Power Allocation Games in Wireless Networks of Multi-antenna Terminals

Transcription

1 Power Aocation Games in Wireess Networs of Muti-antenna Terminas Eena-Veronica Bemega, Samson Lasauce, Mérouane Debbah, Marc Jungers, and Juien Dumont arxiv:0.4598v2 [cs.it] 23 Nov 200 Abstract We consider wireess networs that can be modeed by mutipe access channes in which a the terminas are equipped with mutipe antennas. The propagation mode used to account for the effects of transmit and receive antenna correations is the unitary-invariant-unitary mode, which is one of the most genera modes avaiabe in the iterature. In this context, we introduce and anayze two resource aocation games. In both games, the mobie stations sefishy choose their power aocation poicies in order to maximize their individua upin transmission rates; in particuar they can ignore some specified centraized poicies. In the first game considered, the base station impements successive interference canceation (SIC) and each mobie station chooses his best space-time power aocation scheme; here, a coordination mechanism is used to indicate to the users the order in which the receiver appies SIC. In the second framewor, the base station is assumed to impement singe-user decoding. For these two games a thorough anaysis of the Nash equiibrium is provided: the existence and uniqueness issues are addressed; the corresponding power aocation poicies are determined by expoiting random matrix theory; the sum-rate efficiency of the equiibrium is studied anayticay in the ow and high signa-to-noise ratio regimes and by simuations in more typica scenarios. Simuations show that, in particuar, the sum-rate efficiency is high for the type of systems investigated and the performance oss due to the use of the proposed suboptimum coordination mechanism is very sma. Index Terms theory. MIMO, MAC, non-cooperative games, Nash equiibrium, power aocation, price of anarchy, random matrix E. V. Bemega and S. Lasauce are with LSS (joint ab of CNRS, Supéec, Univ. Paris-Sud ), Supéec, Pateau du Mouon, 992 Gifsur-Yvette Cedex, France, {bemega,asauce}@ss.supeec.fr; M. Debbah is with the Acate-Lucent Chair on Fexibe Radio, Supéec, Pateau du Mouon, 992 Gif-sur-Yvette Cedex, France, merouane.debbah@supeec.fr; M. Jungers is with CRAN, Nancy-Université, CNRS, 2 avenue de a Forêt de Haye, 5456 Vandoeuvre-es-Nancy, France, marc.jungers@cran.uhp-nancy.fr; J. Dumont is with Lycée Chapta, 45 Bouevard des Batignoes, Paris, France, dumont@crans.org

2 2 I. INTRODUCTION In this paper, we consider the upin of a decentraized networ of severa mobie stations (MS) and one base station (BS). This type of networ is commony referred to as the decentraized mutipe access channe (MAC). The networ is said to be decentraized in the sense that each user can freey choose his power aocation (PA) poicy in order to sefishy maximize a certain individua performance criterion, which is caed utiity or payoff. This means that, even if the the BS broadcasts some specified poicies, every user is free to ignore the poicy intended for him if the atter does not maximize his performance criterion. To the best of the authors nowedge, the probem of decentraized PA in wireess networs has been propery formaized for the first time in [], [2]. Interestingy, this probem can be formuated quite naturay as a non-cooperative game with different performance criteria (utiities) such as the carrier-to-interference ratio [3], aggregate throughput [4] or energy efficiency [5], [6]. In this paper, we assume that the users want to maximize information-theoretic utiities and more precisey their Shannon transmission rates. Indeed, the point of view adopted here is cose to the one proposed by the authors of [7] for DSL (digita subscriber ines) systems, which are modeed as a parae interference channe; [8] for the singe input singe output (SISO) and singe input mutipe output (SIMO) fast fading MACs with goba CSIR and goba CSIT (Channe State Information at the Receiver/Transmitters); [9] for MIMO (Mutipe Input Mutipe Output) MACs with goba CSIR, channe distribution information at the transmitters (goba CDIT) and singe-user decoding (SUD) at the receivers; [0], [] for Gaussian MIMO interference channes with goba CSIR and oca CSIT and, by definition of the conventiona interference channe [2], SUD at the receivers. Note that reference [3] where the authors considered Gaussian MIMO MACs with neither CSIT nor CDIT differs from our approach and that of [7], [8], [9], [0], [] because in [3] the MIMO MAC is seen as a two-payer zero-sum game where the first payer is the group of transmitters and the second payer is the set of MIMO sub-channes. The cosest wors to the wor presented here are [9] and [4]. Athough this paper is in part based on these wors, it sti provides significant contributions w.r.t. to them, as expained beow. In [9], the authors consider MIMO mutipe access channes and assume SUD at the BS; the authors formuate the PA probem into a team game in which each user chooses his PA to maximize the networ sum-rate. In [4], the same type of decentraized networs is considered but SIC is assumed at the BS. As each user needs to now his decoding ran in order to adapt his PA poicy to maximize his individua transmission rate, a coordination mechanism has to be introduced: the coordination signa precisey indicates to a the users the decoding order used by the receiver. The present paper differs from these two contributions on at east four

