Utility Maximization for Uplink MU-MIMO: Combining Spectral-Energy Efficiency and Fairness

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1 EuCNC-MngtTech Utlty Maxmzaton for Uplnk MU-MIMO: Combnng Spectral-Energy Effcency and Farness Le Deng, Wenje Zhang, Yun Ru, Yeo Cha Kat Department of Informaton Engneerng, The Chnese Unversty of Hong Kong, Hong Kong School of Computer Scence, Mnnan Normal Unversty, Chna Shangha Advanced Research Insttute, Chnese Academc of Scence, Shangha, Chna School of Computer Engneerng, Nanyang Technologcal Unversty, Sngapore Emal: dl@e.cuhk.edu.hk, zhan@ntu.edu.sg, ruy@sar.ac.cn, asckyeo@ntu.edu.sg Abstract Drven by green communcatons, energy effcency (EE) has become a new mportant crteron for desgnng wreless communcaton systems. However, hgh EE often leads to low spectral effcency (SE), whch spurs the research on EE-SE tradeoff. In ths paper, we focus on how to the utlty n physcal layer for an uplnk mult-user multple-nput multpleoutput (MU-MIMO) system, where we wll not only consder EE-SE tradeoff n a unfed way, but also ensure user farness. We frst formulate the utlty maxmzaton problem, but t turns out to be non-convex. By explotng the structure of ths problem, we fnd a convexzaton procedure to convert the orgnal nonconvex problem nto an equvalent convex problem, whch has the same global optmum wth the orgnal problem. Then, we present a centralzed algorthm to solve the utlty maxmzaton problem, but t requres the global nformaton of all users. Thus we propose a prmal-dual dstrbuted algorthm whch consumes a small amount of overhead. Furthermore, we have proved that the dstrbuted algorthm can converge to the global optmum. Fnally, the numercal results show that our approach can both capture user dversty for EE-SE tradeoff and ensure user farness, and they also valdate the effectveness of our prmal-dual dstrbuted algorthm. Index Terms MU-MIMO, Spectral Effcency, Energy Effcency, Farness, Power Control, Prmal-Dual I. INTRODUCTION Among the total worldwde energy consumpton, communcaton networks have contrbuted ncreasngly from.% n 7 to.8% n, and ths proporton s antcpated to grow contnuously n the comng years []. Ths stmulates the fast development of green communcatons recently []. Compared to spectral effcency (SE), energy effcency (EE), defned as the number of bts that can be transmtted wth per energy consumpton, becomes a new mportant crteron for desgnng green wreless systems. How to obtan optmal EE has become a hot research topc n dfferent wreless communcaton systems []. On the other hand, multple-nput multple-output (MIMO) has been a key technque n modern wreless communcaton systems, because t can sgnfcantly ncrease SE by explotng transmt dversty and spatal multplexng gans []. MIMO system s used for one sngle transmtter and one sngle recever n a pont-to-pont way, so t s often referred to sngleuser MIMO (SU-MIMO). However, n some applcatons, especally n cellular networks, t s often dffcult to nstall many antennas due to the sze lmtatons of many devces such as smartphones and tablets. To ncrease the networkwde SE, mult-user MIMO (MU-MIMO) technque has been proposed. Although dstrbuted users only have a small number of antennas or even just one, they can share the same tmefrequency resource block to form a MU-MIMO system []. In ths paper, we are nterested n the uplnk MU-MIMO because users, such as smartphones and tablets, are often more energysenstve. Recently there are some papers studyng how to EE for uplnk MU-MIMO system. In [7], Mao nvestgates the uplnk MU-MIMO system where each user deploys multantennas and he demonstrates that EE s d when some antennas are turned off. In [8], Ru et al. study the uplnk MU-MIMO system where each user deploys only one antenna and they EE by jontly dong mode selecton and optmal power allocaton. However, t s well-known that SE and EE are two conflctng objectves []. Often hgh EE leads to low SE and vce verse, whch means t s more practcal to consder SE and EE smultaneously. Thus how to study the EE- SE tradeoff has attracted a lot of attenton [9], [] whereas, only a few artcles study uplnk MU-MIMO system. The authors