Energy-Efficient Resource Allocation in Multiuser MIMO Systems: A Game-Theoretic Framework

ENERGY -EFFICIENT RESOURCE ALLOCA TION IN MUL TIUSER MIMO SYSTEMS: A GAME-THEORETIC FRAMEWORK Stefano Buzzi 1 , H. V incent P oor 2 , and Daniela Saturnino 1 1 University of Cassino, D AEIMI 03043 Cassino (FR) - Italy; { buzzi, d.saturnino } @unicas.it 2 Princeton University , School of Engineering and Applied Science Princeton, NJ, 0854 4 - USA; p oor@prince ton.edu ABSTRA CT This paper focuses on the cross-layer issue of resource al- location for energy efficiency in the uplink of a mu ltiuser MIMO wireless commun ication system. Assuming that all of the transmitter s and the u plink receiv er are equipp ed with multiple antennas, the situation considered is that in which each terminal is allowed to vary its transmit power , beam - forming vector, and uplink receiver in or der to maximize its own utility , which is defin ed as the ratio of data throug hput to transmit power; the case in which non-linea r interferen ce cancellation is u sed at the recei ver is also investi gated. Ap- plying a game-theo retic formu lation, several non-co operative games for utility maximization are thus for mulated, and their perfor mance is compared in term s of achie ved average util- ity , ach ie ved average SINR and average tr ansmit power at the Nash equilibr ium. Numerical results show that the use of the proposed cross-layer reso urce allocation p olicies brings remarkab le advantages t o the network perfor mance. 1. INTR ODUCTION The incr easing deman d for new wireless app lications, and the tre mendou s pro gress in the development of smartp hones and handheld d evices with exceptional co mputing capabili- ties requires wireless communication infrastructur es capable of deli vering data at hig her and higher data-r ates. The use of multiple antennas a t b oth e nds of a wir eless link ha s p roved to be a ke y technolo gy to improve the spe ctral efficiency of wireless ne tworks [ 1]. Like wise, in telligent resou rce allo- cation procedur es also will play a prominent ro le to ensure reliability and efficienc y in future wireless data networks. This pa per f ocuses on the u plink of a multiuser multiple-inp ut multiple-outpu t (MI MO) commun ication sys- tem, wherein both the mobile terminals and the c ommon ac- cess po int (AP) are equ ipped with multiple antenna s. W e are interested in the d esign of non- coopera ti ve re source al- location po licies aimed at energy efficiency ma ximization, which is defin ed here as th e numb er of reliably delivered in- formation symbols p er u nit-energy taken fro m the b attery . Energy-efficiency maximization is in deed a crucial prob lem in m obile wireless comm unications, wherein mob ile users are interested in makin g a careful and smart use of the en- ergy stored in their battery . Following a recent trend, we use game theory tools [2] in ord er to o btain non-co operative re- source allocation procedures, maximizing each user’ s energy This research w as sup ported in par t b y t he U. S. Nationa l Scienc e Foun- dation under Grants ANI-03-38807 and CNS-06-25637. efficiency with respect to its o wn transmit po wer , beamfor m- ing vector and uplink receiv er . A game-th eoretic frame work for non-coop erativ e energy efficiency m aximization has bee n widely ap plied in the re- cent past to design resource allocation policies for code di vi- sion multiple access (CDMA) systems [3, 4, 5] and for ultra- wideband (UWB) sy stems [ 6]. On the o ther h and, MIMO commun ication systems h av e r eceiv ed a great d eal o f at- tention in the last decade (see, for instance, the ref erences in the recent textbook [1]) . Among the studies addr essing joint transmitter and recei ver adap tation for improved per for- mance, we cite the pap ers [7, 8], which consider transceiv er optimization for multiuser MIMO systems in cooperative en- vironm ents, i.e. as suming that a central processor a llocates resources a mong a cti ve users, and neglecting th e issue of power con trol. In this paper , we extend th e game-theore tic framework, surveyed in [9], to multiuser MIMO wireless systems. W e consider the case in which en ergy efficiency is to b e m axi- mized with respect to a. the transmit power of each user, ass uming matche d filter - ing at the receiv er; b . the transmit power and the choice o f the uplink linear re- ceiv er for