Joint Receiver and Transmitter Optimization for Energy-Efficient CDMA Communications

TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETECTION FOR ADV ANCED COMMUNICA TION SYSTEMS AND NET WORKS 1 Joint Recei v er and T ransmitter Optimizati on for Ener gy-Ef ficient CDMA Communications Stefano Buzzi, Sen ior Member , IEEE , and H. V incent Poor , F ellow , IEEE Abstract —This paper fo cuses on the cr oss-layer issue of joint multiuser detection and r esource allocation f or energy efficiency in wireless CDMA networks. In particu lar , assuming that a linear multiu ser detector is adopted in the uplink recei ver , th e case co nsidered is that in wh ich each terminal is allowed to vary its transmit power , spreading code, and uplin k rec eiv er in order to maximize its own u tility , which is defined as the ratio of data throughput to transmit power . R esorting to a game-theor etic f ormulation, a non-cooperative game f or utility maximization is form ulated, and it is prov ed that a unique Nash equilibriu m exists, which, under certain conditions, is also Pareto- optimal. Theoretical results concerning the relationship between the problems of S INR maximization and MSE minimization are giv en, and, resorting to the tools of larg e system analysis, a new distributed power control a lgorithm is implemented, based on very little p rior information about the user of interest. The utility p rofile achiev ed by the active users in a large CDM A system is also computed , and, moreov er , the c entralized socially optimum solution is analyzed. Considerations on the extension of the proposed framework to a multi-cell scenario are also briefly detailed. Simulation results confirm that the proposed non- cooperativ e game largely outperforms competing alternativ es, and th at it ex hibits a quite sma ll performance l oss with r esp ect to the socially optimum so lution, and on ly in th e case in which the users n umber exceeds the p rocessing gain. Finally , results also show an excellent agreement between the theoretical closed- fo rm f ormulas based on large system analysis and the outcome of numerical ex periments. Index T erms —Multi user detection, M MSE recei ver , CDMA, power control, large-sy stem analysis, spreading code optimiza- tion, game th eory , Pareto frontier , energy effi ciency . I . I N T R O D U C T I O N A SUBST ANTIAL amou nt of research has bee n carried out on mu ltiuser detection for code division multiple access (CDMA) n etworks over the last thirty year s. Start- ing fro m the pion eering work of V erd ´ u, who derived the optimal, minimu m error pro bability , multiuser receiver alo ng with fu ndamen tal s uboptim al multiuser detectors such as the decorre lating receiver , sign ificant p rogre ss h as been made over the years. Sev eral r elev ant issues such as appr oximate implementatio ns of the optimal multiuser detector, the impact of fading on mu ltiuser detection structur es, th e synthesis of adaptive, possibly blind, multiuser detection algo rithms, and Stefa no Buzzi is with D AE IMI, Univ ersity of Cassino, V ia G. Di Biasio, 43, I-03043 Cassino (FR), Italy (e-mail: buzzi@unic as.it); H. V incent Poor is with the S chool of Engi neering and Ap plied Science, Pri nceton Univ ersity , Princet on, NJ, 08544, USA (e-mail: poor@princeton.ed u). This paper was partly presented at the 2007 Eur opean W ire less Confer ence , Paris, France, April 2007, and at the 2007 IEE E International Symposium on Informatio n Theory , Nice, France, June 2007. This research was supported in part by the U. S. Ai r For ce Research Laborato ry under Co operati ve Agreeme nt No. F A8750-06-1-0252 and in part by the U. S. Defense Adva nced Research Project s Agency under Grant HR0011-06-1-0052. the joint m ultiuser detection and channel equ alization p rob- lem, have been tac kled and thoroug hly inv estigated. Results regarding these research issues are surveyed, amo ng the others, in the textboo ks [1], [2]. In the r ecent past, a new tren d has eme rged, i.e. the so-called cross-layer approac h. Roug hly speaking, the basic idea here is to perfo rm joint optimization of procedu res that are implemented in different lay ers of the network proto col stack, so as to outperfo rm so lutions ba sed on single optimization o f the procedu res of each network layer . Regarding CDMA systems, th e cro ss layer appr oach has mainly focused o n the prob lem of in tegrating physical layer issues, such as multiuser de tection and ch annel estimation, with network level issues, such as call admission co ntrol, power co ntrol, and, mo re ge nerally , resource allocation [3]. I n keeping with this recent trend, this pap er fo cuses on the issue of join t m ultiuser detection an d resource allocation in o rder to achieve e nergy efficiency in wireless CDMA networks. The re sults of th is pa per are m ainly based on two p owerful mathematical to ols, n amely ga me theo ry and lar ge system analysis . Game theory [4] is a bran ch o f mathematics th at has been applied prima rily in econo mics a nd other so cial sciences to study the interactions among se veral auto nomou s sub jects with contrasting interests. More recently , it has been discovered that it can also be used for th e design and ana lysis of co mmuni- cation systems, mostly with application to resou rce alloca tion algorithm s [ 5], and, in particular, to p ower contro l [6]. As examples, th e read er is refe rred to [7]–[ 9]. In these papers, fo r a multiple access wireless data ne twork, no ncoop erative and cooper ati ve games are introduced, wherein each u ser cho oses its tra nsmit p ower in ord er to max imize its own utility , d efined as the ratio o f the throug hput to transmit power . While the above papers consider the issue of p ower contro l assuming that a con vention al matched filter is a vailable at the re ceiv er , the recent paper [10] considers the cross-layer prob lem of joint linear receiver design an d p ower control so as to maximize the utility of each user: it is thus shown in [10] that the inclusion of receiver design in the co nsidered game brin gs remark able advantages. This same u tility f unction is also used in [11] for energy-efficient p ower control in ultra-wid eband (UWB) commun ications, while the sur vey paper [1 2] r evie ws recen t advances in the app lication of a game-th eoretic fr amew ork fo r energy-efficient resou rce allocation . Large system analysis (LSA) is a r elativ ely new ma themat- ical to ol, first intro duced in [13], th at h as r ecently em erged in the an alysis of CDMA systems. In sum mary , [13] h as revealed that in a CDMA system with pro cessing g ain and numb er of users both increasing withou t bo und but with their r atio fixed, and with random ly cho sen, un it-norm , spread ing codes, TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 2 the Sign al-to-Inte rference plus Noise Ratio (SINR) of each user fo r the ca se in which a linear minimu m mean squ are error (MM SE) receiver is ado pted conv erges in pro bability to a non-ran dom co nstant. In par ticular , den oting by K the number of activ e users, by N the system proc essing gain, by N 0 / 2 th e additive therm al noise power sp ectral de nsity (PSD) level, and b y E P [ · ] the expectation with re spect to the limiting em pirical distribution F of the received powers of the interferer s, the SINR of the MMSE receiver for th e k -th user, say γ k , con verges, for K, N → ∞ , K/ N = α = co nstant, in probab ility to γ ∗ k the u nique solu tion o f th e equation γ ∗ k = P k N 0 / 2 + αE P h P P k P k + P γ ∗ k i , (1) with P k the r eceiv ed power for the k -th user . Interesting ly , the limitin g SINR dep ends only on th e limiting empirica l distribution o f the received powers o f the inter ferers, th e load α , the thermal noise le vel and the recei ved po wer of the user of interest, wh ile being ind ependen t of the actual realization of the received powers o f the interf erers a nd of the spreading codes of the activ e users. LSA is now a well-established mathematical too l fo r design and analysis o f com munication systems (see, e.g., [14]–[16], to cite a fe w). A. Su mmary o f the r esults This paper i s the first in this area that considers the cross- layer issue of utility maxim ization with respec t to the choice of linear multiuser detector , spreading code an d transmit po we r . Using game theory and L SA, the follo wing contributions are giv en here. - W e gene ralize the non -coop erative gam e considered in [10] by considering utility m aximization with respect to the linea r up link multiuser receiver , transmit power and spreading co de assignment. W e wi ll show that the newly considered no n-coo perative game admits a un ique Nash equilibriu m, which, for th e case in which the n umber of users doe s n ot exceed the system processing gain, is a lso Pareto-optimal. - As an intro ductory step to the p revious item, we also fo r- mulate a n on-coo perative game for SINR maxim ization with re spect to lin ear m ultiuser dete ctor a nd sprea ding code choice, and show th at this g ame admits a un ique Nash equilibrium point that is also Pareto-op timal. - Using LSA, we design a new d istributed power con trol algorithm th at needs very little pr ior infor mation (i.e. th e channel gain for the u ser of intere st) to be implemented . This algorith m may be integrated in the u tility maximiz- ing n on-co operative gam e of [10]. - Using LSA, we are able to predict the utility and SINR profile a cross u sers in a la rge CDMA system, for both the cases in