3 3 important technica points: (i) when SUD is assumed, the PA game is not formuated as a team game but as a non-cooperative one; (ii) we expoit severa proof techniques that are different from [9]; (iii) whie [9] and [4] assume a Kronecer propagation mode with common receive correation we assume here a more genera mode, the unitary-invariant-unitary (UIU) propagation mode introduced by [2], for which the users can have different receive antenna correation profies. This is usefu in practice since, for instance, it aows one to study propagation scenarios where some users can be in ine of sight with the BS (the receive antenna are strongy correated) whereas other users can be surrounded by many obstaces, which can strongy decorreate the receive antennas for these users; (iv) whie the authors of [4] restricted their attention to either a purey spatia PA probem or a purey tempora PA probem, we tace here the genera space-time PA probem. In this context, our main objective is to study the equiibrium of two power aocation games associated with the two types of decoding schemes aforementioned (namey SIC and SUD). The motivation for this is that the existence of an equiibrium aows networ designers to predict, with a certain degree of stabiity, the effective operating state(s) of the networ. Ceary, in our context, uniqueness is a desirabe feature of the equiibrium. As it wi be seen, it is possibe to prove the existence in both games under investigation. Uniqueness is proven in the case of SUD whie it is conjectured for the case of SIC. In order to estabish the corresponding resuts, the paper is structured as foows. After presenting the genera system mode in Sec. II, we anayze in detai the space-time PA game when SIC and a corresponding coordination mechanism are assumed (Sec. III). For this game, the existence and uniqueness of the NE are proven and the equiibrium is determined by expoiting random matrix theory when the numbers of antennas are sufficienty arge. Its sum-rate efficiency is aso anayzed. In Sec. IV, we anayze the case of SUD since this decoding scheme, athough suboptima in terms of performance (even in the case of a networ with singe-antenna terminas), has some features that can be found desirabe in some contexts: the receiver compexity is ow, there is no need for a coordination signa, there is no propagation error since the data fows are decoded in parae and not successivey and aso it is intrinsicay fair. To anayze the case of the SUD-based PA game, we wi foow the same steps as in Sec. III and we wi see that, the equiibrium anaysis can be deduced, to a arge extent, from the SIC case. Numerica resuts are provided in Sec. V to iustrate our theoretica anaysis and to better assess the sum-rate efficiency of the considered games. Sec. VI corresponds to the concusion. II. SYSTEM MODEL We assume a MAC with arbitrary number of users, K 2. Regarding the origina definition of the MAC by [5] and [6], the system under consideration has two common features: a transmitters send at once and at

4 4 different rates over the entire bandwidth, and the transmitters are using good codes in the sense of the Shannon rate. Our system differs from [5][6] in the sense that mutipe antennas are considered at the termina nodes, channes vary over time and the BS does not dictate the PA poicies to the MSs. Aso, we assume the existence of coordination signa which is perfecty nown to a the terminas. If the coordination signa is generated by the BS itsef, this induces a certain cost in terms of downin signaing but the distribution of the coordination signa can then be optimized. On the other hand, if the coordination signa comes from an externa source, e.g., an FM transmitter, the MSs can acquire their coordination signa for free in terms of downin signaing. However this generay invoves a certain sub-optimaity in terms of upin rate. In both cases, the coordination signa wi be represented by a random variabe denoted by S S. Since we study the K user MAC, S = {0,,...,K!} is a K!+-eement aphabet. When the reaization is in {,...,K!}, the BS appies SIC with a certain decoding order (game ). When S = 0 the BS aways appies SUD (game 2), where a users are decoded simutaneousy (no interference canceation). In a rea wireess system the frequency at which the reaizations woud be drawn woud be roughy proportiona to the reciproca of the channe coherence time (i.e., /T coh ). Note that the proposed coordination mechanism is suboptima because it does not depend on the reaizations of the channe matrices. We wi see that the corresponding performance oss is in fact very sma. We wi further consider that each mobie station is equipped with n t antennas whereas the base station has n r antennas (thus we assume the same number of transmitting antennas for a the users). In our anaysis, the fat fading channe matrices of the different ins vary from symbo vector (or space-time codeword) to symbo vector. We assume that the receiver nows a the channe matrices (CSIR) whereas each transmitter has ony access to the statistics of the different channes (CDIT). The equivaent baseband signa received by the base station can be written as: Y (s) (τ) = K = H (τ)x (s) (τ)+z(s) (τ), () wherex (s) (τ) is the n t-dimensiona coumn vector of symbos transmitted by user at time τ for the reaization s S of the coordination signa, H (τ) C nr nt is the channe matrix (stationary and ergodic process) of user and Z (s) (τ) is a n r -dimensiona compex white Gaussian noise distributed as N(0,σ 2 I nr ). For the sae of carity we wi omit the time index τ from our notations. In order to tae into account the antenna correation effects at the transmitters and receiver, we wi assume the different channe matrices to be structured according to the unitary-independent-unitary mode introduced in [2]: {,...,K}, H = V H W, (2)

5 5 where V and W are deterministic unitary matrices that aow one to tae into consideration the correation effects at the receiver and transmitter. Aso H is an n r n t matrix whose entries are zero-mean independent compex Gaussian random variabes with an arbitrary profie of variances, such that E H (i,j) 2 = σ(i,j) n t. The Kronecer propagation mode for which the channe transfer matrices factorizes as H = R /2 specia case of the UIU mode where the profie of variances is separabe i.e., E H (i,j) 2 = d(r) Θ T /2 (i)d(t)(j) n t is a, with for each : Θ is a random matrix with zero-mean i.i.d. entries, T is the transmit antenna correation matrix, R is the receive antenna correation matrix, {d (T) (j)} j {,...,nt} and {d (R) (i)} i {,...,nr} are their associated eigenvaues. In this paper we wi consider that V = V for a users. The reason for assuming this wi be made cearer a itte further. In spite of this simpification, we wi sti be abe to dea with some usefu scenarios where the users see different propagation conditions in terms of receive antenna correation. III. SUCCESSIVE INTERFERENCE CANCELLATION When SIC is assumed at the BS, the strategy of user {,2,...,K}, consists in choosing the best vector ( ) [ ] of precoding matrices Q = Q (),Q(2),...,Q(K!) where Q (s) = E X (s) X(s),H, for s S, in the sense of his utiity function. For carity sae, we wi introduce another notation which wi be used in the remaining of this section to repace the reaization s of the coordination signa. We denote by P K the set of a possibe permutations of K eements, such that π P denotes a certain decoding order for the K users and π() denotes the ran of user K and π P K denotes the inverse permutation (i.e. π (π()) = ) such that π (r) denotes the index of the user that is decoded with ran r K. We denote by p π [0,] the probabiity that the receiver impements the decoding order π P K, which means that p π =. At ast note that there is a one-to-one mapping between the set of reaizations of the coordination signa S and the set of permutations P K, i.e. ξ : S P such that ξ( ) is a bijective function. This is the reason why the index s can be repaced with the index π without introducing any ambiguity or oss of generaity. The vector of precoding matrices can ( ) be denoted by Q = Q (π) and the utiity function can be written as: where R (π) (Q(π),Q(π) u SIC (Q,Q ) = p π R (π) (Q(π),Q(π) ) (3) ) = Eog 2 I+ρH Q (π) HH +ρ K (π) H Q (π) H H Eog 2 I+ρ K (π) H Q (π) H H (4)

