ACCEPTED FOR PUBLICATION AT THE EURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORKING 1

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1 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 1 A Approach to Optimum Joit Beamformig Desig i a MIMO-OFDM Multiuser System Atoio Pascual-Iserte, Aa I. Pérez-eira, ad Miguel Ágel Laguas Departmet of Sigal Theory ad Commuicatios Telecommuicatios Techological Ceter of Cataloia Techical Uiversity of Cataloia (UPC) (CTTC) - Edifici exus I C/ Jordi Giroa, 1-3 (campus ord UPC - edifici D5), Barceloa (Spai) toip@gps.tsc.upc.es, auska@gps.tsc.upc.es, m.a.laguas@cttc.es Abstract This paper describes a multiuser sceario with several termials accedig simultaeously to the same frequecy chael. The objective is to desig a optimal multiuser system that may be used as a comparative framework whe evaluatig other suboptimal solutios, ad to cotribute to the already published works o this topic. The preset work assumes that a cetralized maager kows perfectly all the chael resposes betwee all the termials. Accordig to this, the trasmitters ad receivers, usig atea arrays ad leadig to the so-called Multiple-Iput-Multiple-Output (MIMO) chaels, are desiged i a joit beamformig approach, attemptig to miimize the total trasmit power subject to Quality of Service (QoS) costraits. Sice this optimizatio problem is ot covex, the use of the Simulated Aealig (SA) techique is proposed to fid the optimum solutio. Idex Terms Multiuser systems, Simulated Aealig, atea arrays, MIMO systems, orthogoal frequecy divisio multiplexig, joit beamformig. I. ITRODUCTIO OE of the most importat problems of the curret ad commercial wireless commuicatio systems is that the umber of users ad the Quality of Service (QoS) are very limited, icludig the bit-rate ad the Bit Error Rate (BER). These limitatios are extremely importat sice the demads for wireless services are icreasig at a very high speed. I this sceario, diversity is a powerful method to icrease the umber of users ad improve the performace. Amog the differet solutios, the spatial diversity, based o the use of multiple ateas at the trasmitter ad/or the receiver, has received much attetio i last years. Thaks to such techiques, the performace ad capabilities of the commuicatio systems ca approach the theoretical limits of the wireless chael. As a illustrative example, we may cite the Space Divisio Multiple Access (SDMA), a advaced medium access protocol that permits the icrease of the umber of users that ca be served simultaeously. I these scearios, the sigals from differet users ca be separated usig array ad multichael processig techiques. Thus, spatial processig ca be adopted as a very powerful tool i the so-called multiuser systems. Although curretly there are several papers related to this topic, further work is ecessary o this research area to fully exploit the capabilities ad beefits provided by the use of multiple ateas i multiuser scearios. I this paper a multiuser wireless sceario is cosidered i which all the termials are assumed to have multiple ateas ad, as a cosequece, several parallel Multiple-Iput-Multiple-Output (MIMO) chaels arise. I the case of a cellular system, the termials correspod to both the Mobile Termials (MTs) ad Base Statios (BSs). It is also assumed that all of them accede simultaeously to the same frequecy radio chael. I this kid of MIMO systems, ad depedig o the quality ad quatity of the Chael State This work was partially supported by the Spaish Govermet uder projects TIC C02-01 (GIRAFA, joitly fiaced by FEDER) ad FIT (MEDEA+ A111 MARQUIS), ad by the Europea Commissio uder projects WIDES (cotract ) ad IST (EWCOM). This paper was preseted i part at the I Europea Sigal Processig Coferece (EUSIPCO) 2002.

2 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 2 Iformatio (CSI) at the trasmitters, several desigs ad architectures are possible. We are iterested i desigig a optimum SDMA strategy that ca be used as a comparative framework whe desigig ad evaluatig other suboptimal desigs for multiuser MIMO systems. Accordig to this objective, we cosider that there exists a cetralized maager with kowledge of the chael resposes betwee all the termials i the etwork. Obviously, this assumptio requires the chael to be slowly time varyig, so that the trasmitter ca have a accurate chael estimate by meas of a feedback chael, for example. Curretly, there are several stadards i which this assumptio is valid. Amog them, some examples ca be cited, such as the Europea Wireless Local Area etwork (WLA) iperla/2 [1] ad the IEEE a [2]. These WLAs use Orthogoal Frequecy Divisio Multiplexig (OFDM) [3] [4] modulatio for the physical layer ad, therefore, the use of OFDM by all the termials has bee cosidered i this paper. ere, a joit beamformig approach is proposed for the multiuser MIMO-OFDM system, that is, all the trasmitters ad receivers exploit a beamformig architecture per carrier, istead of usig a space-time ecoder [5] [6] (the details of the joit beamformig structure are give i Sectio II). Obviously, if aother architecture differet from joit beamformig is used, the a optimum desig will be foud differet from that proposed i this paper. Uder this cosideratio, the receiver is based o a bak of sigle-user detectors ad a joit desig of all the trasmit beamvectors is carried out by the cetralized maager, attemptig to miimize the total trasmit power. This is doe subject to several QoS costraits, which are formulated i terms of the maximum mea BER for each commuicatio or lik ad, possibly, the maximum trasmit power for some MTs. This optimizatio problem is very difficult to