each user; c. the tran smit power , the beamfo rming vector and the choice of the uplink linear receiv er for each user; and d. the transmit power and the choice of the n on-linear serial interferen ce cancellation (SIC) u plink receiver for each user . Note th at con sideration of the se games is not a tri vial exten- sion of the results reported in authors’ pre vious stud ies, since the analysis of the Nash equilibrium (NE) points for some of the above games, an d in particu lar fo r the cases c. and d. poses new mathem atical challeng es. More precisely , we will see that proble ms a. and b . are somewhat equ i valent to those treated in [3 ] a nd [4] for a CDMA system, while, instead proof of the existence of a NE for problem c. requires a new and different approach. Finally , the consideration of problem d. , which assumes the use of a non-linear SIC recei ver , has not yet appeared in the open literature. Results will show that the use of advanced resource allo- cation po licies brings rema rkable improvements in te rms of achieved energy ef ficiency at the equilibrium thus enabling the transmission of larger bulks of data for a gi ven amount of energy stored in the battery . 2. PRELIM INARI ES AND PR OBLEM FORMULA TION Consider the up link of a K -user synch ronous, single-cell, MIMO m ultiuser sy stem subject to flat fading. Den ote b y N T the number of transmit antennas for each user, and by N R the number of receive an tennas at the comm on AP . Collect- ing in an N R -dimension al vector , say r , th e samples at the output of the receiv er front-en d filter and correspon ding to one symbol interval, we have r = K ∑ k = 1 √ p k H k a k b k + n , (1) wherein p k is the tran smit power of the k -th user 1 , b k ∈ {− 1 , 1 } is the information sym bol o f the k -th user , and H k is the real 2 N R × N T matrix channel gain between the k -th user’ s transmitter and the AP; the entries of H k depend on both the distance of the k -th user’ s termin al from the AP and on the fading fluctuatio ns. Th e N T -dimension al vector a k is the be amformin g vector of the k -th user; we assume that a T k a k = k a k k 2 = 1, with ( · ) T denoting transpo se. Finally , n is th e am bient n oise vector, which we a ssume to be a zero- mean white Gau ssian random pr ocess with covariance matr ix ( N 0 / 2 ) I N R , with I N R the identity matrix of order N R . Assume n ow that e ach mobile te rminal sends its data in packets of M bits, an d that it is interested both in h aving its data r eceiv ed with as small as possible e rror pr obability at the AP , an d in making careful use of the energy stored in its battery . O bviously , these are conflictin g goals, since erro r- free re ception may be achieved b y in creasing the transmit power , which of co urse co mes at the expense of battery life . A useful appro ach to q uantify these conflicting goals is to define the utility of the k -th user as the ratio of its throughpu t, defined as the numb er of informatio n bits tha t are received with no error in unit time, to its transmit power [3] - [6], i.e. u k = T k / p k . (2) Note that u k is m easured in bit/Joule, i.e. it r epresents the number o f successful bit transmissions that can be m ade f or each energy-unit drain ed from the battery . Denoting by R the common rate of the network ( exten- sion to th e case in which each user transmits with its own rate R k is quite simple) and assuming that each packet o f M symbols contains L in formation symbols and M − L overhead symbols, reserved, e.g., for channel estimatio n an d/or par ity checks, denoting by γ k the Signal-to-Interf erence-plu s-Noise Ratio (SINR) for the k -th user at the receiver output, and fol- lowing the reasoning of [3, 4] , a faithful and m athematically tractable approxim ation for the utility u k in (2) is the follow- ing u k = R L M f ( γ k ) p k , ∀ k = 1 , . . . , K . (3) In the above equatio n, f ( γ k ) is the so-called efficiency func- tion , appro ximating th e p robability of successful (i.e. erro r- free) packet recep tion. As an example, for BPSK modula- tion, the choice f ( γ k ) = ( 1 − e − γ k ) M is widely accepted. The 1 T o simplify subsequent notation, we assume that the transmitted power p k subsumes also the gain of the transmit and recei ve ante nnas. 2 W e assume here, for simplicity , a real channel m odel; gene raliza tion to practi cal channels, with I and