On the Maximum Weighted Sum-Rate of MIMO Gaussian Broadcast Channels

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1 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2008 proceedngs On the Maxmum Weghted Sum-Rate of MIMO Gaussan Broadcast Channels Ja Lu, Y Thomas Hou, and Hanf D Sheral Bradley Department of Electrcal and Computer Engneerng Grado Department of Industral and Systems Engneerng Vrgna Polytechnc Insttute and State Unversty, Blacksburg, VA 2406 Abstract In ths paper, we nvestgate the maxmum weghted sum-rate problem MWSR of MIMO Gaussan broadcast channels MIMO-BC We propose an effcent algorthm that employs conjugate gradent projectons CGP to solve the MWSR problem The proposed CGP offers provable convergence By deflectng gradent drecton to ts Hessan conjugate, CGP enjoys a superlnear convergence rate Also, CGP has a modest memory requrement It only needs the soluton nformaton from the prevous step More mportantly, CGP s able to solve the MWSR problem wth arbtrary number of antennas on both sdes of a MIMO-BC I INTRODUCTION The capacty regon of multple-nput multple-output broadcast channels MIMO-BC has receved great attenton n recent years MIMO-BC belongs to the class of nondegraded broadcast channels, for whch the capacty regon s a wellknown hard problem [] Recently, Wegarten et al [2] proved that drty paper codng DPC acheves the entre capacty regon of MIMO-BC Moreover, by uplnk-downlnk dualty [3], the nonconvex MIMO-BC capacty regon wth respect to the downlnk nput covarance matrces can be transformed to ts dual MIMO multple access channel MIMO-MAC capacty regon wth a sum power constrant Snce the capacty regon of the dual MIMO-MAC s convex wth respect to the uplnk nput covarance matrces, effcent optmzaton for MIMO-BC becomes possble In ths paper, we nvestgate the maxmum weghted sumrate problem MWSR of MIMO-BC Important applcatons of MWSR arse from cross-layer optmzaton for MIMObased ad hoc networks [4] and stablzng the transmsson buffers to guarantee farness for MIMO-based cellular downlnks [5] The MWSR problem of MIMO-BC s the general case of the maxmum sum-rate problem MSR, whch has been solved by a number of algorthms such as the mnmax method MM [6], the gradent method GD [7], the Lagrangan dual decomposton LDD method [8], and the teratve water-fllng methods IWF [9] However, IWF, MM, and LDD cannot be readly appled to solve MWSR In [5], Kobayash et al extended IWF to solve the MWSR problem and proposed some modfcatons to IWF M-IWF to handle scalablty ssue However, ther algorthm s only vald for the case where each recever s equpped wth sngle antenna For general scenaros where recevers are equpped wth multple antennas, only GD s readly applcable However, GD does not fully take advantage of the gradent nformaton and may not converge under some crcumstances The lmtatons of these exstng algorthms motvate us to desgn an effcent, robust, and scalable algorthm to solve the MWSR problem of large MIMO-BC systems wth arbtrary number of antennas Our man contrbuton n ths paper s that we desgn an effcent algorthm to solve the MWSR problem based on conjugate gradent projecton CGP approach Our algorthm s nspred by [0], where Ye et al used a gradent projecton method to fnd a local optmum of the maxmum sum-rate for Gaussan MIMO Interference Channels MIMO-IC However, unlke [0], we propose to use conjugate gradent drectons nstead of gradent drectons to reduce the zgzaggng phenomenon so as to speed up convergence Also, snce the MWSR problem of MIMO-BC can be transformed nto an equvalent convex problem, our CGP method can determne the global optmum of MIMO-BC For the semdefnte cone projecton subproblem, we develop a rgorous algorthm based on Lagrangan dualty Our proposed CGP has the followng attractve features CGP offers provable convergence Unlke M-IWF, whch s only vald for cases where each recever has a sngle antenna, CGP can handle arbtrary number of antennas on both sdes of a MIMO-BC CGP enjoys a superlnear convergence rate Also, per teraton complexty of CGP s OK, where K s the number of users CGP has a modest memory requrement: t only needs the soluton nformaton from the prevous step The remander of