solve, as the costrait set over which the optimizatio has to be carried out is ot covex [7]. As a cosequece, i this paper the applicatio of the Simulated Aealig (SA) techique [8] is proposed, a very powerful heuristic optimizatio tool able to fid the global optimum desig eve whe the mathematical problem is ot covex. This is the mai differece of this work whe compared to other classical techiques foud i the literature, i additio to geeralizig the already proposed etwork topologies ad desig costraits. Most of the other works are based o Gradiet Search (GS) methods or o Alterate & Maximize (AM) approaches, which may fid a suboptimal desig sice they are ot able to hadle o-covex problems. The otatio used i this paper is quite geeral ad models may commuicatio systems, icludig, but ot limited to, both the uplik ad dowlik trasmissio i cellular etworks. There are some papers i the literature cosiderig similar joit beamformig problems to that preseted i this paper. T. M. Lok preseted i [9] a uplik multiuser Multicarrier - Code Divisio Multiple Access (MC-CDMA) system with oe atea at the trasmitter side ad several ateas at the receiver. The problem cosisted i the desig of the optimum receiver ad the trasmit frequecy sigatures for each user. Accordig to this, the obtaied otatio ad the mathematical optimizatio problem was show to be equivalet to the oe deduced i our paper. There, the QoS costraits were formulated i terms of a miimum Sigal-to-oise plus Iterferece Ratio (SIR) for each user istead of a maximum mea BER, as used i this paper, ad o costraits were applied regardig the maximum idividual trasmit powers. The optimizatio problem was solved by usig a Gradiet Search (GS) techique based o the Lagrage multiplier method ad the pealty fuctios. I [10], K. K. Wog also cosidered a multiuser MIMO OFDM system based o joit beamformig. There, the optimizatio of the trasmit beamvectors was based o the applicatio of the AM techique, i.e., whe desigig the beamvector for oe user, all the other trasmit beamvectors were assumed to be fixed. Oce the desig was fiished, the optimizatio of the beamvector for aother user was performed. This was applied successively util covergece was attaied, although the global optimum was ot guarateed to be foud or were the QoS costrais i terms of a miimum SIR guarateed to be fulfilled. J.-. Chag aalyzed i [11] the case of a uplik flat fadig multiuser MIMO chael, where both the MTs ad BS had multiple ateas. Two differet optimizatio problems were cosidered. I the first oe, the miimizatio of the total trasmit power was addressed, forcig the SIR for each user to be higher tha a prefixed value. I the secod problem, the objective was to maximize the miimum SIR subject to a total trasmit power costrait. I both cases, o idividual trasmit power costrait was applied. I that paper, several iterative algorithms were proposed to desig the beamvectors, although it was show that those techiques might fid a local suboptimum desig istead of the global optimum oe due to the o-covex behaviour of the optimizatio problem.

3 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 3 1 # 5 t(5)=2, r(5)=1 # 1 t(1)=1, r(1)=3 # 2 t(2)=1, r(2)=4 # 4 t(4)=3, r(4)=2 3 # 3 t(3)=3, r(3)=4 desired sigal r(1) = 4 # 4 r(3) = 5 # 5 r(2) = 4 # 1 ;# 2 desired sigal iterferig sigal (MAI) desired sigal # # 1 t(1) = 1 # 2 t(2) = 2 t(3) = 3 # 3 Fig. 1. (a) Geeral cofiguratio for a multiuser system with poit-to-poit cofiguratios. I this example there are 5 simultaeous commuicatios ad 4 termials. (b) Typical cofiguratio i a multiuser MIMO-OFDM sceario with 3 users or commuicatios. There are 5 termials, where 3 of them are MTs ad the other 2 oes are BSs. There are may other papers that aalyze differet multiuser systems cosiderig the use of multiple ateas. I [12] a uplik sceario with oe BS ad several MTs was studied, all of them with multiple ateas. There, the beamformig solutio was show to be optimum i the sese that it achieved the sum capacity for a high umber of users, although o QoS could be guarateed for each user. The same sceario was also cosidered i [13]. I that paper the objective was the miimizatio of the global Mea Square Error (MSE) subject to a trasmit power costrait for each MT. The iterative techique was based o the applicatio of the AM algorithm, which might coverge to local suboptimum solutios. A multiuser dowlik sceario with oe multi-atea BS ad several sigle-atea MTs was aalyzed i [14]. There, the global optimum desig miimizig the total trasmit power subject to miimum SIR costraits was preseted based o the duality betwee the uplik ad dowlik scearios ad, furthermore, the coditios for the existece of a feasible solutio subject to a total trasmit power costrait were deduced. The same problem was aalyzed i [15], where the sceario was afterwards exteded to the case of several multiatea BSs ad multi-atea MTs, as i our paper. The proposed AM iterative algorithm was show to coverge, but ot always to the global optimum solutio, oce agai due to the o-covexity of the problem. Fially, i [16] the same sceario with several multi-atea BSs ad MTs was cosidered. A iterative AM techique for the desig of the beamformers was preseted to miimize the total trasmit power subject to QoS costraits i terms of a miimum SIR for each user, although it was show that it might coverge to a local suboptimum solutio. I all these papers the chael was assumed to be frequecy flat, although i our work we have exteded the desig to the case of a multicarrier modulatio i a frequecy selective chael. This paper is structured as follows. I Sectio II the system ad sigal models for the MIMO-OFDM