n [9] consder how to get the EE-SE tradeoff for a largescale uplnk MU-MIMO system n a system level. They study EE-SE tradeoff n low and hgh SE regme asymptotcally and do not nvolve user farness explctly whch s mportant n mult-user system. Dfferent from [9], our paper nvestgates uplnk MU-MIMO system n the lnk level rather than the system level. More mportantly, we study EE-SE tradeoff n a unfed way and we also guarantee farness among users. Specfcally, our contrbutons are three-fold, We construct a utlty functon of all users whch not only captures the user dversty for EE-SE tradeoff n a unfed way, smlar to [], but also guarantees farness among all users. To the best of our knowledge, we are the frst to study EE-SE tradeoff and user farness together n uplnk MU-MIMO system. We proposed an approach to convert the problem nto an equvalent convex programmng problem whch has the same optmal soluton wth the orgnal problem. Apart from the centralzed algorthm, we further devse a

2 prmal-dual dstrbuted algorthm whch only consumes a small amount of overhead. Moreover, we have proved that the dstrbuted algorthm converges to the global optmal soluton. The rest of ths paper s outlned as follows. We descrbe the system model and formulate the problem n Secton II. In Secton III, we analyze the optmal power allocaton by convertng the optmzaton problem nto a convex programmng problem. Next n Secton IV, we propose a prmal-dual dstrbuted algorthm, whch can acheve the global optmum. The numercal results are shown n Secton V, followed by concluson n Secton VI. Throughout ths paper, we wll use [ ] j to denote the matrx s entry n -th row and j-th column, E[ ] to denote expectaton, I n to denote the n n dentty matrx, and the superscrpt to denote Hermtan transpose. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model Consder a MU-MIMO system wth N users ndexed from ton, and one Node-B n a sngle cell. In ths paper, we assume that each user s only equpped wth one transmttng antenna, and the Node-B s equpped wth M(M N) recevng antennas, as shown n Fg.. In uplnk, all N users share the same tme-frequency resource to transmt data to the Node-B. Denote P as the transmt power for user. Then the receved sgnal vector y C M s, y = Hs+n, () where s C N denotes the transmt sgnal vector wth E[ss ] = dagp,p,,p N }, H C M N denotes the channel matrx, and n C M denotes the addtve whte Gaussan nose (AWGN) wth zero mean and covarance matrx E[nn ] = σ ni M. In ths paper, we assume that Node-B has perfect channel state nformaton (CSI) for all users and the recever at Node- B uses zero forcng (ZF) detecton method. Thus, the decoded sgnal vector s H # y = s+h # n, () where H # = (H H) H denotes the pseudo-nverse of channel matrx H. Then the sgnal-to-nterference-plus-nose rato (SINR) at the Node-B s recever for user s, User ( RF) User ( RF) User N ( RF) Fg.. Node-B M recevng antennas at Node-B System Model P γ = σn[(h H) = P ] σn[(h H). () ] }} δ Then we can obtan SE and EE for user as SE = log(+γ ) = log(+δ P ), () EE = SE P +P c = log(+δ P ) P +P c, () where P c s a postve constant crcut power consumed by the relevant electronc devces for user. B. Problem Formulaton Next, we wll construct the utlty functon n two steps. Frst, we consder the EE-SE tradeoff. Inspred by the wdelyused Cobb-Douglas producton functon n economcs [], we adopt ths model emprcally to get the producton of SE and EE for user, u = (SE ) w (EE ) w, () wherew [,]. More specfcally, we can regard(w, w ) as a pror artculaton of preferences for SE and EE, whch captures EE-SE tradeoff n a unfed way []. Second, we consder the farness among all N users. If we apply the proportonal farness metrc, we can defne the fnal utlty functon for user as U (P) = log(u ) = log[(se ) w (EE ) w ] = log[log(+δ P )] ( w )log(p +P c ) = U (P ), (7) where P = (P,P,,P N ) and the last step shows that the utlty for user s not related to the transmt power of other users. Based on the utlty functon n (7), we then formulate our utlty maxmzaton problem subject to a power constrant for each user and a power sum constrant for all users, (P) subject to U (P ) (8) = P P max, (9) P P max. () In (8), we am at maxmzng the sum of the utlty for all users,.e., the network-wde utlty. Inequalty (9) s the ndvdual power constrants where P max s the maxmal transmt power for user. Inequalty () s the power sum constrant for the total MU-MIMO system where P max s the maxmal transmt power for all users, whch s the power budget of the whole system. =