which spreading cod e optimization is either considered or not considered. - Using LSA, we are a ble to d erive the socially o ptimum solution to the pr oblem o f utility maximization with equal SINR co nstraint; num erical results will show that the perfo rmance lo ss incurred by th e propo sed n on- cooper ati ve g ame with respect to the socially optim um solution is quite ne gligible. - W e a lso consider th e extension of our fr amework to a multi-cell scenario. In par ticular, we co nsider the issue of n on-coo perative utility maxim ization in a multi-cell system with predetermined base station ass ignment, and show th at, as th e n umber of users does n ot exceeds the pro cessing gain, this gam e admits a u nique Nash equilibriu m w hich i s also Pareto-optimal. B. Outlin e of the pa per The r est of this paper is organized as follows. The n ext section contains some preliminar ies on the considered payo ff function and o n th e system mo del o f inter est. Section II I dwells on the definition of Nash equ ilibrium and Pareto optimality , and, also, provides an inter esting r esult on the equilibriu m point of the SINR-maxim izing non -cooper ativ e game. I n section IV the non- cooper ativ e game for u tility maximization with respect to the choic e of linear multiuser detection, p ower contro l and spreading code is d escribed an d analyzed. LSA is used in Section V to d eriv e a d istributed power con trol proced ure th at can be implem ented based o n little prio r infor mation; it is shown that this algorithm ma y be u sed to ob tain distributed im plementation s o f th e non- cooper ati ve gam e pro posed in [10] . In Section VI, LSA is used in order to pred ict the SINR and utility pro file across users in a large CDMA sy stem, while Section VII conta ins the discussion on the socially optimum, e qual SINR, cooperativ e game. Section VIII considers the extension of the considered non-co operative g ames to a mu lti-cell scenario, wherein out- of-cell interfer ence is properly taken into accou nt. Fina lly , numerical results are illu strated in Sectio n IX, while Sectio n X contains the e ventual wra p-up o f th e pap er . I I . S Y S T E M M O D E L A N D P RO B L E M S TA T E M E N T Consider the uplink of a K -user synchr onou s, single-cell, direct-sequ ence code division multiple access ( DS/CDMA) network with pr ocessing ga in N and subject to flat fading. After chip-m atched filtering and sam pling at the c hip-rate, th e N -dimen sional recei ved data vector, say r , co rrespon ding to one s ymbol in terval, can be written as r = K X k =1 √ p k h k b k s k + n , (2) wherein p k is the transmit power of the k -th user 1 , b k ∈ {− 1 , 1 } is the info rmation sym bol of the k -th user, and h k is the real 2 channel gain between the k -th user’ s transmitter and the access point (AP); the actual value o f h k depend s o n bo th the distance of the k -th u ser’ s terminal fr om the AP and the channel fadin g fluctuation s. The N -dimension al vector s k is the spr eading c ode of th e k - th u ser; we assume that the entries 1 T o si mplify subsequent notati on, we a ssume t hat the transmitted po wer p k subsumes also the gain of the transmit and recei ve antennas. 2 W e assume he re, for simpl icity , a real c hannel mod el; generaliz ation to practi cal channels, with I and Q components, is s traight forwar d. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 3 of s k are real an d that s T k s k = k s k k 2 = 1 , with ( · ) T denoting transpose. Finally , n is the am bient n oise vector, which we assume to be a zero-me an white Gaussian ran dom process with covariance matrix ( N 0 / 2) I N , with I N the id entity m atrix of order N . An alternative and comp act representation of (2) is giv en by r = S P 1 / 2 H b + n , (3) wherein S = [ s 1 , . . . , s K ] is the N × K -dimen sional spreadin g code matrix, P and H ar e K × K - dimensiona l d iagonal matrices, whose diagona ls are [ p 1 , . . . , p K ] and [ h 1 , . . . , h K ] , respectively , and, finally , b = [ b 1 , . . . , b K ] T is the K - dimensiona l vector of the data symbols. Assume now th at each m obile termin al sends its data in packets of M bits, and that it is interested both in having its data received with a s small as po ssible error pr obability at the AP , a nd i n making careful use of the energy stored in its battery . Obviously , these are conflictin g go als, since erro r-free reception may be achie ved by increasing the recei ved SNR, i.e. by increasing the transmit power , which of cou rse comes at the expense of battery life 3 . A useful app roach to quantify these conflicting goals is to define th e utility of the k -th user as the ratio of its throughp ut, define d as the number of i nform ation bits that are receiv ed with no err or in unit time, to its tr ansmit power [7], [8], i.e. u k = T k p k . (4) Note that u k is me asured in bit/Joule, i.e. it represents the number of successful bit tra nsmissions that can be m ade for each Joule of energy drain ed fr om the battery . The utility function (4) is widely a ccepted and indeed it h as b een already used in a n umber of previous studies such as [7]– [12]. Of course, there ar e also altern ativ e cho ices that co uld be m ade. For instance, pap ers [17], [18 ] consider an ou tage-based utility suited for fast time-varying channels, while the r ecent stud y [19] consider s an u tility that is the pr oduct o f the transmit power times the interfer ence 4 . Utility (4), howe ver, is b y no doubt the mo st suited one when energy ef ficien cy is to be taken into acc ount, an d fr om n ow on we will defin itely em brace this model. Denoting by R th e com mon rate of the netw ork (extension to the case in which each user transmits with its own rate R k is qu ite simple) and assum ing tha t each packet o f M symbols contains L informa tion symbols and M − L overhead symbols, reserved, e.g., for ch annel estimation and/or par ity checks, the throug hput T k can b e e xpressed as T k = R L M E k (5) wherein E k denotes th e the pro bability that a pac ket from the k -th user is received error-free. In the considered DS/CDMA 3 Of c ourse there are many other stra te gies to lo wer th e dat a e rror prob- abili ty , such as for exa mple the use of error correct ing codes, di versity expl oitat ion, and implementa tion of optimal reception techni ques at the recei ver . Here, howe ver , we are mainly interest ed to energy effici ent data transmission and powe r usage, so we consider only the ef fects of v arying the transmit power , the recei ver and the spreading code on energy effici enc y . 4 Obviou sly in this case we are interested to utility minimizatio n rather than to its maximizat ion. setting, the term E k depend s for mally on a num ber of pa- rameters s uch as the spread ing code s o f all the users and the diagona l entries of the matrices P and H , as well as on the strength of the used error correcting codes. Howe ver, a custom - ary ap proach is to model the multip le access interferen ce as a Gaussian rand om process, an d assume th at E k is an incr easing function of the k -th u ser’ s Signal-to- Interfer ence plus No ise- Ratio (SINR) γ k , which is na turally the case in many pr actical situations. Recall that, for the case in wh ich a linear r eceiv er is used to detect the data symbol b k , accordin g, i.e., to the decision rule b b k = sign h d T k r i , (6) with b b k the estimate of b k and d k the N -dime nsional vector representin g the receive filter for the user k , it is easily seen that the SINR γ k can b e written as γ k = p k h 2 k ( d T k s k ) 2 N 0 2 k d k k 2 + X i 6 = k p i h 2 i ( d T k s i ) 2 . (7) Of related interest is also th e mean squar e err or (MSE) for the user k , which, for a linear recei ver, is defined as MSE k = E   b k − d T k r  2  = 1+ d T k M d k − 2 √ p k h k d T k s k , (8) wherein E {·} deno tes statistical expectation an d M =  S H P H T S T + N 0 2 I N  is the covariance matr ix of the data. The exact shape o f E k ( γ k ) depends on factors such as th e modulatio n and codin g typ e. Howev er , in all cases of relev an t interest, it is an increasing func tion of γ k with a sigmoid al shape, and converges to un ity as γ k → + ∞ ; as an example, for binary phase-shift-keying ( BPSK) modulation coup led with no channel coding, it is easily shown that E k ( γ k ) = h 1 − Q ( p 2 γ k ) i M , (9) with Q ( · ) th e c omplemen tary cum ulative distribution fu nction of a zero-mea n Gau ssian rando m variate with unit variance. A plot of ( 9) is shown in Fig. 1 for the case M = 10 0 . It shou ld be noted that substituting ( 9) into (5 ), and, in turn, in to (4), leads to a stro ng inco ngru ence. Ind eed, for p k → 0 , we have γ k → 0 , but E k conv erges to a sma ll but non-ze ro value (i.e. 2 − M ), thus imp lying that an unbo unded ly large u tility ca n be ach iev ed by tran smitting with zero power , i.e. not transmitting at all an d making blin d g uesses at the receiver on what data we re tran smitted. T o circu mvent this problem , a cu stomary approach [8], [1 0] is to replace E k with an efficiency functio n , say f k ( γ k ) , who se behavior shou ld approx imate as close as po ssible that of E k , except that for γ k → 0 it is req uired that f k ( γ k ) = o ( γ k ) . The fu nction f ( γ k ) = (1 − e − γ k ) M is a widely acce pted su bstitute fo r the true pro bability of correct p acket recep tion, a nd in the following we will adop t this model 5 . This ef ficiency fun ction is increasing and S-sh aped, con verges to unity as γ k approa ches 5 See Fig. 1 for a comparison between the probabili ty E k and the effici enc y functio n. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 4 infinity , and has a continuous first or der deri vativ e. Note that we have omitted the subscript “ k ′′ , i. e. we have used the notation f ( γ k ) in place o f f k ( γ k ) since we assume that the efficiency fu nction is the sam e fo r all the users. Summing up, substituting ( 5) into (4) and replacing the probab ility E k with the above d efined efficiency function, we obtain the following expression fo r the k - th u ser’ s utility: u k = R L M f ( γ k ) p k , ∀ k = 1 , . . . , K . (10) Now , based on the utility d efinition (10), many interesting questions arise c oncern ing how ea ch user may max imize its utility , and how this maxim ization affects utilities achieved by other users. Likewise, it is n atural to qu estion wh at hap pens in a non -cooper ativ e settin g wherein each user au tonom ously and selfishly tries to maxim ize its own utility , with no care for o ther u sers u tilities. In pa rticular, in this latter situation, is th e system able to reach an equilibrium wherein n o user is interested in varying its par ameters since ea ch action it would take would lead to a decrease in its own utility? And , also, what is the p rice to be paid in terms of p erforma nce loss du e to the selfish behavior (i.e ., lack o f cooperatio n) of the users? Game theor y provides means to stud y these inter actions and to provide som e usef ul an d insigh tful answers to th ese question s. Initially , game theory was a pplied in this co ntext main ly as a tool to study n on-co operative scenar ios wher ein mobile users are allo wed to v ary their transmit po wer only (see [7]– [9], for example) to maximize utility , and where conventional matched filterin g is u sed at th e receiver . Recen tly , instead, in [10] such an appr oach ha s b een extended to the cross layer scen ario in which each user may vary its power an d its u plink linear receiver , i.e. the pro blem of join t linear multiuser de tection op timization and power con trol for utility maximization has b een tackled. In th e following, we will go further b y conside ring and analyzing the case of spread ing code cho ice, power contr ol an d linear mu ltiuser detector design for utility maximization. A. The pr opo sed non- cooperative g ame Formally , the g ame G pro posed h ere can be describ ed as the trip let G = [ K , { S k } , { u k } ] , wh erein K = { 1 , 2 , . . . , K } is the set of active users participating in the game, u k is the k -th user’ s utility defined in (10), and S k = [0 , P k, max ] × R N × R N 1 , (11) is th e set of possible actions (strategies) t hat u ser k can take. It is seen that S k is written as the Cartesian product o f three different sets, and indeed [0 , P k, max ] is the range of available transmit p owers fo r the k -th user (n ote that P k, max is the maximum allowed transmit power for user k ), R N , with R the real line, defines the set of all po ssible linear re ceiv e filters, and, finally , R N 1 = n d ∈ R N : d T d = 1 o , defines th e set of the allowed spread ing codes 6 for user k . 6 Here we assume that the spreading codes hav e real entries; the problem of utility maximiza tion with reasonabl e comple xity for the case of discrete - v alued entrie s is a challen ging issue that will be considere d in the future. Summing up, th e proposed non-coo perative g ame to b e considered in the fo llowing can b e cast as the following maximization pr oblem max S k u k = max p k , d k , s k u k ( p k , d k , s k ) , ∀ k = 1 , . . . , K . (12) I I I . P R E L I M I N A RY R E S U LT S A N D C O N C E P T S Before p roceedin g furth er , we r evie w her e some basic definitions on game theor y , and provide som e results on the relationship b etween SI NR max imization an d MSE minimiza- tion in multiuser systems. The co ntent of this section will reveal useful in the sequel of the p aper . A. Nash equilibria an d P a r eto optimality W e giv e her e the definition o f Nash equilibrium . L et ( s 1 , s 2 , . . . , s K ) ∈ S 1 × S 2 × . . . S K denote a certain stra tegy K -tuple f or th e a ctiv e users. The point ( s 1 , s 2 , . . . , s K ) is a Nash eq uilibrium if for a ny u ser k , we ha ve u k ( s 1 , . . . , s k , . . . , s K ) ≥ u k ( s 1 , . . . , s ∗ k , . . . , s K ) , ∀ s ∗ k 6 = s k . Otherwise stated, at a Nash equilibrium , no user can unilate rally improve its own utility by taking a d ifferent strategy . A f ast reading of this definition migh t lead to think that at Nash equilibrium users’ utilities achieve their maximum values. Actually , th is is not the case, since the existence of a Na sh equilibr ium point does not imply that no other strategy K -tuple does exist that can lead to an improvement of the utilities of som e u sers while not decreasing the utilities of the r emaining ones. Th ese latter strategies are usua lly said to b e Pareto- optimal [4] . Other wise stated, at a Nash equilibriu m, each user , provided that the o ther users’ strategies do not chan ge, i s not in terested in ch anging i ts o wn strategy . Howe ver, if some sor t of coop eration would be a vailable, users might agree to simultaneously s witch to a different strategy K - tuple, so as to improve the utility of so me, if not all, active users, wh ile not decreasing the utility of the remaining ones. B. SI NR maximizatio n a nd MSE minimization W e are no w read y to state our first result. Proposition 1: Given the linear decision r ule (6) and the SINR expr ession in eq. (7 ), consider the non-coope rative game max d k , s k γ k , ∀ k = 1 , . . . , K , (13 ) with the c onstraint k s k k = 1 . Th is game admits a uniqu e 7 Nash equ ilibrium point, which co incides with the uniq ue global minimizer (with r espect to spr ea ding code choice and linear receiver choice) o f the total MSE (TMSE) defined as TMSE = K X i =1 MSE i . (14) 7 Here and i n the fol lo wing uniqueness of th e line ar recei ve filter d k is meant up to a positi ve scaling facto r . Uniqueness of the spreading codes is instead intended with respect to the set of eigen value s of the matrix SH P H T S T . TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 5 Mor eover , the Nash equilibrium po int is also P areto-optimal. Proof: This proof is partly based on results that are scattered in other paper s; fo r the sake of concisen ess and to av o id useless reprod uction o f already kno wn mater ial, we u se these results citing th eir o rigin b ut without pr oving them again. First of all, r ecall that amo ng lin ear multiuser detectors, th e MMSE receiver is the on e that maximizes the SINR of each user [2 ], th us implying th at, in o rder to ma ximize its own SINR, each user is to adopt a linea r MMSE recei ver , i.e. we have d k = √ p k h k M − 1 s k . Substituting this last relation in eq. (7), and using stand ard linear algebr a techniqu es, it is easily shown that γ k = p k h 2 k s T k  M − p k h 2 k s k s T k  − 1 s k . (15) Giv en eq. ( 15), it is seen that the k -th u ser SINR is max i- mized taking s k equal to th e eigenvector corr espondin g to th e minimal eigen value of the covariance m atrix of th e k -th user interferen ce  M − p k h 2 k s k s T k  . So far no thing guarantees that this s trategy lead s to a stable equ ilibrium point. On th e other hand, if a linear MMSE rec eiv er is used, th e fo llowing relation is well- known to hold MSE k = 1 1 + γ k , (16) thus im plying th at SINR max imization f or the g eneric k - th user is equivalent to minimizatio n o f its MSE. Moreover, exploiting the results c ontained in the Appen dix I of [2 0], it can be shown th at, in the considered settin g, in dividual MSE minimization is equiv alent to minimization of the TMSE, d e- fined in (14). F o llowing [2 1]–[2 3], letting D = [ d 1 , . . . , d K ] and den oting by ( · ) + Moore-Pen rose pseudoinv ersion, it can be sh own tha t the TM SE adm its a uniq ue global optimu m, and that the iterations d i = √ p i h i  S H P H T S T + N 0 2 I N  − 1 s i ∀ i = 1 , . . . , K s i = √ p i h i  p i h 2 i D D T + µ i I N  + d i ∀ i = 1 , . . . , K (17) admit as uniq ue stable fixed points spread ing code sets that are the glo bal minim izer of th e total MSE. I n the above rela tions, µ i should b e set so that k s i k = 1 , a nd a p rocedur e for efficiently finding the value of µ i for ensuring this constrain t is g iv en in Appendix A. So far , we have shown tha t the n on-coo perative game in (13) can be solved by minimizing , thro ugh iterations (1 7), the to tal MSE, an d that these iterations are guar anteed to conv erge to the uniq ue and stable g lobal optim um, i.e. the non-co operative game admits a Nash equilibriu m. It remain s to show t he Pareto-op timality of this point. T o this end , it suffices to show that, letting ¯ S and ¯ D b e the spreadin g code matrix and the linear receiver matrix th at jo intly ac hieve the g lobal minimum of the total MSE, no strategy of spr eading codes and decoder c an be fou nd to increase the SINR of o ne o r m ore users without d ecreasing th e SINR of at least one other user . T o see this, no te that if ¯ S an d ¯ D ar e the glo bal m inimizers of th e MSE, then ¯ D contains th e M MSE rece i vers resulting from the sp reading code s of ¯ S . Deno te by  γ i ( ¯ S , ¯ D )  K i =1 the SINR v alu es achieved b y the matrices ¯ S and ¯ D . Ass ume now that there exists a spreadin g code matrix S ∗ 6 = ¯ S suc h that γ i ( S ∗ , ¯ D ) > γ i ( ¯ S , ¯ D ) , for at least on e i ∈ { 1 , . . . , K } and γ j ( S ∗ , ¯ D ) ≥ γ j ( ¯ S , ¯ D ) fo r j 6 = i . If this is the case, we can m ake an MMSE upda te and obtain the matrix D ∗ of the MMSE rece i vers c orrespon ding to the cod es in S ∗ . For a gi ven set o f spreading cod es, u sing the MMSE recei ver always yields a maximization of th e SINR and a min imization of th e MSE. W e thus hav e γ i ( S ∗ , D ∗ ) > γ i ( ¯ S , ¯ D ) , and γ j ( S ∗ , D ∗ ) ≥ γ j ( ¯ S , ¯ D ) , ∀ j 6 = i . Con sequently , given relation (16), we have TMSE( S ∗ , D ∗ ) < T MSE ( ¯ S , ¯ D ) , which contradicts the starting ass umption s th at ¯ S and ¯ D are the g lobal minimizers of the MSE. I V . A N O N - C O O P E R AT I V E G A M E F O R C RO S S - L A Y E R R E S O U R C E A L L O C AT I O N Equipp ed with th e above result, we are now ready to resume the non- coopera ti ve g ame in ( 12). Note that, given (10), the above maximization can be a lso written as max p k , d k , s k f ( γ k ( p k , d k , s k )) p k , ∀ k = 1 , . . . , K . (18) Moreover , since the ef ficiency function is monoto ne and no n- decreasing, we also h ave max p k , d k , s k f ( γ k ( p k , d k , s k )) p k = max p k f max d k , s k γ k ( p k , d k , s k ) ! p k , (19) i.e. we can fir st take care of SINR maxim ization with respect to spreading codes and linear receivers, and then focu s on maximization of the resulting u tility with re spect to tra nsmit power . W e a re now read y to expr ess our result on the no n- cooper ati ve game for spreading c ode op timization, linear re- ceiv er design and power co ntrol. Proposition 2: The non- cooperative game defined in (12) admits a un ique Nash e quilibrium po int ( p ∗ k , d ∗ k , s ∗ k ) , for k = 1 , . . . , K , wher ein - s ∗ k and d ∗ k ar e the unique k -th user s pr ea ding code an d r eceive filter resulting fr om iterations (17). Den ote by γ ∗ k the co rr espo nding SINR. - p ∗ k = min { ¯ p k , P k, max } , with ¯ p k the k -th user transmit power such tha t the k -th user maximum SINR γ ∗ k equals ¯ γ , i. e. the uniq ue solution o f the eq uation f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Mor eover , for K ≤ N , th e Nash equilibrium point is P ar eto-optimal. Proof: Th e pro of gen eralizes the o ne provid ed in [10 ], so, for the sake of br evity , we main ly f ocus o n its original part. Sin ce ∂ γ k /∂ p k = γ k /p k , it is easily seen th at each user’ s utility is m aximized if each user is a ble to ac hieve the SINR ¯ γ , th at is the uniq ue 8 solution o f the eq uation f ( γ ) = γ f ′ ( γ ) . By Proposition 1, ru nning iteration s (17) 8 Uniquene ss of ¯ γ is ensured by the fact that the ef ficienc y function is S-shaped [24]. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 6 until conv ergence is rea ched p rovides the set of spreadin g codes and MM SE receiv ers that maximize the SINRs fo r all the users. As a co nsequenc e, the utility of each u ser is maximized b y ad justing tr ansmit powers so tha t the optim ized (with respect to spread ing codes and linear receivers) SI NRs equal ¯ γ . So far , we h av e shown how to set the tr ansmit power, spreading code and r eceiv er design to maximize utility at the Nash equilibrium. In ord er to shown that a Nash equilibrium exists, we can use th e same arguments of [ 8] and state that a unique Nas h equilibrium po int exists since each user’ s utility function is q uasi-concave 9 in the tr ansmit power p k and since the ef ficien cy function is S-shaped. Assume now that K ≤ N ; in this case, the spreading codes resulting from iterations (17) are o rthog onal, i.e. th e multiu ser channel b oils down to K par allel single-u ser chann els. As a consequen ce, the SINR of each user is n o longer affected by the strategies of the other users, and maximization of the utility of each user has n o endang ering effect o n the utility ac hieved by t he other users. In this scenario, th us, the non-co operative game clearly achiev es Pareto o ptimality . In p ractice, the above Nash equilibrium is reached throug h the following iterative algorithm . Given any set of transmit powers, iteration s ( 17) a re run in o rder to minimize system TMSE. A fter th at, u sers adju st their tran smit power in or der to achieve the target SINR, using, e.g., the s tandard power c ontrol iterations as detailed in [6]. These steps are to be repeated until conv ergence is reached. V . A D I S T R I B U T E D P O W E R C O N T RO L A L G O R I T H M BA S E D O N L S A The above argu ments sh ow that implementatio n o f the propo sed n on-coo perative game, as well as of the game in [10], needs a power control algorithm, s uch as the one outline d in [6]. Classical power co ntrol algorithm s require k nowledge of at least the u plink SINR for each user, or, alternatively , are imp lemented thr ough iterative proce dures [25], [2 6] th at suffer from slow conver gence and excess steady-state error . In this paper, instead, we show that LSA may lead to power control algorith ms that may be implemen ted in a d istributed fashion an d that requir e knowledge o f the ch annel fo r the user of inter est only . The results of this section refer to th e case that spreading co de o ptimization is not p erform ed, i. e. the case of utility m aximization with respect to the ch oice o f th e linear uplin k multiuser r eceiv er and of the transmit p ower is considered . As an introdu ctory step to o ur algorithm, we begin b y illustrating a simple power contro l algor ithm derived f rom [13]. W e have m entioned that in a large CDMA system the k -th user’ s SINR conv erges in p robab ility to the so lution to Eq. (1) . Heuristically , this means th at in a large system, and embracing the nota tion of the previous s ection, the SI NR γ k is d eterministic and appro ximately satisfies γ k ≈ h 2 k p k N 0 / 2 + 1 N X j 6 = k h 2 k h 2 j p k p j h 2 k p k + h 2 j p j γ k (20) 9 A functi on is quasi-conca ve if there exists a poi nt belo w which the functio n is nondecreasing, and above which the function is nonincrea sing. Now , a s n oted in [13] , if all the users must achieve th e same common target SINR ¯ γ , it is reaso nable to assume that they are to be r eceived with the same power , i.e. the co ndition h 2 1 p 1 = h 2 2 p 2 = . . . h 2 K p k = P R , is to be fulfilled. Sub stituting th e above constraint in (20) and equating ( 20) to ¯ γ it is straightf orward to come up with th e following relation P R = ¯ γ N 0 / 2 1 − ¯ γ 1 + ¯ γ α , ⇒ p k = 1 h 2 k ¯ γ N 0 / 2 1 − ¯ γ 1 + ¯ γ α , (21) wherein, we r ecall, α = K / N , an d th e relation α < 1 + 1 / ¯ γ must ho ld. Eq . (21), which derives from eq. (16) in [13 ], giv es a simple power contro l algorith m that per mits setting the transmitted p ower for each user b ased on th e kn owledge of the chann el g ain for the user of in terest only . T he ab ove algorithm , howev er , do es not take into account the situation which, due to fading and p ath losses, some users end up in transmitting at their m aximum power withou t achieving the target SINR, an d indeed our numer ical results to b e sh own in the sequel will prove th e inability of eq. (2 1) to predict with good accuracy th e actual po wer p rofile for the a cti ve users. In order to circumvent this d rawback, we first recall that in [27] ( see als o [28]) the follo wing result has been sho wn: Lemma: Deno ting by F ( · ) the cumulative distribution func- tion (CDF) of th e squared fad ing coefficients h 2 i , an d by [ h 2 [1] , h 2 [2] , . . . , h 2 [ K ] ] th e vector of the users’ squared fading coefficients sorted in non -incr easing order , then we have that h 2 [ ℓ ] conver ges, for increasing nu mber of users K , in pr oba bility to F − 1  K − ℓ K  , ∀ ℓ = 1 , . . . , K . The ab ove lemma states that if we sort a large numb er of identically distrib u ted r andom v ariates, we o btain a v ector that is appro ximately equal to the uniform ly sampled version of the inv erse of the common CDF of the random variates. According ly , in a large CDMA system each user may individ- ually build a ro ugh estimate o f the fadin g coefficients in the network and b e ab le to predict the num ber of users, say u 2 , that possibly will en d up transmitting at th e m aximum p ower . Indeed , since, according to (21) each user is to be received with a power P R , the estimate u 2 of the num ber of u sers transmitting at the maximum power is gi ven by u 2 = K X i =1 u   ¯ γ N 0 / 2 F − 1  K − i K   1 − α ¯ γ 1+ ¯ γ  − P max   , ( 22) with u ( · ) the step-function. It is also obvious to assume that th e users transmitting at P max will be the ones with the smallest channel coefficients, i.e. the squar ed channel gains o f the u sers transmitting at the m aximum power are well ap prox imated by the sam ples F − 1  K − ℓ K  , with ℓ = K − u 2 + 1 , . . . , K . As a con sequence, the generic k -th user will be affected by u 1 = K − u 2 users that are r eceiv ed with p ower P R (these are the u 1 users with the strongest chann el gains and that are ab le to achie ve the target S INR ¯ γ ), and by u 2 users that are recei ved with power P max F − 1  K − ℓ K  , with ℓ = K − u 2 + 1 , . . . , K . TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 7 ξ i = ψ i F − 1  K − i K  N 0 / 2 + K − u 2 N ψ i F − 1  K − i K  P k ψ i F − 1  K − i K  + P k ξ i + 1 N K X j = K − u 2 +1 ,j 6 = i ψ i F − 1  K − i K  P max F − 1  K − j K  ψ i F − 1  K − i K  + P max F − 1  K − j K  ξ i . (28) Denoting by P k the received power fo r the k - th user, Eq. ( 20) can b e no w written as γ k = N P k N N 0 2 + u 1 P k P R P k + P R γ k + K X i = K − u 2 +1 P k P max F − 1  K − i K  P k + P max F − 1  K − i K  γ k . (23) Now , assuming for the moment that user k is able to achie ve its target SINR, i.e. that P k = P R , th e seco nd su mmand at the d enomina tor on the RHS of the ab ove e quation can b e approx imated as P k P R P k + P R γ k ≈ P k 1 + γ k . (24) Substituting th e abov e appro ximations into (23) and equating it to the t arget SINR ¯ γ we ha ve N P k N N 0 2 + u 1 P k 1 + ¯ γ + K X i = K − u 2 +1 P k P max F − 1  K − i K  P k + P max F − 1  K − i K  ¯ γ = ¯ γ . (25) The ab ove relation can be n ow numer ically solved in o rder to determine th e r eceiv e power P k for th e k -th u ser 10 ; the actua l transmit power f or the k -th user is finally set according to the rule p k = min  P k /h 2 k , P max  (26) Summing up, the pr oposed algor ithm may b e summarized as follows. First, th e n umber of users tr ansmitting at the maximum power is estima ted acco rding to ( 22). Th en, the desired receive po we r fo r each user is c omputed solving eq . (25). Finally , the transmit po we r for the k -th user is determined accordin g to r elation (26). Note that this algorithm req uires knowledge o nly o f the chann el gain for the user of in terest. In Append ix B we will briefly sketch a method to com pute the in verse of the CDF of th e chan nel gains takin g into account both fading an d path losses d ue to ran dom users’ loc ation with respect to the AP . V I . N E T W O R K P E R F O R M A N C E P R E D I C T I O N I N A L A R G E C D M A S Y S T E M In th is section we show how LSA argumen ts can be used to der iv e the utility , tr ansmit power and ac hieved SINR p rofile across user s in a large CDMA system. Oth erwise stated, we show here that, b ased o n the knowl edge of the parameters K and N , an estimate of the per forman ce enjoyed by the ensem- ble of the users can be obtained. W e begin by considerin g the case in w hich no spre ading code op timization is used , and, then, we will relax this constraint. 10 Actuall y this equation gi ves the desired recei ve power for each user , and thus in a centralize d po wer contro l algorith m needs to be solved just once. A. P o wer co ntr ol and linear MMSE detectio n Assume that no spreading code optimization is perfo rmed and that an MMSE linear multiuser detector is used at the receiver . Eq. ( 26) pr ovides the tr ansmit p ower of the k -th u ser , wherein P k is the so lution of eq . (2 5), and h 2 k is the sq uare of th e c hannel coefficient for the k -th user . Once eq uation (25) has b een so lved (note that th is eq uation is to be so lved just once), the set P of the tran smitted powers by the acti ve terminals i s exp ressed as P =  min  P k /F − 1  K − ℓ K  , P max  K ℓ =1 . (27) W ith regard to th e set of the ach iev ed SINRs, we have already commente d on the f act that K − u 2 users are able to achie ve the target SIN R ¯ γ . De noting by ξ i the SINR achieved by the user whose chann el coefficient is h [ i ] , an d letting ψ i denote th e i - th element of the set P (i.e. ψ i = min  P k /F − 1  K − i K  , P max  ), it is easily shown th at ξ i can b e appr oximately ob tained as the solutio n to Eq . (28) , shown at the top of this page. According ly , the set of the ach iev ed SINRs in the n etwork will contain K − u 2 elements equal to ¯ γ and u 2 elements given by the solution to th e above equation with i = K − u 2 + 1 , . . . , K . Giv en the set of transmit powers and of achieved SINRs, the set o f ach iev ed utilities will co ntain the elements υ i = R L M f ( ξ i ) ψ i , i = 1 , . . . , K . (29) B. Joint transmitter and r eceiver optimization with K ≤ N Let us consider now the case th at joint power contro l, spreading cod e o ptimization and linear r eceiver design is perfor med so as to m aximize each user’ s utility . In this case, iterations ( 17) converge to a set of or thogon al cod es, thus implying th at th e multip le-access chan nel boils down to the superpo sition of K parallel sing le-user channe ls. In this case, the k -th user SINR γ k is expr essed as γ k = p k h 2 k N 0 / 2 , k = 1 , . . . , K . (30) As a co nsequenc e, since ea ch user sh ould achieve a target SINR ¯ γ , we ha ve that the set P of the transmitted powers is expressed as P = ( min ¯ γ N 0 / 2 F − 1  K − i K  , P max !) K i =1 (31) According ly , den oting by { ξ i } K i =1 the set o f the ach iev ed SINRs, we have ξ i = ψ i F − 1  K − i K  N 0 / 2 , i = 1 , . . . , K (32) TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 8 with ψ i = min ¯ γ N 0 / 2 F − 1  K − i K  , P max ! the generic element of P . Given ξ i and ψ i , the ensemble of achieved utilities can be computed as in (29). C. Joint transmitter and r ec eiver optimization with K > N Consider finally the case of an oversaturated CDMA system, i.e. the number o f active users is larger than th e processing gain. The following analysis refers to the case in which each user is ab le to achieve th e target SINR ¯ γ ; it is thus reasonable to assume that all the user s are re ceiv ed with the same power , i.e. p 1 h 2 1 = . . . = p K h 2 K = P R . (33) Since a linear MM SE detector is used at the re ceiv er , using standard linear algeb ra it is e asily shown that th e k -th user SINR c an b e e x pressed as γ k = ¯ γ = p k h 2 k s T k M − 1 s k 1 − p k h 2 k s T k M − 1 s k = P R s T k M − 1 s k 1 − P R s T k M − 1 s k . (34) On the othe r han d, it is well known [ 21], [2 2], [29 ] that in the case in which K > N a nd a ll the users are recei ved with the same power , iter ations (1 7) co n verge to a set of W elch- Bound-E quality (WBE) sequence s, i.e. the lim iting sequences are suc h that S H P H T S T = P R α I N , (35) where, we recall, α = K / N . As a con sequence, th e d ata covariance matrix is e x pressed as M =  P R α + N 0 2  I N . (36) Substituting eq . (36 ) into (34) and solvin g for P R we ha ve P R = ¯ γ N 0 / 2 1 + ¯ γ (1 − α ) , (37) with α < 1 + 1 / ¯ γ . Once P R has be en co mputed fro m (37), the elements of the set of the transmitted powers are expressed as ψ i = P R F − 1  K − i K  , (38) and the elements o f the set of the achieved u tilities can be computed as in (29), with ξ i = ¯ γ , ∀ i = 1 , . . . , K . V I I . S O C I A L LY O P T I M U M S O L U T I O N I N T H E OV E R S AT U R A T E D S C E N A R I O Proposition 2 has shown that the Nash eq uilibrium of the propo sed non-cooper ativ e gam e is also Pareto- optimal in the case in which K ≤ N . For K > N , instead , the resource allocation strategy resulting fr om the said g ame is not o n th e Pareto-optimal frontier ; the q uestion thus arises on how m uch is the Nash e quilibrium point far from th e optimal f rontier . Usually , t he P are to-optima l f rontier cannot be easil y computed, and an altern ativ e and viable app roach is to con sider the following social problem max S 1 ,..., S K K X i =1 u i , (39) subject to the constrain t o f equal SINR, i.e. γ 1 = . . . = γ K = γ , so that fairness amon g users can be ensure d. The above problem can be th us written as max S 1 ,..., S K K X i =1 u i = max p 1 ,...,p K K X i =1 1 p i max s 1 , . . . , s k c 1 , . . . , c k f ( γ ) , (40) with γ the com mon ou tput SIN R. Now , g iv en the co ndition of eq ual SINR acro ss users, it is n atural to assume that the received powers ar e th e sam e f or all th e u sers, i.e. e q. (33) holds. A ssuming that the optimal transmit power for all th e users are smaller than P max , from eq. (33) we h av e p i = P R /h 2 i , thus implying that the social pro blem beco mes max P R 1 P R max s 1 , . . . , s k c 1 , . . . , c k f ( γ ) K X i =1 h 2 i = max P R 1 P R f    max s 1 , . . . , s k c 1 , . . . , c k γ    K X i =1 h 2 i . (41) Since th e rece i ved p owers ar e the same f or all the users, accordin g to [21], [29 ], iteration s (17) converge to a set of WBE sequences, which m inimize the TMSE and , con se- quently , max imize the co mmon SINR. Maxim ization of the common SINR with respect to the spreadin g cod es an d linear receivers of all the users can be thus carried out using iterations (17) after that the condition (3 3) has b een imposed. Let us denote by γ ∗ the cor respond ing max imum com mon SINR; γ ∗ is thu s the SINR achieved by each user in a CDMA system wherein the spreadin g codes are WBE sequen ces, the receivers are MMSE detec tors and e ach user is received with power P R . According ly , we hav e γ ∗ = P R s T k M − 1 s k 1 − P R s T k M − 1 s k , ∀ k = 1 , . . . , K (42) Using eq . (3 6), we obtain, after some algebra P R = γ ∗ N 0 / 2 1 − γ ∗ ( α − 1) , α < 1 + 1 γ ∗ (43) Substituting the above relation into eq . (4 1) the socia lly optimum problem is finally written as max γ f ( γ ) γ N 0 / 2 [1 − γ ( α − 1)] . (44) T aking th e first order deriv a ti ve of th e above functio n with respect to γ we ha ve γ f ′ ( γ ) [1 − γ ( α − 1)] = f ( γ ) . (45) The solution of th is eq uation r epresents the u tility-maximiz ing target SINR for each user in a soc ial optim um con text. V I I I . E X T E N S I O N S T O T H E M U LT I - C E L L S C E N A R I O So far , we have considered the uplin k o f a single- cell scenario, i.e. out-o f-cell interfe rence has been eith er neglected or inc luded in the ad ditive thermal noise. Howev er , real wireless networks are usually multi-cell, a nd users’ u tilities are affected also by th e strategies of out-of -cell interfer ence TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 9 [30]. In what follows, we thus g iv e a brief look at the m ulti- cell case showing how the results of the pr evious sections may be extend ed