multiuser sceario are preseted, i additio to deducig the expressio of the optimal receive beamvectors as a fuctio of the trasmit beamvectors. The applicatio of the SA algorithm i order to joitly desig all the trasmit beamvectors is preseted i Sectio III, whereas i Sectio IV other classical suboptimum desigs based o GS ad AM algorithms are proposed. Fially, i Sectios V ad VI some simulatio results ad coclusios are show, respectively. II. SYSTEM AD SIGAL MODELS Cosider a wireless sceario i which several termials coexist i the same area. Amog these termials K commuicatios or liks are established ad access the commo chael at the same time ad i the same frequecy bad. As previously stated, the adopted modulatio techique is a -carriers OFDM. All the termials i the system are allowed to have multiple ateas ad each of them is able to trasmit ad/or receive. We cosider that each commuicatio or lik is assiged to two termials, where oe of them is the trasmitter ad the other oe is the receiver. A. MIMO-Multiuser System ad Sigal Models As it has bee stated previously, the system model for the K commuicatios is based o a joit beamformig approach at the trasmitter ad the receiver, where the beamvectors correspodig to differet

4 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 4 iput data S/P coverter pre-beamformig b 0-1 IFFT (OFDM modulator) + cyclic prefix + P/S IFFT (OFDM modulator) + cyclic prefix + P/S cyclic prefix removal + S/P + FFT (OFDM demodulator) cyclic prefix removal + S/P + FFT (OFDM demodulator) post-beamformig a 0 a -1 P/S coverter output data Fig. 2. Architecture of the trasmitter ad the receiver for the kth commuicatio or lik based o joit beamformig. commuicatios or liks are allowed to be differet. I this sceario there exists a set of termials, where we have ot differetiated betwee BSs ad MTs sice all the termials are allowed to trasmit ad/or receive simultaeously. All the termials i the system are umbered ad the quatity of termials may be differet from the umber of established liks (see Fig. 1). Let t represet the termial resposible for trasmittig the iformatio correspodig to the kth lik, whereas r is the termial receivig this iformatio. I Fig. 1 we show some examples of these kids of systems (a geeric example ad a more cocrete oe). Eq. (1) represets the sigal model for the received sapshot vector for the kth lik, i.e., it is the received sigal model at the rth termial ad the th carrier [17] (see Fig. 2 i which we represet the architecture of the trasmitter ad the receiver for the kth lik based o joit beamformig): y (r) (t) = K l=1 (t(l);r) b (l) s(l) (t) +(r) (t) (1) where we have assumed that the legth of the cyclic prefix is higher tha or equal to the chael order [3]. The size of the vectors y (r) (t) ad b (l) is equal to the umber of ateas at the rth ad the t(l)th termials, respectively. The trasmit beamvector applied to s (l) (t) is represeted by b (l), where s (l) (t) is the trasmitted data at the th carrier durig the tth OFDM symbol for the lth lik. The trasmitted symbols are assumed to have a ormalized eergy: Efjs (l) (t)j 2 g =1(Ef g stads for the mathematical expectatio). The matrix (t(l);r) represets the MIMO chael respose at the th carrier betwee the t(l)th ad the rth termials. We have also cosidered that (i;i) = 0; 8i, which meas that the ith termial is ot receivig the sigal trasmitted by itself. Fially, the vector (r) (t) models the cotributio of oise plus iterfereces from outside the system at ρthe rth receiver ad ff the th carrier. The associated covariace matrix is represeted by Φ (r) = E (r) (t) (r) (t), where ( ) stads for complex cojugate traspose. This sigal model is quite geeral ad ca easily fit i with may kow systems icludig, but ot limited to, the cellular eviromets, both for uplik ad dowlik. B. Sigle-User Receiver Optimizatio I this subsectio the attetio is focused o the desig of the receive beamvectors. For every lik ad carrier, a liear combier a is applied to the set of received samples collected i the sapshot vector y (r) (t). The hard estimate of the trasmitted symbol s (t) for the kth lik durig the tth OFDM symbol is, therefore, based o a hard mappig applied to the output of the receive beamvector, i.e., bs (t) = dec ρ a (r) y (t) ff. The optimum receive beamvector a is the oe maximizig the output SIR. The expressio of the optimum beamvector is widely kow ad correspods to the Wieer matched filter [4] [17], which ca be formulated as follows assumig that the trasmit beamvectors are kow: a = ff R R = Φ (r) + 1 (t;r) b K l=1;l6=k (2) (t(l);r) b (l) b(l) (t(l);r) (3)

5 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 5 where R is the total iterferece plus oise covariace matrix see at the receiver for the kth lik, ad ff is a scalar factor that does ot affect the SIR ad ca be calculated to have a equalized equivalet chael: 1 a (t;r) b =1; ff = b (t;r) 1 (t;r) R b. As it ca be see i (2), the optimum receive beamvector for the kth lik depeds o both the trasmit beamvector for the same lik b ad all the other oes fb (l) g l=1;:::;k l6=k, sice the covariace matrix R depeds o the trasmit beamvectors for all the other liks differet from k. This produces a couplig effect that makes difficult the optimizatio of the trasmit beamvectors. I the followig sectio we explicitly focus the attetio o the joit desig of all the trasmitters. By usig this desig criterio for the receivers, the SIR at the output of the receive beamformer for the kth lik ad the th carrier ca be show to be as follows [17]: SIR = b (t;r) R 1 (t;r) b (4) Takig ito accout this result, i OFDM the effective or mea BER is defied as the ucoded BER P