Non-atomic Games for Multi-User Systems
Nicolas Bonneau, Mérouane Debbah, Eitan Altman, Are Hjørungnes
aa r X i v : . [ c s . I T ] J un Non-atomic Games for Multi-User Systems
Nicolas Bonneau ∗ , M´erouane Debbah † , Eitan Altman ∗ , and Are Hjørungnes ‡∗ MAESTRO, INRIA Sophia Antipolis, 2004 Route des Lucioles, B.P. 93, 06902 SophiaAntipolis, FranceEmail: { nicolas.bonneau,eitan.altman } @sophia.inria.fr † Mobile Communications Group, Institut Eurecom, 2229 Route des Cretes, B.P. 193,06904 Sophia Antipolis, FranceEmail: [email protected] ‡ UniK–University Graduate Center, University of Oslo, Instituttveien 25, P. O. Box 70,N-2027 Kjeller, NorwayEmail: [email protected]
Abstract — In this contribution, the performance of amulti-user system is analyzed in the context of frequencyselective fading channels. Using game theoretic tools, auseful framework is provided in order to determine theoptimal power allocation when users know only theirown channel (while perfect channel state information isassumed at the base station). We consider the realisticcase of frequency selective channels for uplink CDMA.This scenario illustrates the case of decentralized schemes,where limited information on the network is available atthe terminal. Various receivers are considered, namely theMatched filter, the MMSE filter and the optimum filter.The goal of this paper is to derive simple expressions forthe non-cooperative Nash equilibrium as the number ofmobiles becomes large and the spreading length increases.To that end two asymptotic methodologies are combined.The first is asymptotic random matrix theory which allowsus to obtain explicit expressions of the impact of allother mobiles on any given tagged mobile. The secondis the theory of non-atomic games which computes goodapproximations of the Nash equilibrium as the number ofmobiles grows. I. I
NTRODUCTION
Resource allocation is of major interest in thecontext of multi-user systems. In the uplink multi-user systems, it is important for users to transmitwith enough power to achieve their requested qual-ity of service, but also to minimize the amount ofinterference caused to other users. Thus, an efficient This work was supported by the BIONETS project and by the Research Councilof Norway through the OPTIMO project “Optimized HeterogeneousMulti-user MIMO Networks”. power allocation mechanism allows to prevent anexcessive consumption of the limited ressources ofthe users.The most straightforward way to design a powerallocation (PA) mechanism is as a centralized pro-cedure, with the base station receiving trainingsequences from the users and signaling back the op-timal power allocation for each user. Power controlschemes in cellular systems were first introduced forTDMA/FDMA [1], [2]; more recently an optimalscheme was derived for Code Division MultipleAccess (CDMA) [3]. In order to achieve the optimalcapacity, the users may also be sorted according tosome rule of precedence [4]. However, this involvesa non negligible overhead and numerous non infor-mational transmissions. In addition, the complexityof centralized schemes increases drastically with thenumber of users. As discussed in [5], centralizedalgorithms generally do not have a practical usefor real systems, but provide useful bounds onthe performance that can be attained by distributedalgorithms.A way to avoid the constraints of a centralizedprocedure is to implement a decentralized one whereeach user calculates its estimation of the optimaltransmission power according to its local knowledgeof the system. This is, for example, the case inad-hoc networks applications. Most of the time,a distributed algorithm means an iterative versionof a centralized one. Mobiles update their powerallocation according to some rule based on thelimited information they retrieve from the system.
Supposing that an optimal power allocation exists,a distributed iterative algorithm is derived from adifferential equation in [6] and its convergence isproven analytically. A distributed version of the al-gorithm of [2] is presented in [7]. Building on theseresults, a general framework for power control incellular systems is given in [8]. A review of differentmethods of centralized and distributed power controlin CDMA systems is given in [5].In this context, a natural framework is gametheory, which studies competition (as well as co-operation) between independent actors. Tools ofgame theory have already been frequently used asa central framework for modeling competition andcooperation in networking, see for example [9] andreferences therein. Building on the framework of[8], a game theoretic approach was introduced in[10], [11]. Numerous works on power allocationgames have followed since, a selection of which wepresent in Sec. II.Game theory can be used to treat the case ofany number of players. However, as the size ofthe system increases, the number of parametersincreases drastically and it is difficult to gain insighton the expressions obtained.In order to obtain expressions depending onlyon few parameters, we consider the system in anasymptotic setting, letting both the number of usersand the spreading factor tend to infinity with a fixedratio. We use tools of random matrix theory [12] toanalyze the system in this limit. Random matrix the-ory is a field of mathematical physics that has beenrecently applied to wireless communications to an-alyze various measures of interest such as capacityor Signal to Interference plus Noise Ratio (SINR).Interestingly, it enables to single out the main pa-rameters of interest that determine the performancein numerous models of communication systems withmore or less involved models of attenuation [13],[14], [15], [16]. In addition, these asymptotic resultsprovide good approximations for the practical finitesize case, as shown by simulations.In the asymptotic regime, the non-cooperativegame becomes a non-atomic one, in which theimpact (through interference) of any single mobileon the performance of other mobiles is negligible.In the networking game context, the related solu-tion concept is often called Wardrop equilibrium[17]; it is often much easier to compute than theoriginal Nash equilibrium [9], and yet, the former equilibrium is a good approximation for the latter,see details in [18]. In this paper, we derive the non-atomic equilibrium, which generally corresponds toa non-uniform PA for the users.The non-atomic Nash equilibrium is studied inthis paper for several linear receivers, namely thematched filter and the MMSE filter, as well as non-linear filters, such as the successive interferencecancellation (SIC) [19] version of those filters.However, in order to perform SIC, the users needto know their decoding order, in order to adjusttheir rates. In this paper, we introduce ways ofobtaining an ordering of the users in a distributedmanner. The ordering can be determined simply ina distributed manner under weak hypotheses. Thisgives rise to a different kind of power allocation,that depend explicitly on the order in which theusers are decoded.Moreover, we quantify the gain of the non-uniform PA with respect to uniform PA, accordingto the number of paths. The originality of the paperlies in the fact that we show that as the numberof paths increases, the optimal PA becomes moreand more uniform due to the ergodic behavior ofall the CDMA channels. This is reminiscent ofan effect (“channel hardening”) already revealed inMIMO [20]. The highest gain (in terms of utility) isobtained in the case of flat fading (which also favorsdis-uniform power allocation between the users).The layout of this paper is the following. First, adetailed account of related works is made in Sec. II.In order to be self-contained, we introduce usefulnotations and concepts of random matrix theory inSec. III. The communication model that will be usedthroughout the paper is detailed in Sec. IV. Asymp-totic SINR and capacity expressions are given inSec. V. The particular game played between usersis introduced in Sec. VI, along with the existenceof a Nash equilibrium. Finally, theoretical resultsfor the power allocation are derived in Sec. VIIfor unordered users and Sec. VIII when there isan ordering of the users. Analytical results arematched with simulations in Sec. IX. Conclusionsare provided in Sec.II. R
ELATED W ORK
This section is dedicated to present some of theworks that use game theory for power control.We remind that a Nash equilibrium is a stable solution, where no player has an incentive to de-viate unilaterally, while a Pareto equilibrium is acooperative dominating solution, where there is noway to improve the performance of a player withoutharming another one. Generally, both concepts donot coincide. Following the general presentation ofpower allocation games in [10], [11], an abundanceof works can be found on the subject.In particular, the utility generally considered inthose articles is justified in [21] where the authordescribes a widely applicable model “from firstprinciples”. Conditions under which the utility willallow to obtain non-trivial Nash equilibria (i.e.,users actually transmit at the equilibrium) are de-rived. The utility consisting of throughput-to-powerratio (detailed in Sec. VI) is shown to satisfy theseconditions. In addition, it possesses a propriety ofreliability in the sense that the transmission occursat non-negligible rates at the equilibrium. This kindof utility function had been introduced in previousworks, with an economic leaning [22], [23].Unfortunately, Nash equilibria often lead to in-efficient allocations, in the sense that higher rates(Pareto equilibria) could be obtained for all mo-biles if they cooperated. To alleviate this problem,in addition to the non-cooperative game setting,[23] introduces a pricing strategy to force usersto transmit at a socially optimal rate. They obtaincommunication at Pareto equilibrium.In [24], defining the utility as advised in [21]as the ratio of the throughput to the transmissionpower, the authors obtain results of existence andunicity of a Nash equilibrium for a CDMA system.They extend this work to the case of multiple carri-ers in [25]. In particular, it is shown that users willselect and only transmit over their best carrier. Asfar as the attenuation is concerned, the considerationis restricted to flat fading in [24] and in [25] (eachcarrier being flat fading in the latter). However,wireless transmissions generally suffer from theeffect of multiple paths, thus becoming frequency-selective. The goal of this paper is to determine theinfluence of the number of paths (or the selectivityof the channel) on the performance of PA.This work is an extension of [24] in the caseof frequency-selective fading, in the framework ofmulti-user systems. We do not consider multiplecarriers, as in [25], and the results are very differentto those obtained in that work. The extension isnot trivial and involves advanced results on random matrices with non-equal variances due to Girko[26] whereas classical results rely on the work ofSilverstein [27]. A part of this work was previouslypublished as a conference paper [28].Moreover, in addition to the linear filters studiedin [24], we study the enhancements provided by theoptimum and successive interference cancellationfilters.III. R
ANDOM M ATRIX T HEORY N OTATIONS AND C ONCEPTS
The following definitions and theorem can befound in [12] and will be used in the followingsections. In this section, N and K are positiveintegers. Definition 1:
Let ν be a probability measure. The Stieltjes transform m ν associated to ν is given by m ν ( z ) = Z t − z ν ( dt ) . Definition 2:
Let v = [ v , . . . , v N ] be a vector.Its empirical distribution is the function F v N : R → [0 , 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 elementsof v that are inferior or equal to x . In particular, if v is the vector of eigenvalues of a matrix V , F v N iscalled the empirical eigenvalue distribution of V . Definition 3:
Let V be a N × K random matrixwith independent columns and entries v ij . Denote by ⌊·⌋ the closest smaller integer. V is said to behaveergodically if, as N, K → ∞ with
K/N → α , for x ∈ [0 , , the empirical distribution of h(cid:12)(cid:12) v ⌊ xN ⌋ , (cid:12)(cid:12) , . . . , (cid:12)(cid:12) v ⌊ xN ⌋ ,K (cid:12)(cid:12) i converges almost surely to a non-random limit dis-tribution denoted F V x ( · ) and, for y ∈ [0 , α ] , theempirical distribution of h(cid:12)(cid:12) v , ⌊ yN ⌋ (cid:12)(cid:12) , . . . , (cid:12)(cid:12) v N, ⌊ yN ⌋ (cid:12)(cid:12) i converges almost surely to a non-random limit dis-tribution denoted F V y ( · ) . Definition 4:
Let V be a N × K random matrixthat behaves ergodically as in Def. 3, such as F V x ( · ) and F V y ( · ) have all their moments bounded. The two-dimensional channel profile of V is the function ρ V ( x, y ) : [0 , × [0 , α ] → R such that, if therandom variable X is uniformly distributed in [0 , , then the distribution of ρ V ( X, y ) equals F V y ( · ) and,if the random variable Y is uniformly distributedin [0 , α ] , then the distribution of ρ V ( x, Y ) equals F V x ( · ) . Theorem 1:
Let Y = V ⊙ W be a N × K matrix,where ⊙ is the Hadamard (element-wise) productand V and W are independent N × K randommatrices. Assume that V behaves ergodically withchannel profile ρ V ( x, y ) as in Def. 4 and that W has i.i.d. entries with zero mean and variance N .Then, as N, K → ∞ with
K/N → α , the empiricaleigenvalue distribution of YY H converges almostsurely to a non-random limit distribution functionwhose Stieltjes transform is given by: m YY H ( z ) = lim N →∞ N Trace (cid:16)(cid:0) YY H − z I (cid:1) − (cid:17) = Z u ( x, z ) dx and u ( x, z ) satisfies the fixed point equation: u ( x, z ) = 1 R α ρ V ( x,y ) dy R ρ V ( x ′ ,y ) u ( x ′ ,z ) dx ′ − z . (1)The solution to equation (1) exists and is uniquein the class of functions u ( x, z ) ≥ , analytic forIm ( z ) > , and continuous on x ∈ [0 , .IV. M ODEL
We consider a single uplink multi-user sys-tem cell, i.e., inter-cell interference free case. Thespreading length is denoted N . The number of usersin the cell is K . The load is α = K/N . The generalcase of wide-band CDMA is considered where thesignal transmitted by user k has complex envelope x k ( t ) = X n s kn v k ( t − nT ) .v k ( t ) is a weighted sum of elementary modulationpulses which satisfy the Nyquist criterion with re-spect to the chip interval T c ( T = N T c ): v k ( t ) = N X ℓ =1 v ℓk ψ ( t − ( ℓ − T c ) . The signal is transmitted over a frequency selectivechannel with impulse response c k ( τ ) . Under theassumption of slowly-varying fading, the continuous time received signal y ( t ) at the base station has theform: y ( t ) = X n K X k =1 s kn Z c k ( τ ) v k ( t − nT − τ ) dτ + n ( t ) where n ( t ) is zero-mean complex white Gaussiannoise with variance σ . The signal (after pulsematched filtering by ψ ∗ ( − t ) ) is sampled at the chiprate to get a discrete-time signal 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 Toeplitz matrices representingthe frequency selective fading for the k -th user, v k isa N × vector representing the spreading code ofthe k -th user, and n is an N × Additive WhiteGaussian Noise (AWGN) vector with covariancematrix σ I N .We consider the case of a multipath channel.Under the assumption that the number of paths fromuser k to the base station is given by L k , the modelof the channel is given by c k ( τ ) = L k − X ℓ =0 η k ( ℓ ) ψ ( τ − τ k ( ℓ )) . (3)where we assume that the channel is invariant duringthe time considered. In order to compare channelsat the same signal to noise ratio, we constrain thedistribution of the i.i.d. fading coefficients η k ( ℓ ) such as: E [ η k ( ℓ )] = 0 and E (cid:2) | η k ( ℓ ) | (cid:3) = ̺L k . (4)Usually, fading coefficients η k ( ℓ ) are supposedto be independent with decreasing variance as thedelay increases. In all cases, ̺ is the averagepower of the channel, such as E (cid:2) | c k ( τ ) | (cid:3) = P L k − ℓ =0 E (cid:2) | η k ( ℓ ) | (cid:3) = ̺ , for all channels consid-ered. For each user k , let h ik be the DiscreteFourier Transform of the fading process c k ( τ ) . Thefrequency response of the channel at the receiver isgiven by: h k ( f ) = L k − X ℓ =0 η k ( ℓ ) e − j πfτ k ( ℓ ) | Ψ( f ) | . (5)where we assume that the transmit filter Ψ( f ) andthe receive filter Ψ ∗ ( − f ) are such that, given the bandwidth W , Ψ( f ) = ( if − W ≤ f ≤ W otherwise. (6)Sampling at the various frequencies f = − W , f = − W + N W , . . . , f N = − W + N − N W , weobtain the coefficients h ik , ≤ i ≤ N , as h ik = h k ( f i ) = L k − X ℓ =0 η k ( ℓ ) e − j π iN W τ k ( ℓ ) e jπW τ k ( ℓ ) . (7)Note that E (cid:2) | h ik | (cid:3) = ̺ .Since the users are supposed to be synchronizedwith the base station and for sake of simplicity, wewill consider in all the following that users add acyclic prefix of length equal to the channel impulseresponse length to their code sequence. This caseis similar to uplink MC-CDMA [30], [31]. As aconsequence, matrices { C k } are circulant [32] andcan all be diagonalized in the Fourier basis F [29].Model (2) simplifies therefore 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 with diagonal ele-ments { h ik } i =1 ...N . For each user k , the coefficients h ik are the discrete Fourier transform of the channelimpulse response.We make the hypothesis that the users employGaussian i.i.d. codes with zero mean and variance /N [33]. This hypothesis enables us to state simplyour results, however almost all of the results arevalid for any distribution of the codes as longas it has mean zero and variance /N [16]. Inparticular, since every unitary tranformation of aGaussian i.i.d. vector is a Gaussian i.i.d. vector (sothat w i = F H v i has the same distribution as v i forany i ), we multiply y in (8) with F H and obtainwithout any change in the statistics: y = K X k =1 H k w k p P k s k + n = (cid:0) H √ P ⊙ W (cid:1) s + n (9)where ⊙ is the Hadamard (element-wise) product. Note that in the asymptotic case (when N → ∞ ), the result holdswithout the need of a cyclic prefix as long as the channel is absolutelysummable [29]. In (9), H is the frequency selective fading matrix,of size N × K : H = h h . . . h K ... ... ... h N h N . . . h NK . √ P is the root square of the diagonal powercontrol matrix, of size K × K . W is an N × K random spreading matrix: W = (cid:2) w | w | · · · | w K (cid:3) where w k = w k ... w Nk . Note that asymptotically (as N → ∞ ), for agiven multipath channel of length L , model (9)is also valid for the case of uplink DS-CDMAsince all Toeplitz matrices can be asymptoticallydiagonalized in a Fourier Basis [29], [34].In the following, we will assume that the fre-quency selective fading matrix H behaves er-godically, as in Def. 3. The two-dimensionalchannel profile of H √ P is denoted ρ ( f, x ) = P ( x ) | h ( f, x ) | , f ∈ [0 , , x ∈ [0 , α ] . f is thefrequency index and x is the user index. Thisenables us to use Th. 1 in order to obtain expressionsfor the SINR.It is also assumed that the power of all users isupper bounded by P max and the square norm of thefading, on all paths, for all users, is upper boundedby h max .V. A SYMPTOTIC
SINR E
XPRESSIONS
Let h k be the k -th column of H , and H ( − k ) be H with h k removed. Similarly, let w k be the k -thcolumn of W , and W ( − k ) be W with w k removed.Let √ P ( − k ) be √ P with the k -th column and lineremoved. Finally, let G ( − k ) = H ( − k ) √ P ( − k ) ⊙ W ( − k ) . A. Matched Filter
Supposing perfect CSI at the receiver, thematched filter for the k -th user is given by g k = √ P k ( h k ⊙ w k ) . This leads to the following expres-sion for the SINR of user k SINR k = (cid:12)(cid:12) g Hk g k (cid:12)(cid:12) σ g Hk g k + g Hk (cid:16) G ( − k ) G H ( − k ) (cid:17) g k . Proposition 1: [16] As
N, K → ∞ with
K/N → α , the SINR of user k at the output ofthe matched filter is given by SINR k = β MF (cid:18) kN (cid:19) where β MF : [0 , α ] → R is given by β MF ( x ) = P ( x ) · ( H ( x )) σ H ( x ) + R α R P ( y ) | h ( f, y ) | | h ( f, x ) | df dy (10)and H ( x ) = R | h ( f, x ) | df .Denoting SINR k = β MF k , Prop. 1 enables us toextract an approximation of the value of the SINRof user k in the finite size case β MF k = P k (cid:16) N P Nn =1 | h nk | (cid:17) σ N P Nn =1 | h nk | + N P j = k P Nn =1 P j | h nj | | h nk | . (11)We observe that P k ∂β MF k ∂P k = β MF k . B. MMSE Filter
Supposing perfect CSI at the receiver,the MMSE filter for the k -th user isgiven by g MMSE k = R − g k , where R = (cid:18)(cid:16) H √ P ⊙ W (cid:17) (cid:16) H √ P ⊙ W (cid:17) H + σ I N (cid:19) . Thisleads to the following expression for the SINR ofuser k [14] SINR k = g Hk (cid:0) G ( − k ) G H ( − k ) + σ I N (cid:1) − g k . (12) Proposition 2: [16] As
N, K → ∞ with
K/N → α , the SINR of user k at the output ofthe MMSE receiver is given by: SINR k = β MMSE (cid:18) kN (cid:19) where β MMSE : [0 , α ] → R is a function defined bythe implicit equation β MMSE ( x ) = P ( x ) Z | h ( f, x ) | dfσ + R α P ( y ) | h ( f,y ) | dy β MMSE ( y ) . (13) Denoting SINR k = β MMSE k , Prop. 2 enables us toextract an approximation of the value of the SINRof user k in the finite size case β MMSE k = P k N N X n =1 | h nk | σ + N P j = k P j | h nj | β MMSE j . (14)From (12), we observe that P k ∂β MMSE k ∂P k = β MMSE k .From Prop. 2, we have the capacity of user kC MMSE k = 1 N log (cid:0) β MMSE k (cid:1) . The global capacity of the system is C MMSE = Z α log (cid:0) β MMSE ( x ) (cid:1) dx. (15) C. Optimal Filter
The term optimal filter designates a filter capa-ble of decoding the received signal at the boundgiven by Shannon’s capacity. Hence it is difficultto define an SINR associated to it. However, resultsof random matrix theory can still be applied. Let Y = (cid:16) H √ P ⊙ W (cid:17) . The definition of Shannon’scapacity per dimension for our system is C OPT ( N ) = 1 N log det (cid:18) I N + 1 σ YY H (cid:19) . (16)As N, K → ∞ with
K/N → α , C OPT ( N ) → Z log (cid:18) σ t (cid:19) ν ( dt ) (17)where ν is the empirical eigenvalue distributionof YY H , as in Def. 2. If we differentiate theasymptotic value C OPT of (17) with respect to σ ,we obtain ∂C OPT ∂σ = log ( e ) Z − σ t σ t ν ( dt )= log ( e ) Z σ (cid:0) − σ t − σ + σ (cid:1) σ (cid:0) σ t (cid:1) ν ( dt )= log ( e ) (cid:18)Z t + σ ν ( dt ) − σ Z ν ( dt ) (cid:19) = log ( e ) (cid:18) m ν ( − σ ) − σ (cid:19) (18)where m ν ( · ) is the Stieltjes transform of the empir-ical eigenvalue distribution of YY H . From Th. 1, m ν ( · ) is given by m ν ( z ) = Z u ( f, z ) df where u ( f, z ) is given by (1) with ρ H √ P ( f, x ) = ρ ( f, x ) = P ( x ) | h ( f, x ) | . Given that if σ = + ∞ , C OPT = 0 , it is immediate to obtain C OPT from (18)as C OPT = log ( e ) Z + ∞ σ m ν ( − z ) − z dz. (19) Proposition 3: C OPT and C MMSE are relatedthrough the following equality C OPT = C MMSE − log ( e ) Z α β MMSE ( x )1 + β MMSE ( x ) dx + Z log (cid:18) σ Z α ρ ( f, x )1 + β MMSE ( x ) dx (cid:19) df. (20) Proof:
See Appendix XI-A.The additional term in the right-hand side of (20)corresponds to the non-linear processing gain. Itquantifies the gain in terms of capacity that can beachieved between pure linear MMSE and non-linearfiltering.Assuming perfect cancellation of decoded users,successive interference cancellation with MMSEfilter achieves the optimum capacity [35]. The fol-lowing proposition ensues from this fact.
Proposition 4: [16] As
N, K → ∞ with
K/N → α , the optimal capacity is given by: C OPT = Z α log (cid:0) β SIC ( x ) (cid:1) dx where β SIC : [0 , α ] → R is a function defined by theimplicit equation β SIC ( x ) = P ( x ) Z | h ( f, x ) | dfσ + R x P ( y ) | h ( f,y ) | dy β SIC ( y ) . (21)Prop. 4 enables us to extract an expression that isanalog to the SINR for the optimal filter. Similarlyto the case of β MMSE in Sec. V-B, the derivative ofthis expression obeys the property P k ∂β SIC k ∂P k = β SIC k .VI. G AMES AND E QUILIBRIA
From now on, we denote
SINR k = β k , whicheverfilter is actually used. A. Power Allocation Game
A game with a unique strategy set for all users isdefined by a triple { S, P , ( u k ) k ∈ S } where S is theset of players , P is the set of strategies , and ( u k ) k ∈ S is the set of utility functions , u k : P | S | → R . In our setting, the players are simply the users,indexed by the set S K = { , . . . , K } . The strategyfor a mobile is its power allocation P k , which wewill assume belongs to a compact interval P =[0 , P max ] ⊆ R . The utility measures the gain of auser as a result of the strategy this user plays. In[21], the author derives what he calls Throughputto Power Ratio (TPR) under minimal requirements.The utility of user k is expressed u k = γ k P k . (22)We denote γ k = γ ( β k ) , where γ ( · ) is the samefunction for all users. In (22), γ is at least C andshould satisfy conditions detailed in [21] in order toobtain an “interesting” equilibrium.For example, in the simulations, we considerthe goodput γ ( β k ) , which is proportional to (cid:0) − e − β k (cid:1) M where M is the number of bits trans-mitted in a CDMA packet. Remark that the usualdefinition of goodput would rather be consideredproportional to q ( β k ) = (1 − BER k ) M , where BERis the bit error rate. However, this quantity is notzero when the transmitted power is zero. Using thisfunction in the utility would lead to the unsatisfyingconclusion that mobiles should not transmit at all,since the (improbable) event of a correct guess givesthem infinite utility [10]. Therefore, an adaptedversion of the goodput is adopted, where a factor 2is added before the BER. The performance measureconsidered is hence proportional to q ( β k ) = (1 − BER k ) M , leading to the expression above. Thisfunction has the desirable property q (0) = 0 and itsshape follows closely the shape of the original good-put q ( · ) . This is a relevant performance measure,as each mobile wants to use its (limited) batterypower to transmit the maximum possible amount ofinformation.This utility is expressed in bits per joule . In thenon-cooperative game setting, each user wants toselfishly maximize its utility. A Nash equilibrium isobtained when no user can benefit by unilaterallydeviating from its strategy.To obtain the maximum utility achievable by user k , we differentiate u k with respect to the power P k and equate to 0. We obtain P k ∂β k ∂P k γ ′ ( β k ) − γ ( β k ) = 0 . (23)For all filters under consideration, (10), (13) and(21) imply P k ∂β k ∂P k = β k , thus (23) reduces an equation on β k β k γ ′ ( β k ) − γ ( β k ) = 0 . (24)Eq. (24) is particularly interesting in the casewhen there exists a unique solution β ⋆ .The existence of a solution to (24) is guaranteedas long as the function γ ( · ) is a quasiconcavefunction of the SINR, i.e., there exists a point belowwhich the function is non-decreasing, and abovewhich the function is non-increasing [23], [21]. Inaddition, we assume that the function γ ( · ) takesvalue γ (0) = 0 , so that users cannot achieve aninfinite utility by not transmitting. This occurs forseveral functions γ ( · ) of interest, in particular thegoodput [24], which we will use for simulations.Unfortunately, the capacity can not be used as afunction γ ( · ) , since it leads to the trivial result β ⋆ = 0 for this utility function. The uniqueness ofthe solution β ⋆ to (24) is due to fact that the SINRof each user is a strictly increasing function of itstransmit power. Given the target SINR β ⋆ , we obtainthe strategy of users in the next section.VII. P OWER A LLOCATION IN THE N ASH E QUILIBRIUM
A. Flat Fading
In this subsection, we show that the results of[24] for Matched and MMSE filters are a specialcase of our setting when L = 1 (flat fading case).In addition, we derive the power allocation for theOptimum filter. When there is only one path, foreach user k , denoted by its index kN = x ∈ [0 , α ] , h ( f, x ) does not depend on f . Given the target SINR β ⋆ , we have explicit expressions of the power withwhich user k transmits for the various receivers.In Appendix XI-B, we show that the influence ofthe strategy of a player on the payoffs of other play-ers is (asymptotically) “small”. It justifies the factthat we can obtain an equilibrium in the asymptoticsetting, without the need for players to possess allthe information on the system. Their local informa-tion is sufficient. In the asymptotic limit, we obtainresults similar to Wardrop equilibrium: the strategyused by each user does not influence the strategy ofother users.
1) Matched filter:
From Prop. 1, the continuousformulation is P ( x ) = β ⋆ (cid:0) σ + R α P ( y ) | h ( y ) | dy (cid:1) | h ( x ) | or equivalently in a discrete form P k = β ⋆ (cid:16) σ + N P Kj =1 ,j = k P j | h j | (cid:17) | h k | . (25)Summing (25) over k = 1 , . . . , K , we obtain aclosed form expression for the minimum power withwhich user k transmits when using the matched filter P k = 1 | h k | σ β ⋆ − αβ ⋆ for α < β ⋆ . (26)
2) MMSE filter:
From Prop. 2, the continuousformulation is P ( x ) = β ⋆ (cid:16) σ + β ⋆ R α P ( y ) | h ( y ) | dy (cid:17) | h ( x ) | . or equivalently in a discrete form P k = β ⋆ (cid:16) σ + β ⋆ N P Kj =1 ,j = k P j | h j | (cid:17) | h k | . (27)Summing (27) over k = 1 , . . . , K , we obtain aclosed form expression for the minimum power withwhich user k transmits when using the MMSE filter P k = 1 | h k | σ β ⋆ − α β ⋆ β ⋆ for α < β ⋆ . (28)Both (26) and (28) are the same results as in [24].
3) Optimum filter:
Each user maximizes its util-ity for a SINR equal to β ⋆ . However, in the case ofthe optimum filter, the SINR is not defined directly.It is nevertheless possible to extract an equivalentquantity from the expression of the capacity, sincethe value of the capacity of user k at the equilibriumis given by C ⋆ = N log (1 + β ⋆ ) . Proposition 5:
The power allocation is given by P k = 1 | h k | σ β + − α β + β + for α < β + (29)where β + is the solution to α log (cid:0) β + (cid:1) − α log ( e ) β + β + +log β + αβ + − α β + β + ! = α log (1 + β ⋆ ) . (30) Proof:
See Appendix XI-C.
B. Frequency Selective Fading
In the context of frequency selective fading, foreach user k , denoted by its index kN = x ∈ [0 , α ] ,there are L > paths with respective attenua-tions h ℓ ( x ) , ℓ = 1 , . . . , L , which are i.i.d. ran-dom variables with some known distribution. Wesuppose that h ℓ ( x ) has mean zero, and the dis-tributions of the real part and imaginary part of h ℓ ( x ) are even functions, as for example the Gaus-sian distribution, which we consider in the simu-lations. h ( f, x ) depends on f through h ( f, x ) = P Lℓ =1 h ℓ ( x ) e − πif ( ℓ − . Given the target SINR β ⋆ ,the Nash equilibrium power allocation is determinedby implicit equations for the various receivers.
1) Matched filter:
The continuous formulation is P ( x ) = β ⋆ · σ H ( x ) + R R α P ( y ) | h ( f, y ) | | h ( f, x ) | df dy ( H ( x )) or equivalently in a discrete form P k = β ⋆ · σ N P Nn =1 | h nk | + N P Nn =1 | h nk | N P Kj = k P j | h nj | (cid:16) N P Nn =1 | h nk | (cid:17) . (31)In (31), h nk = h (cid:0) n − N , kN (cid:1) .In this expression, the power allocation of user k seems to depend on the power allocation and fadingrealization of all the other users. However, when thenumber of users tends to infinity, the strategy of anysingle user does not have any influence on the payoffof user k , as shown in Appendix XI-B. Hence,the appropriate framework is non-atomic games.The expression N P Kj =1 P j | h nj | is asymptoticallya constant (not depending on n ), denoted Ω . Ω = αβ ⋆ σ K P Kj =1 | h nj | E j − αβ ⋆ K P Kj =1 | h nj | E j (32)where E j = N P Nm =1 | h mj | .As K → ∞ , we can apply the Central LimitTheorem to the sum of random variables K K X j =1 | h nj | E j . (33)It tends to its expectation, which is equal to (seeAppendix XI-D). It follows that asymptotically Ω = αβ ⋆ σ − αβ ⋆ (andsimulations in Sec. IX prove that this approximationis valid for moderate finite values of N ). From (31),we obtain a formula similar to (26) P k = 1 E k σ β ⋆ − αβ ⋆ for α < β ⋆ . (34)
2) MMSE filter:
The continuous formulation is P ( x ) = β ⋆ R | h ( f,x ) | dfσ + β⋆ R α P ( y ) | h ( f,y ) | dy (35)or equivalently in a discrete form P k = β ⋆ N P Nn =1 | h nk | σ + β⋆ N P Kj =1 ,j = k P j | h nj | . (36)In (36), h nk = h (cid:0) n − N , kN (cid:1) .As previously, when the number of users tendsto infinity, N P Kj =1 P j | h nj | is asymptotically aconstant (not depending on n ), denoted Ω . Ω = αβ ⋆ σ K P Kj =1 | h nj | E j − αβ ⋆ β ⋆ K P Kj =1 | h nj | E j (37)where E j = N P Nm =1 | h mj | .It follows that asymptotically Ω = αβ ⋆ σ − α β⋆ β⋆ , weobtain a formula similar to (28) P k = 1 E k σ β ⋆ − α β ⋆ β ⋆ for α < β ⋆ . (38)
3) Optimum filter:
Each user maximizes its util-ity for a SINR equal to β ⋆ . However, in the case ofthe optimum filter, the SINR is not defined directly.It is nevertheless possible to extract an equivalentquantity from the expression of the capacity, sincethe value of the capacity of user k at the equilibriumis given by C ⋆ = N log (1 + β ⋆ ) . Proposition 6:
Asymptotically, as
N, K → ∞ ,the power allocation is given by P k = 1 E k σ β + − α β + β + for α < β + (39)where β + is the solution to α log (cid:0) β + (cid:1) − α log ( e ) β + β + +log β + αβ + − α β + β + ! = α log (1 + β ⋆ ) . (40) Proof:
The proof is similar to the proof ofProp. 5.We observe that for all filters considered, theoptimal PA is a constant times the inverse of the total energy of the channel E j . Via Parseval’s The-orem, E j = P Lℓ =1 (cid:12)(cid:12) h ℓ (cid:0) jN (cid:1)(cid:12)(cid:12) . It is a sum of i.i.d.random variables. As the number of paths increases,the optimal PA tends to a uniform PA. This isan effect similar to “channel hardening” [20]: asthe number of paths increases, the variance of thedistribution of the channel energy decreases andthe Nash equilibrium PA becomes more and moreuniform for all users.VIII. S UCCESSIVE I NTERFERENCE C ANCELLATION
The optimal filter gives a bound on the perfor-mance that can be achieved through (non-linear)filtering at the base station. In order to improve theperformance of the system, we introduce SuccessiveInterference Cancellation (SIC) [19] at the basestation. Under the assumption of perfect decoding,SIC improves immensely the performance of lin-ear filters (Matched Filter or MMSE Filter). TheMMSE SIC filter actually achieves the optimum fil-ter bound, under the assumption of perfect decoding.The principle of SIC receivers is quite simple: usersare ordered and are decoded successively. At eachstep, supposing that the user has been encoded at theappropriate decoding rate, the signal is decoded andits contribution to the interference is then perfectlysubtracted. This removes some of the inter-userinterference and therefore increases the
SINR of thefollowing decoded users.The challenge is that the users must transmitat the appropriate rate to avoid the catastrophicoccurrence of imperfect decoding. Usually, the or-dering of users is done in a centralized way, at thebase station which then advertises it to the users.However, for the protocol to remain distributed,users should be able to decide, based on their localinformation, at which rate to transmit.At equilibrium, the rate is determined by theSINR β ⋆ , and it is the transmission power of theuser that is determined according to its rank ofdecoding. The equilibrium PA can be determinedin a simple manner when the number of multipathsis finite ( L < ∞ ) and the number of users is veryhigh ( K → ∞ ). In Sec. VIII-A, we make use of the fact that the whole law of E j is realized in thiscase, so that users automatically know their rankof decoding. Another manner to give a (random)ordering of decoding is to introduce an additionaldegree of liberty in the system. In Sec. VIII-B, wedevelop a correlated game framework that enablesusers to learn their rank of decoding in a simpleway. In the following, we assume that each userhas a unique has a unique i.d. number j rangingbetween 1 to K . A. Ordering when K → ∞ If the number of users K → ∞ , with L fixed,the whole law of the total channel energy will berealized. Assume the base station advertises to theusers that they will be decoded by decreasing totalchannel energy. Each user knows, according to therealization of its fading, its rank in the decodingorder given by K times minus the cumulative dis-tribution function D ( · ) of the total channel energy E j . rank j = K (1 − D ( E j )) . In case that the base station advertises to the usersthat they will be decoded by increasing total channelenergy, user j will have rank rank j = KD ( E j ) . B. Correlated Equilibrium
We wish to introduce a simple mechanism thatenables players to coordinate and to know in whichorder they will be decoded. We place ourselvesin the context of correlated games. The notion ofcorrelated equilibrium was introduced by R. Au-mann in [36] and further studied in [37], [38], [39].They represent a generalization of Nash equilibrium.The important feature of correlated games is thepresence of an arbitrator . An arbitrator needs nothave any intelligence or knowledge of the game,it needs only to send random (private or public)signals to the players that are independent of allother data in the game. In the context of non-cooperative games, each player has the possibilitynot to consider the signal(s) it receives. Coordina-tion between players turns out to be useful also inthe case of cooperative optimization. The signalsenable joint randomization between the strategiesof the players, possibly resulting in equilibria with Prof. R. Aumann has received in 2005 the Nobel prize in economyfor his contributions to game theory, together with Thomas Schelling. higher payoffs. The concept of correlated gameswas recently introduced in a networking context in[40], where the authors consider a simple ALOHAsetting.The simplest and most intuitive coordinationmechanism is given by a common signal whichusers as well as the base station overhear beforeeach transmission. There are K ! possible permu-tations of K users. Hence, the arbitrator broad-casts a signal to the users belonging to the set { , . . . , K ! − } . Each of these numbers correspondsto a permutation π of { , . . . , K } that gives the(random) ordering of decoding as rank j = π ( j ) . Theusers can then adjust their transmit power accordingto this ordering. In terms of size of the message,this is equivalent to the case when the base stationdecides the decoding order and broadcasts it to theusers, or sends K individual messages of ln( K ) bits containing the rank, since ln( K !) = K ln( K ) + o ( K ln( K )) . However, there is no need of either anyknowledge of the system or computations at the basestation in the case of the correlated mechanism. C. SIC Power Allocations
In both cases, once the users know their order,they can calculate their transmit power according tothe filter that is used. The equilibrium still occurswhen all users reach the SINR β ⋆ . A single user willnot benefit by deviating, since it would decrease itsutility. From now on, index k denotes the rank ofdecoding.In the case of the matched filter with SIC, theSINR of the user decoded at rank k is β MF k = P k (cid:16) N P Nn =1 | h nk | (cid:17) σ N P Nn =1 | h nk | + N P j>k P Nn =1 P j | h nj | | h nk | . (41)From (41), we get the equilibrium PA of user k as P k = β ⋆ · σ N P Nn =1 | h nk | + N P j>k P Nn =1 P j | h nj | | h nk | (cid:16) N P Nn =1 | h nk | (cid:17) . (42) In the case of the MMSE filter with SIC, the SINRof the user decoded at rank k is β MMSE k = P k N N X n =1 | h nk | σ + N P j>k P j | h nj | β MMSE j . (43)From (43), we get the equilibrium PA of user k as P k = β ⋆ N P Nn =1 | h nk | σ + β⋆ N P Kj>k P j | h nj | . (44)For flat fading, a simple recursion gives theequilibrium PA (see Appendix XI-E). We obtainrespectively P MF k = σ β ⋆ | h k | (cid:18) N β ⋆ (cid:19) K − k , (45) P MMSE k = σ β ⋆ | h k | (cid:18) N β ⋆ β ⋆ (cid:19) K − k . (46)As far as frequency-selective fading is concerned,this gives us the form of the asymptotic expressions.Asymptotically, the power allocation of one userwill not depend on the PA of the other users, asshown in Appendix XI-B. With a similar reasoningas in Sec. VII, the expressions mimic (45) and (46)with the total channel energy E k replacing | h k | ,i.e., P MF k = σ β ⋆ E k (cid:18) N β ⋆ (cid:19) K − k , (47) P MMSE k = σ β ⋆ E k (cid:18) N β ⋆ β ⋆ (cid:19) K − k . (48)These expressions are also validated by simulations.Since MMSE SIC with perfect decoding is equiv-alent to the optimum filter, we thus obtain a secondpossible equilibrium PA for the optimum filter. InSec. IX, we investigate which is the PA which min-imizes total amount of power needed to transmit atequilibrium SINR. In the case of automatic orderingof the users, one question is whether it is best toorder the users by increasing or decreasing totalfading energy. The answer is the following: it isalways best to decode the users by decreasing totalchannel energy E < · · · < E k (see AppendixXI-F).An interesting feature of equilibrium PA (47) and(48) is that there is no limitation on the numberof users than can be accomodated by the system,contrary to the previous case of (34), (38) and (39). The limitation is only imposed by the increasingpower needed for each new user decoded last, whichgrows without bound as an exponential.IX. N
UMERICAL R ESULTS
In all the following, we consider that P max ischosen sufficiently high so that users can actuallytransmit at the equilibrium PA values. For the simu-lations, we consider the usual case of Rayleigh fad-ing. Although Rayleigh distribution is not boundedfrom above, simulations show that the results stillhold.We consider a CDMA system with K = 32 users and a spreading factor N = 256 . The noisevariance is σ = 10 − . For a number of bitsin a CDMA packet M = 100 , the goodput is γ ( β ) = (cid:0) − e − β (cid:1) (see [24]), and β ⋆ = 6 . .The capacity achieved at the Nash Equilibrium is C = α log (1 + β ⋆ ) = 0 . bits/s. Unfortunately,the capacity itself cannot be used as a relevantperformance measure in the definition of the utility,because in this case the maximal utility is obtainedwhen not sending.We have performed simulations over 10000 re-alizations. Fig. 1 shows the good fit of theoreticvalues calculated directly from (34), (38) and (39)with those simulations. The values of the utility donot depend on the number of multipaths. We seethat optimum filter requires the minimal power, andmatched filter the maximal power to achieve therequired goodput.In Fig. 2 we have plotted the average utilityversus the number of multipaths L . Multipaths aresupposed to be i.i.d. Rayleigh distributed with vari-ance /L , in order for the channels to have thesame energy. Two cases are considered: the utilityobtained in the Nash equilibrium, according to thePA given by (31) and (36), and the utility in thecase where all nodes transmit at the same power.For comparison purposes, the sum of the uniformpowers is equal to the sum of the powers usedin the Nash equilibrium. In addition, simulations(not reproduced here) show that this value gives thehigher average utility for a uniform PA. The utilitydoes not vary with L in the Nash equilibrium: theCentral Limit Theorem applies to the utility, whichis a constant times the random variable E k in theNash equilibrium. The utility with uniform powersis always inferior to the utility in the Nash equilib-rium. However, as L increases, the gap decreases, as the variance of E k decreases, and the equilibriumPA becomes uniform.In Fig. 3 we have plotted the average of theinverse power of the users in the Nash equilibriumfor each of the investigated schemes. We plot theaverage inverse power because of the direct relationto the utility for the users. The higher this average,the higher the utility for the user. The SIC filters arealways more efficient than their linear counterparts.However, for a load α < . and optimum filter ,it is better to use the first variation of PA (39)than use MMSE SIC (48). This relation is reversedwhen α > . . In addition to the theoreticalcurves, Monte-Carlo simulations were performedboth with random ordering (circles) and ordering bydecreasing total channel energy (crosses), for L = 8 multipaths. Simulations show that the optimal order-ing improves the power efficiency of the successiveinterference cancellation filters.In Fig. 4, we investigate the amelioration pro-vided by optimal ordering as a function of thenumber of multipaths. The simulations are done for K = 128 users, in order to be in the “interesting”zone α > . . As expected, as the number of pathsincreases, the total channel energy is more and morethe same for each channel and the gain providedby ordering the users decreases. However, when thenumber of users is very large and they benefit fromautomatic ordering, we see that the utility with theMMSE SIC equilibrium PA is the maximal utilitythat can be obtained in the non-cooperative setting.X. C ONCLUSION
Using tools of random matrices, we have de-rived the equilibrium power allocation in a game-theoretic framework applied to asymptotic CDMAwith cyclic prefix, under frequency-selective fad-ing. Three receivers are considered: matched filter,MMSE and optimum filter (given by Shannon’scapacity). In addition, distributed ordering mecha-nisms are introduced and the successive interferencecancellation variants of the linear filters are studied.For each user, this power allocation depends onlyon the total energy of the channel of the userunder consideration. For a frequency-flat channel,the power allocation among users is dis-uniform,whereas when the number of multipaths increases, The value of α is obtained as solution of the equation αβ ⋆ β ⋆ β ⋆ (1 − α β + β + ) = β + (1 − exp( − α β ⋆ β ⋆ )) . Number of Multipaths L U t ili t y = G oodpu t/ P o w e r Matched FilterMMSE FilterOptimum Filter
Fig. 1. Comparison of theoretic values and simulations for utilitiesin the Nash equilibrium. Number of Multipaths L U t ili t y = G oodpu t/ P o w e r MFMMSEOptMFwMMSEwOptw
Fig. 2. Simulation of utilities in the Nash equilibrium and constantpower allocations versus L . the power allocation tends more and more to auniform one. XI. A PPENDIX
A. Proof of Prop. 3
Notice that when σ → ∞ , C OPT = 0 , C MMSE = 0 and β MMSE ( x ) = β ( x ) = 0 . Thus we only have toprove that the derivatives of either side of (20) areequal.Using ρ ( f, x ) = P ( x ) | h ( f, x ) | , (13) can be Load α A v e r age i n v e r s e po w e r MFMMSEOptMFsicMMSEsic
Fig. 3. Average inverse power used by the different filters. Number of Multipaths L U t ili t y = G oodpu t/ P o w e r MFsicMMSEsicMFsortedMMSEsorted
Fig. 4. Simulation of utilities in the Nash equilibrium with SICfilter with and without optimal ordering, versus L . rewritten β ( x ) = Z ρ ( f, x ) dfσ + R α ρ ( f,y )2 dy β ( y ) . (49)From (1), R ρ ( f, x ) u ( f, − σ ) df satisfies thesame implicit equation (49) as β ( x ) and thus u ( f, − σ ) = 1 R α ρ ( f,y ) dy β ( y ) + σ . (50) Using (49) and (50), we can rewrite Z u ( f, − σ ) df − σ = Z R α ρ ( f,y ) dy β ( y ) + σ df − Z σ df = Z − R α ρ ( f,x )1+ β ( x ) dxσ (cid:16)R α ρ ( f,y ) dy β ( y ) + σ (cid:17) df = Z α − β ( x )) σ Z ρ ( f, x ) df R α ρ ( f,y ) dy β ( y ) + σ dx = − Z α β ( x ) σ (1 + β ( x )) dx. Thus from (18) ∂C OPT ∂σ = − log ( e ) Z α β ( x ) σ (1 + β ( x )) dx. (51)Differentiating (15) with respect to σ , we obtain ∂C MMSE ∂σ = log ( e ) Z α
11 + β ( x ) ∂β∂σ ( x ) dx. (52)Let π ( x ) = σ (1+ β ( x )) . From (51) and (52), weobtain ∂C OPT ∂σ − ∂C MMSE ∂σ = − log ( e ) Z α (cid:18) β ( x ) + σ ∂β∂σ ( x ) (cid:19) π ( x ) dx. (53)From (13), we have Z α σ β ( x ) ∂π∂σ ( x ) dx = Z α Z σ ρ ( f, x ) dfσ + R α σ ρ ( f, y ) π ( y ) dy ∂π∂σ ( x ) dx = Z R α ρ ( f, x ) ∂π∂σ ( x ) dx R α ρ ( f, y ) π ( y ) dy df = 1log ( e ) ∂∂σ Z log (cid:18) Z α ρ ( f, y ) π ( y ) dy (cid:19) df. Observing that Z α (cid:18) β ( x ) + σ ∂β∂σ ( x ) (cid:19) π ( x )+ σ β ( x ) ∂π∂σ ( x ) dx = ∂∂σ Z α σ β ( x ) π ( x ) dx we obtain (20) from Prop. 3. B. Influence of Other Players’ Strategies
We want to prove that asymptotically, in the game { S K , P , ( u k ) k ∈ S K } , the strategy of a single playerdoes not have any influence on the payoff of theother players. In other words, for all k = i ∈ S K ,for all p = ( P , . . . , P K ) ∈ P K , for all P ′ i ∈ P , (cid:12)(cid:12) u k ( p ) − u k ( P ′ i , p ( − i ) ) (cid:12)(cid:12) → , as N → ∞ . Remember that u k = γ ( β k ) P k , and γ is at least C . Let ( β , . . . , β K ) be the SINRs associated withthe power allocation p and ( β ′ , . . . , β ′ K ) the SINRsassociated with the power allocation ( P ′ i , p ( − i ) ) .Then a simple Taylor expansion of γ in β ′ k gives γ ( β ′ k ) = γ ( β k )+( β ′ k − β k ) ∂γ∂β ( β k )+ o ( β ′ k − β k ) . (54)According to (54), it is sufficient to show that (cid:12)(cid:12)(cid:12)(cid:12) β ′ k − β k P k (cid:12)(cid:12)(cid:12)(cid:12) → , as N → ∞ . (55) a) Matched Filter: For the matched filter,the inequality is obtained directly from (11).The denominator of (11) is always greater than σ N P Nn =1 | h nk | . Hence, (cid:12)(cid:12)(cid:12)(cid:12) β ′ k − β k P k (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P k N ( P ′ i − P i ) N P Nn =1 | h ni | | h nk | P k σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ P max h max σ N . b) MMSE Filter:
For the MMSE filter, theinequality is obtained from (12), Lemma 1 from [33]and Lemma 2.1 from [41], which we both reproducebelow for convenience.
Lemma 1: [33] Let C be a N × N complexmatrix with uniformely bounded spectral radius forall N : sup N ( | C | ) < ∞ . Let w = √ N [ w , . . . , w N ] T where { w i } i =1 ...N are i.i.d. complex random vari-ables with zero mean, unit variance and finite eighthmoment. Then: E "(cid:12)(cid:12)(cid:12)(cid:12) w H Cw − N tr C (cid:12)(cid:12)(cid:12)(cid:12) ≤ CN where C is a constant that does not depend on N or C . Lemma 2: [41] Let σ > , A and B N × N with B Hermitian nonnegative definite, and q ∈ C N .Then tr (cid:0)(cid:0) ( B + σ I ) − − ( B + qq H + σ I ) − (cid:1) A (cid:1) ≤ k A k σ . In Lemma 2, k A k is the spectral norm of A , i.e.,the square root of the largest singular value of A .From (7), we can write β k = P k w kH H Hk (cid:0) G ( − k ) G H ( − k ) + σ I N (cid:1) − H k w k ,β ′ k = P k w kH H Hk (cid:16) G ( − k ) ′ G H ( − k ) ′ + σ I N (cid:17) − 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 Hk (cid:16) G ( − k ) G H ( − k ) + σ I N (cid:17) − H k or matrix C = H Hk (cid:16) G ( − k ) ′ G H ( − k ) ′ + σ I N (cid:17) − H k , we obtain[33] (cid:12)(cid:12)(cid:12)(cid:12) w kH Cw k − N tr C (cid:12)(cid:12)(cid:12)(cid:12) → , as N → ∞ . Matrix B = G ( − k ) G H ( − k ) is Hermitian nonnega-tive definite, as for all w ∈ C N , w H G ( − k ) G H ( − k ) w = (cid:13)(cid:13) G ( − k ) w (cid:13)(cid:13) ≥ . Diagonal matrix A = H k H Hk hasspectral norm (cid:13)(cid:13) H k H Hk (cid:13)(cid:13) ≤ h max . Using Lemmas 1and 2, as N → ∞ , we obtain (cid:12)(cid:12)(cid:12)(cid:12) β ′ k − β k P k (cid:12)(cid:12)(cid:12)(cid:12) → , as N → ∞ . c) Optimum and Successive Interference Can-cellation Filters: The analog of the SINR derivedfor the optimum filter stems from the MMSE filterwith SIC. The SINR for SIC filters have similarexpressions with less interfering users appearing inthe denominator. Hence the result is immediate.
C. Proof of Prop. 5
Given C ⋆ , we can use (20) to obtain a Nashequilibrium power allocation in the following way.We rewrite (20) assuming that the target SINR forthe MMSE filter is β + . α log (cid:0) β + (cid:1) − α log ( e ) β + β + + log (cid:18) σ (1 + β + ) Z α P ( y ) | h ( y ) | dy (cid:19) = α log (1 + β ⋆ ) . (56)In the left-hand side of (56), P ( y ) is given by aMMSE power allocation similar to the one given by (28). Hence, the term R α P ( y ) | h ( y ) | dy in (56)does not depend on the actual realizations of thechannels. Replacing β ⋆ by β + in (27), we obtainthat R α P ( y ) | h ( y ) | dy = ασ β + − α β +1+ β + , which gives us(30). Replacing β ⋆ by β + in (28), we obtain thepower allocation (29). D. Expectation of the random variable (33)For each user j , there are L > paths withrespective attenuations h ℓ (cid:0) jN (cid:1) , ℓ = 1 , . . . , L , whichare i.i.d. complex random variables with mean zeroand even distributions of the real and imaginaryparts. The Fourier transform of those attenuationsis h nj = h (cid:0) nN , jN (cid:1) = P Lℓ =1 h ℓ (cid:0) jN (cid:1) e − πi nN ( ℓ − . Thetotal energy of the paths is E j = P Lℓ =1 (cid:12)(cid:12) h ℓ (cid:0) jN (cid:1)(cid:12)(cid:12) .We want to show that the expectation of therandom variable K P Kj =1 | h nj | E j is equal to 1. Byexpanding the expression of h nj , this is equivalent toshowing that the expectation of the random variable h ℓ (cid:0) jN (cid:1) h ℓ ′ (cid:16) j ′ N (cid:17) E j is equal to 0. Denoting by p ( · ) the distribution of h ℓ = h ℓ (cid:0) jN (cid:1) , this expectation is equal to the L -dimensional integral of h ℓ h ℓ ′ | h ℓ | + | h ℓ ′ | + P k = ℓ,ℓ ′ | h k | p ( h ℓ ) p ( h ℓ ′ ) Y k = ℓ,ℓ ′ p ( h k ) which is an odd function of h ℓ . Its integral istherefore 0, which proves the desired result. E. Proof of (45) and (46)Denote m k = P K − k | h K − k | . From (42), with flatfading, the sequence { m k } k ∈ S K satisfies m = β ⋆ σ and m k +1 = β ⋆ σ + β ⋆ N P kj =0 m j . Using the factthat P ki = j (cid:0) ij (cid:1) = (cid:0) k +1 j +1 (cid:1) , it is immediate to prove byrecurrence that m k = β ⋆ σ k X j =0 (cid:18) kj (cid:19) N j β ⋆j = β ⋆ σ (cid:18) N β ⋆ (cid:19) k . Hence formula (45). The demonstration is exactlysimilar for (46) from the recursion m = β ⋆ σ and m k +1 = β ⋆ σ + β ⋆ (1+ β ⋆ ) N P kj =0 m j . F. Optimal Ordering of Users
We determine the ordering that makes use of theleast total power for equilibrium PA (45) (the caseis similar for (46), (47) and (48)). Let the orderingof the users be such as | h | < · · · < | h K | . Let π be any permutation of { , . . . , K } . Let a ij = (cid:0) N β ⋆ (cid:1) K − i − (cid:0) N β ⋆ (cid:1) K − j .Then showing that the optimal ordering is suchas | h | < · · · < | h K | is equivalent to showing thatfor any π K X k =1 | h k | a kπ ( k ) > . (57)Consider first a cyclic permutation. By the def-inition of a ij , the sum of the a kπ ( k ) is equal tozero: P Kk =1 a kπ ( k ) = 0 . The first coefficient a π (1) is positive. It is affected coefficient | h | , which isthe greatest coefficient in the sum in (57). Hencethe sum in (57) is positive in this case.Permutation π can be decomposed as a product ofdisjoint permutation cycles. Each cycle determines asubset of indexes k , these subsets form a partition of { , . . . , K } . With a similar reasoning as precedently,replacing index with the smallest index in thecycle, the sum over the indexes k pertaining to acycle of | h k | a kπ ( k ) is positive. Hence the global sumof (57) is also positive.It can be proven in a similar way that the sameordering maximizes the sum of inverse powers ofthe users. XII. A CKNOWLEDGEMENTS
The authors would like to thank Prof. J. Silver-stein for pointing us to reference [41].R
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