Non-atomic Games for Multi-User Systems

1 Non-atomic Games for Mu lti-User Systems Nicolas Bonneau ∗ , M ´ erouane Debbah † , Eitan Altman ∗ , and Are Hjørungnes ‡ ∗ MAESTR O, INRIA Sophia Antip olis, 2004 Route des Lucioles, B.P . 93, 06902 Sophia Antipolis, France Email: {nicolas.bon neau, eitan.al tman}@sophia.inria.fr † Mobile Communications Group, Institut Eurecom, 222 9 Route des Cretes, B.P . 193, 06904 Sophia Antipolis, France Email: merouane .debbah@e urecom.fr ‡ UniK–Uni versity Graduate Center , Uni versity of Osl o, Instituttv eien 25, P . O. Box 70, N-2027 Kjeller , Norway Email: arehj@un ik.no Abstract — In this contribution, the performance of a multi-user system is analyzed in the context of f requency selective f ading channels. Using g a me theoretic too ls, a useful framework is provided in order to determine the optimal power allocation when users know only their own channel (while perfect channel state information is assumed at t he base sta tion). W e consider the realistic case of frequency selective cha nnels for uplink CDMA. This scenario illustrates the case of decentralized schemes, where limited information o n t he net work is available a t the terminal. V arious receivers are co nsidered, namely the Matched filter , the MMSE filter and the opt imum filter . The g oal of this paper is to derive simple ex pressions for the non-c ooperative Nash equilibrium as the number o f mobiles becomes la rge and t he spreading length increases. T o that end two asymptot ic methodolog ies are co mbined. The first is asymptotic random matrix theory which allows us to o bt ain explicit expressions of the impa c t of all other mobiles on any given tagged mobile. The second is the theory of non-a tomic games which computes good approximations of the Nash equilibrium as the number of mobiles grows. 1 I . I N T RO D U C T I O N Resource allocation is of maj or interest in the context of multi -user sy stems. In t he up link multi- user systems, it is important for users to t ransmit with enough power to achie ve their requested qual- ity of service, but also t o min imize the amount of interference caused t o other users. Thus, an ef ficient 1 This work was supp orted by the BIONETS project http://www.bi onets.org/ and by the Research Council of Norway through the OPTIMO project “Optimized Heterogeneous Multi-user MIMO Networks”. power allocation m echanism allows to prev ent an excessi ve consumption of the limited ressources of the users. The m ost s traightforward way to design a power allocation (P A) mechanism is as a centralized pro- cedure, with the base station recei vi ng training sequences from the users and signali ng back the op- timal power allocatio n for each user . Po wer control schemes in cellular systems we re first introduced for TDMA/FDMA [1], [2]; mo re recently an o ptimal scheme was deri ved for Code Division Multiple Access (CDMA) [3]. In order to achiev e t he optimal capacity , the users may also be sort ed according to some rule of precedence [4]. Howe ver , t his in volv es a non negligible overhead and n umerous non infor- mational transmissions. In addit ion, th e com plexity of centralized s chemes increases drastically with the number of users. As discussed in [5], centralized algorithms generally do not h a ve a practical use for real systems, but provide useful bounds on the performance that can be attained by distributed algorithms. A way to avoid the con straints of a centralized procedure is to implement a decentralized one where each user calculates its est imation of the optimal transmissio n power according to its local kno wledge of the system. This is, for example, the case in ad-hoc networks application s. Most of the time, a d istributed algo rithm means an iterative version of a centralized one. M obiles u pdate their power allocation according to som e rule based on the limited informatio n they retrie ve from the system. 2 Supposing that an optimal power allocation exists, a distri buted iterative algorit hm is derived from a diffe rential equation i n [6] and it s con ver gence is proven analyti cally . A distributed version o f th e al- gorithm of [2] i s presented i n [7]. Building on t hese results, a general framew ork for power control in cellular systems i s gi ven in [8]. A re view of diffe rent methods of centralized and distributed power control in CDMA systems is g iv en in [5]. In thi s cont ext, a natural framework is game theory , which studi es competition (as well as co- operation) bet ween i ndependent actors. T ools of game theory have already b een frequently used as a central framew ork for mo deling competition and cooperation in networking, see for example [9] and references therein. Building on the framew ork of [8], a g ame theoretic approach was in troduced in [10], [11]. Numerous works on power allocatio n games have followed since, a selecti on of which we present in Sec. II. Game theory can be used to treat t he case of any number o f players. Ho wever , as the size of the sys tem i ncreases, the number of parameters increases d rastically and it is difficult to gain i nsight on the expressions obtained. In order to obt ain expressions d epending only on few parameters, we consider the system i n an asymptotic setting , letting bo th th e number of users and the sp reading factor tend to infinity with a fixed ratio. W e use to ols of rando m m atrix theory [12] to analyze the syst em in this limit. Random matrix the- ory is a field of m athematical physics that has been recently applied to wireless communication s to an- alyze v arious measures of interest such as capacity or Signal to Int erference plus Noi se Ratio (SINR). Interestingly , it enables to single out the m ain pa- rameters of interest that d etermine the performance in numerous models of comm unication systems with more or less inv olved mo dels of attenuation [13], [14], [15], [16]. In addition, thes e asymptot ic results provide good approximati ons for the practical finit e size case, as sho wn by sim ulations. In th e asymptoti c regime, the non-cooperativ e game becomes a non-atomic one, in which the impact (through interference) of any single mobile on the performance of other mob iles is negligible. In the networking game context, the related sol u- tion concept is often called W ardrop equi librium [17]; it is often much easier t o comp ute than the original Nash equilibrium [9], and yet, the former equilibrium is a good approximati on for th e latter , see details in [18]. In this paper , we derive t he non- atomic equ ilibrium, which generally correspon ds to a non-uniform P A for the users. The non-atom ic Nash equili brium is stu died in this paper for several lin ear recei vers, namely the matched filter and the MMSE filter , as well as non- linear filters, such as the successive interference cancellation (SIC) [19] version of those filters. Howe ver , i n order to perform SIC, the users need to k now their decodi ng order , in order to adjust their rates. In this paper , we in troduce ways of obtaining an ordering of the users in a dis tributed manner . The