averaged over all the subcarriers: BER = 1 1 k m SIR, where we have assumed that all =0 Q q R the iterfereces are approximately Gaussia distributed, Q(x) = p 1 1 =2 2ß dt, ad k x e t2 m is a parameter depedig o the modulatio applied to each subcarrier (for BPSK, k m =2). III. SIMULATED AEALIG BASED TRASMITTER OPTIMIZATIO The last sectio was devoted to the optimum desig of the receive beamvectors assumig that the trasmit beamvectors were kow, obtaiig the closed form solutio correspodig to the Wieer matched filter [4]. ow, the attetio is focused o the joit desig of all the trasmit beamvectors for all the users ad all the OFDM carriers. Whe desigig the trasmit beamvectors, a objective fuctio or optimizatio criterio has to be idetified, as well as a set of desig costraits. Obviously, a desirable objective is the miimizatio of the total trasmit power, sice i wireless etworks, high trasmit powers imply a shorter lifetime of the MTs. I the case of usig several ateas, the power used for trasmittig the iformatio symbol correspodig to the th carrier of the kth user is proportioal to kb k 2. Takig this ito accout, the total trasmit power P T ca be expressed as: P T fb g k=1;:::;k =0;:::; 1 = K 1 k=1 =0 kb k 2 = K 1 k=1 =0 b b (5) Besides the objective fuctio, additioal costraits are ecessary i order to avoid the trivial solutio miimizig the trasmit power: b = 0. I this paper two kids of costraits are proposed. The first oes refer to the miimum QoS for each commuicatio or lik ad are madatory, whereas the other oes are related to the maximum idividual trasmit powers for a cocrete set of termials. This set of termials ca be empty ad, therefore, the idividual trasmit power costraits are optioal. ffl QoS costraits: these costraits are formulated i terms of the maximum mea BER for each lik ad ca be expressed as follows: BER» fl ; k =1;:::;K (6) where fl is the maximum permitted BER for the kth lik ad, therefore, is a iput parameter of the optimizatio problem. This formulatio geeralizes the results preseted i [9] for a MC-CDMA system, ad i [11] [14] [15] [16] for flat fadig chaels, where the QoS costraits were formulated i terms the SIR istead of the mea BER. I [10] [12] [13] the trasmit power was stated to be a prefixed value ad the goal was the optimizatio of the mea quality of all the users i terms of capacity, miimum MSE, etc. ad, therefore, o QoS could be guarateed for each lik. I all cases, the proposed algorithms for the most geeral sceario, comprisig several BSs ad MTs with multiple ateas, were show to be iefficiet i the sese that they might fid local suboptimum solutios istead of the global optimum oe because of the o-covex behaviour of the optimizatio problems.

6 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 6 ffl Idividual trasmit power costraits: i additio to the QoS costraits, optioal costraits ca also be icluded regardig the maximum idividual trasmit powers for some termials. This is specially useful for MTs with a power-limited battery i a uplik trasmissio. Let be the set of termials to which these costraits are applied. They ca be formulated as: P (i) T = K 1 k=1;t=i =0 kb k2» P (i) max ; i 2 (7) where P max (i) represets the maximum trasmit power for the ith termial. These kids of costraits have ot bee cosidered i ay of the works refereced i this paper. Curretly, there exists o closed form solutio for this extremely complicated costraied optimizatio P 1 =0 kb P K problem, sice it is ot covex [7]. Although i this case the objective fuctio k=1 k 2 is covex i the optimizatio variables b, the costrait set is ot. I order to prove this last statemet, let us cosider the simplest example correspodig to oly oe user usig a OFDM modulatio with oly oe carrier. For this simple case, the maximum BER costrait is equivalet to a miimum SIR costrait. Let us assume that = I ad that Φ = I (we obviate the sub- ad super-idexes to facilitate the otatio). Accordig to this, the QoS costrait ca be formulated as: b b SIR mi. This costrait ca be represeted geometrically as the exterior of a sphere i the variable vector b, which, obviously, is ot covex. Due to the o-covex behaviour of the problem, if a classical GS or AM method is applied to fid the optimal desig, a local miimum may be foud istead of the global optimum i the costrait set. Sice we are iterested i fidig the global optimum desig i order to provide a referece system to be used as a comparative framework for other suboptimal desigs, we have decided to exploit the Simulated Aealig (SA) algorithm. SA is a very powerful heuristic tool able to fid the global optimum desig eve whe the objective fuctio or the costrait set is ot covex. As stated i the itroductio, some previous works have proposed GS techiques, such as i [9], or AM methods [10] [11] [13] [15] [16], amog others. The mai problem of these techiques is that they are ot able to fid the global optimum desig due to the o-covex behaviour of the problem, as it was clearly show i [11] ad other works. Besides, i GS ad AM techiques, it may be extremely difficult to iclude ay kid of costrait, although i the case of SA this ca be doe easily, as will be show later i this sectio. Specifically, for the case of GS, the costraits are required to be differetiable, although this is ot ecessary i SA. I this paper the existece of a feasible solutio is assumed, that is, a collectio of trasmit beamvectors that satisfies all the costraits simultaeously. I case that a feasible solutio does ot exist, the the algorithm will ot coverge to ay acceptable desig. The SA algorithm has aalogies with the aealig of solids i physics ad thermodyamics, as has bee explaied