Q components, is straightforwa rd. results of this paper, howev er , hold not only fo r this particu- lar choic e, but for any efficiency fu nction f ( · ) th at is increas- ing, S-shap ed, app roaching unity as γ k → + ∞ , and such that f ( γ k ) = o ( γ k ) for vanishing γ k . Now , based on the utility definition (3), many interesting questions arise concernin g how each user may maximize its utility , and how this max imization af fects utilities achie ved by othe r users. Game th eory provides means to study these interactions and to provide som e usef ul an d insightf ul an- swers to the se questions. It has b een applied in this co ntext mainly as a tool to study non-co operative resource allocation proced ures for CDMA systems and for UWB communica - tions. In th e following, instead , we address the p roblem of non-co operative en ergy efficiency maximization in multiuser MIMO systems. 3. NON-COOPERA TIVE RESOURCE ALLOCA TION: LINEAR RECEIVER A linear receiver detects the data symbol b k , according to the decision rule b b k = sign  d T k r  , (4) with b b k the estimate of b k and d k ∈ R N R the N R -dimension al vector representing the recei ve filter for the user k (the set R is th e real field). It is easily seen that the SI NR γ k can b e written as γ k = p k ( d T k H k a k ) 2 N 0 2 k d k k 2 + ∑ i 6 = k p i ( d T k H i a i ) 2 . (5) In the following, we consider non-co operative resource allo- cation gam es aimed at energy ef ficiency maximization with respect to (a) the transmit power , assumin g that a matched filter is used at the recei ver; (b) the transmit power a nd the linear uplink receiver; and (c) the transmit power , the beam- forming vector and the choice of the linear uplink receiv er . 3.1 T ransmit power control with matched filtering Assume that the k - th user’ s beamforming vector is taken par- allel to the eigenvector co rrespond ing to the max imum eigen- value of the matrix H T k H k , and that a classical matched filter is u sed at the AP , i.e. we have d k = H k a k . The k -th user SINR is now e xpressed as γ k = p k k H k a k k 4 N 0 2 k H k a k k 2 + ∑ i 6 = k p i ( a T k H T k H i a i ) 2 . (6) Considering the non-c ooperative g ame max p k ∈ [ 0 , P k , max ] f ( γ k ) p k , k = 1 , . . . , K , (7) with P k , max the maximum allowed transmit po wer for the k -th user , the following result can be proven. Proposition 1: The non- cooperative ga me defin ed in ( 7) admits a u nique NE point ( p ∗ k ) , fo r k = 1 , . . . , K , wher ein p ∗ k = m in { ¯ p k , P k , max } , with ¯ p k the k-th user transmit power such that the k- th user maximum SINR γ ∗ k equals ¯ γ , i.e. the unique solution of the equation f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Proof: The pr oof is here sketched as a lead-in to the exposi- tion o f the full M IMO case of th e forth coming Section 3.3. According to theo rem 1 1 in [3], a NE in a non-coope rativ e game exists if th e strategy set S k is a n onempty , conv ex, and compact subset of an Eu clidean space, and if the utility function of each player of the game is quasi-conc a ve in its own po wer (this means that there exists a point below which the function is no n-decreasin g, and above wh ich the func- tion is no n-increasing ). In the consider ed gam e, we have S k = [ 0 , P k , max ] , so th e fo rmer cond ition is obvio usly ful- filled; to verify the latter cond ition, it suffices to show that the utility functio n u k is increasing in an ε -neighb orhoo d of p k = 0 and that the first or der partial deriv ativ e of u k with respect to p k has only one zer o for p k > 0. Note that the utility u k equals zero fo r p k = 0 , and is positive for p k = 0 + , thus implying that it is an increasing fu nction for p k ∈ [ 0 , ε ] . Consider now the par tial deri vati ve of u k ( · ) with respect to p k and equ ate it to zer o; since, gi ven Eq. ( 6), it is seen that γ k = p k d γ k / d p k , each user’ s utility is maximized if each user is ab le to achieve the SI NR ¯ γ , that is the un ique 3 solution of th e equation f ( γ ) = γ f ′ ( γ ) . The existence of an NE is thus proven. Given the uniq ueness of the utility-max imizing SINR ¯ γ , and the b i-injective corr espondenc e between the achieved SINR and the tra nsmit power for each user, th e above NE is also unique . In practice, th e a bove NE is reached th rough the follow- ing iterative alg orithm. Giv en any set of transmit p owers, the standard power con trol itera tions as detailed in [10] are used so that each user may either achiev e its target SINR or , should this be not possible, transmit at its maximum allowed power . 