhen 119870 = 5 the performance gap regarding the optimumuser selection is about 11 For 119870 = 10 IPP achieves 90of the optimum users selectionrsquos sum rate and outperformsSUS (120572th = 1) It is worth mentioning that the parameter 120572thhas the function of dropping users whose correlation factor isbelow its value as described in Subsection 51 In this case weselect 120572th = 1 in order to guarantee that the set constraint in(11) is not violatedTheobjective of IPP algorithm is to achievethe performance of the greedy user selection based on thenull space projection (NSP) The performance of the IPPalgorithm has an asymptotic behavior regarding the NSPapproach as 119870 grows For 119870 = 20 IPP achieves roughly 97of the sum rate of the NSP based algorithms [8 9]

A comparison of the IPP algorithm to the ILP optimiza-tion shows that the latter exploits more efficiently the userdiversity as 119870 grows It is interesting that for 119870 ge 20 theILP optimization achieves better performance than the NSPapproach in Figure 2 This result suggests that there exists acritical value of 119870 for which the user selection of the ILP

The Scientific World Journal 9

Optimal solutionProposed IPPILP optimization

NSP [8 9]MCGRRS

14

16

18

20

22

24

26

Aver

age s

um ra

te (b

psH

z)

21215

22225

23235

24

101 1021012 1013

Number of users (K)

SUS [5] (120572th = 1)

Figure 2 Average sum rate as a function of the number of users 119870for the ZFBF scheme with SNR = 18 [dB] and119873

119905= 4

optimization overcomes the selection performed using themetric defined in (13) For 119870 = 20 the performance gapbetween the optimum user selection and the ILP optimiza-tion is less than 5 This means that for given deployment119873119905 there exists a finite value 119870

0for which forall119870 gt 119870

0the

sum rate gap between the exhaustive search and the model(30) is negligible However the complexity of computingthe solution of (30) grows exponentially with 119870 which isimpractical (infeasible) for online implementations but it isstill an appealing approximation to (11) compared to the largesearch space size of the optimumsolution formoderate valuesof119870

The performance of the IPP is determined by the pre-coding scheme that is used For ZFDP in Figure 3 it can beobserved that IPP performs as well as SUS but there is still aperformance gap compared to theNSP approach For119870 = 20IPP achieves the same performance of the greedy selectionof [11] and 98 and 99 of the sum rate of the optimumselection and the NSP approach respectively For ZFDP and119870 ge 8 the ILP optimization achieves better performancethan IPP but is not effective enough to reach the performanceof the NSP approach for low values of 119870 Nevertheless for119870 = 20 the ILP optimization achieves 98 of the sum rateof the optimum selection IPP shows an asymptotic perfor-mance as119870 rarr infin with respect to the NSP approach and theoptimum selection for both precoding schemes

62 Throughput (119877) versus SNR (119875) For zero-forcing-basedbeamforming we know from [4] that for a given SNR (119875) themaximum throughput R under the constraint |S| le 119873

119905in

(11) might be achieved by a set of selected users of cardinalitystrictly less than rank(H(S)) Nevertheless from the proper-ties of water-filling power allocation in (5) there exists a finitevalue 119875

0(which depends on H(S)) for which forall119875 ge 119875

0 R is

19

20

21

22

23

24

25

26

27

28

Aver

age s

um ra

te (b

psH

z)

Greedy [10]Greedy [11]

24

244242

246248

25

Optimal solutionProposed IPPILP optimization

NSP [8 9]

MCGRRS

101 102

1012 1013

Number of users (K)

SUS [5] (120572th = 1)

Figure 3 Average sum rate as a function of the number of users 119870for the ZFDP scheme with SNR = 18 [dB] and119873

119905= 4

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

SNR (dB)

Aver

age s

um ra

te (b

psH

z)

9 10 11 12 131011121314

Greedy [10]Greedy [11]