i [8]. The mai objective of the aealig process i physics is to obtai a solid with a perfect particles arragemet, i.e., a perfect structure, so that the eergy of the liks betwee these particles is miimized. I order to obtai this perfect structure, iitially the solid has to be melted by heatig it, i.e., util all the particles have total freedom of movig. Oce this hot state is attaied, the temperature has to be lowered util the perfect state is obtaied, i which the particles have o movemet. If the coolig process is doe very quickly, the obtaied state may be ot the oe with the miimum eergy ad, therefore, is ot perfect. If the miimum eergy is desired, the the system has to be cooled very slowly, so that the particles have eough time to be placed i their optimal positios. I our problem, i each step of the iterative algorithm there is a collectio of trasmit beamvectors, which is called the curret solutio. Give the curret solutio, which is equivalet to a cocrete particles arragemet or a state i the aealig process i physics, a ew solutio or collectio of beamvectors is proposed. If it is better tha the origial oe, the it is retaied as the curret oe. O the cotrary, if it is worse, the the proposed solutio is accepted with a certai probability. That meas that worse solutios may be accepted. This mechaism, which is called hill-climbig, is extremely importat so as to avoid a suboptimal solutio or local miimum. The parameter that cotrols this acceptace probability is the temperature T, as i the case of the aealig i physics. The higher the temperature, the higher the acceptace probability. The temperature is lowered step by step, so that asymptotically, oly better solutios are accepted ad a miimum is approached. The meaig of better ad worse is based o the fb g k=1;:::;k =0;:::; 1

7 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 7 defiitio of a cost fuctio f ( ) that depeds o the trasmit beamvectors ad is directly related to the total trasmit power. This fuctio correspods to the eergy of a state i physics ad its miimizatio is the goal of the aealig process. As i the thermodyamics aealig process, if the temperature is lowered very slowly, the optimum state with the miimum eergy, that is, the global miimum of the total trasmit power, ca be achieved, as desired iitially. ere we provide the descriptio ad all the basic ideas of the SA algorithm proposed to solve the already stated optimizatio problem: ffl ffl ffl Cost fuctio defiitio: f fb g = P T fb g + ff T ψ K k=1 log BER fl! +2 + ff T iffl! ψlog P (i) + 2 T P max (i) where (x) + = max(x; 0). This cost fuctio, which also depeds o the temperature T, is equal to the total trasmit power plus a quadratic pealty term. This pealty term takes ito accout whether the BERs are greater tha the maximum permitted oes, ad whether the idividual trasmit powers are greater tha those specified. Besides, this pealty term is iversely proportioal to the temperature, sice i the simulatios it has bee show that this rule performs quite well i terms of covergece speed. As T is lowered, the pealty term is icreased ad, therefore, the acceptace of solutios that do ot fulfill the costraits is asymptotically avoided. The parameter ff is a proportioal factor for the pealty term ad its value has bee adjusted by simulatios to ff = 100 i order to have good covergece properties. The pealty term is based o relative comparisos of the BERs ad the trasmit powers with the maximum permitted values by meas of the log( ) fuctio. These kids of comparisos have bee chose, sice it has bee observed experimetally that they behave better tha absolute comparisos. ote, however, that other kids of pealty fuctios could have bee used. Proposed solutio geeratio: eb = b + w ; w οc(0;ffb 2 I); =0;:::; 1; k =1;:::;K (9) The proposed solutios are geerated by applyig idepedet complex circularly symmetric Gaussia oise with variace ffb 2 to the compoets of the trasmit beamvectors. This oise is used to geerate ay possible collectio of trasmit beamvectors i a cotiuous solutio space. ote that there is a differece whe compared to the problems for which SA was iitially applied, i which the solutio space was discrete [8]. The acceptace ratio is moitored for every value of T. I case that it is lower tha 0:1 for 5 times, the the variace of the Gaussia oise is lowered by meas of a expoetial profile (ffb 2 ψ 0:95ff2 b ). This is doe i this way as it has bee show experimetally that this rule improves the covergece speed of the algorithm. Probability of acceptace of the proposed solutio: ρ Prob = exp 1 + ff f fe b g f fb g (10) T (8) ffl This acceptace probability correspods to the Metropolis criterio, as described i [8], ad is related to the Maxwell-Boltzma approximatio of the Fermi-Dirac distributio describig the eergy of a electro i differet levels. This criterio was iitially used i thermodyamics i order to simulate a thermal equilibrium process. It was show that, usig this criterio, the system could arrive at the miimum possible eergy, i.e., to the optimum state, if the temperature was lowered slowly. This philosophy was afterwards adopted i the SA algorithm, as show i this paper, as a efficiet criterio to fid the global miimum of o-covex problems. System coolig : T ψ fit; fi ο = 0:99 (11) As described i this equatio, the temperature is lowered very slowly by meas of a decreasig expoetial rule, as described i [8]. This value of fi has bee chose sice i the simulatios it has bee show to provide good covergece properties, while still guarateeig that the global optimum solutio is attaied. As see i Eq. (10), the hotter the system, the higher the acceptace probability.