ths paper s organzed as follows In Secton II, we present the network model and the problem formulaton Secton III ntroduces the key components n of CGP, ncludng the computaton of conjugate gradents and solvng projecton subproblem In Secton IV, we analyze the complexty of CGP and present numercal results Secton V concludes ths paper II SYSTEM MODEL AND PROBLEM FORMULATION We begn wth ntroducng notaton We use boldface to denote matrces and vectors For a complex-valued matrx A, A denote the conjugate transpose of A and Tr{A} denotes the trace of A WeletI denote an dentty matrx wth dmenson determned from the context A 0 represents that A s Hermtan and postve semdefnte PSD Dag{A A n } /08/$ IEEE

2 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2008 proceedngs denotes the block dagonal matrx wth matrces A,,A n on ts man dagonal Suppose that a MIMO Gaussan broadcast channel has K users, the transmtter has n t antennas, and each of the K users s equpped wth n r antennas The channel matrx for user s denoted as H C nr nt It has been shown n [2] that the capacty regon of a MIMO-BC s equal to the drtypaper codng DPC rate regon Suppose that users,,k are encoded sequentally, then the DPC rate of user can be computed as [3] K I + H C DPC j= S j H S = log K I + H j=+ S, j where S C nt nt, =,,K,arethedownlnk nput covarance matrces and S {S,S K } denotes the collecton of all the downlnk covarance matrces The MWSR problem can be wrtten as follows: Maxmze w C DPC S subject to S 0, =,,K 2 TrS P, where w s the weght assgned to user, P represents the maxmum transmt power It s evdent that 2 s a nonconvex optmzaton problem snce the DPC rate equaton n s a nonconvex functon of the nput covarance matrces S,,S K However, from the uplnk-downlnk dualty theorem [3], we know that the rates achevable n a MIMO- BC are also achevable n ts dual MIMO-MAC That s, gven a feasble S, there exsts a set of feasble uplnk nput covarance matrces for ts dual MIMO-MAC, denoted by Q, such that C MAC Q =C DPC S Thus, 2 s equvalent to the followng MWSR problem of the dual MIMO-MAC wth a sum power constrant: Maxmze w C MAC Q subject to Q 0, =,,K C MAC Q C MAC P, H 3, =,,K TrQ P, where Q C nr nr, =,,K, are the uplnk nput covarance matrces, Q {Q,Q K } represents the collecton of all the uplnk covarance matrces, and C MAC P, H represents the capacty regon of the dual MIMO-MAC, and can be determned by C MAC P, H = S C Q log I + S H Q H, Conv C,,C K S {,,K},, 4 TrQ P, Q 0, where Conv represents the convex hull operaton When the dual MIMO-MAC s Gaussan, the convex hull operaton can be dropped [] H The capacty regon of a MIMO-MAC can be acheved by successve decodng [] However, n order to determne the capacty regon of a MIMO-MAC, K! possble successve decodng orders may need to be enumerated, whch makes the problem ntractable f the number of users s large We gve the followng result, whch shows that such an enumeraton can ndeed be avoded Theorem The MWSR problem n 3 can be solved by the followng equvalent optmzaton problem: Maxmze subject to w π w π log I + K j= H Q H TrQ P max Q 0, =,,K, where w π0 0, π s a permutaton of the set {,,K} such that w π w πk e, π =j represents the th poston n permutaton π s user j Proof Snce the objectve functon s monotoncally ncreasng, the optmal soluton of 3 must be acheved on the boundary of the capacty regon We assume that the weghts are not dentcal Otherwse, the MWSR s reduced to a scaled MSR problem, where the optmal soluton s trvally acheved at any of the K! corner ponts of the capacty regon As a result, the optmal soluton s acheved n a subregon on the boundary of the capacty regon under one of the K! decodng order [] Furthermore, snce such a subregon conssts of all corner ponts wth the same decodng order under all feasble power allocatons, the optmal soluton must be acheved at a corner pont correspondng to some decodng order and some power allocaton Suppose that π s the optmal decodng order It s easy to see that, for the MWSR problem, the objectve gradent at every pont on the boundary of the capacty regon s [ wπ w π2 w πk ] T Snce the objectve functon of 3 s lnear and the capacty regon s convex, 3 s a convex programmng problem wth non-empty feasble regon, whch means Slater condtons holds cf [] As a result, KKT condton s necessary and suffcent for optmalty For smplcty, we drop the superscrpt MAC and smply refer the rates n the dual MIMO-MAC as C Note that n successve decodng wth decodng order π, the actve rate vector constrants at a corner pont correspondng to π are C πk log I + H πk Q πk H πk, C πk + C πk log I +, =K H π Q π H π C π + + C πk + C πk log I +, H π Q π H π 5 6