to this case, poin ting out som e differences with the single-cell scenario, and revealing some interesting ope n issues f or f uture in vestigations. Let us thu s consider the up link of a multi-cell DS/CDMA wireless d ata network. Denote by B the n umber o f acce ss points, and let h i,j be the real ch annel between the j -th user and the i - th AP; moreover, denote b y a ( j ) the in dex of the AP assigned to the j -th user 11 . After chip-match ed filter ing and chip-rate samp ling, the N -d imensional received data vector at the ℓ -th AP , say r ℓ , is written as r ℓ = K X k =1 √ p k h ℓ,k b k s k + n ℓ , ℓ = 1 , . . . , B . (46) The gen eric k -th user da ta is th us decod ed at the a ( k ) -th AP , based on the decision rule b b k = sgn h d T k r a ( k ) i , ( 47) and the k -th user utility is now expre ssed as u k = R L M f ( γ a ( k ) ,k ) p k , (48) where, here, γ a ( k ) ,k is the k -th user SINR at the outpu t o f its linear recei ver in its assigne d AP , and is expressed as γ a ( k ) ,k = p k h 2 a ( k ) ,k ( d T k s k ) 2 N 0 2 k d k k 2 + X j 6 = k p j h 2 a ( k ) ,j ( d T k s j ) 2 . (49) Of related interest is also the k -th user MSE achieved by th e detection r ule (47 ); it is easy to sho w that it is expressed as MSE k = 1 + d T k M a ( k ) d k − 2 √ p k h a ( k ) ,k d T k s k , (50) with M a ( k ) = K X m =1 p m h 2 a ( k ) ,m s m s T m + N 0 2 I N the covariance matrix of the data received that the a ( k ) -th AP . Based on the above d efinitions, non-coopera ti ve gam es fo r energy-efficient resource allocatio n can be co nsidered in a multi-cell setting. Leaving a side for the moment the issue of spreading code allocation , the follo win g result holds. Proposition 3: Con sider a n on-co operative g ame wher ein the k -th u ser utility (48) is maximized with r espect to the choice of the transmit po wer p k ∈ [0 , P k, max ] a nd of th e linear r eceiver d k ∈ R N . A unique Nash equilibrium point ( p ∗ k , d ∗ k ) fo r k = 1 , . . . , K , e xists, wher ein - d ∗ k is the ve ctor corresponding to a linear MMS E r e- ceiver; - p ∗ k = min { ¯ p k , P k, max } , with ¯ p k the k -th user transmit power such that the k -th user SINR γ a ( k ) ,k equals ¯ γ , i.e. the u nique solution of the equ ation f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Proof: T he pro of fo llows alon g the same lines of that of Proposition 2 and is omitted for the sake of bre v ity . 11 Note that we are assuming here t hat each user is assigned to a certain AP , i.e. AP assignments hav e already take n place. The above result states that, if transmit p ower and linear receiver are to be allocated, a unique Nash-e quilibrium point does exist a lso in a multi-cell system. Unfortu nately , thing s are mo re in volved as op timization with resp ect to sp reading codes too c omes into p lay . If the total number K of users in the network does no t exceed the system processing gain N , then the following result holds. Proposition 4 : Consider a non- cooperative game wher ein the k - th user u tility (4 8) is maximized with respect to th e choice of the transmit power p k ∈ [0 , P k, max ] , of the lin ear r eceiver d k ∈ R N and of the spreading co de s k ∈ R N 1 ; assume that K ≤ N . A Nash eq uilibrium point ( p ∗ k , d ∗ k , s ∗ k ) for k = 1 , . . . , K , exist s, w her e in - s ∗ k and d ∗ k ar e the k -th user spreading code a nd receive filter minimizing th e total MSE and can be obtain ed as fixed po ints of the iter a tions d k = √ p k h a ( k ) ,k M − 1 a ( k ) s k , ∀ k = 1 , . . . , K , s k = d k / k d k k , ∀ k = 1 , . . . , K . (51) Denote by γ ∗ k the co rr espo nding SINR. - p ∗ k = min { ¯ p k , P k, max } , with ¯ p k the k -th user transmit power such tha t the k -th user maximum SINR γ ∗ k equals ¯ γ , i. e. the uniq ue solution o f the eq uation f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . This N ash equilibrium point is P ar eto-optima l. Proof: The proof is omitted for the sake of brevity . Basically , the above result states th at if K ≤ N a Nash equilibriu m point does exist wh ich is also Pareto-optimal; this point correspon ds to the globa l minimu m of the total MSE, which is a fixed p oint of iterations (51). H owe ver, further in vestigation is needed to establish if oth er Nash equilibria may exist an d, also, if iteratio ns (51) h av e some other fixed points correspond ing to lo cal minima of the total MSE. Likewise, the case in which K > N , which is the m ost relev an t o ne in a multi-cell network , also merits som e further in vestigation. Th ese tasks are howev er beyond the scope of this paper, an d a thorou gh investigation o f the multi-cell scena rio, which is certainly w orthwhile, is left f or future work. I X . N U M E R I C A L R E S U LT S In this section we illu strate some simulation results that give insight i nto the perfo rmance of the p roposed non-coopera ti ve games, and, also, co rrobo rate th e validity of the ana lytical results of the p revious sections. W e c onsider an uplink DS/CDMA system with processing gain N = 16 , an d assume that the packet length is M = 1 20 ; for this value o f M the equatio n f ( γ ) = γ f ′ ( γ ) can be shown to admit the so lution ¯ γ = 6 . 689 = 8 . 25 dB. A sing le- cell system is consider ed, wherein user s may have rand om positions with a d istance from the AP ranging from 10 m to 1000m . The chan nel coefficient h k for the gen eric k -th u ser is assumed to be Rayleigh distributed with mean equal to d − 1 k , with d k being the distance of user k fr om th e AP . W e take the amb ient noise level to be N 0 = 10 − 9 W/Hz, wh ile the maximum a llowed p ower P k, max is − 25 dBW . W e presen t the TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 10 results o f averaging over 10 0 00 indepen dent r ealizations for the users lo cations, fadin g chann el coefficients and starting set of spreadin g codes. More precisely , for each iteration we random ly generate an N × K -d imensional spreadin g code matrix with entries in the set n − 1 / √ N , 1 / √ N o ; this matr ix is then used as the starting poin t f or th e games that in clude spreading cod e op timization, a nd as the spreadin g code matrix for the games that do not perform spreading code optim ization. Figs. 2 - 4 report the achie ved av erage utility (measured in bits/Joule), the av erage user transmit power and the av erage achieved SINR at the receiver outpu t versus the num ber of users, f or th e g ame in [10] , the game in [8] and for th e non-co operative ga me conside red in Section IV of this paper . Inspecting the cur ves, it is seen that th e p roposed appro ach largely outperform s th e games of [8], [10]. As an example, it is seen that for K = 10 user s the utility achieved by the proposed game is abou t twice that achieved by the game in [10] , i.e. the same am ount of energy c an be used to transmit a doubled bulk of data. In particu lar , it is seen th at for K ≤ N a very substantial perf ormance gain can be obtaine d by resorting to spreading code o ptimization; ind eed, when K ≤ N , u sers can be given orthogonal spreading codes, so that the m ultiaccess channel reduces to a superp osition of K separate single-u ser A WGN channels. It is also seen fr om Fig. 4 th at receivers achieve on the average an output SINR th at is smaller than the target SINR ¯ γ : indeed, du e to fading and distan ce path losses, achieving the tar get SINR would require for some users a tr ansmit power larger than th e m aximum allowed p ower P k, max , a nd so these u sers are not able to achieve the optimal target SINR. A s a con firmation of this, in Fig. 5 w e repor t the fr action of users transmitting at the ma ximum p ower: as expected, th e smaller fraction correspo nds to the proposed game, but it is s een that this fraction is larger than zer o. In order to validate the LSA-ba sed distrib u ted power control algorithm o f Section V , we con sider a system w ith pro cessing gain N = 128 . Fig . 6 repor ts the transmitted power pro file across users for the proposed distributed power control algo- rithm, fo r the algorith m derived by Eq. (1 6) in [13] ( i.e. eq . (21)), and for the conventional power contro l algo rithm of [6], that is non- adaptive and requires a substantial am ount of prior informa tion. It is seen that the propo sed algorith m is capable of reprod ucing the optima l power pro file with very goo d accuracy , while, o n the contra ry , the algo rithm descendin g from p aper [13] overestimates th e require d transmit p owers and d oes no t achieve a goo d p erforma nce. While Fig. 6 shows the resu lt o f just on e simulation trial (n ote howe ver that a similar behavior has been observed in any considered case) the subsequen t three fig ures r eport results com ing f rom an av erage over 1000 in depend ent realization s of th e spread ing codes, ch annel co efficients an d users’ location s. Figs. 7 - 9 show the achieved average utility ( measured in b its/Joule), th e av erage user transmit power and the average achieved SINR at the receiver output versus the numbe r of active users, for the conventional p ower control algo rithms (i.e. for the no n- cooper ati ve ga me o f [10]) , for the pr oposed algor ithm, an d for the power contr ol alg orithm d erived by p aper [13 ]. Results show th at the pr oposed algorithm ach iev es a per forman ce lev el practically indistinguish able f rom that of the standard algorithm , while the algo rithm (21) ac hieves an utility much smaller . Fro m Fig. 9 it is howev er seen th at the alg orithm (21) achieves an ou tput SINR larger than that of the other algorithm s: this should not be interprete d as a sign of go od perfor mance. Ind eed, in the c onsidered scenario the aim of the power contr ol algo rithm is to make each user operate at a SINR e qual to ¯ γ . Finally , we consider an oversaturated system with pro cess- ing g ain N = 64 , and number o f users K = 70 , so th at K > N . I n Fig . 