8 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 8 TABLE I IITIALIZATIO I TE SA ALGORITM Its objective is to fid a iitial value of the temperature T, so that the umber of accepted solutios is higher tha 95 %. 1) T =1, ff 2 b is set equal to the mea power ecessary at the trasmitters to attai the required QoS assumig o iterferece amog the users (experimetally it has bee show to have good covergece properties). The iitial trasmit beamvectors are set equal to all zero vectors. 2) Propose 100 solutios. Measure the umber of o-accepted solutios: a. 3) If a < 95, the T ψ 2T ad go to step 2. If a = 100, the T ψ 0:9T ad go to step 2. I ay other case, ed. TABLE II MAI ITERATIOS OF TE SA ALGORITM They correspod to the applicatio of the SA algorithm. 1) Lar =0: iitializatio of the couter correspodig to the umber of times that the acceptace ratio is lower tha 10 %. 2) Propose 100 solutios. Measure the umber of o-accepted solutios: a. Update the temperature: T ψ 0:99T. 3) If a < 10, the Lar ψ Lar +1.IfLar =5, the ff 2 b ψ 0:95ff 2 b ad go to step 1. I ay other case, go to step 2. The algorithm fiishes whe the value of the cost fuctio has stabilized ad a miimum has bee achieved. As a cosequece, whe the temperature is high, most of the proposed trasmit beamvectors are accepted, which meas they are searchig over the rage of all the possible spatial directios. Whe the temperature is lowered, this rage is reduced ad the accepted trasmit beamvectors begi to look for the best spatial directios, i.e., for the spatial directios that couple the maximum power towards the desired termial while reducig the iterferece towards the other oes. I the SA algorithm, iitially the temperature T has to be high eough so that most of the proposed solutios are accepted. The iitial trasmit beamvectors are set equal to all zero vectors. ote, however, that the iitializatio of the beamvectors is ot importat sice i the first iteratios most of the proposed solutios are accepted ad the variace of the oise to geerate ad propose ew solutios is very high. I this paper, 100 iteratios are ru for every value of T. As a summary, the mai steps of the algorithm are preseted ad briefly detailed i Tables I ad II. IV. OTER CLASSICAL SUBOPTIMUM TECIQUES I the last sectio the SA techique has bee proposed to fid the global optimum desig of the stated costraied optimizatio problem. I this sectio we preset two alterative algorithms based o the Gradiet Search (GS) ad the Alterate & Maximize (AM) methods. A. Lagrage-Gradiet Search (GS) Trasmitter Optimizatio A classical approach differet from the SA cosists i the utilizatio of a gradiet techique, although, as it has bee already said, the mai drawback of this family of algorithms is that they may coverge to local suboptimum desigs. I order to compare the SA with other classical approaches, i this sectio we propose a iterative gradiet techique based o the classical Lagrage multipliers method ad the quadratic pealty term [9] [18]. This techique is based o the defiitio of a Lagragia expressio L. Whe formulatig the Lagragia expressio ad the pealty term, we take ito accout the fact that the optimal solutio implies that the QoS are fulfilled with equality. Uder this assumptio, that ca be show easily, the Lagragia expressio ca be formulated as: L = P T + 2 ψ 4 K j=1 log BER(j) fl (j)! 2 + i2! 3 ψlog P (i) + 2 T 5 (12) P max (i) The equatios that show how to update the trasmit beamvectors ad the pealty factor correspod to the well kow gradiet descet ad ascet techiques, as also used i [9]: b ψ b μr b 2 ψ L! (13) ψ + μ4 K 2! 3 log BER(j) fl (j) + ψlog P (i) + 2 T 5 (14) P max (i) j=1 i2

9 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 9 where μ is the step-size parameter that has to be adjusted to cope with the tradeoff betwee the covergece speed ad the covergece itself. The iitial beamvectors ca be calculated assumig that there is o iterferece betwee users, as show i [10] ad [17]. The iitial value for the pealty factor is set equal to 0. ere, the ecessary expressios to calculate the gradiet r b L are provided. I order to facilitate the otatio, we assume a uplik sceario with several MTs trasmittig to a sigle BS, which is resposible for the detectio of the symbols trasmitted by all the MTs. The modulatio of the subcarriers is BPSK. I this sceario, the matrix represets the respose of the MIMO chael at the th carrier betwee the kth MT ad the BS. The extesio to other kids of scearios is quite simple by usig very similar expressios. The fuctio ffi is defied as ffi =1; k 2 ad ffi =0; k 62. ψ!! K r b L = b +2 1 BER (j) log BER(j) r fl (j) b BER(j) + ffi 2 b ψlog P + T (15) j=1 P max The expressio of r b BER(j) depeds o j. Firstly, we give the expressio for the case j = k: r b BER = 1 p 2ß exp SIR 1 q R 2SIR P T 1 b (16) For the case j 6= k the expressio is as follows, where the matrix iversio lemma has bee used : r b BER(j) = p 1 2ß exp SIR (j) 1 q r b SIR(j) (17) r b SIR(j) = 1+b b(j) R (j;k) = Φ + 1+b R (j;k) 1 R (j;k) b 1 2SIR (j) 2 b (j) (j;k) 1 R b (j;k) 1 R b K l=1;l6=j;l6=k (l) b(l) b(l) (l) fi fi fib (j) (j) R (j;k) 1 b R (j;k) 1 (j) b(j) fi fi fi 2 (18) (19) As previously stated, oe of the mai drawbacks of the GS techique is that a local suboptimum desig may be foud. This could be solved by usig differet iitial sets of beamvectors, selected radomly. ote, however, that this icreases the computatioal load ad does ot guaratee a successful result. (20) B. Alterate & Maximize (AM) Trasmitter Optimizatio Fially, aother classical solutio that has bee used previously by may authors i papers such as [10] [13] [15] [16], amog others, is the AM algorithm. I our problem, the SIR for a cocrete user ad carrier depeds, ot oly o the beamvector for the cosidered user, but also o all the trasmit beamvectors for all the other users