ordering can be determined simp ly in a distributed manner under weak hypotheses. Th is giv es ris e to a different k ind of power allocation, that depend explicitl y on the order in which the users are decoded. Moreover , we quantify th e gain of the non- uniform P A with respect t o u niform P A, according to the numb er of paths. The origin ality of the paper lies in the fact that we show that as the number of paths increases, the optim al P A becomes more and more uniform due to the ergodic behavior of all the CDMA channels. Thi s i s reminiscent of an effect (“channel hardening ”) already reve aled in MIMO [20]. The highest gain (in terms o f utilit y) is obtained in the case of flat fading (which also fa vors dis-uniform po wer allocati on between the users). The layout of this paper is t he following. First, a detailed account of related works is m ade in Sec. II. In order to be self-contained, we i ntroduce useful notations and concepts of random matrix theory in Sec. III. The communication mo del that wi ll be us ed throughout the paper is detailed in Sec. IV. Asy mp- totic SINR and capacity e xpressions are gi ven in Sec. V. Th e particular game played between users is introduced in Sec. VI, along wit h the existence of a Nash equilibrium. Finally , theoretical results for the power allocatio n are derived in Sec. VII for unordered users and Sec. VIII when there is an ordering of the users. An alytical results are matched with simulat ions i n Sec. IX. Conclus ions are provided in Sec. I I . R E L A T E D W O R K This section is dedicated to p resent some of the works t hat use game theory for power con trol. W e remind that a Nash equilibrium is a st able 3 solution, where no player has an i ncentiv e to de- viate unil aterally , whil e a Pareto equi librium is a cooperativ e dominating so lution, where there is no way to improve the performance of a player without harming ano ther one. Generally , b oth concepts do not coincide. Follo wi ng the general p resentation of power all ocation gam es in [10], [11 ], an abundance of works can be found on the subject. In particular , the utilit y generally considered in those articles is justified in [21] where the author describes a widely applicable model “from first principles”. Conditions under which the utility will allow to obtain non-trivial Nash equili bria (i.e., users actually transmit at the equilibrium) are d e- riv ed. T he u tility cons isting of th roughput-to-power ratio (detailed in Sec. VI) is s hown to satisfy these conditions. In addition, it possesses a propriet y of reliability in the sense that the t ransmission occurs at non-negligibl e rates at t he equili brium. Th is kind of utility fun ction had been i ntroduced in p re v ious works, wit h an econom ic leaning [22], [23]. Unfortunately , Nash equilibria often lead to in- ef ficient allocatio ns, in the sens e th at higher rates (Pare to equi libria) could be obtai ned for all mo- biles if they cooperated. T o alleviate this probl em, in additi on to the non-cooperative game s etting, [23] introduces a p ricing strategy to force users to transmit at a socially optimal rate. They obtain communication at Pare to equilibrium. In [24], defining t he utilit y as advised in [21] as the ratio of the th roughput to the transmissi on power , the authors obtain resul ts of existence and unicity of a Nash equilibrium for a CDMA system. They extend thi s work to the case of mul tiple carri- ers in [25]. In particular , it is sh own that users will select and onl y transmit ov er th eir best carrier . As far as the attenuation is concerned, the consideration is restricted to flat fa ding in [24] and in [25] (each carrier being flat fading in the latter). Howe ver , wireless transmiss ions generally suf fer from the ef fect of multiple paths, thus becoming frequency- selectiv e. The goal of this paper is to determine the influence of the number o f paths (or t he selectivity of the chann el) on the p erformance of P A. This work is an extension of [24] in the case of frequenc y -selectiv e fading, in the frame work o f multi-user systems. W e do not consider m ultiple carriers, as i n [25], and the results are very d iff erent to those obtained in that work. The extension is not trivial and in volv es advanced results on random matrices with n on-equal variances d ue to Girko [26] whereas classical result s rely on the work of Silverstein [27]. A part of t his work was previously published as a conference paper [28]. Moreover , in addition to the linear filters studied in [24], we s tudy th e enhancements provided by t he optimum and successiv e interference cance llation filters. I I I . R A N D O M M A T R I X T H E O RY N O T A T I O N S A N D C O N C E P T S The following definitions and theorem can b e found in [12 ] and wi ll be used in the foll owing sections. In this section, N and K are posit iv e integers. Definition 1: Let ν b e a probabilit y measure. The Stieltjes transform m ν associated to ν is g iv en by m ν ( z ) = Z 1 t − z ν ( dt ) . Definition 2: Let v = [ v 1 , . . . , v N ] be a vector . Its empirical distribution i s the function F v N : R → [0 , 1] defined by: F v N ( x ) = 1 N # { v i ≤ x | i = 1 . . . N } . In other words, F v N ( x ) is the fraction of element s of v that are inferior or equ al t o x . In particular , if v is the vector of eigen values of a matrix V , F v N is called the empirical ei gen value distri bution of V . Definition 3: Let V be a N × K rando m matrix with independent columns and entries v ij . Denote by ⌊·⌋ the closest smal ler integer . V is said to behave er godically if, as N , K → ∞ with K / N → α , for x ∈ [0 , 1] , the empirical distribution o f h   v ⌊ xN ⌋ , 1   2 , . . . ,   v ⌊ xN ⌋ ,K   2 i con ver ges almost surely to a non-random limit dis- tribution denoted F V x ( · ) and, for y ∈ [0 , α ] , the empirical distribution of h   v 1 , ⌊ y N ⌋   2 , . . . ,   v N , ⌊ y N ⌋   2 i con ver ges almost surely to a non-random limit dis- tribution denoted F V y ( · ) . Definition 4: Let V be a N × K rando m matrix that behaves ergodically as in Def. 3, such as F V x ( · ) and F V y ( · ) ha ve all their moments bounded. The two-dimensional channel pr ofile of V is the function ρ V ( x, y ) : [0 , 1] × [0 , α ] → R such that, if the random variable X is uniformly di stributed in [0 , 1] , 4 then th e distribution o f ρ V ( X , y ) equals F V y ( · ) and, if the random variable Y i s uni formly distri buted in [0 , α ] , then the distribution of ρ V ( x, Y ) equals F V x ( · ) . Theor em 1: Let Y = V ⊙ W be a N × K m atrix, where ⊙ is the Hadamard (element-wise) produ ct and V and W are in dependent N × K random matrices. Assume t hat V beha ves ergodically with channel profile ρ V ( x, y ) as in Def. 4 and that W has i.i .d. entries with zero mean and va riance 1 N . Then, as N , K → ∞ wit h K / N → α , the emp irical eigen value d istribution of YY H con ver ges almost surely to a non-random limit distribution function whose Stieltjes transform is g iv en by: m YY H ( z ) = lim N → ∞ 1 N T race   YY H − z I  − 1  = Z 1 0 u ( x, z ) dx and u ( x, z ) satis