10 we repo rt th e utility profile across users for the n on-co operative game prop osed in Section IV , in comp arison with th e utility pr ofile p redicted accord ing to the content of Section VI.C and with the u tility pro file correspo nding to the socially optim um solu tion with eq ual SINR con straint. It is seen that the performance loss incurred by t he n on-co operative g ame in comparison with the socially optimum solution is q uite negligible, and , also, that the LSA- based p rofile follows with g ood accu racy the actual utility profile. As a conseq uence, this plot corrob orates the validity of ou r asympto tic an alysis, that it is seen to b e useful a lso when the system is actually “not so large”. X . C O N C L U S I O N In th is paper th e cross-layer issue of joint multiuser de- tection, power con trol, and spread ing cod e o ptimization for wireless data networks has been add ressed. First of all, build- ing on the stud y [10 ], we ha ve prop osed a more general non- cooper ati ve ga me wherein also spre ading code optim ization can be used to further increa se the en ergy e fficiency of CDMA-based wireless networks. W e have shown that this game ad mits a uniq ue Nash equilibrium point, that, fo r un- saturated systems, is also Pareto-optimal. For oversaturated CDMA systems, instead, we hav e shown that the socially optimum solution with equa l SINR constraint exhibits a per - forman ce le vel practically co incident with that of the pr oposed non-co operative game. Using LSA, an d assum ing that no spreading code optimizatio n is pe rforme d, a new d istributed power control alg orithm that can b e imp lemented based o n the knowledge of the channel for the user o f interest only has b een propo sed. Ad ditionally , through LSA results we have been able to der iv e the network utility profile for a large CDMA system, for both the cases that e ither spr eading code optimizatio n is carried out or it is not. Moreover , as an introducto ry step to the prop osed non-co operative game, w e have clarified the relationship between th e pro blems of SINR maximization and TMSE minimiza tion in a syn chron ous CDMA system. Fin ally , we have also given a brief look at the mu lti-cell scenar io, an d, while extending some of our results to this case too, we have highligh ted op en issues worth b eing investigated in a f uture work. Numer ical results have c onfirmed the superior ity of the propo sed n on-co operative game with r espect to competin g alternatives, as well as that the LSA-based theoretical formulas describe with g ood accur acy the actual network perf orman ce. A C K N O W L E D G M E N T S The authors wish to thank Dr . Husheng Li for insightful comments o n a prelim inary version o f this pa per . They are TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 11 also g rateful to the Associate Editor, pr of. L ars Rasmu ssen, for his excellent manag ement o f the re view p rocess. A P P E N D I X A Giv en the relation s k = √ p k h k  p k h 2 k D D T + µ k I N  + d k , we s how here ho w to choose th e co nstant µ k so that k s k k = 1 . Let U Λ U T be the eig endecom position o f the matr ix p k h 2 k D D T . Obviou sly , U is an orth onorm al matr ix whose columns are the eig en vectors of p k h 2 k D D T , a nd Λ is the correspo nding diago nal e igenv alue m atrix. Note that some of these eigenv alue w ill be zero for K < N . Now , letting u i and λ i denote the i -th column o f U and the i -th diagonal eleme nt of Λ , respecti vely , and z ( λ i , µ k ) =    1 λ i + µ k if λ i + µ k 6 = 0 0 if λ i + µ k = 0 , (52) it is easy to show that the ab ove spreading cod e up date can be r ewritten as s k = √ p k N X i =1 z ( λ i , µ k ) u i u T i d k . (53) From (53) it is seen that, as µ k → + ∞ , k s k k → 0 , thus implying that there exists a finite constan t Q u such th at k s k k < 1 for any µ k ≥ Q u . Now , let λ m = min i λ i (note that λ m may be 0 if K < N or in general i f DD T is not of full ran k). It is easy to show that, as µ k → λ + m , k s k k → + ∞ . According ly , there e x ists a finite constant Q l > λ m such that k s k k > 1 f or µ k ∈ ] λ m , Q l ] . Since k s k k is mon otonically decreasing f or µ k ∈ [ Q l , Q u ] an d sin ce k s k k > 1 for µ k = Q l and k s k k < 1 fo r µ k = Q u , there exists just one value of µ k , say µ ∗ k , such that k s k k = 1 for µ k = µ ∗ k . The value of µ ∗ k can be f ound u sing s tandard methods. A P P E N D I X B In this appendix we sho w how th e in verse CDF of the f ading coefficients can be co mputed. In ord er to acc ount for bo th fading and path lo ss, we assume that h 2 k is g iv en by th e ratio of two random variables, i. e. h 2 k = α 2 k d n k , wherein α k is an exponential random v ariable (this correspond s to con sidering a Rayleigh fading channel) , while d k is the distance of th e k -th user from the B S; we assume th at d k is unifo rmly distributed in the interval [ R a , R b ] ; typical values may b e R a = 10 m and R b = 50 0 m . Finally n is a non -rando m exponen t; in urba n en vironme nts n is usually taken in the interval [2 , 5] . I t is easy to sho w that the CDF o f h 2 k is gi ven b y F h 2 k ( x ) = Prob  h 2 k ≤ x  = E d k h 1 − e − xd n k i (54) For the case n = 2 , straightforward computatio ns lead to F h 2 k ( x ) = 1 − 1 R b − R a r π x  1 2 erfc ( R a √ x ) − 1 2 erfc ( R b √ x )  , (55) where erfc ( · ) is the comple mentary error fu nction. Th e ab ove equation shou ld now be in verted n umerically in or der to obtain th e in verse CDF . Howe ver, such a n inv ersion may be co mputation ally deman ding, and, mor eover , c losed fo rm expressions for the case n 6 = 2 are no t available. An effecti ve alternative ap proach is the following. The interval [ R a , R b ] can b e p artitioned in a given n umber, say P , of smaller intervals, and the pr obability d ensity f unction o f d k can b e approximated as f d k ( d ) ≈ 1 P P X i =1 δ ( d − R a − i ∆) , with ∆ = R b − R a P . (56) As a conseque nce, we ha ve F h 2 k ( x ) ≈ 1 − 1 P P X i =1 e − xd n i , (57) with d i = R a + i ∆ . Equ ation (57) is nu merically invertible. Indeed , u pon letting e − x = z , we have 1 − 1 P P X i =1 z d n i = y ⇒ P X i =1 z d n i = (1 − y ) P . (58) It is easy to see th at the above eq uation admits a uniq ue solution z > 0 , and a ny standard nume rical equation so lver routine can be used to fin d it; af ter that, we h av e x = − ln( z ) , and this equals F − 1 h 2 k ( y ) . In o ur simulation s we h ave assumed R a = 10 m , R b = 1000 m and P = 200 . R E F E R E N C E S [1] S. V erd ´ u, Multiuser Detecti on , Cambridge Uni versi ty Press, 1998. [2] X. W ang and H. V . Poor , W irel ess Communication Systems: Advanced T echniqu es for Signal Reception . Upper Saddle Ri ver , NJ : Prentice-Ha ll, 2004. [3] C. Comanic iu, N. B. Mandayam and H. V . Poor , W ireless Ne tworks: Multiu ser Detecti on in Cro ss-Layer Design , Springer , 2005. [4] D. Fudenberg and J. Tirole, Game Theory , Cambridge , MA: MIT Press, 1991. [5] A. B. MacK enzie and S. B. Wi cke r , “Game theory in communications: Moti vati on, expla nations, and applicati ons to po wer control, ” P r oc. IEEE Global T elecommun . Confer ence , San Antonio, TX, 2001. [6] R. D. Y ates, “ A framew ork for upli nk power control in cel lular radio systems, ” IEEE J. Sel. A re as Comm. , V ol. 13, pp. 1341-1347, Sep. 1995. [7] D. J. Goodman and N. B. Mandayam, “Power control for wireless data, ” IEEE P ers. Commun. , vol. 7, pp. 48-54, Apr . 2000. [8] C. U. Saraydar , N. B. Mandayam and D. J . Goodman, “Efficie nt power control via pricing in wireless data networks, ” IEEE T rans. Commun. , vol. 50, pp. 291-303, Feb . 2002. [9] C. U. Saraydar , “Prici ng and power control in wireless data networ ks, ” Ph.D. dissertati on, Dept. Elect. Comput. Eng., Rutgers Univ ersity , Pis- cata way , NJ , 2001. [10] F . Meshkati, H. V . Poor , S. C. Schw artz and N. B. Mand ayam, “ An ener gy-ef ficient approac h to powe r control and recei ver design in wire- less data network s, ” IEEE T rans. Comm. , V ol. 53, pp. 1885-1894, Nov . 2005. [11] G. Bacci, M. Luise, H. V . Poor and A. Tul ino, “Energy ef ficient power control in impulse radio UWB networks, ” IE EE J. of Selec ted T opics in Sig. Pro c. , V ol. 1, pp. 508-520, Oct. 2007. [12] F . Meshkati, H. V . Poor and S. C. Schwartz , “Energy-ef ficient re- source alloc ation i n wirele ss ne tworks: an overvie w of game-theoretic approac hes, ” IEEE Signal Proc. Magazine , V ol. 24, pp. 5 8 - 68, May 2007. [13] D. N. C. Tse and S. V . Hanly , “Linear multiuser recei vers: ef fecti ve interfe rence, ef fecti ve bandwidth and user capa city , ” IEEE Tr ans. Inf. Theory , V ol. 45, pp. 641-657, March 1999. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 12 [14] L. G. F . Trichard , J. S. Evans, I. B. C ollings, “Large system analysis of linear multistage paralle l interfer ence cancella tion, ” IEEE T rans. Commun. , V ol. 50, pp. 1778-1786, Nov . 