3 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2008 proceedngs where Q π, =, 2,,K represent the optmal nput covarance matrces that acheve the maxmum weghted sumrate Thus, by KKT condton, we have w π w πk w πk 0 = u 0 0 +u 2 + +u K where u 0, Solvng for u n 7, we have u K = w π u K = w π+ w π,, 2,,K Snce u 0, t then follows that w π w π2 w πk, 7 Snce the constrants n 6 are tght at optmalty, we have C πk = log I + H πk Q πk H πk, 8 C π = log I + H πk Q πk H πk j= log I + H πk Q πk H πk, 9 j=+ for =, 2,,K Summng up all w π C π and after rearrangng the terms, t can be readly verfed that w π C π = w π w π log I + H Q H 0 j= It then follows that the MWSR problem of the dual MIMO- MAC s equvalent to maxmzng 0 wth the sum power constrant, e, the optmzaton problem n 5 One mportant observaton from 5 s that, snce log s a concave functon for postve semdefnte matrces [], 5 s a convex optmzaton problem wth respect to the uplnk nput covarance matrces Q π,,q πk However, although standard convex optmzaton tools can be used to solve 5, t s consderably more complex than a customdesgned method that explots the specal structure of the problem III SOLVING MWSR USING CONJUGATE GRADIENT PROJECTION To solve 5, we propose an effcent algorthm based on conjugate gradent projecton CGP, whch utlzes an mportant concept called Hessan conjugate to deflect the gradent drecton In dong so, we can acheve an asymptotc superlnear convergence rate [], whch s close to that of quas- Newton methods, eg, BFGS method The convergence proof of CGP reles on provng the closedness of the algorthmc maps for fndng conjugate gradent drectons and performng projectons, respectvely Due to space lmtaton, we refer readers to [] for detals The CGP pseudo-code for solvng 5 s shown n Algorthm Algorthm CGP Method for Solvng MWSR Intalzaton: Choose the ntal condtons Q 0 =[Q 0, Q0 2,,Q0 K ]T Let k =0 Man Loop: Calculate the gradents G k, =, 2,,K as follows: Ḡ =2H [ j wπ w π I + H πk Q πkh πk H 2 Deflect the gradents usng Fletcher and Reeves choce of deflecton: ρ k = Ḡk 2 Ḡk 2 3 Choose an approprate step sze s k LetQ k = Q k + s k G k, for =, 2,,K 4 Let Q k be the projecton of Q k onto Ω + P, whereω + P {Q, =,,K Q 0, Tr{Q } P } 5 Choose approprate step sze α k LetQ k+ Q k, =, 2,,K = Q k + α k k Q 6 k = k + If the maxmum absolute value of the elements n Q k Q k <ɛ,for =, 2,,L, then stop; else go to step Due to the complexty of the objectve functon n 5, we adopt Armjo Rule nexact lne search to avod excessve objectve functon evaluatons, whle stll enjoyng provable convergence [] We now consder two major components n the CGP framework: how to compute the conjugate gradent drecton G ; and 2 how to project Q k onto the set Ω + P {Q,,,K : Q 0, Tr{Q } P } A Computng the Conjugate Gradents For convenence, we denote the objectve functon of 5 as JQ To compute the gradent Ḡ Q JQ, the frst step s to compute the partal dervatve of JQ wth respect to Q The computaton of partal dervatves of JQ reles on the followng equaton from matrx dfferental calculus ln A+BXC X = [ CA + BXC B ] T [0], [2] Frst of all, we can compute the partal dervatve of the th term n the summaton of JQ wth respect to Q, j, as follows: K w π w π log I + H πk Q πkh πk Q = w π w π H I + H πk Q πkh πk H Note that for gradent Ḡ, only the frst j terms n JQ nvolve Q From the defnton z fz =2 fz/ z T