3 III. OPTIMAL POWER ALLOCATION In the prevous secton, we have formulated the problem to the network-wde utlty n (8), whch however s not a concave functon snce EE n () s nether convex nor concave []. Therefore, n ths secton, we wll explot the nner structure of (P) and fnd that we can narrow down the feasble regon wthout changng the global optmum. A. Convexzaton Procedure To narrow down the feasble regon n (P), we frst consder the ndvdual power constrants n (9). Snce the optmzaton problem can be changed as, max P P max, = U (P ) = = max P P max U (P ), () we just need to fnd the maxmal ndvdual utlty,.e., U (P ) for any user,,,n}. For the ndvdual utlty functon U (P ) n (7), we have the followng proposton. Proposton : For any user under ndvdual power constrant n (9), there exsts one and only one pont P u ] that s U (P ). The functon U (P ) s strctly ncreasng and strctly concave over the nterval [,P u] whle strctly decreasng over the nterval (P u,pmax ]. In addton, P u can be derved as follows, P u P max f w > β(p max ) = P f w β(p max () ) (,P max where β(p ) = δ (P +P c ) (+δ P )log(+δ P ), () and P s the unque soluton to the followng equaton when w β(p max ), β(p ) = w. () Proof: We can prove ths proposton by analyzng the frst and second dervatve of U (P ) wth respect to P. For full proof, please see Appendx A n our techncal report []. Let us denote optmal soluton under ndvdual power constrants as P u = P u,p u,,pn u}. Now we consder the power sum constrant n (). In (P), snce the feasble regon s a compact set and the objectve functon s contnuous, a global optmal soluton can be attaned. Let us denote the global optmal soluton as P = P,P,,PN }. Then we have the followng proposton. Proposton : P P u,.e., P P u,,,,n}. Proof:,,,N}, suppose P > P u. Then k P k + Pu < N k= P k P max, whch means P = P,,P,Pu,P +,,P N } s a feasble soluton to (P). Accordng to the Proposton, we have U (P ) < U (P u). So N k= U k(pk ) < k U k(pk )+U (P u ), whch s a contradcton to the fact that P s the optmal soluton to (P). Ths completes the proof. Proposton shows that for any user, the optmal transmt power P cannot be greater than P u. Therefore we have the followng man result of ths secton. Theorem : (P) s equvalent to the followng problem, U (P ) = (P) subject to P P u, () P P max. () In addton, (P) s a convex programmng problem. Proof: Followng from Proposton, we mmedately conclude that (P) s equvalent to (P). In addton, from Proposton, we know that U (P ) s strctly concave at P [,P u ]. Thus, (P) s a problem to a strctly concave functon n a convex regon, whch means t s a convex problem now. Ths completes the proof. B. Some Analyss = Next we wll gve some analyss for the optmal soluton P n the followng two cases. Case : N = Pu P max In ths case, P u s feasble for (P), so t s also the optmal soluton for (P),.e., P = P u. Case : N = Pu > P max In ths case, we can further narrow down the feasble regon for (P) and acheve the followng proposton. Proposton : If N = Pu > P max, (P) s equvalent to the followng convex optmzaton problem, U (P ) = (P) subject to P P u, (7) P = P max. (8) Proof: Suppose N = P < P max. Snce N = Pu > P max, there exsts at least one,,,n} such that P < P u (Otherwse, N = P = N = Pu > P max, whch s a contradcton). Therefore, there exsts a ǫ > such that P + ǫ P u and k P k + (P + ǫ) P max. So P = P,,P,P +ǫ,p +,,P N } s a feasble soluton for (P). Accordng to the Proposton, we have U (P ) < U (P +ǫ). Then N k= U k(pk ) < k U k(pk )+U (P + ǫ), whch s a contradcton to the fact that P s the optmal = soluton to (P). Therefore, we must have N = P = P max, whch completes the proof. C. Centralzed Algorthm Based on the above analyss, we can readly get the optmal power allocaton P for (P) wth a centralzed algorthm, as shown n Algorthm. In practce, we can mplement such a centralzed algorthm as follows. Frst, each user