3.2 T ransmit power control and choice of the linea r re- ceiver Consider now the following non-coo perative game: max p k ∈ [ 0 , P k , max ] , d k ∈ R N R f ( γ k ) p k , k = 1 , . . . , K . (8) W e have no w max p k , d k f ( γ k ) p k = max p k f ( max d k γ k ) p k , (9) i.e. we can first take care of SINR maximiz ation with respect to d k and then co nsider th e problem o f utility m aximization with respect to p k . It is well-known that, among linea r re- ceiv ers, the m inimum mean square er ror (MMSE ) receiver is th e o ne th at m aximizes th e SI NR. As a consequence, we have the following result. Proposition 2: The non-coop erative game defined in (8) ad- mits a unique 4 NE point ( p ∗ k , d ∗ k ) , for k = 1 , . . . , K , wher ein - d ∗ k = √ p k M − 1 H k a k is the MMSE r eceiver fo r the k-th user , with M = ( ∑ K k = 1 p k H k a k a T k H T k + N 0 2 I N R ) the data covariance matrix. Denote by γ ∗ k the corr esponding SINR. - p ∗ k = min { ¯ p k , P k , max } , with ¯ p k the k- th user transmit power such that the k -th user maximum SI NR γ ∗ k equals 3 Uniquene ss of ¯ γ is ensured by the fact that the ef ficienc y function is S-shaped [3]. 4 Here and in the follo wing uniqueness with respect to the recei ver d k is meant up to a positi ve s cali ng factor . ¯ γ , i.e. the unique solutio n of the equa tion f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Proof: For the sake of b revity , we just sk etch the key parts of this proo f. W e ha ve alre ady discu ssed t he f act that SINR ma ximization re quires th at th e receiver d k is the MMSE detecto r . It is easy to show that, with MMSE de- tection, the SINR for the k -th u ser is expr essed as γ k = p k a T k H T k M − 1 k H k a k , with M k = M − p k H k a k a T k H T K the co - variance matrix of the interference seen by the k -th user . No te that the relatio n γ k = p k d γ k / d p k still holds h ere, thus imply - ing that th e argu ments of the proof o f Proposition 1 can be easily borr owed in orde r to show existence and uniqueness of the NE for the game (8). In practice, th e above NE is reached thro ugh the follow- ing iterative alg orithm. Given any set of tra nsmit powers, each user sets its uplink receiver eq ual to the MMSE re- ceiv er . Af ter that, users ad just their transmit power in order to achieve the target SINR, using the standard power con trol iterations of [10]. Th ese steps are repeated until conv ergence is reached. 3.3 T ransmit power co ntrol, beamforming, and choice of the linear receiver Finally , let us now consid er th e mo re challeng ing case in which utility maximization is perfor med with respect to the transmit power , b eamform ing vector and choice of the uplink linear receiver , i.e. max p k ∈ [ 0 , P k , max ] , d k ∈ R N R , a k ∈ R N T 1 f ( γ k ) p k , k = 1 , . . . , K , (10) with R N T 1 the set of unit-norm N T -dimension al vectors with real entries. No te that max p k , d k , a k f ( γ k ) p k = max p k f ( max d k , a k γ k ) p k . (11) Giv en the above equation, we ha ve to consider first the prob- lem of SINR m aximization with respect to the vecto rs d k and a k . Again, th e SINR-maximizing linear r eceiv er is the MMSE receiver . Sin ce, as already d iscussed, the k - th user SINR for MMSE detection is γ k = p k a T k H T k M − 1 k H k a k , it is easily seen that the SI NR-maximizing beam forming vector a k is the eigen vecto r correspond ing to the maximum eigen- value of th e ma trix H T k M − 1 k H k . Of course, the question now arises if, when such beamforming vecto r update is cycli- cally perf ormed by the acti ve users, a stab le equilibriu m is reached. The follo wing result ho lds. Theorem: Assume tha t the active users cyclically up date their beamfo rming vec tors in or der to m aximize their own achieved SINR at the output of a linear MMSE r eceiver . This pr oc edur e con verges to a fixed point. Proof: Denote b y a 1 , . . . , a K the set of cu rrent bea mformer s for the active users. T he system sum capacity is well-known to be expressed as C SUM = 1 2 log ( det ( M )) − 1 2 log  det  N 0 2 I N R  = 1 2 log ( det ( M k + p k H k a k a T k H T k )) − 1 2 log  det  N 0 2 I N R  . (12) Exploiting the relation det ( A + xy T ) = det ( A )( 1 + y T A − 1 x ) , the sum capa city is also written as C SUM = 1 2 log det ( M k )  2 N 0  N R ! + 1 2 log  1 + p k a T k H T k M − 1 k H k a k  . (13) The und erlined term in the ab ove equation is the k -th user SINR at the output of its MMSE receiv er . According ly , if the k -th user updates its beamformin g v ector with th e eigenvec- tor correspon ding to the m aximum eigenv alu e of the matrix H T k M − 1 k H k , the system sum cap acity is increased. Iterating this reasoning, it can be shown that ev ery time that a user updates its own beamfo rming vector this leads to an increase of the system sum capacity . Since sum capacity is obviously upper bounded , this pro cedure must admit a fixed point. Equipp ed with the above result, and assuming 5 that the equilibrium SINR resulting for th e n on-coo perative SINR maximizatio n game with re spect to th e vectors d k and a k is co ntinuous with r espect to the transmit p owers p 1 , p 2 , . . . , p K , we are now r eady to state ou r result on the game (10). Proposition 3: Th e non- cooperative ga me defin ed in (10) admits a NE point ( p ∗ k , d ∗ k , a ∗ k ) , for k = 1 , . . . , K , wher ein - a ∗ k and d ∗ k ar e th e equilibrium k- th user beamfo rm- ing vector and r eceive filter resulting fr om the non- cooperative SI NR ma ximization game. Deno te b y γ ∗ k the corr esponding SINR. - p ∗ k = min { ¯ p k , P k , max } , with ¯ p k the k- th user transmit power such that the k -th user maximum SI NR γ ∗ k equals ¯ γ , i.e. the uniq ue solution of the equa tion f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Proof: The complete proof is omitted due to lack o f space. Note that the considered gam e can be seen as the com position of two separab le games, name ly the power control game and the beamformer plus receive filter game. The former ga me admits a unique NE based on Proposition 1 , while the latter game admits a NE based on th e previous theor em on su m capacity . Exp loiting the results of [11], the e xistence of a NE for the two subgames implies the existence of a NE for the game (10). 4. NON-COOPERA TIVE RESOURCE ALLOCA TION: NO N-LINEAR SIC RECEIVER Consider now the case in wh ich a non-linear decision feed- back recei ver is used at the receiver . W e assume that th e users are indexed ac cording to a no n-increasin g sorting o f their channel- induced sig natures, i.e. we assume that k H 1 a 1 k > k H 2 a 2 k > . . . , k H K a K k . W e consider a SIC recei ver wherein detection o f the bit fr om th e k - th user is made accord ing to the following rule b b k = sign " d T k r − ∑ j < k √ p j H j a j b b j !) . (14) Otherwise stated , when detecting a certain symbol, the con- tribution fr om the info rmation symb ols that have bee n al- ready detected is subtra cted from the received data. If past 5 Actuall y , this assumption has been numerical ly tested ; its formal proof ho we ver , appears a little in vol ved and is the object of current in ve stigati on. 2 3 4 5 6 7 8 9 10 11 12 10 6 10 7 10 8 10 9 10 10 10 11 AVERAGED ACHIEVED UTILITY (bits/Joule), Nt = 4 users number MMSE with beamforming (Nr = 4) MMSE with beamforming (Nr = 8) MMSE (Nr = 4) MMSE (Nr = 8) SIC−MMSE (Nr = 4) SIC−MMSE (Nr = 8) Matched Filter (Nr = 4) Matched Filter (Nr = 8) Figure 1: Achiev ed average utility at the NE versus the users’ number for the propo sed non- coopera ti ve g ames. decisions are correct, u sers that are detected later enjoy a considerab le reduction of multiple access interf erence, and indeed the SINR for user k , und er the assumption of corre ct- ness of past decisions, is written as γ k = p k ( d T k H k a k ) 2 N 0 2 k d k k 2 + ∑ j > k p j ( d T k H j a j ) 2 . (15) Now , gi ven rec ei ver (14) and the SINR expression (15), we consider here the problem of utility m aximization with re- spect to the transmit po wer , and t o t he choice of t he recei vers d 1 , . . . , d K , i.e.: max p k , d k f ( γ k ( p k , d k )) p k , ∀ k = 1 , . . . , K . (16) The following result can be shown to hold. Proposition 4: D efine ˜ H k = [ H k a k , . . . , H K a K ] , and P k = diag ( p k , . . . , p K ) . The non-coop erative game defined in (16) admits a uniqu e NE poin t ( p ∗ k , d ∗ k ) , for k = 1 , . . . , K , wher ein - d ∗ k = √ p k ( ˜ H k P k ˜ H T k + N 0 2 I N ) − 1 H k a k is th e unique k -th user r eceive filter 6 that maximizes the k-th user’ s SINR γ k given in (15). Deno te γ ∗ k = max d k γ k . - p ∗ k = min { ¯ p k , P k , max } , with ¯ p k the k- th user transmit power such tha t the k-th u ser max imum SINR γ ∗ k equals ¯ γ , i.e. the unique solutio n of the equa tion f ( γ ) = γ f ′ ( γ ) , with f ′ ( γ ) the derivative of f ( γ ) . Proof: The proo f is om itted her e du e to lack of sp ace. It can be ma de alon g the same track that le d to the pro of o f Proposition 2. 5. NUMERICAL RESUL TS In this section we present some simulation results that give insight into the perform ance of the propo sed no n-coop erative resource allocation policies. 6 Uniquene ss is here up to a positiv e scaling fact or . 2 3 4 5 6 7 8 9 10 11 12 −42 −40 −38 −36 −34 −32 −30 −28 −26 −24 AVERAGED TRANSMIT POWER (dB), Nt = 4 users number maximum power MMSE with beamforming (Nr = 4) MMSE with beamforming (Nr = 8) MMSE (Nr = 4) MMSE (Nr = 8) SIC−MMSE (Nr = 4) SIC−MMSE (Nr = 8) Matched Filter (Nr = 4) Matched Filter (Nr = 8) Figure 2: A verage transmit power a t the NE versus the users’ number for the propo sed non- coopera ti ve g ames. W e consider an uplink mu ltiuser MI MO system u sing un- coded BPSK and conside r the correspon ding efficiency f unc- tion f ( γ k ) = ( 1 − e − γ k ) M . W e consider N T = 4 transmit an - tennas fo r each user, and assume tha t the packet leng th is M = 1 20; for th is value of M the equa tion f ( γ ) = γ f ′ ( γ ) can be shown to admit the solution ¯ γ = 6 . 689 = 8 . 25dB. The system d ata rate is R = 10 5 bps. A sing le-cell system is con- sidered, wherein users m ay ha ve rando m positions with a d is- tance fro m the AP ranging from 10m to 1000m. The chan- nel matrix H k for the gen eric k -th user is assumed to have Rayleigh distributed entries with mean equal to d − 1 k , with d k being the distance of user k from the AP . W e take the ambi- ent n oise level to be N 0 = 10 − 9 W/Hz, while the maximum allowed power P k , max is − 25 dBW . W e p resent the results of av eraging over 3000 independ ent realization s for the u sers locations and fading chan nel coefficients. T he beamf orm- ing vector of the gen eric k -th user is chosen as the eigen- vector correspondin g to the max imum eigen value of the ma- trix H T k H k ; this vector is the n used as the starting poin t for the games th at in clude beamforme r optimization, and as the (constant) beamforme r for the remain ing gam es. Figs. 1 - 3 sho w the achie ved average u tility (measured in bits/Joule), the av erage user transmit power and the average achieved SINR at the receiv er output versus the number of users, for th e sev eral considered games, and for a 4 × 4 and 4 × 8 MIMO system. Inspecting the curves, it is seen that smart resou rce allocation algor ithms may bring very remark - able performan ce improvements. As an example, for K = 10 users and N R = 8 the utility achieved by the SIC-MM SE game and by the MMSE +Beamformin g gam e is about 660 and 330 times larger (!! ) than the utility achieved b y th e power allocation gam e coupled with a matched filter , respec- ti vely . Interestingly , it is seen that the SIC-MMSE game out- perfor ms the MMSE+ Beamforming game for low num ber of users an d for large number of user s: indeed, in a lightly loaded system beamfor ming may no t yield substantial per- forman ce imp rovements, while, for heavily loaded systems, SIC pro cessing is e xtremely beneficial. It is also seen from Fig. 3 that in m any instances receivers achieve on the a ver - age an output SINR that is smaller th an the target SINR ¯ γ : indeed, due to fading and distance path losses, achie ving the target SINR would requir e some users to a transmit at high er 2 3 4 5 6 7 8 9 10 11 12 −8 −6 −4 −2 0 2 4 6 8 10 AVERAGED ACHIEVED SINR (dB), Nt = 4 users number SINR target MMSE with beamforming (Nr = 4) MMSE with beamforming (Nr = 8) MMSE (Nr = 4) MMSE (Nr = 8) SIC−MMSE (Nr = 4) SIC−MMSE (Nr = 8) Matched Filter (Nr = 4) Matched Filter (Nr = 8) Figure 3: Achieved average SINR at the NE versus the users’ number for the propo sed non- coopera ti ve g ames. power than the maximu m allowed power P k , max , and so these users are n ot able to achieve the op timal target SINR. Of course, th e use of cr oss-layer resource alloca tion procedu res help reducing the gap between the a verag e and t arget SINRs. 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