through the covariace matrix, as show i Eq. (3) ad (4). The AM algorithm is a iterative techique, so that i each step the beamvectors associated to a cocrete user are desiged assumig that the beamvectors of all the other users are fixed, i.e., assumig that the oise plus iterfereces covariace matrix is kow. Obviously, whe a beamvector for a user is desiged, the covariace matrix for the other users chage ad, therefore, the techique has to be applied iteratively util covergece is attaied. I this subsectio we provide the descriptio of a AM algorithm i which we oly take ito accout the QoS costraits, but ot the idividual trasmit power costraits, sice their iclusio i the algorithm is extremely difficult. I each step, the optimum trasmit beamvector maximizig the SIR correspods to the eigevector associated to the maximum eigevalue of the matrix (t;r) 1 (t;r) R (see a complete proof of this i [10] ad [17]). Besides this, a adequate power allocatio amog the carriers of the OFDM modulatio has to be calculated, so that the QoS costrait i terms of the maximum BER is

10 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 10 TABLE III APPLICATIO OF TE AM ALGORITM 1) Iitializatio: set all the trasmit beamvectors proportioal to the maximum eigevectors of the matrices (t;r) (r) Φ 1 (t;r), i.e., without takig ito accout the iterfereces from other users. Calculate the power allocatio (either uiform or maxmi) to satisfy the QoS costraits. 2) Repeat util covergece: ffl Calculate all the covariace matrices (see Eq. (3)). ffl Calculate all the trasmit beamvectors as the maximum eigevectors of (t;r) R 1 (t;r) ad the correspodig power allocatio satisfyig the QoS costraits Simulated Aealig. MTs power. Sceario Simulated Aealig. MTs power. Sceario power (db) Simulated Aealig. Mea BER. Sceario 1 power (db) Simulated Aealig. Mea BER. Sceario 1 mea BER mea BER Fig. 3. Part (a): Performace of the Simulated Aealig i sceario 1. Part (b): Performace of the Simulated Aealig i sceario 1. Power costrait i MT 1. fulfilled. I this paper we have used two differet power allocatio policies: the uiform ad the maxmi techiques, as completely described i [17]. Table III shows the mai steps of the AM techique, icludig the beamvectors iitializatio. The mai disadvatage of this algorithm, as commeted previously ad i papers such as [13] [15] [16], is that the obtaied solutio may be a local suboptimum desig istead of the global optimum oe, sice the optimizatio problem is ot covex. Besides, there is o a-priori guaratee of covergece. A possible solutio would cosist i usig differet radom iitializatios for the trasmit beamvectors. ote, however, that this is a adhoc procedure that does ot cotrol ad guaratee that the global optimum desig is obtaied. V. SIMULATIO RESULTS I this sectio we simulate a uplik sceario with 3 MTs ad 1 BS. The OFDM modulatio cosists of =16carriers ad both the MTs ad BS have 5 ateas. The QoS costraits i terms of the mea BER are 10 3, 10 3 ad 10 2, ad ff =100, as stated i Sectio III. The oise is assumed to be white both i the time ad space domais, with a ormalized variace equal to 1, i.e., Φ (r) = I. The simulatios ad algorithms are applied to a sigle realizatio of the multiple OFDM-MIMO chaels, although we do ot provide the umerical expressios of the chael matrices for the sake of clarity. I the first sceario it is assumed that the path-loss is very similar for all the users. I Fig. 3(a) we show the evolutio of the powers allocated to the three users ad the mea BERs as the iteratios of the SA algorithm ru, cocludig that the proposed techique is able to fid a desig fulfillig the costraits whe o idividual power restrictios are applied. The optimum power correspodig to the first user is 8.45 W, ad the total power is 20.1 W. If a power costrait is applied to the first user equal to 8 W, the the results are those show i Fig. 3(b). The mai coclusio is that, i this case, the SA algorithm allocates 7.6 W to the first user, whereas the other oes icrease their correspodig power cosumptio. As it is also show, the global trasmit power has icreased up to 20.8 W. This icrease of the total trasmit power is ormal,

11 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG Simulated Aealig. MTs power. Sceario Gradiet based method. MTs power. Sceario 2 mea BER power (db) Simulated Aealig. Mea BER. Sceario power (db) mea BER Gradiet based method. Mea BER. Sceario , Fig. 4. Part (a): Performace of the Simulated Aealig i sceario 2. Part (b): Performace of the gradiet based algorithm i sceario 2. TABLE IV POWER AD BER FOR SIMULATED AEALIG (SA), GRADIET SEARC (GS), AD ALTERATE &MAIMIZE (AM) MT 1 power MT 2 power MT 3 power Total power MT 1 BER MT 2 BER MT 3 BER SA 10.2 W 8.7 W 54.5 W 73.4 W GS 9.6 W 8W 46.5 W 64.1 W 1: : : AM-maxmi 8.3 W 6.8 W 63.6 W 78.7 W AM-uiform 9.1 W 7.9 W 65.5 W 82.5 W as i the secod example, a more restrictive costrait has bee applied ad, therefore, the optimizatio has to be carried out over a more limited set of trasmit beamvectors fulfillig the costraits. I Fig. 4 a set of results are preseted for the case of a sceario i which the third user has a path loss with respect to the first two users equal to 12 db. I this example o idividual trasmit power costrait has bee cosidered. Fig. 4(a) correspods to the applicatio of SA, whereas Fig. 4(b) correspods to the GS algorithm with a μ parameter, i.e., the step size, equal to The mai coclusio is that with the same computatioal load or umber of floatig poit operatios, the SA algorithm ca fulfill the costraits, whereas the GS techique decreases importatly the covergece speed as the solutio approaches these costraits. This is because the pealty terms applied i the Lagragia expressio (12) are quadratic ad, therefore, whe calculatig the derivatives i a poit ear from the fulfillmet of the costraits, these