Optimal solutionProposed IPPILP optimization

NSP [8 9]

MCGRRSSUS [5] (120572th = 1)

P0

Figure 4 Average sum rate as a function of the SNR for ZFBFscheme with 119870 = 10 and119873

119905= 4

achieved by a subset of cardinality 119873119905 Notice that since the

greedy class-A algorithms in [10 11] obey the constraint |S| le

119873119905 the sum rate that they achieve for 119875 lt 119875

0is higher than

the capacity of the optimal solution in (11) but the numberof scheduled users is less This phenomenon can be observedin Figure 4 where for a given number of user 119870 = 10 thevalue of 119875

0asymp 10 [dB] and the optimum solution of (11) are

always better than the solution of the algorithms in [10 11] Itis worthy to point out that the optimum user selection here

10 The Scientific World Journal

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

SNR (dB)

Aver

age s

um ra

te (b

psH

z)

18 185 19 195 2023

23524

24525

25526

Greedy [10]Greedy [11]

Optimal solutionProposed IPPILP optimization

NSP [8 9]

MCGRRSSUS [5] (120572th = 1)

Figure 5 Average sum rate as a function of the SNR for ZFDPscheme with 119870 = 10 and119873

119905= 4

presented is found in a search space of size ( 119870119873119905) in a class-

B algorithm whilst the search space in class-A algorithms[10 11] has a size ofsum119873119905

119899=1( 119870119899) which has no constraints on the

minimum number of selected users Therefore the optimumsolution shown in our results is valid only for class-B algo-rithms and presenting class-A algorithms have as objectiveto highlight the difference between classes Considering thehigh SNR regime (10 le 119875 le 20) in Figure 4 the performancegap between IPP and the optimum solution ranges from 14to 9 and for the NSP approach the performance gap goesfrom 9 to 4 in the same SNR range For the case ofZFBF the ILP optimization achieves a better approximationto NSP than the IPP approach However in the case ofZFDP in Figure 5 the performance gap between IPP andthe ILP optimization is about 1 and both approachesachieve roughly 98 of the optimum selection capacity forSNR of 20 dB An interesting fact is that the MCG selectionachieves 93 the optimum selection capacity for 119870 = 10

and 119875 = 20 dB under ZFDP This indicates that for thehigh SNR regime channel gains play a more important rolefor the user selection process in scenarios where nonlinearprecoding can be implemented This can result in the designof novel low-complexity user selection algorithms for specificnonlinear precoding schemes Still the performance of aclass-B algorithm depends on the multiuser diversity and theSNR regime

63 Cardinality of S and Ω The cardinality of the set S isconditioned by the class of the algorithm that is implementedits parameters and the type of precoding that is used InFigure 6 we analyze in percentage the average value of theratio |S|119873

119905for (a) ZFBF and (b) ZFDP Such ratio indicates

if full spatial multiplexing is achieved In the case of ZFBF we

can see that both class-A algorithms [10 11] require119870 ge 20 inorder to achieve the maximum cardinality ofS To exemplifythe inconvenience of designing an algorithm dependent ofnondynamic parameters notice that setting a wrong valueto the parameter 120572th of the SUS algorithm might lead to adegradation of both the cardinality of the set of selected usersand the sum rate For the case of ZFDP we can see that therobustness of the precoder allows us to schedule 119873

119905user in

both classes of algorithms This has a direct impact in theachieved fairness owing to the large cardinality ofS The ratedistribution among the users is improved since more usersachieve a portion of the sum rate regardless of the fact thatthroughput maximization is the main objective of (11)

With the reduction of the set Ω each iteration becomesrelevant for high values of 119870 and 119873

119905 The effects of (25) on

the cardinality of the set of unselected users Ω per iterationsare presented in Figure 7 for (a) 119873

119905= 3 and (b) 119873

119905= 4 The

figures show the average number of users kept in the setΩ ofeach iteration of Algorithm 1 for different number of usersThe first iteration always considers all 119870 users to find theinitial selected user As the size of S increases the numberof required users to achieve |S| = 119873