6 6 with ρ = σ 2 and K (π) = { K π() π()} represents, for a given decoding order π, the subset of users that wi be decoded after user. Aso, we use the standard notation, which stands for the other payers than. An important point to mention here is the power constraint under which the utiities are maximized. Indeed for user {,...,K}, the strategy set is defined as foows: { } ( ) A SIC = Q = Q (π) π P K,Q (π) 0, p π Tr(Q (π) ) n tp. (5) In order to tace the existence and uniqueness issues for Nash equiibria in the genera space-time PA game, we expoit and extend the resuts from Rosen [7], which we wi briefy state here beow in order to mae this paper sufficienty sef-contained. Theorem : [7] Let G = (K,{A } K,{u } K ) be a game where K = {,...,K} is the set of payers, A,...,A K the corresponding sets of strategies andu,...,u the utiities of the different payers. If the foowing three conditions are satisfied: (i) each u is continuous in the a the strategies a j A j, j K; (ii) each u is concave in a A ; (iii) A,...,A K are compact and convex sets; then G has at east one NE. Theorem 2: [7] Consider the K-payer concave game of Theorem. If the foowing (diagonay strict concavity) condition is met: for a K and for a (a,a ) A2 such that there exists at east one index K j K for which a j a j, (a [ a )T a u (a,a ) a u (a,a )] > 0; then the uniqueness of the NE is insured. = In the space-time power aocation game under investigation, the obtained resuts are stated in the foowing theorem. Theorem 3: [Existence of an NE] The joint space-time power aocation game described by: the set of payers K; the sets of actions A SIC and the utiity functions u SIC (Q,Q ) given in (3), has a Nash equiibrium. Proof: It is quite easy to prove that the strategy sets A SIC are convex and compact sets and that the utiity functions u SIC (Q,Q ) are concave w.r.t. Q and continuous w.r.t. to (Q,Q ) and by Theorem at east one Nash equiibrium exists. For more detais, the reader is referred to Appendix A. Theorem 4: [Sufficient condition for uniqueness] If the foowing condition is met K ( Tr {(Q (π) Q (π) ) u SIC (Q,Q ) Q (π) = ) ) Q (π) u SIC (Q,Q ) )} > 0 (6) ( ( for a Q = Q (π),q = Q (π) A SIC such that (Q π P,...,Q K ) (Q,...,Q K ), then the K Nash equiibrium in the power aocation game of Theorem 3 is unique. This theorem corresponds to the matrix generaization of the diagonay strict concavity (DSC) condition of [7] and is proven in Appendix B. To now whether this condition is verified or not in the MIMO MAC one

7 needs to re-write it in a more expoitabe manner. It can be checed that C expresses as C = for each π P K, T π is given by: where A (π) r T π = K Tr = K = E Tr r= ( ( = E {(Q (π) )[ Q (π) Q (π) R (π) (Q(π),Q (π) {ρh π (r)(q (π) π (r) Q(π) π (r) )HH π (r) I+ρH π (r)q (π) π (r) HH π (r) +ρ K I+ρH π (r)q (π) π (r) HH π (r) +ρ K K Tr r= E[F π (H)] ( A (π) r ) ( A (π) r I+ = ρh π (r)q (π) π (r) HH π (r), A(π) r ) Q (π) R (π) (Q(π) H π (s)q (π) π (s) HH π (s)) s=r+ H π (s)q (π) π (s) HH π (s) s=r+ K s=r A (π) s ) ( I+ K s=r = ρh π (r)q (π) π (r) HH π (r) A (π) s ) ) 7 p π T π where,q (π) ) ]} (7) and the users have been ordered using their decoding ran rather than their index. Notice that since the expectation operator is inear we can switch between the trace and expectation. ) Let us denote by H = [H,...,H K ], Q = (Q ) K,, Q = (Q ) K, = (Q (π) Q(π) ) ) ),, K Q(π), K A = (A (π), π P A = (A (π). K, K, K In order to prove that the DSC condition hods we have to prove that for a Q Q we have C > 0. (Q (π) Let us give a very usefu resut. Lemma : For any positive definite matrices A, B, and any positive semi-definite matrices A i, B i, i {2,...,K}, we have that K i Tr (A i B i ) i= j= B j i where the equaity hods if and ony if A j = B j for a j {,...,K} j= A j 0 (8) The proof can be found in [27], for K = 2, and in [28] for arbitrary K 2. Using this resut, we can prove that for any channe reaization, any Q,Q and any π P K : ( F π (H) = Tr A (π) r ) ( A (π) r I+ K s=r A (π) s ) ( I+ K s=r A (π) s ) = 0 (9)

8 8 impying that T π 0 and that C 0. Let us consider now two arbitrary covariance matrices such that Q Q. This means that there is at east one decoding order ϑ P K such that Q (ϑ) Q (ϑ). We wi prove that T ϑ > 0 which wi impy the desired resut C > 0. Remar: Assuming that ran(h H H ) = n t, for a K, and n t n r +, then Q Q impies that A A. This means that for any channe reaization we have F ϑ (H) > 0 which impies directy T ϑ > 0 and C > 0. For the genera proof, et us define the foowing sets: } A H (Q (ϑ),q (ϑ) (ϑ) ) = {H D H K : H (Q (ϑ) Q )H H = } 0 (0) à H (Q (ϑ),q (ϑ) ) = {H D H K : H (Q (ϑ) Q (ϑ) )H H 0 We now that: T ϑ = E[F ϑ (H)] = F ϑ (H)L(H)dH D H = F ϑ (H)L(H)dH+ A H(Q (ϑ),q (ϑ) ) = F ϑ (H)L(H)dH à H(Q (ϑ),q (ϑ) ) à H(Q (ϑ),q (ϑ) ) F ϑ (H)L(H)dH where L(H) > 0 stands for the p.d.f. of H D H C nr Knt. The second equaity foows since D H = A H (Q (ϑ),q (ϑ) ) ÃH(Q (ϑ),q (ϑ) ). The third equaity foows since for a H A H (Q (ϑ),q (ϑ) ) we have that F ϑ (H) = 0 from Lemma 8. We now that for a H ÃH(Q (ϑ),q (ϑ) ) we have that F ϑ (H) > 0. It suffices to prove that à H (Q (ϑ),q (ϑ) ) is a subset of non-zero Lebesgue measure to impy that T ϑ > 0 and thus that C > 0. It turns out that we can prove the existence of a compact set U H ÃH(Q (ϑ),q (ϑ) ) for arbitrary Q (ϑ) Q (ϑ). Thus, we have the desired resut C > 0. Determination of the Nash equiibrium. In order to find the optima covariance matrices, we proceed in the same way as described in [9]. First we wi focus on the optima eigenvectors and then we wi determine the optima eigenvaues by approximating the utiity functions under the arge system assumption. Theorem 5: [Optima eigenvectors] For a K, Q A SIC there is no oss of optimaity by imposing the structure Q = (Q (π) ) π PK, Q (π) = W P (π) W H, in the sense that: where S SIC = Diag(P (π) (),...,P (π) (n t )). max Q A SIC u SIC (Q,Q ) = max Q S SIC { Q = (Q (π) ) π P A SIC Q (π) = W P (π) WH u SIC (Q,Q ), () }, s S, mode from (2) and P (s) =

9 9 The detaied proof of this resut is given in Appendix C. This resut, athough easy to obtain, it is instrumenta in our context for two reasons. First, the search of the optimum precoding matrices bois down to the search of the eigenvaues of these matrices. Second, as the optimum eigenvectors are nown, avaiabe resuts in random matrix theory can be expoited to find an accurate approximation of these eigenvaues. Indeed, the eigenvaues are not easy to find in the finite setting. They might be found using numerica techniques based on extensive search. Here, our approach consists in approximating the utiities in order to obtain expressions which are not ony easier to interpret but aso easier to be optimized w.r.t. the eigenvaues of the precoding matrices. The ey idea is to approximate the different transmission rates by their arge-system equivaent in the regime of arge number of antennas. The corresponding approximates can be found to be accurate even for reativey sma number of antennas (see e.g., [8][9] for more detais). Since we have assumed V = V, we can expoit the resuts in [20][2] for singe-user MIMO channes, assuming the asymptotic regime in terms of the number of antennas: n r, n t, corresponding approximated utiity for user is: n r n t β. The where R (π) (P(π),P(π) ) = n r ũ SIC ({P (π) } K, ) = n r n r K (π) n r i= n t {} j= og 2 + K (π) n r K (π) n r n t {} j= n t j= og 2 + n r i= n r K (π) n r j= p R(π) π (P(π),P(π) ( ) og 2 +(N (π) +)ρp (π) (j)γ (π) (j) + (N (π) +)n t K (π) γ (π) (j)δ (π) (j)og 2 e {} j= ) (2) n t og 2 ( +N (π) ρp (π) (j)φ (π) (j) N (π) n t K (π) φ (π) (j)ψ (π) (j)og 2 e n t j= σ (i,j)δ (π) (j) ) σ (i,j)ψ (π) (j) + (3)

10 0 where N (π) of: = K (π) and the parameters γ(π) (j) and δ(π) (j) j {,...,n t}, K, π P K are the soutions γ (π) (j) = δ (π) (j) = j {,...,n t }, K (π) {} : n r (N (π) +)n t i= + (N (π) +)nt (N (π) +)ρp (π) (j) +(N (π) +)ρp (π) (j)γ (π) (j), σ (i,j) n t σ r (i,m)δ r (π) (m) r K (π) {} m= and φ (π) (j), ψ (π) (j), j {,...,n t } and π P K are the unique soutions of the foowing system: j {,...,n t }, K (π) φ (π) (j) = ψ (π) (j) = N (π) n t The corresponding water-fiing soution is: n r i= : + N (π) nt N (π) ρp π) (j) +N (π) ρp (π) (j)φ (π) P (π),ne (j) = [ n2n r λ r K (π) (j). σ (i,j) n t m= σ r (i,m)ψ (π) r (m) N (π) ργ (π) (j) where λ 0 is the Lagrangian mutipier tuned in order to meet the power constraint: n t j= [ p π n2n r λ N (π) ργ (π) (j) (4) (5) ] +, (6) ] + = n t P. Note that to sove the system of equations given above, we can use the same iterative power aocation agorithm as the one described in [9]. At this point, an important point has to be mentioned. The existence and uniqueness issues have be anayzed in the finite setting (exact game) whereas the determination of the NE is performed in the asymptotic regime (approximated game). It turns out that arge system approximates of ergodic transmission rates have the same properties as their exact counterparts, as shown recenty by [23], which therefore ensures the existence and uniqueness of the NE in the approximated game. Nash Equiibrium efficiency. In order to measure the efficiency of the decentraized networ w.r.t. its centraized counterpart we introduce the foowing quantity: SRE = RNE sum C sum, (7)

11 where SRE stands for sum-rate efficiency; the quantityrsum NE represents the sum-rate of the decentraized networ at the Nash equiibrium, which is achieved for certain choices of coding and decoding strategies; the quantity C sum corresponds to the sum-capacity of the centraized networ, which is reached ony if the optimum coding and decoding schemes are nown. Note that this is the case for the MAC but not for other channes ie the interference channe. Obviousy, the efficiency measure we introduce here is strongy connected to the price of anarchy [24] (POA). The difference between SRE and POA is subte. In our context, information theory provides us with fundamenta physica imits on the socia wefare (networ sum-capacity) whie in genera no such upper bound is avaiabe. In our case, the sum-capacity is given by: K C sum = max I+ρ H Ω H H, (8) with (Ω,...,Ω K) A (C) Eog = A (C) = { (Ω,...,Ω K ) K,Ω 0,Ω = Ω H,Tr(Ω ) n t P }. (9) In genera, it is not easy to find a cosed-form expression of the SRE. This is why we wi respectivey anayze the SRE in the regimes of high and ow signa-to-noise ratio (SNR), and for intermediate regimes simuations wi compete our anaysis. It turns out that the SRE tends to in the two mentioned extreme regimes, which is the purpose of what foows. In the high SNR regime, where ρ, we observe from (4) that δ (π) (j) is easy to chec that by setting the derivatives of L w.r.t. P (s) poicy at the NE is the uniform power aocation P (π),ne γ (π) (j). Under this condition, it (j) to zero, we obtain that the power aocation = P I, regardess the reaization of the coordination signa S. Furthermore, in the high SNR regime, the sum-capacity is achieved by the uniform power aocation. Thus, we obtain that the gap between the NE achievabe sum-rate and the sum-capacity is optima, SRE = for any distribution of S. In the ow SNR regime, whereρ 0, from (4) we obtain thatδ (π) (j) 0 and thatγ (π) n r (j) = σ (N (π) (i,j). +)nt i= By approximating n(+x) x when x <<, the power aocations poicies at the NE are the soutions of the foowing inear programs: {P (π) max (j)} j nt { } n t p π P (π) n r (j) σ (i,j) j= i= n t, (20) s.t. (j) P n t P (π) j=

12 2 given by: n r p π P (π),ne n t P if j = arg max σ (i,m) (j) = m n t i= 0 otherwise. (2) The optima power aocation that achieves the sum-capacity is equa to the equiibrium power aocation, P = p π P (π),ne (j) Thus, the achievabe sum-rate at the NE is equa to the centraized upper bound and thus SRE = for any distribution of S. In concusion, when either the ow or high SNR regime is assumed, the sum-capacity of the fast fading MAC is achieved at the NE athough a sub-optimum coordination mechanism is assumed and aso regardess of the distribution of the coordination channe. IV. SINGLE USER DECODING In this section the coordination signa is deterministic (namey Pr[S = s] = δ(s), δ being the Kronecer symbo) and therefore the amount of downin signaing the BS needs in order to indicate to the MSs that it is using SUD can be made arbitrary sma (by etting the frequency at which the reaizations of the coordination signa are drawn tend to zero). In this framewor, each user has to optimize ony one precoding matrix. Indeed, [ ] the strategy of user K, consists in choosing the best precoding matrix Q (0) = E X (0) X(0)H, in the sense of his utiity function obtained with SUD: u SUD (Q (0),Q(0) ) = Eog I+ρH Q (0). The strategy set of user becomes A SUD = HH +ρ H Q (0) H H Eog I+ρ H Q (0) H H (22) { Q (0) 0,Q (0) = Q (0),H,Tr(Q (0) ) n tp }. (23) It turns out that the equiibrium anaysis in the game with SUD can be, to a arge extent, deduced from the game with SIC. For this reason, we wi not detai the corresponding proofs. The existence and uniqueness issues are given in the foowing theorem. Theorem 6: [Existence and uniqueness of an NE] The space power aocation game described by: the set of payers K; the sets of actions A SUD unique Nash equiibrium. and the payoff functions u SUD (Q (0),Q(0) ) given in (22), has a To prove the existence of a Nash equiibrium we aso expoit Theorem and the four necessary conditions on the utiity functions and strategy sets can be verified using the same toos as described in Appendix A. Uniqueness of the Nash equiibrium. Here we can speciaize Theorem 4, which is the matrix extension of Theorem 2. When the strategies sets are not sets of pairs of matrices but ony sets of matrices, the diagonay strict

13 3 concavity condition in (6) can be written as foows. For a Q (0),Q (0) A SUD such that (Q (0),...,Q (0) K ) (Q (0),...,Q (0) K ): C = K Tr = [ {(Q (0) Q (0) ) Now we can evauate C and obtain that: K C = E Tr [ρh (Q (0) = ( K I+ρ = Q (0) u (Q (0),Q (0) Q (0) )H H ) H Q (0) H H = ETr{(B B )[(B ) (B ) ]}, = E[F 0 (H)] which is positive for any B = I+ K H Q (0) H H, = I+ B = to prove that for any (Q (0),...,Q (0) K ) (Q(0),...,Q (0) K ) Q (0) ] ( I+ρ K = ]} u (Q (0),Q (0) ). (24) H Q (0) H H ) (25) K H Q (0) H H from (8) for K = 2. We need = ) we have C > 0. Remar: Assuming that ran(h H H) = Kn t and Kn t n r + K, then Q Q impies that B B. This means that for any channe reaization we have F 0 (H) > 0 which impies directy that C > 0. For the genera proof, we define the foowing sets: { B H (Q (0),Q (0) ) = H D H } K = H (0) (Q (0) Q )H H { = 0 B H (Q (0),Q (0) ) = H D H } (26) K = H (0) (Q (0) Q )H H 0 We now that: C = E[F 0 (H)] = F 0 (H)L(H)dH D H = F 0 (H)L(H)dH+ F 0 (H)L(H)dH B H(Q (0),Q (0) ) B H(Q (0),Q (0) ) = F 0 (H)L(H)dH B H(Q (0),Q (0) ) (27) The second equaity foows since D H = B H (Q (0),Q (0) ) B H (Q (0),Q (0) ). The third equaity foows because F 0 (H) = 0 for a H B H (Q (0),Q (0) ) from Lemma. We aso now that F 0 (H) > 0 for a H B H (Q (0),Q (0) ). It suffices to prove that BH (Q (0),Q (0) ) is a subset of non-zero Lebesgue measure to impy that C > 0. Here as we, the existence of the compact set can be proved (simiary to the proof for the SIC decoding technique).

14 4 Determination of the Nash equiibrium. As for the optima eigenvectors of the covariance matrices, we foow the same ines as in Appendix C. In this case aso there is no oss of optimaity by choosing the covariance matrices Q (0) = W P (0) WH, where W is the same unitary matrix as in (2) and P is the diagona matrix containing the eigenvaues of Q (0). Here aso we further expoit the asymptotic resuts for the MIMO channe given in [20] [2]. The approximated utiity for user is: ũ SUD (P (0),P(0) ) = K n t og n 2 (+KρP (0) (j)γ (j))+ r = j= n r og n 2 + K n t σ (i,j)δ (j) r Kn i= t = j= K n t γ (j)δ (j)og 2 e n r = j= n r n r i= n r n t og 2 (+(K )ρp (0) (j)φ (j)) j= n r n t og 2 + σ (i,j)ψ (j) + (K )n t n r j= φ (j)ψ (j)og 2 e j= where the parameters γ (j) and δ (j) j {,...,n t }, {,2} are soution of: γ (j) = j {,...,n t }, K : n r σ (i,j) Kn t K n t i= + σ (i,m)δ (m) Kn t = m= KρP (0) δ (j) = (j) +KρP (0) (j)γ (j). and φ (j), ψ (j), j {,...,n t } are the unique soutions of the foowing system: φ (j) = ψ (j) = j {,...,n t }, K\{} : n r σ (i,j) (K )n t n t i= + (K )n t σ r (i,m)ψ r (m) (K )ρp (0) (j) +(K )ρp (0) (j)φ (j). r m= (28) (29) (30)

15 5 The corresponding water-fiing soution is: [ P (0),NE (j) = n2n r λ ] +, (3) Kργ (j) [ n t whereλ 0 is the Lagrangian mutipier tuned in order to meet the power constraint: n t P. j= n2n r λ ] + = Kργ (j) In what the efficiency of the NE point is concerned, we aready now that the SUD decoding technique is sub-optima in the centraized case (SUD does aow the networ to operate at an arbitrary point of the centraized MAC capacity region) and it is impossibe to reach the sum-capacity C sum even if the high and ow SNR regime are assumed. V. SIMULATION RESULTS In what foows, we assume the regime of arge numbers of antennas. From [9], [20], [2], we now that the approximates of the ergodic achievabe rates in the asymptotic regime are accurate even for reativey sma number of antennas. For the channe matrices, we assume the Kronecer mode H = R /2 Θ T /2 mentioned in Sec. II, where the receive and transmit correation matricesr, T foow an exponentia profie characterized by the correation coefficients (see e.g., [25], [26]) r = [r,r 2 ] and t = [t,t 2 ] such that R (i,j) = r i j, T (i,j) = t i j. By assuming that the receive antenna is a uniform inear array (ULA) and nowing that, when the dimensions of Toepitz matrices increase they can be approximated by circuar matrices we obtain that a the receive correation matrices R can be diagonaized in the same vector basis (i.e., the Fourier basis). Thus the considered mode is incuded in the UIU mode that we studied where V = V. Fair SIC decoding versus SUD decoding. First we compare the resuts of the genera space-time PA game considered in Sec. III, where SIC decoding is used at the receiver, and the game described in Sec. IV, where SUD decoding is used. Fig. depicts the achievabe sum-rate at the equiibrium as a function of the transmit power P = P 2 = P, for the scenario n r = n t = 0, r = [0.5,0.2], t = [0.5,0.2], ρ = 3dB. In order to have a fair comparison we assume that p = 2 (on average each user is decoded second haf of the time when SIC is assumed). We observe that, even in this scenario, which was thought to be a bad one in terms of sub-optimaity, the sum-rate obtained with the first game is very cose to the sum-capacity upper bound. Aso, the sum-rate reached when the BS uses SUD is ceary much ower than the sum-rate obtained by using SIC. SIC decoding, comparison between the joint space-time PA and the specia cases of spatia PA and tempora PA. Now we want to compare the resuts of the genera space-time PA with the two particuar cases that were

16 6 studied in [4]: the spatia PA, where the users are forced to aocate their power uniformy over time (regardess of their decoding ran) but are free to aocate their power over the transmit antennas; the tempora PA, where the users are forced to aocate their power uniformy over their antennas but they can adjust their power as a function of the decoding ran at the receiver. Fig. 2 represents the sum-rate efficiency as a function of the coordination signa distribution parameter p [0,] when n r = n t = 0, r = [0.3,0], t = [0.5,0.2], ρ = 4dB, P = 5, P 2 = 50. We observe that the three types of power aocation poicies perform very cose to the upper bound. What is most interesting is the fact that the performance of the networ at the equiibrium is better by using a purey spatia PA instead of the most genera space-time PA. This has been confirmed by many other simuations and iustrates a Braess paradox: athough the sets of strategies for the space-time case incude those of the purey spatia case, the performance obtained at the NE are not better in the space-time case. SIC decoding, spatia PA, achievabe rate region. In Fig. 3, we observe that the rate region achieved at the NE of the space PA as a function of the distribution of the coordination signa p for the scenario n r = n t = 0, r = [0.4,0.2], t = [0.6,0.3], ρ = 3dB, P = 5, P 2 = 50. It is quite remarabe that in arge MIMO MACs, the capacity region comprises a fu cooperation segment just ie the SISO MACs. The coordination signa precisey aows one to move aong the corresponding ine. This shows the reevance of arge systems in decentraized networs since they aow to determine the capacity region of certain systems whereas it is unnown in the finite setting. Furthermore, they induce an averaging effect, which maes the users behavior predictabe. VI. CONCLUSIONS Interestingy, the existence and uniqueness of the Nash equiibrium can be proven in mutipe access channes with muti-antenna terminas for a genera propagation channe mode (namey the unitary-invariant-unitary mode) and the most genera case of space-time power aocation schemes. In particuar, the uniqueness proof requires a matrix generaization of the second theorem of Rosen [7] and proving a trace inequaity [28]. For a the types of power aocation poicies (purey tempora PA, purey spatia PA, space-time PA), the sum-rate efficiency of the decentraized networ is cose to one when SIC is assumed and the networ is coordinated by the proposed suboptimum coordination mechanism. Quite surprisingy, the space-time power aocation performs a itte worse than its purey spatia counterpart, which puts in evidence a Braess paradox in the types of wireess networs under consideration. One of the interesting extensions of this wor woud be to anayze the impact of a non-perfect SIC on the PA probem. Indeed, the effect of propagation errors coud then be assessed (which does not exist with SUD).

17 7 APPENDIX A A. Concavity of the utiity functions u SIC Let us focus on user K. We want to prove that u SIC (Q,Q ) is concave w.r.t. Q A SIC. We observe that the term R (π) (Q(π),Q(π) ) in (3) depends ony on Q(π) and Q (π) and not on the covariance matrices Q(τ), Q (τ) for any other possibe decoding rue τ P K \{π}. Thus, in order to prove that u SIC (Q,Q ) is stricty concave w.r.t. to Q = (Q (π) ), it suffices to prove that R (π) (Q(π) π P K.,Q(π) To this end, we study the concavity of the function f(λ) = R (π) (λq(π) [0,] for any pair of matrices (Q (π) 2 f λ (λ) = ETr ρ 2 H H 2 H H,Q (π) ). The second derivative of f is equa to: I+ρH Q (π) H H +ρλh Q (π) HH +ρ I+ρH Q (π) H H +ρλh Q (π) HH +ρ = ETr[A Q (π) A Q(π) ] with A = ρ 2 H H I+ρH Q (π) H H +ρλh Q (π) HH +ρ to be a Hermitian positive definite matrix, Q (π) = Q (π) with B = A /2 Q (π) A/2. K (π) K (π) ) is concave w.r.t. Q(π) for a +( λ)q (π) ) over the interva K (π) H Q (π) H H H Q (π) H H 2 f λ (λ) = ETr[A /2 Q (π) 2 A/2 A /2 Q (π) A/2 ] = ETr[BB H ] < 0 H Q (π) H H H Q (π) H Q (π) H, which can be proven Q (π) aso a Hermitian matrix, and ρ = σ 2.,, B. Continuity of the utiity functions u SIC Considering the Leibniz formua, the determinant of a matrix can be expressed as a weighted sum of products of its entries. Knowing that the product and the sum of continuous functions are continuous, we concude that the determinant function is continuous. Aso, it is we nown that the ogarithmic function is a continuous function. Thus, for any π P K, the function R (π) (Q(π) continuous functions which is aso continuous w.r.t. (Q (π) is continuous w.r.t. (Q,Q ).,Q(π),Q(π) ) is nothing ese but the composition of two ). This suffices to prove that usic (Q,Q )

18 8 C. Convexity of the strategy sets A SIC In order to prove that the set A SIC A SIC A SIC, we have: is convex, we need to verify that, for any two matrices (Q,Q ) αq +( α)q A SIC, for a α 0. For any Q,Q A(SIC), the matrices Q (π) are Hermitian which impies that αq (π) +( α)q (π) aso Hermitian matrices, for a π P K. Furthermore, for any Q,Q ASIC, we have that Q (π), Q (π) that αq (π) +( α)q (π) are aso non-negative matrices, for a π P K. Finay, nowing that the trace is a inear appication we have that: ) p π Tr (αq (π) +( α)q (π) = π P = α p π Tr(Q (π) )+( α) p π Tr(Q (π) π P π P αn t P +( α)n t P = n t P. are are non-negative matrices which impies ) Thus αq +( α)q ASIC and the set is convex. D. Compactness of the strategy sets A SIC To prove that the strategy sets are compact sets we use the fact that, in finite dimension spaces, a cosed and bounded set is compact. First et us prove that A SIC is a cosed set. We define the function g : A SIC f(q ) = p π Tr(Q (π) ). [0,n t P ], with We see that g( ) is a continuous function and that its image is a compact and thus cosed set. Knowing that the continuous inverse image of a cosed set is cosed, we concude that A SIC Now we want to prove that the set A SIC the foowing norm Q = is cosed. is a bounded set. We associate to the tupe of matrices (Q (π) ) Q (π) 2 2 where. 2 is is the spectra norm of a matrix. Q (π) 2 = max{λ Q (π)h Q (i)} n (π) i=.

19 9 Since for a Q A SIC, Q (π) is a non-negative, Hermitian matrix we have that: max{λ (π) Q (i)} n i= Tr(Q(π) ), and thus: Q (π) 2 = max{λ Q (π)2 (i)} n i= max{λ = (π) Q (i) 2 } n i=. In concusion the associated norm Q. APPENDIX B We suppose that there exist two different equiibrium strategy profies: ( Q, Q ) A SIC ( Q, Q ) A SIC A SIC and A SIC, such that ( Q, Q ) ( Q, Q ). Then the condition given in the theorem, C > 0 is met for the particuar choice of (Q,Q ) = ( Q, Q ) and (Q,Q ) = ( Q, Q ). By the definition of the Nash Equiibrium, the strategies Q, K, are the soutions of the foowing maximization probems: max Q A SIC u (Q, Q ). Thus, Q satisfy the foowing Kuhn-Tucer optimaity conditions: ) Q A SIC, which means that: Q (π) = ( Q (π) )H 0 p π Tr( Q (π) ) n tp,, π P K 2) There exist λ 0, and the foowing Hermitian non-negative matrices of ran, such that: [ ] λ p π Tr( Q (π) ) n tp = 0 Tr( Φ (π) Q (π) ) = 0, π P K, Φ (π), for a π P K, 3) π P K : (π) Q u ( Q, Q (π) ) = p π λ I Φ,

20 20 Having assumed that ( Q, Q ) is aso a Nash Equiibrium, Q, with K are the soution of: max Q A SIC u (Q, Q ), and thus Q satisfy the foowing Kuhn-Tucer optimaity conditions: 4) Q A SIC, which means that: Q (π) = ( Q (π) )H 0 p π Tr( Q (π) ) n tp,, π P K 5) There exist λ 0, K and the foowing non-negative, Hermitian matrices of ran, π P K such that: [ ] λ p π Tr( Q (π) ) n tp = 0 Tr( Φ (π) Q (π) ) = 0, π P K, (π) Φ, for a 6) π P K : Q (π) u ( Q, Q (π) ) = p π λ I Φ Using the third and the sixth optimaity conditions, the condition given in (6) becomes: C = K = Tr( Q (π) K = 0. { p π λ Tr( Q (π) )+p πˆλ Tr( Q (π) ) p π λ Tr( Q (π) ) p πˆλ Tr( Q (π) ) Φ (π) { λ [ } (π) ) Tr( Q Φ (π) (π) )+Tr( Q Φ (π) (π) )+Tr( Q Φ (π) ] [ ) ]} p π Tr( Q (π) ) n tp + ˆλ p π Tr( Q (π) ) n tp From the other four K-T conditions, we obtain that a the terms on the right are negative and thus C 0. But this contradicts the diagonay strict concavity condition and so the Nash Equiibrium is unique.. APPENDIX C We want to prove that there is no optimaity oss when restricting the search for the optima covariance matrices to Q A SIC such that Q (π) = W P (π) WH, for a π P K. Let us consider user K. We have

21 2 that: arg max Q A SIC = arg max Q A SIC = arg max Q A SIC = arg max Q A SIC = arg max Q A SIC where we denoted with X (π) u (Q,Q ) p π Eog 2 I+ρH Q (π) HH +ρ H Q (π) H H K (π) p π Eog 2 I+ρV H W H Q(π) W H H VH +ρ V H W H Q(π) W HH V H, K (π) p π Eog 2 I+ρ H W H Q(π) W H H +ρ H W H Q(π) W HH K (π) Eog 2 I+ρ H X (π) H H +ρ H W H Q(π) W HH π K (π) K (π) (32) W H Q(π) W. Knowing that the utiity function is concave w.r.t. the new defined matrices X (π), and the channe matrix H has independent entries, we can directy appy the resuts given in [22] to prove that annuing the non-diagona entries of X (π) can ony increase the vaues of the functions Eog 2 I+ρ H X (π) H H +ρ H W H Q(π) W HH. In concusion the optima matrices X(π) are diagona, that we wi denote with P (π) Q (π) = W P (π) WH. K (π). The spectra decomposition of the optima covariance matrices are: REFERENCES [] S. A. Grandhi, R. Vijayan and D. J. Goodman, Distributed agorithm for power contro in ceuar radio systems, Proc. of Annua Aerton Conf. on Comm. Contro and Computing, Sep [2] S. A. Grandhi, R. Vijayan and D. J. Goodman, Distributed power contro in ceuar radio systems, IEEE Trans. on Comm., Vo. 42, No. 234, pp , Feb/Mar/Apr 994. [3] H. Ji and C.-Y. Huang, Non-cooperative upin power contro in ceuar radio systems, Wireess Networs, Vo. 4, No. 3, pp , 998. [4] S.-J. Oh, T. L. Osen and K. M. Wasserman, Distributed power contro and spreading gain aocation in CDMA data networs, IEEE Proc. of INFOCOM, Vo. 2, March 2000, pp [5] D. J. Goodman and N. B. Mandayam, Power Contro for Wireess Data, IEEE Person. Comm., Vo. 7, No. 2, pp , Apri [6] F. Meshati, M. Chiang, H. V. Poor and S. C. Schwartz, A game-theoretic approach to energy-efficient power contro in muti-carrier CDMA systems, IEEE Journa on Seected Areas in Communications, Vo. 24, No. 6, pp. 5 29, June [7] W. Yu, G. Ginis and J. M. Cioffi, Distributed mutiuser power contro for digita subscriber ines, IEEE Journa of Seected Areas in Communications, Vo. 20, No. 5, June 2002, pp

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