fies the fixed point equation: u ( x, z ) = 1 R α 0 ρ V ( x,y ) dy 1+ R 1 0 ρ V ( x ′ ,y ) u ( x ′ ,z ) dx ′ − z . (1) The solution to equation (1) exists and is uni que in the class of functions u ( x, z ) ≥ 0 , analytic for Im ( z ) > 0 , and continuous on x ∈ [0 , 1] . I V . M O D E L W e con sider a single uplink multi -user sys- tem cell, i.e., inter-cell i nterference free case. The spreading leng th is denoted N . Th e nu mber of users in the cell is K . The load is α = K / N . Th e general case of wide-band CDMA is consi dered where the signal transmitted by user k h as complex en velope x k ( t ) = X n s k n v k ( t − nT ) . v k ( t ) is a weigh ted sum of elem entary modulation pulses which satisfy t he N yquist criterion with re- spect to the chip i nterva l T c ( T = N T c ): v k ( t ) = N X ℓ =1 v ℓk ψ ( t − ( ℓ − 1) T c ) . The signal is transmitt ed over a frequenc y selectiv e channel with impulse response c k ( τ ) . Under the assumption of slo wly-varying fading, the cont inuous time received signal y ( t ) at t he base s tation has the form: y ( t ) = X n K X k =1 s k n Z c k ( τ ) v k ( t − nT − τ ) dτ + n ( t ) where n ( t ) is zero-mean compl ex white Gaussian noise with va riance σ 2 . The signal (after pulse matched filtering by ψ ∗ ( − t ) ) is s ampled at the chip rate to g et a discrete-time s ignal that has the form: y = K X k =1 C k v k p P k s k + n (2) where C k are N × N T oeplitz m atrices representing the frequenc y selecti ve fading for the k -th user , v k is a N × 1 vector representing the spreading code of the k -th user , and n is an N × 1 Additive White Gaussian Noise (A WGN) vector with cov ariance matrix σ 2 I N . W e cons ider the case of a multipath channel. Under the assum ption that the num ber of paths from user k to t he base station is giv en by L k , the model of the chann el is gi ven by c k ( τ ) = L k − 1 X ℓ =0 η k ( ℓ ) ψ ( τ − τ k ( ℓ )) . (3) where we assume that the channel is in variant durin g the tim e considered. In order to compare channels at the same signal to noise ratio, we constrain the distribution of the i.i.d . fading coefficients η k ( ℓ ) such as: E [ η k ( ℓ )] = 0 and E  | η k ( ℓ ) | 2  =  L k . (4) Usually , fading coef ficients η k ( ℓ ) are supposed to be independent wi th decreasing variance as the delay in creases. In all cases,  i s the av erage power of the channel, su ch as E  | c k ( τ ) | 2  = P L k − 1 ℓ =0 E  | η k ( ℓ ) | 2  =  , for all channels consid- ered. For each user k , let h ik be the Discrete Fourier Transform of the fading process c k ( τ ) . The frequency response of the channel at t he receiver i s giv en by: h k ( f ) = L k − 1 X ℓ =0 η k ( ℓ ) e − j 2 πf τ k ( ℓ ) | Ψ( f ) | 2 . (5) where we assum e th at t he transm it filter Ψ( f ) and the receiv e filter Ψ ∗ ( − f ) are such that, given the 5 bandwidth W , Ψ( f ) = ( 1 if − W 2 ≤ f ≤ W 2 0 otherwise. (6) Sampling at the various frequencies f 1 = − W 2 , f 2 = − W 2 + 1 N W , . . . , f N = − W 2 + N − 1 N W , we obtain the coef ficients h ik , 1 ≤ i ≤ N , as h ik = h k ( f i ) = L k − 1 X ℓ =0 η k ( ℓ ) e − j 2 π i N W τ k ( ℓ ) e j πW τ k ( ℓ ) . (7) Note that E  | h ik | 2  =  . Since the users are su pposed to be synchronized with the b ase station and for sake of simplicity , we will consider in all the following that users add a cyclic prefix o f length equal to the channel impulse response length to their code sequence. 2 This case is similar to uplink M C-CDMA [30], [31]. As a consequence, matrices { C k } are circulant [32] and can all b e di agonalized in th e Fourier basis F [29]. Model (2) simplifies th erefore to: y = K X k =1 FH k F H v k p P k s k + n (8) where H k is a diagonal matrix wit h diagonal ele- ments { h ik } i =1 ...N . For each user k , the coeffi cients h ik are the discrete Fourier transform of the channel impulse response. W e make the hypothesis that th e u sers employ Gaussian i .i.d. codes with zero mean and variance 1 / N [33]. This hypothesis enables us to state simply our results, howe ver almost all of the results are valid for any distribution of the codes as long as it has mean zero and variance 1 / N [16]. In particular , since ev ery unitary tranformation of a Gaussian i.i.d. vector is a Gaussian i.i.d. ve ctor (so that w i = F H v i has the same distribution as v i for any i ), we multip ly y in (8) with F H and obtain without any change in the statisti cs: y = K X k =1 H k w k p P k s k + n =  H √ P ⊙ W  s + n (9) where ⊙ is the Hadamard (element-wise) product. 2 Note t hat i n t he asymptotic case (when N → ∞ ), the result holds without the need of a cyclic prefix as long as t he channel is absolutely summable [29]. In (9), H is t he frequency s electiv e fading m atrix, of size N × K : H =   h 11 h 12 . . . h 1 K . . . . . . . . . h N 1 h N 2 . . . h N K   . √ P is the roo t square of the diagonal p ower control matrix, of size K × K . W is an N × K random spreading mat rix: W =  w 1 | w 2 | · · · | w K  where w k =   w 1 k . . . w N k   . Note that asymptotically (as N → ∞ ), for a giv en multipath channel of length L , model (9) is also valid for the case o f uplink DS-CDMA since all T oeplitz m atrices can be asym ptotically diagonalized in a Fourier Basis [29], [34]. In the following, we will assume that the fre- quency selective fading matrix H b eha ves er - godically , as in Def. 3. The two-dimensional channel profile of H √ P is denoted ρ ( f , x ) = P ( x ) | h ( f , x ) | 2 , f ∈ [0 , 1] , x ∈ [0 , α ] . f is the frequency index and x is the user index. This enables us to use Th. 1 in order to obtain e x pressions for the SINR. It is also assumed th at the powe r of all users is upper bounded by P max and the square norm of the fading, on all paths, for all users, is upper bounded by h max . V . A S Y M P T O T I C S I N R E X P R E S S I O N S Let h k be the k -th col umn of H , and H ( − k ) be H wi th h k removed. Simil arly , let w k be the k -th column of W , and W ( − k ) be W with w k removed. Let √ P ( − k ) be √ P with the k -th column and line removed. Finally , let G ( − k ) = H ( − k ) √ P ( − k ) ⊙ W ( − k ) . A. Matched F il ter Supposing perfect CSI at the recei ver , the matched filter for the k -th u ser is giv en by g k = √ P k ( h k ⊙ w k ) . This leads to t he following expres- sion for the SINR of user k SINR k =   g H k g k   2 σ 2 g H k g k + g H k  G ( − k ) G H ( − k )  g k . 