4 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2008 proceedngs [3], we obtan [ j Ḡ =2H wπ w π I + H πk Q πkh πk H Although the s seemngly qute complex, we can n fact explot the specal summaton structure to reduce ts computatonal complexty when mplementng CGP Note that the most computatonally heavy part n the expresson of Ḡ s the summaton of the terms n the form of H πk Q πkh πk Under drect computaton, we wll have j2k + j/2 tmes of such addtons for these terms Fortunately, most of these terms n the summaton occur repeatedly when j vares Therefore, we can store a runnng sum n the form of I + H πk Q πkh πk Then, startng out from j = K and reducng j by one subsequently, we only need to compute such addton once n each teraton The conjugate gradent drecton n the k th teraton can be computed as G k = Ḡk + ρ kg k We adopt the Fletcher and Reeves choce of deflecton [], whch can be computed as ρ k = Ḡk 2/ Ḡk 2 After such deflecton, we obtaned the so-called Hessan-conjugate of The beneft of usng Hessan conjugate deflecton s that we can reduce the zgzaggng phenomenon encountered n the conventonal gradent projecton method, and acheve an asymptotc superlnear convergence rate [] Also, n CGP, we do not need to store a Hessan approxmaton matrx as n quas-newton methods, whose sze s usually large G k B Constraned Semdefnte Cone Projecton The goal of the projecton subproblem n CGP s to fnd a projecton on a constraned semdefnte cone for Q, Snce G s Hermtan, we have that Q k = Qk + s kg k s Hermtan as well Then, the projecton problem becomes how to smultaneously project K Hermtan matrces onto the set Ω + P max {Q l : l Tr{Q l} P max, Q l 0, l =,,K} We construct a block dagonal matrx D = Dag { } Q π Q πk C K n r K n r It s easy to recognze that Q Ω + P max, j =,,K, f and only f TrD = j= Tr Q Pmax and D 0 In our projecton, gven a block dagonal matrx D n,wewsh to fnd a matrx D n Ω + P max such that Dn mnmzes D n D n F, where F denotes Frobenus norm, e, equvalently, we solve the followng optmzaton problem Mnmze 2 D D 2 F subject to Tr D P max, D 0 2 Note that ths problem s a convex mnmzaton problem and we can solve ths mnmzaton problem by solvng ts Lagrangan dual Assocatng Hermtan matrx X to the constrant D 0 and µ to the constrant Tr D Pmax, we can wrte the Lagrangan as gx,µ = mn D{/2 D D 2 F TrX D +µtr D Pmax } Snce gx,µ s an unconstraned convex quadratc mnmzaton problem, we can compute the mnmzer of the Lagrangan by smply settng ts frst dervatve wth respect to D to zero, e, D D X + µi =0 Notng that X = X, wehave D = D µi + X Substtutng D back nto the Lagrangan and after some algebrac smplfcatons, we can rewrte the Lagrangan dual problem as Maxmze 2 D µi + X 2 F µp max + 2 D 2 subject to X 0,µ 0 3 In semdefnte programmng, 3 s referred to as matrx nearness problems [4], [5] Generc matrx nearness problems are hard to solve Fortunately, thanks to the pecewse quadratc structure n 3, t s possble to solve 3 effcently Due to space lmtaton, we refer readers to [6] for more detals IV PERFORMANCE AND COMPLEXITY COMPARISONS In ths secton, we compare CGP wth other exstng algorthms Among these algorthms, the mnmax method MM [6] s more complex than the others havng the lnear complexty and s not readly applcable for MWSR The Lagrangan dual decomposton method LDD [8] conssts of nested teratve loops n solvng the Lagrangan dual n each teraton and therefore has a non-determnstc complexty per teraton The teratve water-fllng methods IWF n [9] do not scale well as the number of users ncreases because the most recently updated soluton n each teraton only accounts for a fracton of /K n the effectve channels computaton Also, IWF cannot be drectly used to solve MWSR Kobayash et al proposed modfcatons of IWF M-IWF for solvng MWSR They also came up wth a new averagng update scheme to address the scalablty ssue of IWF However, as ndcated by Kobayash et al, M-IWF can only handle the case when each recever n a MIMO-BC has only one antenna The gradent method GD n [7] also uses the gradent nformaton to gude the search of optmal soluton Let v and λ be the prncpal egenvector of unt norm and prncpal egenvalue for Q, respectvely, =, 2,,KLetj = arg maxλ,,λ K The terate of GD s updated as Q k+ where the movng drecton s = Q k + t d k, d k =[ Q k, Qk 2, Q k j + P v j v j, Qk K ]T, and the step-sze t s determned by the followng lne search t = arg max J tqk,, tqk j + tp v 0 t j v j,, tq k K, where the nterval 0 t ensures that the searchng stays n the feasble regon The drecton d k s obtaned by v j v j projected onto the hyperplane TrQk =P, whch, although related to the gradent at Q k, s very dfferent to