2002. [15] J. Evans and D. N. C. Tse, “Large s ystem performance of linear multiuser recei vers in multipath fading channels, ” IEEE T rans. Inf. Theory , V ol. 46, pp. 2059-2078, Sept. 2000. [16] J. Zhang a nd X. W ang, “Large-syst em performanc e analysis of b lind and group-blind multiuser recei vers, ” IEEE T rans. Inf. T heory , V ol. 48, pp. 2507-2523, Sept. 2002. [17] S. Dey and J. Evans, “Optimal power control in wireless data netw orks with outage-base d utilit y guarantees, ” Proc. of the 42nd IEEE Conf. on Decision and Contro l , Maui (HI), USA, Dec. 2003. [18] T . Alpcan, T . Basar , and S. Dey , “ A power control game based on outage probabil ities for multice ll wireless data networks, ” IEE E T rans. W ir . Commun. , V ol. 5, pp. 890-899, April 2006. [19] C. Lacatus and D. C. Popescu, “ Adapti ve interference av oidance for dynamic wireless systems: a game theore tic approach, ” IEEE J . of Selec ted T opics in Sig. Proc . , V ol. 1, pp. 189-202, June 2007. [20] G. S. Rajappan and M. L. Honig, “Signature sequence a daptat ion for DS/CDMA with multipath, ” IEEE J . Sel. Area s Commun. , V ol. 20, pp. 384-395, Feb . 2002. [21] S. Ulukus and A. Y ener , “Iterati ve transmitte r and recei ver optimiz ation for CDMA networks, ” IEEE Tr ans. W ireless Commun. , V ol. 3, pp. 1879- 1884, Nov . 2004. [22] P . Anigstein and V . Ananthara m, “Ensuring conv ergence of the MMSE iterat ion for interfere nce avoidan ce to the global optimum, ” IEEE Tr ans. Inform. Th. , V ol. 46, pp. 873-885, Sept. 2000. [23] C. Rose, “CDMA code word optimiza tion: Interferenc e avoida nce an d con verg ence via class warfare , ” IEE E T rans. Inf. Th. , vol. 47, pp. 2368- 2382, Sept. 2001. [24] V . Rodriguez, “ An analyti cal foundatio n for resource manage ment in wireless communication, ” Proc. IE EE Global T elecommun. Confer ence , San Francisco, CA, Dec. 2003. [25] S. Ulukus and R. D. Y ates, “Stochastic po wer control for cellul ar radio systems, ” IEEE T rans Commun. , V ol. 46, pp. 784-798, June 1998. [26] J. L uo, S. Ulukus and A. Ephremides, “Standard and quasi-standard stochasti c power control algorithms, ” IEEE T rans. Inf. T heory , V ol. 51, pp. 2612-2624, July 2005. [27] S. Shamai (Shitz) and S. V erd ´ u, “Decoding only the s trongest CDMA users, ” Codes, Graphs and Systems , R. Blahut and R. Koet ter , E ds., pp. 217-228, Kluwer , 2002. [28] H. Li and H. V . Poor , “Power allocat ion and spectral effic ienc y of DS- CDMA syste ms in fading ch annels with fixed QoS-pa rt I: single-rate case, ” IE EE T rans. W ire less Commun. , V ol. 5, pp. 2516-2528, September 2006. [29] P . V iswanath and V . Anantharam, “Optimal sequences and sum capacity of synchrono us CDMA systems, ” IEEE T rans. Inform. Th. , V ol. 45, pp. 1984 - 1991, September 1999. [30] C. U. Saraydar , N. B. Mandayam and D. J. Goodman, “Pricing and po wer control in a multicell wireless data network, ” IEE E J . Sel. Area s Commun. , V ol. 19, pp. 1883-1892, Oct. 2001. Stefano Buzzi (M’98 - SM ’07) was born in Piano di Sorrento , Italy on December 10, 1970. He recei ved wit h honors the Dr . Eng. degree in 1994, and the Ph. D. degre e in Electronic E ngineer ing and Computer Science in 1999, bot h from the Uni versity of Napl es ”Fede rico II”. In 1996 he spent six months at CSELT (Centro Studi e Laboratori T elecomunicaz ioni), T urin, Italy , while he has had short-term visiting appoint ments at the Dept. of Electric al Engineering, Princeton Universit y , in 1999, 2000, 2001 and 2006. He is curre ntly an Associate Professor at the Unive rsity of Cassino, Italy . His curr ent resea rch and st udy inte rests lie in the area of statistical signal processing and resource allocation for wirele ss communications and radar applic ations. Dr . Buzzi was aw arded by the A EI (Associazione E lettrot ecnic a ed Ele ttronic a Italiana) the ”G. Oglietti” scholarship in 1996, and wa s the recipi ent of a N A TO/CNR advanc ed fell o wship in 1999 and of three CNR short-term m obilit y grants. He is currently serving as an Associate Editor for the IEEE Communications Letters . H. Vi ncent Poor (S72, M77, SM82, F87) recei ved the Ph.D. degree in EE CS from Princet on Univ ersity in 1977. From 1977 until 1990, he w as on the fac ulty of the Uni versity of Illinois at Urbana- Champaign . Since 1990 he has been on the fac ulty at Princeton, where he is the Michae l Henry Strater Uni ver sity Professor of Electrical Engineeri ng and Dean of the School of Engineeri ng and Applied Scienc e. Dr . Poor’ s research interests are in the areas of stochasti c analysis, statistical signal processing and their applicati ons in wireless netwo rks and related fields. Among his publica tions in these areas is the recent book MIMO W irel ess Commun icatio ns (Cambridge Univ ersity Press, 2007). Dr . Poor is a member of the National A cademy of E ngineeri ng, a Fello w of the American Academy of Arts and Science s, and a former Guggenhei m Fello w . He is also a Fello w of the Institute of Mathematica l Statist ics, the Optical Society of America, and other organiza tions. In 1990, he served as President of the IEE E Information Theory Society , and in 2004- 2007 he ser ved as the Edit or-in- Chief of the IEEE T ransacti ons on Information Theory . Recent rec ognitio n of his work includ es the 2005 I EEE Educ ation Medal and the 2007 IE EE Marconi Prize Paper A ward. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 13 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ k E k and efficiency function, M=100 Prob. of error−free reception efficiency function Fig. 1. Comparison of probability of error-fre e packet reception and ef ficienc y functio n versus recei ve SINR and for packet size M = 100 . Note the S-shape of both function s. 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 x 10 9 UTILITY (bit/Joule) USERS NUMBER Proposed non−cooperative game Meshkati et al., IEEE TCOM, Nov. 2005 Saraydar et al., IEEE TCOM, Feb. 2002 Fig. 2. Achie ved a verage utili ty v ersus number of acti ve user s for the proposed noncooperati ve game and for the games in references [8] and [10]. The system processing gain is N = 15 . TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 14 0 5 10 15 20 25 −28.5 −28 −27.5 −27 −26.5 −26 −25.5 −25 REQUIRED TRANSMIT POWER (dB) USERS NUMBER Proposed non−cooperative game Meshkati et al., IEEE TCOM, Nov. 2005 Saraydar et al., IEEE TCOM, Feb. 2002 Fig. 3. A verage transmit power versus number of acti ve users for the proposed noncooper ati ve game and for the game in references [8] and [10]. The system processing gain is N = 15 . 0 5 10 15 20 25 −1 0 1 2 3 4 5 6 7 8 9 USERS NUMBER ACHIEVED SINR (dB) Proposed non−cooperative game Meshkati et al., IEEE TCOM, Nov. 2005 Saraydar et al., IEEE TCOM, Feb. 2002 Fig. 4. Achie ved avera ge output SINR versus number of acti ve users for the proposed noncoopera ti ve game and for the game in references [8] and [10]. The system processing gain is N = 15 . TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 15 0 5 10 15 20 25 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 USERS NUMBER FRACTION OF USERS TRANSMITTING AT MAXIMUM POWER Proposed non−cooperative game Meshkati et al., IEEE TCOM, Nov. 2005 Saraydar et al., IEEE TCOM, Feb. 2002 Fig. 5. A verag e fraction of users transmitting at their m aximum allowe d powe r versus number of acti ve users for the proposed noncooperat i ve game and for the game in reference s [8] and [10]. The system processing gain is N = 15 . 0 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 User index power profile across users N=128, K=120, Rayleigh fading, linear MMSE receiver Proposed algorithm Conventional algorithm Hanly & Tse formula Fig. 6. T ransmitte d power profile across users for the proposed distribute d algori thm based on LSA, the con vent ional po wer control algorithm [6] and the profile deriv ed according to the algorithm in [13]. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 16 0 20 40 60 80 100 120 140 0 0.5 1 1.5 2 2.5 x 10 9 UTILITY (bit/Joule) USERS NUMBER Linear MMSE: Hanly & Tse algorithm Linear MMSE: proposed algorithm Linear MMSE: standard algorithm Fig. 7. A verage utilit y versus number of users for the proposed distribute d algorithm based on LSA, for the centraliz ed implementati on of reference [10] and for the distribut ed algorithm based on the power control algorithm of reference [13]. 0 20 40 60 80 100 120 140 −28.5 −28 −27.5 −27 −26.5 −26 −25.5 REQUIRED TRANSMIT POWER (dB) USERS NUMBER Linear MMSE: Hanly & Tse algorithm Linear MMSE: proposed algorithm Linear MMSE: standard algorithm Fig. 8. A verage transmit powe r ve rsus number of users for the proposed distrib uted algorithm based on LSA, for the centra lized implementat ion of reference [10] and for the distrib uted algori thm based on the po wer control algorithm of reference [13]. TO APPEAR ON IEEE J SA C - SPECIAL ISSUE ON MUL TIUSER DETE CTION FOR ADV ANCED COMMUNICA TION SYSTE MS AND NETWORKS 17 0 20 40 60 80 100 120 140 6 6.5 7 7.5 8 8.5 9 9.5 10 ACHIEVED SINR (dB) USERS NUMBER Linear MMSE: Hanly & Tse algorithm Linear MMSE: proposed algorithm Linear MMSE: standard algorithm Fig. 9. A verage achie ved SINR versus number of users for the proposed distrib uted algorithm based on LSA, for the centraliz ed implementa tion of reference [10] and for the distrib uted algori thm based on the po wer control algorithm of reference [13]. 10 20 30 40 50 60 70 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 user index utility profile proposed non−cooperative game utility profile estimate social optimum Fig. 10. Utilit y profile for the proposed non-cooperat i ve game, in comparison with the social optimum and with the utilit y profile predic ted by large s ystem analysi s. Here the processing gain is N = 64 and the number of users is K = 70 .