119905reduces and (24) takes

into account such decrement to give more or less priority to119872119886(119894lowast)

64 Complexity Analysis and Implementation LimitationsThe complexity of solving (11) can be analyzed in two partsThe first one is the complexity required to implement eachone of the precoders and the second one is the complexity ofIPP For the case of ZFBF the precoding requires an 119873

119905times 119873119905

matrix inversion W = Hdagger and for ZFDP the evaluation ofthe beamforming weights requires a QR-type decompositionFor both coding schemes this process is carried out after IPPfinished the user selection processThemost costly operationin IPP is the evaluation of (1198702minus119870)2 inner products to definethe correlation coefficients that can be done in time O(119870)Since this values does not change along the selection processthey must be computed once and can be stored in memoryNotice that the evaluation of (16) requires a time O(119870) sinceonlymultiplications of real positive numbers are required anda sort operation (ordering) performed in time O(119870 log

2(119870))

For the case where the set Ω reduces in one element periteration and a total of 119873

119905iterations are required the total

complexity is O(119870119873119905+ 119873119905119870 log2(119870)) asymp O(119870119873

119905) However

for the following iterations the time complexity of computing(22) is a function of the set of unselected user that ismodified according the statistics of the projection powergiven by119872

119886(119894lowast)and theweight119908

(119873119905SΩ)This implies that each

iteration will require a time O(|Ω|(1 + log2(|Ω|))) asymp O(|Ω|)

andΩ changes for each iteration according to (25)The solution of (30) requires the optimization over 119871 ILP =

(12)119870(119870+3) binary variables in the objective functionThismeans that a total of 2119871ILP configurations of those variables areavailable and the number of valid configurations depends onthe constraints imposed over the binary variables Regardlessof the existence of pseudopolynomial algorithms that solveinteger programs avoiding the evaluation of all configura-tions [24] real time computation of the solution of (30) is

The Scientific World Journal 11

65

70

75

80

85

90

95

100

101 102

Number of users (K)

SUS 120572th = 05

SUS 120572th = 03

E||Nt

()

119982

(a)

65

70

75

80

85

90

95

100

101 102

Number of users (K)

SUS 120572th = 05

SUS 120572th = 03

Greedy [10]Greedy [11]

Optimal solutionProposed IPPILP optimization

NSP [8 9]

MCGRRSSUS [5]

E||Nt

()

119982

(b)

Figure 6 The metric E|S|119873119905 measures the degree of spatial multiplexing that is exploited for each scheduling algorithm considering

SNR = 18 [dB] (a) ZFBF and (b) ZFDP

10 20 40 60 80 1000

5

10

15

20

25

30

35

40

3rd iteration

2nd iteration

Nt = 3

Number of users (K)

E|Ω|

per i

tera

tion

(a)

10 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

50

3rd iteration

4th iteration

2nd iteration

Nt = 4

Number of users (K)

E|Ω|

per i

tera

tion

(b)

Figure 7 Average cardinality of the set of unselected users (E|Ω|) for each iteration of the IPP algorithm with SNR = 18 [dB] (a) 119873119905= 3

and (b)119873119905= 4

12 The Scientific World Journal

Table 1 Complexity comparison of user selection algorithms

Class A Class B[10] [11] SUS [5] NSP [9] IPPO (1198701198733

119905) O (1198701198732

119905) O (1198701198733

119905) O (1198701198733

119905) O (119870119873

119905)

prohibited for large values of119870 Table 1 summarizes the timecomplexity of different user selection algorithms