derivatives ted to zero. Simulatios cocerig the applicatio of the AM algorithm have also bee doe for two differet power allocatio techiques: uiform ad maxmi [17]. From the simulatios, it is cocluded that AM has a high covergece speed. Table IV shows a summary of the results for all the techiques. The coclusio is that GS does ot fid a solutio fulfillig the costraits, whereas AM does ot have this problem, as i the case of SA. The mai drawback is that the ecessary trasmit power is higher for AM tha for SA, cocludig that a local suboptimum desig has bee foud. Ideed, ad as explaied i [15], the o-covexity ad the umber of local miima icreases as more BSs ad MTs are coexistig i the same area. VI. COCLUSIOS As a geeral coclusio, i this paper a MIMO-OFDM multiuser system based o a joit beamformig approach has bee proposed. The objective was the joit desig of the beamvectors associated to all the established commuicatios or liks, takig as the optimizatio criterio the miimizatio of the total trasmit power subject to maximum mea BER ad idividual trasmit power costraits. It has bee show that this problem is ot covex ad, therefore, the applicatio of the SA techique has bee proposed, i additio to classical GS ad AM methods. The SA has bee show to be able to fid the optimum solutio ad,

12 ACCEPTED FOR PUBLICATIO AT TE EURASIP JOURAL O WIRELESS COMMUICATIOS AD ETWORKIG 12 therefore, the obtaied desig may be used as a comparative framework for other suboptimum solutios. Other classical techiques such as GS ad AM, also preseted i this paper, may have problems related to the covergece speed ad the fact that local suboptimum desigs may be foud. Besides, GS ad AM caot always iclude every kid of costrait, whereas i SA this ca be easily doe by usig adequate pealty fuctios. Although SA has bee show to be a powerful tool to cope with the optimizatio of o-covex problems, such as the oe preseted i this paper, there exist other heuristic approaches that should be also cosidered as possible strategies. Amog these techiques, some examples ca be give such as the Geetic Algorithms (GA) [19] or Taboo Search (TS) approaches. I both cases, the techiques are based o a radom geeratio of possible solutios, such as i the SA algorithm, ad are also able to fid the optimum solutio, eve if the problem is ot covex. The mai differece betwee SA ad the GA-TS strategies is that the last two techiques trasform the solutio space, i.e., the set of possible trasmit beamvectors, ito a space composed of bits by meas of a ecodig process. Oce this trasformatio has bee performed, the optimizatio problem is solved i this ew trasformed solutio space. Fially, the solutio i terms of trasmit beamvectors should be foud by trasformig or decodig the solutio i the coded space. Further work is to be doe o the applicatio of these techiques i order to evaluate whether the computatioal load of the optimizatio problem ca be decreased while still guarateeig that the global optimum desig is foud. VII. ACKOWLEDGEMET The authors would like to thak the aoymous reviewers that have cotributed to improve this paper with their helpful commets. REFERECES [1] ETSI TS v1.1.1: Broadbad Radio Access etworks (BRA); IPERLA Type 2; Physical (PY) layer, ETSI, April [2] Part 11: Wireless LA Medium Access Cotrol (MAC) ad Physical Layer (PY), IEEE Std a, IEEE, December [3] Z. Wag ad G. B. Giaakis, Wireless Multicarrier Commuicatios, IEEE Sigal Processig Magazie, vol. 17, o. 3, pp , May [4] J. G. Proakis, Digital Commuicatios, 3rd ed. McGraw-ill, [5] V. Tarokh,. Seshadri, ad A. R. Calderbak, Space-Time Codes for igh Data Rate Wireless Commuicatio: Performace Criterio ad Code Costructio, IEEE Tras. Iformatio Theory, vol. 44, o. 2, pp , March [6] V. Tarokh,. Jafharkai, ad A. R. Calderbak, Space-Time Block Codes from Orthogoal Desigs, IEEE Tras. Iformatio Theory, vol. 45, o. 5, pp , July [7] S. Boyd ad L. Vadeberghe, Itroductio to Covex Optimizatio with Egieerig Applicatios. Course otes. Staford Uiversity, [8] P. J. M. va Laarhove ad E.. L. Aarts, Simulated Aealig: Theory ad Applicatios. Kluwer Academic Publishers, [9] T. M. Lok ad T. F. Wog, Trasmitter ad Receiver Optimizatio i Multicarrier CDMA Systems, IEEE Tras. o Commuicatios, vol. 48, o. 7, pp , July [10] K.-K. Wog, R. S. K. Cheg, K. B. Letaief, ad R. D. Murch, Adaptive Ateas at the Mobile ad Base Statios i a OFDM/TDMA System, IEEE Tras. o Commuicatios, vol. 49, o. 1, pp , Jauary [11] J.-. Chag, L. Tassiulas, ad F. Rashid-Farrokhi, Joit Trasmitter Receiver Diversity for Efficiet Space Divisio Multiaccess, IEEE Tras. o Wireless Commuicatios, vol. 1, o. 1, pp , Jauary [12] W. Rhee, W. Yu, ad J. M. Cioffi, The Optimality of Beamformig i Uplik Multiuser Wireless Systems, IEEE Tras. o Wireless Commuicatios, vol. 3, o. 1, pp , Jauary [13] S. Serbetli ad A. Yeer, Trasceiver Optimizatio for Multiuser MIMO Systems, IEEE Tras. o Sigal Processig, vol. 52, o. 1, pp , Jauary [14]. Boche ad M. Schubert, A Geeral Duality Theory for Uplik ad Dowlik Beamformig, i Proc. IEEE Vehicular Techology Coferece, VTC 02, September 2002, pp [15] E. Visotsky ad U. Madhow, Optimum Beamformig Usig Trasmit Atea Arrays, i Proc. IEEE Vehicular Techology Coferece, VTC 99, May 1999, pp [16] M. Begtsso, A Pragmatic Approach to Multi-User Spatial Multiplexig, i Proc. IEEE Sesor Array ad Multichael Sigal Processig Workshop, SAM 02, August 2002, pp [17] A. Pascual-Iserte, A. I. Pérez-eira, ad M. A. Laguas, O Power Allocatio Strategies for Maximum Sigal to oise ad Iterferece Ratio i a OFDM-MIMO System, accepted at IEEE Tras. o Wireless Commuicatios, April [18] D. P. Bertsekas, Costraied Optimizatio ad Lagrage Multiplier Methods. Computer Sciece ad Applied Mathematics, Academic Press, [19] D. E. Goldberg, Geetic Algorithms i Search, Optimizatio, ad Machie Learig. Addiso-Wesley, 1988.