5 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2008 proceedngs our approach In essence, v j v j s a rank-one update to Qk j, as opposed to the full-rank update n CGP In fact, GD s a varant of Zoutendjk method, for whch the convergence s not guaranteed as the algorthmc map of Zoutendjk method s not closed [] Fg shows a equal-weghted 0-user equalweghted MIMO-BC example where GD fals to converge to the optmal soluton Sum Rate nats/s/hz GD CGP Number of Iteratons Fg A 0-user MIMO-BC channel wth n t = n r =4where GD fals to converge to the optmal soluton It can also be seen from Fg that CGP s very effcent It only takes 20 teratons to reach to the global optmum In fact, t can be shown that, by usng conjugate gradent drectons, CGP acheves an asymptotc superlnear convergence rate V CONCLUSION In ths paper, we nvestgated the maxmum weghted sumrate problem MWSR of MIMO Gaussan broadcast channels MIMO-BC We proposed an effcent algorthm called conjugate gradent projectons CGP to solve the MWSR problem The proposed CGP has provable convergence Wth approprate deflectons for gradents, CGP enjoys an asymptotc superlnear convergence rate Also, CGP has a modest memory requrement, whch only needs the soluton nformaton from the prevous step Another attractve feature of CGP s that t can handle arbtrary number of antennas on both sdes of a MIMO-BC [3] S Vshwanath, N Jndal, and A Goldsmth, Dualty, achevable rates, and sum-rate capacty of MIMO broadcast channels, IEEE Trans Inf Theory, vol 49, no 0, pp , Oct 2003 [4] J Lu, Y T Hou, and H D Sheral, Routng and power allocaton for mesh networks wth MIMO Gaussan broadcast channels, n Proc IEEE ICC, Bejng, Chna, May9-23, 2008 [5] M Kobayash and G Care, An teratve water-fllng algorthm for maxmum weghted sum-rate of Gaussan MIMO-BC, IEEE J Sel Areas Commun, vol 24, no 8, pp , 2006 [6] T Lan and W Yu, Input optmzaton for mult-antenna broadcast channels and per-antenna power constrants, n Proc IEEE GLOBECOM, vol, Dallas, TX, USA, Nov 2004, pp [7] H Vswanathan, S Venkatesan, and H Huang, Downlnk capacty evaluaton of cellular networks wth known-nterference cancellaton, IEEE J Sel Areas Commun, vol 2, no 5, pp 802 8, Jun 2003 [8] W Yu, Sum-capacty computaton for the gaussan vector broadcast channel va dual decomposton, IEEE Trans Inf Theory, vol 52, no 2, pp , Feb 2006 [9] N Jndal, W Rhee, S Vshwanath, S A Jafar, and A Goldsmth, Sum power teratve water-fllng for mult-antenna Gaussan broadcast channels, IEEE Trans Inf Theory, vol 5, no 4, pp , Apr 2005 [0] S Ye and R S Blum, Optmzed sgnalng for MIMO nterference systems wth feedback, IEEE Trans Sgnal Process, vol 5, no, pp , Nov 2003 [] M S Bazaraa, H D Sheral, and C M Shetty, Nonlnear Programmng: Theory and Algorthms, 3rd ed New York, NY: John Wley & Sons Inc, 2006 [2] J R Magnus and H Neudecker, Matrx Dfferental Calculus wth Applcatons n Statstcs and Economcs New York: Wley, 999 [3] S Haykn, Adaptve Flter Theory Englewood Clffs, NJ: Prentce-Hall, 996 [4] J Malck, A dual approach to semdefnte least-squares problems, SIAM Journal on Matrx Analyss and Applcatons, vol 26, no, pp , Sep 2005 [5] S Boyd and L Xao, Least-squares covarance matrx adjustment, SIAM Journal on Matrx Analyss and Applcatons, vol 27, no 2, pp , Nov 2005 [6] J Lu, Y T Hou, and H D Sheral, Conjugate gradent projecton approach for MIMO Gaussan broadcast channels, n Proc IEEE Internatonal Symposum on Informaton Theory ISIT, Nce, France, Jun24-29, 2007, pp [7] J Lu, Y T Hou, Y Sh, and H Sheral, On the capacty of multuser MIMO networks wth nterference, IEEE Trans Wreless Commun, vol 7, no 2, pp , Feb 2008 ACKNOWLEDGEMENTS The work of YT Hou and J Lu has been supported n part by the Natonal Scence Foundaton NSF under Grant CNS and Offce of Naval Research ONR under Grant N The work of HD Sheral has been supported n part by NSF Grant CMMI REFERENCES [] T M Cover and J A Thomas, Elements of Informaton Theory New York: John Wley & Sons, Inc, 99 [2] H Wengarten, Y Stenberg, and S Shama, The capacty regon of the Gaussan multple-nput multple-output broadcast channel, IEEE Trans Inf Theory, vol 52, no 9, pp , Sep 2006