The proposed algorithms assume perfect CSIT Howeverin practical systems it is difficult to guarantee this conditionEven if channel estimation is very accurate there is anerror in the channels at the transmitter due to mobilityand feedback delays Several works (eg [5 8 27]) showedthat outdated CSIT destroys the quasiorthogonality of theselected channels which degrades the performance of zero-forcing-based transmission schemes Orthogonality can befully exploitedwhen there is near to perfect CSITThe authorsin [27] showed that a significant fraction of the sum ratewith perfect CSIT can be achieved if the ratio between theoutdated channel at the transmitter and the estimation erroris kept above a threshold Therefore as the frame lengths aredesigned so thatmagnitude of the real channels and the errorsdue to outdated estimates maintain a given average ratio theproposed user selection techniques are effective

7 Conclusions

In this paper we presented a low-complexity algorithm thatfinds a quasiorthogonal set of users that maximizes thesystem throughput forMIMOBC channels using linear ZFBFand nonlinear ZFDP beamforming schemes We exploiteda fundamental relation between the projection power lossfactors related to the correlation coefficients and the orthog-onal complement projector matrix related to the null spaceof the selected channels Our algorithm approximates theprojected power using a metric that is based exclusively onthe physical characteristics of the channels whose accuracyincreases with the number of competing users However thedependence of the multiuser diversity is not critical and fora moderate number of users the algorithm achieves a goodtrade-off between performance and complexity We com-pared the proposed algorithm to different state-of-the-artalgorithms and numerical results show a small performancegap between the optimum user selection and the proposedalgorithm We also presented an integer program model thatapproximates the performance of the exhaustive search whenthe number of users is large and it provides an upper boundof the performance of the proposed algorithm The resultsobtained by numerical simulation indicate that an efficientand low-complexity cross-layer scheduling design can profitfrom fundamental information that characterizes the relationbetween wireless channels without implementing extensivematrix operations for the user selection process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work presented in this paper was supported bythe Projects CROWN (PTDCEEA-TEL11582809) ADIN(PTDCEEI-TEL29902012) COPWIN (PTDCEEI-TEL14172012) and the FCTGrant (SFRHBD701302010) for thefirst author

References

[1] E Telatar ldquoCapacity ofmulti-antennaGaussian channelsrdquoEuro-pean Transactions on Telecommunications vol 10 no 6 pp585ndash595 1999

[2] S Vishwanath N Jindal and A Goldsmith ldquoDuality achiev-able rates and sum-rate capacity of Gaussian MIMO broadcastchannelsrdquo IEEE Transactions on InformationTheory vol 49 no10 pp 2658ndash2668 2003

[3] MHMCosta ldquoWriting on dirty paper (corresp)rdquo IEEETrans-actions on InformationTheory vol 29 no 33 pp 439ndash441 1983

[4] G Caire and S Shamai ldquoOn the achievable throughput of amultiantenna Gaussian broadcast channelrdquo IEEE Transactionson Information Theory vol 49 no 7 pp 1691ndash1706 2003

[5] T Yoo and A Goldsmith ldquoOn the optimality of multiantennabroadcast scheduling using zero-forcing beamformingrdquo IEEEJournal on Selected Areas in Communications vol 24 no 3 pp528ndash541 2006

[6] A Bayesteh and A K Khandani ldquoOn the user selection forMIMO broadcast channelsrdquo IEEE Transactions on InformationTheory vol 54 no 3 pp 1086ndash1107 2008

[7] A Wiesel Y C Eldar and S Shamai ldquoZero-forcing precodingand generalized inversesrdquo IEEE Transactions on Signal Process-ing vol 56 no 9 pp 4409ndash4418 2008

[8] L C Wang and C J Yeh ldquoScheduling for multiuser MIMObroadcast systems transmit or receive beamformingrdquo IEEETransactions on Wireless Communications vol 9 no 9 pp2779ndash2791 2010

[9] Z Tu and R S Blum ldquoMultiuser diversity for a dirty paperapproachrdquo IEEE Communications Letters vol 7 no 8 pp 370ndash372 2003