4 transmts ts parameters, ncludng P max,p c and w to Node- B. After collectng all the nformaton of all users, Node-B runs Algorthm to obtan the optmal power allocaton P, and then updates the optmal transmt power P to each user. Fnally, each user transmts data at the optmal transmt power. Algorthm Centralzed Algorthm for (P) : for N do : f w > β(p max ) then : P u = P max ; : else : Get P wth Newton-Raphson teraton method for the equaton (); : P u = P ; 7: end f 8: end for 9: f N = Pu P max then : P = P u ; : else : Get P wth gradent projecton method for (P); : end f IV. DISTRIBUTED PRIMAL-DUAL IMPLEMENTATION In the prevous secton, we provde Algorthm to solve the utlty maxmzaton problem n a centralzed manner. However, t requres Node-B to have knowledge of all the global nformaton of all the users. Furthermore, the centralzed algorthm stll ncurs some computatonal complexty and s not robust aganst temporary varaton of system parameters, such as nstantaneous CSI. Hence, we hope to mplement the algorthm n a dstrbuted manner. Inspred by the dstrbuted algorthm n network flow optmzaton problem [], we desgn the followng prmal-dual dstrbuted algorthm to acheve the optmal power allocaton P, P = k [U (P ) λ] Pu + P,,,,N} λ = g[ N = P P max ] + λ, (9) where and [f] + z = [f] a+ z = max(f,), z f, z > max(f,), z mn(f,), z a f, < z < a () () and k and g are postve stepsze. From (9), each user does not need the nformaton of others but just the penalty λ. The only overhead s that Node- B broadcasts λ to all users and each user updates P to Node-B untl convergence. Therefore, such mplementaton only consumes a small amount of overhead between each user and Node-B. Also, our proposed dstrbuted algorthm reduces computatonal complexty compared to Algorthm. Moreover, we further prove that ths prmal-dual dstrbuted algorthm n (9) can converge to the global optmal, as shown n the followng theorem. Theorem : The dstrbuted algorthm n (9) s globally asymptotcally stable and the only equlbrum s P. Proof: We can prove ths theorem by constructng the followng Lyapunov functon, V(P,λ) = (P P ) + (λ λ ), () k g = where (P,P,,P N,λ ) satsfy the KKT condtons for (P) and λ s the multpler for (). For full proof, please see Appendx B n our techncal report []. V. NUMERICAL RESULTS In ths secton, smulaton results are provded to valdate our theoretc analyss. Throughout ths secton, we wll set the maxmal transmt power to be W (dbm) for all users, by adoptng the transmtter s power level for 8/9 MHz moble phones []. The crcut power s set to be.w for all users. A. User Dversty for EE-SE Tradeoff As shown n (), dfferent users can have dfferent preferences for SE and EE accordng tow. In ths part, we wll show how to capture user dversty wth w. Fg. shows the mpact of users dfferent preferences,.e., w and w. From Fg. (a), we can see the optmal transmt power for user. Under the optmal transmt power for user, Fg. (b) demonstrates that SE ncreases as w ncreases whle Fg. (c) demonstrates that EE decreases as w ncreases. Such results comply wth our ntuton for the effect ofw n (), and therefore verfy that our utlty functon can capture user dversty for EE-SE tradeoff very well wth the preference w. B. User Farness In ths part, we wll show that our utlty functon n (7) can ensure farness among all users. As shown n Fg., the closer the channel condtons are, the better the farness s. In addton, when δ = δ, the ndex s (the best farness) whch means no bas exsts and two users have the same utlty. Furthermore, even though when the channel condton s worst for user wth δ = db and the channel condton s best for user wth δ = db, the ndex stll does not touch. (the worst farness) exactly and t s actually.7. Ths means user can stll transmt data wth a postve transmt power. Therefore, user farness can be guaranteed under our proposed utlty functon n (7). C. Prmal-Dual Dstrbuted Algorthm In ths part, we wll valdate the effectveness of our prmaldual dstrbuted algorthm n (9). From Fg., we can see that the dstrbuted algorthm can converge to the global optmum, whch verfes Theorem.

5 P.8... SE(P ) EE(P ) w w w w w w (a) Optmal transmt power P (b) SE wth P (c) EE wth P Fg.. User dversty for EE-SE tradeoff wth N =,M =,P max = P max = W,P max =.W,P c = Pc =.W,δ = δ = db. Jan s Farness Index δ = db δ = db δ = db. δ (db) Fg.. User farness wth N =,M =,P max = P max = W,P max =.W,P c = Pc =.W,w = w =.. Fg.. Utlty Sum 7 Optmal Value Dstrbuted Algorthm Number of Iteratons Convergence of prmal-dual dstrbuted algorthm. VI. CONCLUSION AND FUTURE WORK In ths paper, we consder utlty maxmzaton for the uplnk MU-MIMO system. We defne the utlty functon combnng both EE-SE tradeoff and user farness. After formulatng the u- tlty maxmzaton problem wth ndvdual power constrants and sum power constrant, we analyze the optmal power allocaton scheme. Although the orgnal optmzaton problem s not convex, we propose a convexzaton procedure to convert t nto an equvalent convex programmng problem, whch has been proven to have the same global optmal soluton as the orgnal problem. Moreover, we have proposed two algorthms to obtan the optmal soluton: one s the centralzed algorthm whch requres knowledge of all the global nformaton; the other s the prmal-dual dstrbuted algorthm whch only needs a small amount of overhead between each user and Node-B. Furthermore, we have proved that our proposed dstrbuted algorthm can converge to the global optmal soluton. ACKNOWLEDGMENTS Ths work was supported by the Scentfc Research Foundaton for the Returned Overseas Chnese Scholars, State Educaton Mnstry, and No.JK7, No.MJ. REFERENCES [] S. Lambert, W. Van Heddeghem, W. Vereecken, B. Lannoo, D. Colle, and M. Pckavet, Worldwde electrcty consumpton of communcaton networks, Optcs Express, vol., no., pp. B-B,. [] F. R. Yu, X. Zhang, and V. C. M. Leung, Green Communcatons and Networkng. CRC Press,. [] D. Feng, C. Jang, G. Lm, L. J. Cmn, Jr., G. Feng, and G. Y. L, A survey of energy-effcent wreless communcatons, IEEE Commun. Surveys & Tutorals, vol., no., pp. 7-78,. [] Y. Chen, S. Zhang, S. Xu, and G. Y. L, Fundamental trade-offs on green wreless networks, IEEE Commun. Mag., vol. 9, no., pp. -7, Jun.. [] D. Tse and P. Vswanath, Fundamentals of Wreless Communcatons. Cambrdge Unversty Press,. [] GPP TR.8 (V7..), Physcal layer aspects for evolved unversal terrestral tado access (UTRA), Sept.. [7] G. Mao, Energy-effcent uplnk mult-user MIMO,, IEEE Trans. Wreless Commun., vol., no., pp. -, May. [8] Y. Ru, Q. T. Zhang, L. Deng, P. Cheng, and M. L, Mode selecton and power optmzaton for energy effcency n uplnk vrtual MIMO systems, IEEE J. Sel. Areas Commun., vol., no., pp. 9-9, May.. [9] X. Hong, Y. Je, C. X. Wang, J. Sh, and X. Ge, Energy-spectral effcency trade-off n vrtual MIMO cellular systems, IEEE J. Sel. Areas Commun., vol., no., pp. 8-, Oct.. [] L. Deng, Y. Ru, P. Cheng, J. Zhang, Q. T. Zhang, and M. L, A unfed energy effcency and spectral effcency tradeoff metrc n wreless networks, IEEE Commun. Lett., vol. 7, no., pp. -8, Jan.. [] C. W. Cobb and P. H. Douglas, A theory of producton, Amercan Economc Revew, vol. 8, no., pp. 9-, Mar. 98. [] S. Shakkotta and R. Srkant, Network optmzaton and control, Foundatons and Trends n Networkng, vol., no., pp. 7-79, 7. [] GPP TS. (V8..), Rado transmsson and recepton, Nov.. [] L. Deng, W. Zhang, Y. Ru, and Y. C. Kat, Utlty Maxmzaton for U- plnk MU-MIMO: Combnng Spectral-Energy Effcency and Farness, Techncal Report.