6 Pr oposit ion 1: [16] As N , K → ∞ with K/ N → α , the SINR of user k at the output of the matched filter i s gi ven by SINR k = β MF  k N  where β MF : [0 , α ] → R is given by β MF ( x ) = P ( x ) · ( H ( x )) 2 σ 2 H ( x ) + R α 0 R 1 0 P ( y ) | h ( f , y ) | 2 | h ( f , x ) | 2 d f dy (10) and H ( x ) = R 1 0 | h ( f , x ) | 2 d f . Denoting SINR k = β MF k , Prop. 1 enabl es us to extract an approximation of th e value of the SINR of user k in the finite si ze case β MF k = P k  1 N P N n =1 | h nk | 2  2 σ 2 N P N n =1 | h nk | 2 + 1 N 2 P j 6 = k P N n =1 P j | h nj | 2 | h nk | 2 . (11) W e observe that P k ∂ β MF k ∂ P k = β MF k . B. MMSE F ilter Supposing perfect CSI at th e recei ver , the MMSE filter for the k -th us er is giv en by g MMSE k = R − 1 g k , where R =   H √ P ⊙ W   H √ P ⊙ W  H + σ 2 I N  . This leads to the following expression for t he SINR of user k [14] SINR k = g H k  G ( − k ) G H ( − k ) + σ 2 I N  − 1 g k . (12) Pr oposit ion 2: [16] As N , K → ∞ with K/ N → α , the SINR of user k at the output of the MMSE recei ver is gi ven by: SINR k = β MMSE  k N  where β MMSE : [0 , α ] → R is a function defined by the implicit equation β MMSE ( x ) = P ( x ) Z 1 0 | h ( f , x ) | 2 d f σ 2 + R α 0 P ( y ) | h ( f ,y ) | 2 dy 1+ β MMSE ( y ) . (13) Denoting SINR k = β MMSE k , Prop. 2 enables u s to extract an approximation of the value of t he SINR of user k in the finite si ze case β MMSE k = P k 1 N N X n =1 | h nk | 2 1 σ 2 + 1 N P j 6 = k P j | h nj | 2 1+ β MMSE j . (14) From (12), we observe that P k ∂ β MMSE k ∂ P k = β MMSE k . From Prop. 2, we ha ve the capacity of user k C MMSE k = 1 N log 2  1 + β MMSE k  . The global capacity of the sys tem is C MMSE = Z α 0 log 2  1 + β MMSE ( x )  dx. (15) C. Optimal F ilter The term optim al filter designates a filter capa- ble of decoding the rece iv ed signal at the bound giv en by Shannon’ s capacity . Hence it is diffic ult to define an SINR asso ciated to it. Howev er , results of random matrix theory can st ill be appli ed. Let Y =  H √ P ⊙ W  . The definition of Shannon’ s capacity per dimension for ou r sy stem is C OPT ( N ) = 1 N log 2 det  I N + 1 σ 2 YY H  . (16) As N , K → ∞ wi th K/ N → α , C OPT ( N ) → Z log 2  1 + 1 σ 2 t  ν ( dt ) (17) where ν is the empirical eigen value dist ribution of YY H , as in Def. 2. If we dif ferentiate the asymptotic va lue C OPT of (17) with respect to σ 2 , we obtain ∂ C OPT ∂ σ 2 = log 2 ( e ) Z − 1 σ 4 t 1 + 1 σ 2 t ν ( dt ) = log 2 ( e ) Z σ 2  − 1 σ 4 t − 1 σ 2 + 1 σ 2  σ 2  1 + 1 σ 2 t  ν ( dt ) = log 2 ( e )  Z 1 t + σ 2 ν ( dt ) − 1 σ 2 Z ν ( dt )  = log 2 ( e )  m ν ( − σ 2 ) − 1 σ 2  (18) where m ν ( · ) is the Stieltjes transform o f the empir- ical eigen value distribution of Y Y H . From Th. 1, m ν ( · ) is giv en by m ν ( z ) = Z 1 0 u ( f , z ) d f 7 where u ( f , z ) is given by (1) wi th ρ H √ P ( f , x ) = ρ ( f , x ) = P ( x ) | h ( f , x ) | 2 . Given that if σ 2 = + ∞ , C OPT = 0 , it i s immedi ate to obtain C OPT from (18) as C OPT = lo g 2 ( e ) Z + ∞ σ 2 m ν ( − z ) − 1 z dz . (19) Pr oposit ion 3: C OPT and C MMSE are re lated through the fol lowing equality C OPT = C MMSE − log 2 ( e ) Z α 0 β MMSE ( x ) 1 + β MMSE ( x ) dx + Z 1 0 log 2  1 + 1 σ 2 Z α 0 ρ ( f , x ) 1 + β MMSE ( x ) dx  d f . (20) Pr oof: See Appendix XI-A. The addi tional term in the right-hand side of (20) corresponds to the non-linear p rocessing gain. It quantifies the gain in terms of capacity that can b e achie ved between pure linear M MSE and non-li near filtering. Assuming perfect cancellation of d ecoded users, successiv e interference cancellation w ith MMSE filter achie ves the o ptimum capacity [35 ]. The fol- lowing propositio n ensues from this fact. Pr oposit ion 4: [16] As N , K → ∞ with K/ N → α , the optimal capacity i s g iv en by: C OPT = Z α 0 log 2  1 + β SIC ( x )  dx where β SIC : [0 , α ] → R is a function defined by the implicit equation β SIC ( x ) = P ( x ) Z 1 0 | h ( f , x ) | 2 d f σ 2 + R x 0 P ( y ) | h ( f ,y ) | 2 dy 1+ β SIC ( y ) . ( 21) Prop. 4 enables u s t o extract an expression that is analog to the SINR for the opti mal filter . Similarly to the case of β MMSE in Sec. V -B, t he deri va tiv e o f this expression obeys the property P k ∂ β SIC k ∂ P k = β SIC k . V I . G A M E S A N D E Q U I L I B R I A From now on, we deno te SINR k = β k , whichever filter is actually used. A. P ower Allocation Game A game with a unique strategy s et for all users is defined by a triple { S, P , ( u k ) k ∈ S } where S is the set of players , P is the set of strate gies , and ( u k ) k ∈ S is the set of uti lity functions , u k : P | S | → R . In our setti ng, the players are si mply the users, indexed by the set S K = { 1 , . . . , K } . The st rategy for a m obile is its power all ocation P k , which we will assume belongs to a compact interv al P = [0 , P max ] ⊆ R . The u tility measures the gain of a user as a result of the s trategy this user pl ays. In [21], the author deriv es what he calls Throughput to Power Ratio (TPR) under minimal requirements. The utility of user k i s expressed u k = γ k P k . (22) W e denote γ k = γ ( β k ) , where γ ( · ) is the same function for all u sers. In (22), γ is at least C 2 and should satisfy condi tions detailed in [21] in order to obtain an “interesting” equilibrium. For example, in the simulations, we con sider the goodput γ ( β k ) , which is proportional to  1 − e − β k  M where M is the nu mber of bits trans- mitted in a CDMA packet. Remark t hat the usual definition of goodput would rather be considered proportional to q ( β k ) = (1 − BER k ) M , where BER is th e bi t error rate. Howe ver , this quantity is not zero when the transm itted power is zero. Using this function in the utili ty would lead t o the unsatis fying conclusion that mobi les should not t ransmit at all, since the (improbable) ev ent of a correct guess gives them i nfinite ut ility [10]. Therefore, an adapted version of t he g oodput is adopted, where a fac tor 2 is added before t he BER. The performance m easure considered is hence proportional to q 2 ( β k ) = (1 − 2 BER k ) M , leading to t he expression above. This function has the desirable property q 2 (0) = 0 and its shape follo ws closely the shape of the original good- put q ( · ) . This is a relev ant performance m easure, as each mobi le wants to use its (limited) battery power t o t ransmit t he maximum pos sible amou nt of information. This utility is expressed in bits per joule . In the non-cooperativ e g ame setting , each user wa nts to selfishly maximize i ts utility . A Nash equilibrium is obtained wh en no user can b enefit by unilaterally deviating from its strategy . T o obt ain the maxim um u tility achiev able by user k , we differentiate u k with respect to t he power P k and equate t o 0. W e obtain P k ∂ β k ∂ P k γ ′ ( β k ) − γ ( β k ) = 0 . (23) For all filters under consideratio n, (10), (13) and (21) imply P k ∂ β k ∂ P k = β k , thus (23) reduces an 8 equation on β k β k γ ′ ( β k ) − γ ( β k ) = 0 . (24) Eq. (24) is particularly interesting in the case when there exists a unique solution β ⋆ . The existence of a soluti on to (24) is guaranteed as long as the function γ ( · ) is a quasiconcav e function o f t he SINR, i.e., there exists a point below which the fun ction is non-decreasing, and above which the function is non-increasing [23], [21]. In addition, we assume that the functi on γ ( · ) takes value γ (0) = 0 , so that users cannot achiev e an infinite utility by not t ransmitting . This occurs for sev eral functions γ ( · ) of int erest, in particular the goodput [24], which we will use for simulatio ns. Unfortunately , the capacity can not be used as a function γ ( · ) , since it leads to t he trivial result β ⋆ = 0 for thi s u tility function. The uniqueness of the solution β ⋆ to (24) is du e to fact that the SINR of each user i s a strictly increasing function of its transmit po wer . Given the tar get SINR β ⋆ , we obtain the strategy of users in the next section. V I I . P OW E R A L L O C A T I O N I N T H E N A S H E Q U I L I B R I U M A. Flat F ading In this subsection, we sho w that the results of [24] for Matched and MMSE filters are a s pecial case of our setting when L = 1 (flat fading case). In addi tion, we derive t he power allo cation for the Optimum filter . When there i s onl y one path, for each us er k , denoted by its index k N = x ∈ [0 , α ] , h ( f , x ) does not depend on f . Gi ven the tar get SINR β ⋆ , we have explicit expressions of the po wer wit h which user k transmits for t he v arious recei vers. In Appendix XI-B, w e show that the influence of the strategy of a p layer o n t he payoffs of other play- ers is (asymptotically) “small”. It justifies the fact that we can obt ain an equilibriu m in the asympto tic setting, without the need for players to possess all the inform ation on the sy stem. Their local in forma- tion is sufficient. In the asymptotic l imit, we obtain results si milar to W ardrop equilibrium : the st rategy used by each user does not i nfluence the strategy of other users. 1) Matched fil ter: From Prop. 1, the continuous formulation is P ( x ) = β ⋆  σ 2 + R α 0 P ( y ) | h ( y ) | 2 dy  | h ( x ) | 2 or equiv alently in a dis crete form P k = β ⋆  σ 2 + 1 N P K j =1 ,j 6 = k P j | h j | 2  | h k | 2 . (25) Summing (25) ov er k = 1 , . . . , K , we obt ain a closed form e xpression for the mi nimum po wer with which us er k transmits when us ing the matched filter P k = 1 | h k | 2 σ 2 β ⋆ 1 − α β ⋆ for α < 1 β ⋆ . (26) 2) MMSE fil ter: From Prop. 2, the continuous formulation is P ( x ) = β ⋆  σ 2 + 1 1+ β ⋆ R α 0 P ( y ) | h ( y ) | 2 dy  | h ( x ) | 2 . or equiv alently in a dis crete form P k = β ⋆  σ 2 + 1 1+ β ⋆ 1 N P K j =1 ,j 6 = k P j | h j | 2  | h k | 2 . (27) Summing (27) ov er k = 1 , . . . , K , we obt ain a closed form e xpression for the mi nimum po wer with which user k transmits when usi ng the M MSE filter P k = 1 | h k | 2 σ 2 β ⋆ 1 − α β ⋆ 1+ β ⋆ for α < 1 + 1 β ⋆ . (28) Both (26) and (28 ) are t he same results as in [24]. 3) Optimum filter: Each user maximizes its ut il- ity for a SINR equal t o β ⋆ . Howe ver , in the case of the optimu m filter , the SINR is no t d efined directly . It is nev ertheless possible to extract an equiv alent quantity from the expression of the capacity , since the value of the capacity o f user k at th e equilibri um is gi ven by C ⋆ = 1 N log 2 (1 + β ⋆ ) . Pr oposit ion 5: The power allocation is giv en b y P k = 1 | h k | 2 σ 2 β + 1 − α β + 1+ β + for α < 1 + 1 β + (29) where β + is the sol ution to α log 2  1 + β +  − α log 2 ( e ) β + 1 + β + +log 2 1 + 1 1 + β + αβ + 1 − α β + 1+ β + ! = α lo g 2 (1 + β ⋆ ) . (30) Pr oof: See Appendix XI-C. 9 B. F r equency Selective F ad i ng In the context of frequency selecti ve fading, for each us er k , denoted by its index k N = x ∈ [0 , α ] , there are L > 1 paths with respective attenua- tions h ℓ ( x ) , ℓ = 1 , . . . , L , which are i.i.d. ran- dom variables with some kno wn distribution. W e suppose that h ℓ ( x ) has mean zero, and the dis- tributions of the real part and imagi nary part of h ℓ ( x ) are even fun ctions, as for example th e Gaus- sian distribution, which we consider i n the simu- lations. h ( f , x ) depends on f through h ( f , x ) = P L ℓ =1 h ℓ ( x ) e − 2 π if ( ℓ − 1) . G iv en the target SINR β ⋆ , the Nash equilibrium po wer allocation is determined by implicit equations for th e v arious receiv ers. 1) Matched filter: The con tinuous formulat ion is P ( x ) = β ⋆ · σ 2 H ( x ) + R 1 0 R α 0 P ( y ) | h ( f , y ) | 2 | h ( f , x ) | 2 d f dy ( H ( x )) 2 or equiv alently in a dis crete form P k = β ⋆ · σ 2 N P N n =1 | h nk | 2 + 1 N P N n =1 | h nk | 2 1 N P K j 6 = k P j | h nj | 2  1 N P N n =1 | h nk | 2  2 . (31) In (31), h nk = h  n − 1 N , k N  . In this expression, the power allocation of user k seems to depend on the p ower allocation and fading realization of all the other users. Howe ver , when the number of users tends t o i nfinity , the s trategy of any single user does not hav e any influence on the payoff of user k , as shown in Appendix XI-B. Hence, the approp riate framew ork is non-atomic games. The expression 1 N P K j =1 P j | h nj | 2 is asymptotically a constant (not dependi ng on n ), denoted Ω . Ω = αβ ⋆ σ 2 1 K P K j =1 | h nj | 2 E j 1 − αβ ⋆ 1 K P K j =1 | h nj | 2 E j (32) where E j = 1 N P N m =1 | h mj | 2 . As K → ∞ , we can apply the Central Limit Theorem to the sum of random variables 1 K K X j =1 | h nj | 2 E j . (33) It tends to its expectation, which is equal to 1 (see Appendix XI-D). It follo ws that asymptoti cally Ω = αβ ⋆ σ 2 1 − αβ ⋆ (and simulatio ns in Sec. IX prove that this approximat ion is valid for moderate finite values o f N ). From (31), we obtain a formula sim ilar to (26) P k = 1 E k σ 2 β ⋆ 1 − α β ⋆ for α < 1 β ⋆ . (34) 2) MMSE filter: The continuous form ulation is P ( x ) = β ⋆ R 1 0 | h ( f ,x ) | 2 d f σ 2 + 1 1+ β ⋆ R α 0 P ( y ) | h ( f ,y ) | 2 dy (35) or equiv alently in a dis crete form P k = β ⋆ 1 N P N n =1 | h nk | 2 σ 2 + 1 1+ β ⋆ 1 N P K j =1 ,j 6 = k P j | h nj | 2 . (36) In (36), h nk = h  n − 1 N , k N  . As previously , when the numb er of users tends to infinity , 1 N P K j =1 P j | h nj | 2 is asymptotically a constant (not depending o n n ), denoted Ω . Ω = αβ ⋆ σ 2 1 K P K j =1 | h nj | 2 E j 1 − αβ ⋆ 1+ β ⋆ 1 K P K j =1 | h nj | 2 E j (37) where E j = 1 N P N m =1 | h mj | 2 . It follows that asymptotically Ω = αβ ⋆ σ 2 1 − α β ⋆ 1+ β ⋆ , we obtain a formula si milar to (28 ) P k = 1 E k σ 2 β ⋆ 1 − α β ⋆ 1+ β ⋆ for α < 1 + 1 β ⋆ . (38) 3) Optimum filter: Each user maximizes its ut il- ity for a SINR equal t o β ⋆ . Howe ver , in the case of the optimu m filter , the SINR is no t d efined directly . It is nev ertheless possible to extract an equiv alent quantity from the expression of the capacity , since the value of the capacity o f user k at th e equilibri um is gi ven by C ⋆ = 1 N log 2 (1 + β ⋆ ) . Pr oposit ion 6: Asym ptotically , as N , K → ∞ , the po wer allocatio n is given by P k = 1 E k σ 2 β + 1 − α β + 1+ β + for α < 1 + 1 β + (39) where β + is the sol ution to α log 2  1 + β +  − α log 2 ( e ) β + 1 + β + +log 2 1 + 1 1 + β + αβ + 1 − α β + 1+ β + ! = α lo g 2 (1 + β ⋆ ) . (40) 10 Pr oof: The proof is similar to the proof of Prop. 5. W e observe that for all filters considered, t he optimal P A is a cons tant times the in verse of the total energy of the channel E j . V ia Parse va l’ s T he- orem, E j = P L ℓ =1   h ℓ  j N    2 . It is a sum of i.i.d . random variables. As the num ber of p aths in creases, the optim al P A tends to a u niform P A. This is an eff ect similar to “channel hardening” [20]: as the number of paths increases, the variance of the distribution of the channel energy decreases and the Nash equi librium P A becomes more and more uniform for all us ers. V I I I . S U C C E S S I V E I N T E R F E R E N C E C A N C E L L A T I O N The optimal filt er giv es a bound on the perfor- mance that can be achieved through (non-li near) filtering at the base station. In order to improve the performance of the sy stem, we i ntroduce Successiv e Interference Cancellation (SIC) [19] at th e base station. Under the assumptio n of perfect decoding, SIC i mproves immensely t he performance of lin- ear filters (M atched Filt er or MMSE Filter). The MMSE SIC filter actually achieves t he opt imum fil- ter bound, under the assumpt ion of p erfect decoding. The principle of SIC receiv ers is quite simple: users are ordered and are decoded successive ly . At each step, supposing that the us er has b een encoded at the appropriate decoding rate, the signal is d ecoded and its contribution to the interference is then perfectly subtracted. This remov es som e of the inter-user interference and therefore increases the SINR of the following decoded users. The challenge is th at the users must transmit at th e appropriate rate to a void the catastrophic occurrence of imperfect decodi ng. Usually , the or- dering of users is done in a centralized way , at the base station which then advertises it to th e users. Howe ver , for the prot ocol to rem ain distributed, users shoul d be able to decide, based on their local information, at which rate to transmit. At equilib rium, the rate is determined by the SINR β ⋆ , and it is the transmiss ion power of the user that is determi ned acc ording to its rank of decoding. The equilibrium P A can be determined in a sim ple manner when the n umber of multip aths is finite ( L < ∞ ) and the number of users i s very high ( K → ∞ ). In Sec. VIII-A, we make use of the fact t hat the whole law of E j is realized in thi s case, so that users automatically know their rank of decoding. Another manner to give a (random) ordering of decodin g is t o introduce an add itional degree of liberty in the system. In Sec. VIII-B, we dev elop a correlated game frame work that enables users to l earn their rank of decoding in a simpl e way . In the following, we assume that each user has a unique h as a u nique i. d. num ber j rangi ng between 1 t o K . A. Or dering when K → ∞ If the number of users K → ∞ , with L fixed, the whole law of the to tal channel energy will be realized. Assum e the base station advertises to the users that they will be decoded by decreasing total channel energy . Each user knows, according to th e realization of its fading, its rank in t he d ecoding order given by K times 1 minus the cum ulative dis- tribution functi on D ( · ) of t he total channel ener gy E j . rank j = K (1 − D ( E j )) . In case that the base station advertises t o the users that they will be decoded by increasing total channel ener g y , user j will ha ve rank rank j = K D ( E j ) . B. Corr elated Equili b rium W e wish to introduce a simple mechanism that enables players t o coordinate and to know in which order they will be decoded. W e p lace ourselves in the context of correlated games. Th e notion of correlated equili brium was introduced by R. Au- mann 3 in [36] and further stud ied in [37], [38], [39]. They represent a generalization of Nash equilibrium. The important feature of correlated games is the presence of an ar bitrator . An arbitrator needs not hav e any intelligence o r kno wledge of the g ame, it needs only to send random (priv ate or pub lic) signals to the players that are independent of all other dat a in t he game. In th e context of non- cooperativ e games, each player has the possibili ty not t o consid er the signal(s) it receiv es. Coordina- tion betw een players t urns out t o be useful also in the case of cooperative optim ization. T he signals enable joint randomization between the strategies of t he players, po ssibly resulti ng in equilibria wit h 3 Prof. R. Auman n has receiv ed in 2005 the Nobel prize in economy for his contribution s to game theory , together with Thomas Schelling. 11 higher payoffs. The concept of correlated games was recently i ntroduced in a networking context i n [40], where the authors consider a simpl e ALOHA setting. The simp lest and most intu itive coordi nation mechanism is given by a commo n sign al which users as well as th e base station ov erhear before each transmission . There are K ! possible permu- tations of K users. Hence, the arbitrator broad- casts a sign al to the users belong ing to the set { 0 , . . . , K ! − 1 } . E ach of these nu mbers corresponds to a perm utation π of { 1 , . . . , K } that gives t he (random) ordering of decoding as r ank j = π ( j ) . The users can then adj ust their transmit power according to this ordering. In terms of size of the message, this is equiv alent to the case when the base stati on decides the decoding order and broadcasts it to the users, or sends K individual m essages of ln( K ) bits containing the rank, sin ce ln( K !) = K ln( K ) + o ( K ln( K )) . Howe ver , there is no n eed of either any knowledge of the system or comput ations at the base station in the case o f the correlated mechanism. C. SIC P ower Allocati o ns In both cases, once the users know their order , they can calculate th eir transmit power according t o the filter that is used. The equili brium still occurs when all users reach the SINR β ⋆ . A single user will not benefit by d e viating, since it would decrease its utility . From now on, index k denotes the rank of decoding. In the case of the matched filter with SIC, the SINR of the us er decoded at rank k is β MF k = P k  1 N P N n =1 | h nk | 2  2 σ 2 N P N n =1 | h nk | 2 + 1 N 2 P j >k P N n =1 P j | h nj | 2 | h nk | 2 . (41) From (41), we get th e equilibrium P A of user k as P k = β ⋆ · σ 2 N P N n =1 | h nk | 2 + 1 N 2 P j >k P N n =1 P j | h nj | 2 | h nk | 2  1 N P N n =1 | h nk | 2  2 . (42) In the case of th e M MSE filter with SIC, t he SINR of the u ser decoded at rank k is β MMSE k = P k 1 N N X n =1 | h nk | 2 1 σ 2 + 1 N P j >k P j | h nj | 2 1+ β MMSE j . (43) From (43), we get th e equilibrium P A of user k as P k = β ⋆ 1 N P N n =1 | h nk | 2 σ 2 + 1 1+ β ⋆ 1 N P K j >k P j | h nj | 2 . (44) For flat fading, a simpl e recursion gives th e equilibrium P A (see Appendix XI-E). W e obtain respectiv ely P MF k = σ 2 β ⋆ | h k | 2  1 + 1 N β ⋆  K − k , (45) P MMSE k = σ 2 β ⋆ | h k | 2  1 + 1 N β ⋆ 1 + β ⋆  K − k . (46) As far as frequency-selecti ve fading is concerned, this gives u s the form of th e asymptotic expressions. Asymptoti cally , the power allocation of one user will not depend on the P A of th e other users, as shown in Append ix XI-B. W ith a similar reasoning as in Sec. VII, the expressions mimic (45) and (46) with the total channel ener gy E k replacing | h k | 2 , i.e., P MF k = σ 2 β ⋆ E k  1 + 1 N β ⋆  K − k , (47) P MMSE k = σ 2 β ⋆ E k  1 + 1 N β ⋆ 1 + β ⋆  K − k . (48) These expressions are also validated by sim ulations. Since M MSE SIC with perfect decoding is equiv- alent to the optimum filter , we t hus obtain a second possible equilibrium P A for t he optimum filter . In Sec. IX, we in vestigate which i s t he P A which min- imizes total amount of po wer needed to transmit at equilibrium SINR. In t he case of automatic ordering of the users, one question is whether i t is best to order the us ers by increasing or decreasing t otal fading energy . The answer is the following: it is alwa ys best to decode the users by decreasing total channel energy E 1 < · · · < E k (see App endix XI-F). An interestin g feature of equil ibrium P A (47) and (48) i s that there is no limitation on the num ber of users than can be accomodated by t he s ystem, contrary to the previous case of (34), (38) and (39). 12 The l imitation is only imposed b y t he i ncreasing power needed for each new us er decoded last, which grows without bound as an e xponential. I X . N U M E R I C A L R E S U L T S In all the following, we consid er that P max is chosen sufficiently high so that users can actuall y transmit at t he equilibriu m P A values. For the s imu- lations, we consider the us ual case of Rayleigh fad- ing. Although Rayleigh distribution is not bounded from above, si mulations sho w that the results s till hold. W e consider a CDMA system with K = 32 users and a spreading factor N = 256 . The noi se var iance is σ 2 = 1 0 − 10 . For a number of bits in a CDMA packet M = 100 , th e goodp ut i s γ ( β ) =  1 − e − β  100 (see [24]), and β ⋆ = 6 . 48 . The capacity achiev ed at the Nash Equilibriu m is C = α log 2 (1 + β ⋆ ) = 0 . 39 b its/s. Unfortunately , the capacity itself cannot be us ed as a relev ant performance measure i n the definition of the uti lity , because in t his case the maximal ut ility is ob tained when not sending. W e hav e performed simul ations over 1000 0 re- alizations. Fig. 1 shows the good fit of theoretic values calculated directly from (34), (38) and (39) with t hose sim ulations. The v al ues of the u tility do not d epend on the number of multip aths. W e see that optimum filter requires the minimal power , and matched filter the maxim al powe r to achiev e the required goodput. In Fig. 2 we have pl otted the average utility versus the number of multi paths L . Mu ltipaths are supposed to be i. i.d. Rayleigh distri buted with vari- ance 1 / L , in order for the channels to hav e the same energy . T wo cases are considered: the utili ty obtained in the Nash equili brium, according to the P A given by (31) and (36), and the u tility i n the case where all nodes transmit at the same power . For comparison p urposes, the sum of the un iform powers is equal to the sum of the powe rs used in the Nash equili brium. In additio n, simul ations (not reproduced here) s how that this value gives the higher a verage utility for a uniform P A. The ut ility does not vary with L in the Nash equilibrium: the Central Lim it Theorem applies to the util ity , wh ich is a const ant times the random v ariable E k in the Nash equ ilibrium. The utility wi th uniform powers is always inferior t o the utility in the Nash equi lib- rium. Howe ver , as L increases, the gap decreases, as t he variance of E k decreases, and the equilibrium P A becomes uniform. In Fig. 3 we have plotted the a verage of the in verse po wer of the users in the Nash equi librium for each of the in vestigated schemes. W e plot t he a verage in verse power because of the direct relation to the utility for the users. The high er this a verage, the higher the utility for the user . Th e SIC filters are alwa ys m ore efficient than their linear counterparts. Howe ver , for a l oad α < 0 . 12 and opti mum filter 4 , it is better to use the first v ariation of P A (39) than use MMSE SIC (48). This relation is rev ersed when α > 0 . 12 . In addi tion to the theoretical curves, Monte-Carlo simul ations were performed both wi th random ordering (circles) and ordering b y decreasing total channel ener gy (crosses), for L = 8 multipaths . Simul ations sho w that the optimal o rder - ing improves th e power ef ficiency of the successiv e interference cancellation filters. In Fig. 4, we in vestigate the amelioration pro- vided by optim al ordering as a function of the number of mult ipaths. Th e simu lations are done for K = 128 users, in order to be in the “int eresting” zone α > 0 . 12 . As expected, as th e nu mber o f paths increases, the tot al channel ener gy is m ore and more the same for each channel and the gain p rovided by ordering the users decreases. Howe ver , when t he number of users is very large and they benefit from automatic ordering, we see that t he utilit y with t he MMSE SIC equilibrium P A is the maximal utility that can be obtained in the non-cooperati ve setting. X . C O N C L U S I O N Using tools of random matrices, we ha ve de- riv ed t he equilib rium power allocation in a gam e- theoretic frame work applied to asymptotic CDMA with cyclic prefix, under frequency-selectiv e fad- ing. Three receive rs are consi dered: m atched filter , MMSE and optim um filter (given by Shannon’ s capacity). In addition, d istributed ordering mecha- nisms are introdu ced and the successive i nterference cancellation variants of the linear filters are studi ed. For each user , this power allocation depends only on th e t otal energy of the channel of the user under consideration. For a frequency-flat channel, the po wer allocation among users i s dis-uniform, whereas when the num ber of multipaths increases, 4 The v alue of α is obtained as solution of the equation αβ ⋆ β ⋆ 1+ β ⋆ (1 − α β + 1+ β + ) = β + (1 − exp( − α β ⋆ 1+ β ⋆ )) . 13 0 5 10 15 20 25 30 35 2 4 6 8 10 12 14 x 10 5 Number of Multipaths L Utility = Goodput/Power Matched Filter MMSE Filter Optimum Filter Fig. 1. Comparison of theoretic values and simulations for util ities in the Nash equilibrium. 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 14 x 10 5 Number of Multipaths L Utility = Goodput/Power MF MMSE Opt MFw MMSEw Optw Fig. 2. Simulation of utilities in the Nash equilibrium and constant po wer allocations versus L . the power al location tends more and more to a uniform one. X I . A P P E N D I X A. Pr oof of Pr op. 3 Notice that when σ 2 → ∞ , C OPT = 0 , C MMSE = 0 and β MMSE ( x ) = β ( x ) = 0 . Thus we only have to prove that the deriv atives of eit her si de of (20) are equal. Using ρ ( f , x ) = P ( x ) | h ( f , x ) | 2 , (13) can be 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −2 0 2 4 6 8 10 12 14 16 18 x 10 5 Load α Average inverse power MF MMSE Opt MFsic MMSEsic Fig. 3. A verage inv erse powe r used by t he different filters. 0 5 10 15 20 25 30 35 3 4 5 6 7 8 9 10 11 12 x 10 5 Number of Multipaths L Utility = Goodput/Power MFsic MMSEsic MFsorted MMSEsorted Fig. 4. Simulation of utilities in the Nash equilibrium with SIC filter with and without optimal ordering, versus L . re w ritten β ( x ) = Z 1 0 ρ ( f , x ) d f σ 2 + R α 0 ρ ( f ,y )2 dy 1+ β ( y ) . (49) From (1), R 1 0 ρ ( f , x ) u ( f , − σ 2 ) d f satisfies the same implicit equ ation (49) as β ( x ) and thus u ( f , − σ 2 ) = 1 R α 0 ρ ( f ,y ) dy 1+ β ( y ) + σ 2 . (50) 14 Using (49) and (50), we can rewrite Z 1 0 u ( f , − σ 2 ) d f − 1 σ 2 = Z 1 0 1 R α 0 ρ ( f ,y ) dy 1+ β ( y ) + σ 2 d f − Z 1 0 1 σ 2 d f = Z 1 0 − R α 0 ρ ( f ,x ) 1+ β ( x ) dx σ 2  R α 0 ρ ( f ,y ) dy 1+ β ( y ) + σ 2  d f = Z α 0 − 1 (1+ β ( x )) σ 2 Z 1 0 ρ ( f , x ) d f R α 0 ρ ( f ,y ) dy 1+ β ( y ) + σ 2 dx = − Z α 0 β ( x ) σ 2 (1 + β ( x )) dx. Thus from (18) ∂ C OPT ∂ σ 2 = − log 2 ( e ) Z α 0 β ( x ) σ 2 (1 + β ( x )) dx. (51) Diffe rentiating (15) with respect to σ 2 , we obtain ∂ C MMSE ∂ σ 2 = log 2 ( e ) Z α 0 1 1 + β ( x ) ∂ β ∂ σ 2 ( x ) dx. (52) Let π ( x ) = 1 σ 2 (1+ β ( x )) . From (51) and (52), we obtain ∂ C OPT ∂ σ 2 − ∂ C MMSE ∂ σ 2 = − log 2 ( e ) Z α 0  β ( x ) + σ 2 ∂ β ∂ σ 2 ( x )  π ( x ) dx. (53) From (13), we ha ve Z α 0 σ 2 β ( x ) ∂ π ∂ σ 2 ( x ) dx = Z α 0 Z 1 0 σ 2 ρ ( f , x ) d f σ 2 + R α 0 σ 2 ρ ( f , y ) π ( y ) dy ∂ π ∂ σ 2 ( x ) dx = Z 1 0 R α 0 ρ ( f , x ) ∂ π ∂ σ 2 ( x ) dx 1 + R α 0 ρ ( f , y ) π ( y ) dy d f = 1 log 2 ( e ) ∂ ∂ σ 2 Z 1 0 log 2  1 + Z α 0 ρ ( f , y ) π ( y ) dy  d f . Observing that Z α 0  β ( x ) + σ 2 ∂ β ∂ σ 2 ( x )  π ( x )+ σ 2 β ( x ) ∂ π ∂ σ 2 ( x ) dx = ∂ ∂ σ 2 Z α 0 σ 2 β ( x ) π ( x ) dx we obtain (20) from Prop. 3. B. Influence of Other Players’ Strate gies W e want to prov e that asymptoti cally , in the game { S K , P , ( u k ) k ∈ S K } , the strategy of a single player does not hav e any influence o n the p ayof f of the other players. In other words, for all k 6 = i ∈ S K , for all p = ( P 1 , . . . , P K ) ∈ P K , for all P ′ i ∈ P ,   u k ( p ) − u k ( P ′ i , p ( − i ) )   → 0 , as N → ∞ . Remember t hat u k = γ ( β k ) P k , and γ i s at least C 2 . L et ( β 1 , . . . , β K ) b e th e SINRs associated with the power all ocation p and ( β ′ 1 , . . . , β ′ K ) the SINRs associated wi th t he po wer allocation ( P ′ i , p ( − i ) ) . Then a si mple T aylor expansion of γ in β ′ k giv es γ ( β ′ k ) = γ ( β k )+( β ′ k − β k ) ∂ γ ∂ β ( β k )+ o ( β ′ k − β k ) . (54) According to (54), it i s suf ficient to show that     β ′ k − β k P k     → 0 , as N → ∞ . (55) a) Matched F ilter: For the m atched filter , the inequality is obtained directly from (11). The denominato r of (11) is always greater than σ 2 N P N n =1 | h nk | 2 . Hence,     β ′ k − β k P k     ≤      P k 1 N ( P ′ i − P i ) 1 N P N n =1 | h ni | 2 | h nk | 2 P k σ 4      ≤ P max h 2 max σ 4 N . b) MMSE F ilter: For the MMSE filter , the inequality is obtained fr om (12), Lemma 1 fr om [33] and Lemma 2.1 from [41], which we both reproduce below for con venience. Lemma 1: [33] Let C be a N × N complex matrix with uniformely bounded spectral radius for all N : sup N ( | C | ) < ∞ . Let w = 1 √ N [ w 1 , . . . , w N ] T where { w i } i =1 ...N are i.i.d. complex random vari- ables with zero mean, unit variance and finite eight h moment. Then: E "     w H Cw − 1 N tr C     4 # ≤ C N 2 where C i s a constant that does not depend on N or C . Lemma 2: [41] Let σ 2 > 0 , A and B N × N with B Hermitian non negati ve definite, and q ∈ C N . Then tr  ( B + σ 2 I ) − 1 − ( B + qq H + σ 2 I ) − 1  A  ≤ k A k σ 2 . 15 In Lemma 2, k A k is the spectral norm of A , i .e., the square roo t of the l ar gest singular va lue of A . From (7), we can wri te β k = P k w k H H H k  G ( − k ) G H ( − k ) + σ 2 I N  − 1 H k w k , β ′ k = P k w k H H H k  G ( − k ) ′ G H ( − k ) ′ + σ 2 I N  − 1 H k w k where G ( − k ) ′ G H ( − k ) ′ = G ( − k ) G H ( − k ) + ( P ′ i − P i )( h i ⊙ w i )( h i ⊙ w i ) H . A corollary of Lemma 1 is that for either matrix C = H H k  G ( − k ) G H ( − k ) + σ 2 I N  − 1 H k or m atrix C = H H k  G ( − k ) ′ G H ( − k ) ′ + σ 2 I N  − 1 H k , we obtain [33]     w k H Cw k − 1 N tr C     → 0 , as N → ∞ . Matrix B = G ( − k ) G H ( − k ) is Hermitian nonnega- tiv e definite, as for all w ∈ C N , w H G ( − k ) G H ( − k ) w =   G ( − k ) w   2 ≥ 0 . Diagonal matrix A = H k H H k has spectral norm   H k H H k   ≤ h 2 max . Using Lemmas 1 and 2, as N → ∞ , we obtain     β ′ k − β k P k     → 0 , as N → ∞ . c) Optimum and Successive Interfer ence Can- cellation F ilters: The analog of the SINR derive d for the optimum filter stems from the MMSE filter with SIC. The SINR for SIC filters have simil ar expressions with less interfering users appearing in the denominator . Hence the result is immediate. C. Pr oof of Pr op. 5 Giv en C ⋆ , we can use (20) to obtain a Nash equilibrium power allocation in the following wa y . W e rewrite (20) assum ing that the target SINR for the MMSE filter is β + . α log 2  1 + β +  − α log 2 ( e ) β + 1 + β + + log 2  1 + 1 σ 2 (1 + β + ) Z α 0 P ( y ) | h ( y ) | 2 dy  = α log 2 (1 + β ⋆ ) . (56) In the left-hand side of (56), P ( y ) is given by a MMSE power allocation similar to t he one giv en by (28). Hence, the term R α 0 P ( y ) | h ( y ) | 2 dy in (56) does not depend on the actual realizatio ns of the channels. Replacing β ⋆ by β + in (27), we o btain that R α 0 P ( y ) | h ( y ) | 2 dy = ασ 2 β + 1 − α β + 1+ β + , which giv es us (30). Replacing β ⋆ by β + in (28), we obtain t he power allocation (29). D. Expectation of the random variable (33) For each us er j , there are L > 1 p aths with respectiv e attenuations h ℓ  j N  , ℓ = 1 , . . . , L , which are i.i .d. comp lex random var iables wit h mean zero and ev en distributions o f the real and im aginary parts. The Fourier t ransform of those attenuations is h nj = h  n N , j N  = P L ℓ =1 h ℓ  j N  e − 2 π i n N ( ℓ − 1) . The total ener gy of th e paths i s E j = P L ℓ =1   h ℓ  j N    2 . W e want to show that the expectation of th e random va riable 1 K P K j =1 | h nj | 2 E j is equal to 1. By expanding the expression of h nj , this is equiv al ent to showing that the expectation of the random variable h ℓ  j N  h ℓ ′  j ′ N  E j is equal to 0. Deno ting by p ( · ) the dist ribution of h ℓ = h ℓ  j N  , this expectation is equal to the L - dimensional integral of h ℓ h ℓ ′ | h ℓ | 2 + | h ℓ ′ | 2 + P k 6 = ℓ,ℓ ′ | h k | 2 p ( h ℓ ) p ( h ℓ ′ ) Y k 6 = ℓ,ℓ ′ p ( h k ) which i s an odd functio n of h ℓ . Its i ntegral i s therefore 0, which proves the desired result. E. Pr oof of (45) a n d (46 ) Denote m k = P K − k | h K − k | . From (42), with flat fading, the sequence { m k } k ∈ S K satisfies m 0 = β ⋆ σ 2 and m k +1 = β ⋆ σ 2 + β ⋆ N P k j =0 m j . Using the fact that P k i = j  i j  =  k +1 j +1  , i t is immediate to prove by recurrence that m k = β ⋆ σ 2 k X j =0  k j  1 N j β ⋆ j = β ⋆ σ 2  1 + 1 N β ⋆  k . Hence formu la (45). The demonstrati on is exactly similar for (46) from the recursion m 0 = β ⋆ σ 2 and m k +1 = β ⋆ σ 2 + β ⋆ (1+ β ⋆ ) N P k j =0 m j . 16 F . Optimal Or dering of Us ers W e determine t he ordering that m akes use of the least total power for equi librium P A (45) (the case is similar for (46), (47) and (48)). Let the ordering of the users be such as | h 1 | 2 < · · · < | h K | 2 . Let π be any permu tation of { 1 , . . . , K } . Let a ij =  1 + 1 N β ⋆  K − i −  1 + 1 N β ⋆  K − j . Then showing that the optimal ordering is s uch as | h 1 | 2 < · · · < | h K | 2 is equiv alent to showing that for any π K X k =1 1 | h k | 2 a k π ( k ) > 0 . (57) Consider first a cyclic permu tation. By the def- inition of a ij , the sum of the a k π ( k ) is equal to zero: P K k =1 a k π ( k ) = 0 . 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