[10] G Dimic and N D Sidiropoulos ldquoOn downlink beamformingwith greedy user selection performance analysis and a simplenew algorithmrdquo IEEE Transactions on Signal Processing vol 53no 10 pp 3857ndash3868 2005

[11] S Karachontzitis and D Toumpakaris ldquoEfficient and low-complexity user selection for themultiuserMISO downlinkrdquo inProceedings of the IEEE 20th Personal Indoor and Mobile RadioCommunications Symposium (PIMRC rsquo09) pp 3094ndash3098 Sep-tember 2009

[12] T Ji C Zhou S Zhou and Y Yao ldquoLow complex user selectionstrategies formulti-userMIMOdownlink scenariordquo in Proceed-ings of the IEEEWireless Communications and Networking Con-ference (WCNC rsquo07) pp 1532ndash1537 March 2007

[13] V K N Lau ldquoOptimal downlink space-time scheduling designwith convex utility functionsmdashmultiple-antenna systems withorthogonal spatial multiplexingrdquo IEEE Transactions on Vehicu-lar Technology vol 54 no 4 pp 1322ndash1333 2005

[14] Z Shen R Chen J G Andrews R W Heath Jr and B LEvans ldquoLow complexity user selection algorithms for multiuserMIMO systems with block diagonalizationrdquo IEEE Transactionson Signal Processing vol 54 no 9 pp 3658ndash3663 2006

The Scientific World Journal 13

[15] R C Elliott and W A Krzymien ldquoDownlink scheduling viagenetic algorithms for multiuser single-carrier andmulticarrierMIMO systems with dirty paper codingrdquo IEEE Transactions onVehicular Technology vol 58 no 7 pp 3247ndash3262 2009

[16] T Yoo and A Goldsmith ldquoSum-rate optimal multi-antennadownlink beamforming strategy based on clique searchrdquo inProceedings of the IEEE Global Telecommunications Conference(GLOBECOM rsquo05) vol 3 pp 1510ndash1514 December 2005

[17] H Yanai K Takeuchi and Y Takane Projection MatricesGeneralized InverseMatrices and SingularValueDecompositionSpringer 2011

[18] J Gentle Matrix Algebra Theory Computations and Applica-tions in Statistics Springer 2007

[19] E Driouch and W Ajib ldquoA graph theory based schedulingalgorithm forMIMO-CDMA systems using zero forcing beam-formingrdquo in Proceedings of the 13th IEEE Symposium on Com-puters and Communications (ISCC rsquo08) pp 674ndash679 July 2008

[20] S Sigdel and W A Krzymien ldquoEfficient user selection andordering algorithms for successive zero-forcing precoding formultiuser MIMO downlinkrdquo in Proceedings of the IEEE 69thVehicular Technology Conference (VTC rsquo09) pp 1ndash6 April 2009

[21] H C Yang and M S Alouini Order Statistics in Wireless Com-munications Diversity Adaptation and Scheduling in MIMOand OFDM Systems Cambridge University Press 2011

[22] P Lu and H C Yang ldquoSum-rate analysis of multiuser MIMOsystem with zero-forcing transmit beamformingrdquo IEEE Trans-actions on Communications vol 57 no 9 pp 2585ndash2589 2009

[23] T Cormen C Leiserson R Rivest and C Stein Introduction toAlgorithms MIT Press 2001

[24] S Rao Engineering Optimization Theory and Practice JohnWiley amp Sons 2009

[25] CVX Research ldquoCVX Matlab software for disciplined convexprogrammingrdquo version 20 2012 httpcvxrcomcvx

[26] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the IEEE International Sympo-sium on Computer Aided Control System Design pp 284ndash289September 2004

[27] T FMacIel andA Klein ldquoOn the performance complexity andfairness of suboptimal resource allocation formultiuserMIMO-OFDMA systemsrdquo IEEE Transactions on Vehicular Technologyvol 59 no 1 pp 406ndash419 2010

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal