Polite Water-filling for Weighted Sum-rate Maximization in MIMO B-MAC Networks under. Multiple Linear Constraints

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1 2011 IEEE Internatona Symposum on Informaton Theory Proceedngs Pote Water-fng for Weghted Sum-rate Maxmzaton n MIMO B-MAC Networks under Mutpe near Constrants An u 1, Youjan u 2, Vncent K. N. au 3, Hage Xang 1, Wu uo 1 1 State Key aboratory of Advanced Optca Communcaton Systems & Networks, Schoo of EECS, Pekng Unversty 2 Department of Eectrca and Computer Engneerng, Unversty of Coorado at Bouder 3 Department of Eectronc and Computer Engneerng, Hong Kong Unversty of Scence and Technoogy Abstract The agorthms n ths paper expot optma nput structure n nterference networks and s a major advance from the state-of-the-art. Optmzaton under mutpe near constrants s mportant for nterference networks wth ndvdua power constrants, per-antenna power constrants, and/or nterference constrants as n cogntve rados. Whe for snge-user MIMO channe transmtter optmzaton, no one uses genera purpose optmzaton agorthms such as steepest ascent because water-fng s optma and much smper, ths s not true for MIMO mutaccess channes MAC, broadcast channes BC, and the non-convex optmzaton of nterference networks because the tradtona water-fng s far from optma for networks. We recenty found the rght form of water-fng, pote waterfng, for some capacty/achevabe regons of the genera MIMO nterference networks, named B-MAC networks, whch ncude BC, MAC, nterference channes, X networks, and most practca wreess networks as speca cases. In ths paper, we use weghted sum-rate mzaton under mutpe near constrants n nterference tree networks, a natura extenson of MAC and BC, as an exampe to show how to desgn hghy effcency and ow compexty agorthms. Severa tmes faster convergence speed and orders of magntude hgher accuracy than the state-of-the-art are demonstrated by numerca exampes. Index Terms Pote Water-fng, MIMO Networks, Weghted Sum-rate Maxmzaton, Mutpe near Constrants I. INTRODUCTION adtona water-fng can be empoyed for sum-rate, not weghted sum rate, mzaton of MIMO mutaccess channes/broadcast channes MAC/BC [1], [2], because the sumrate can be wrtten as a snge-user rate. However, for other capacty boundary ponts and genera networks, the optma nput structure has been open and drecty appyng the tradtona water-fng s far from optma [3]. In [4], we recenty found the rght form of water-fng for genera nterference networks, named B-MAC networks, for practcay mportant achevabe rate regons. In a B-MAC network [4], [5], each transmtter may send ndependent data to mutpe recevers and each recever may coect ndependent data from mutpe transmtters. Thus, t s a combnaton of mutpe nterferng BC s, from transmtter pont of vew, and MAC s, from recever pont of vew. It ncudes BC, MAC, nterference channes, X networks [6] and most practca wreess networks, such as ceuar networks, WF networks, DS, as speca cases. The new water-fng s named pote waterfng because t strkes an optma baance between reducng nterference to others and mzng a nk s own rate. The The work was supported n part by NSFC Grant No , and n part by US-NSF Grant CCF and ECCS pote water-fng s satsfed by a Pareto optma nput, not ony the sum-rate optma nput, of the achevabe rate regons of the B-MAC networks, ncudng the MAC/BC capacty regons as speca cases, under a snge near constrant. It can be used to desgn better agorthms for reated network optmzaton probems, e.g., those n [4], [5]. Ths paper extends the pote water-fng resuts from a snge near constrant to mutpe near constrants, whch appears n systems wth ndvdua power constrants, per-antenna power constrants, and nterference constrants as n cogntve rados. The extenson s non-trva and the proof uses a dfferent approach than that n [4]. As an appcaton, we use t to desgn hghy effcent agorthms for weghted sum-rate mzaton under mutpe near constrants WSRM_MC n MIMO nterference tree ee networks, a subcass of MIMO B-MAC networks ncudng MIMO MAC/BC as speca cases. The theoretca mportance of ee networks, where the nterference has no oops, has been dscussed n [4]. Based on the agorthms n ths paper, hghy effcent agorthms for networks wth nterference oops have been deveoped n [7]. We adopt the same achevabe scheme as n [4], [5], whch assumes Gaussan nput and that each sgna s decoded by no more than one recever. It s optma for MIMO MAC/BC and ncudes most nterference management technques as speca cases [4]. Our agorthm s the frst to expot the optma nput structure, reatve to the achevabe regons, and have ower compexty and hgher performance than the start-of-the-art agorthms n [8], [9]. In [8], WSRM_MC s soved for MIMO BC by a varaton of the agrange duaty method [10]. Effcent nfeasbe Newton start agorthm was aso apped for mutpe-nput snge-output MISO BC n [9]. Dfferent from the above agorthms whch are based on genera purpose convex programmng, the agorthm n ths paper mposes the optma nput structure at each teraton, resutng n ower compexty, severa tmes faster convergence speed, and orders of magntude hgher accuracy than the state-of-the-art. It works for more genera cases and can be easy modfed to sove the optmzaton of genera MIMO B-MAC networks [7]. The rest of the paper s organzed as foows. Secton II defnes the achevabe scheme and formuates the probem. The optma pote water-fng structure for WSRM_MC s derved n Secton III. Secton IV presents the pote waterfng based agorthm for ee networks. The performance s verfed by smuaton n Secton V. Secton VI concudes /11/$ IEEE 2378

2 II. SYSTEM MODE AND PROBEM FORMUATION Consder a generazed MIMO nterference network named a MIMO B-MAC network [4], [5], where each transmtter may have ndependent data for dfferent recevers and each recever may want ndependent data from dfferent transmtters. There are data nks. Assume the set of physca transmtter abes s T = {TX 1, TX 2, TX 3,...} and the set of physca recever abes s R = {RX 1, RX 2, RX 3,...}. Defne transmtter recever T R of nk as a mappng from to nk s physca transmtter recever abe n T R. For exampe, n a 2-user MAC, the sets are T = {TX 1, TX 2 }, R = {RX 1 }. And the mappngs coud be T 1 = TX 1, T 2 = TX 2, R 1 = RX 1, R 2 = RX 1. The numbers of antennas at T and R are T and respectvey. The receved sgna at R s R y = H,k x k + w, 1 where x k C T 1 k s the transmt sgna of nk k and s assumed to be crcuary symmetrc compex Gaussan CSCG; H,k C R Tk s the channe matrx between T k and R ; and w C R 1 s a CSCG nose vector wth non-snguar covarance matrx W. To hande a wde range of nterference canceaton, we defne a coupng matrx Φ {0, 1} + as a functon of the nterference canceaton scheme [4], [5]. It specfes whether nterference s competey canceed or treated as nose: f x k, after nterference canceaton, st causes nterference to x, Φ,k = 1 and otherwse, Φ,k = 0. The coupng matrces vad for the resuts of ths paper are those for whch there exsts a transcever scheme such that each sgna s decoded and possby canceed by no more than one recever. For exampe, n a BC MAC empoyng DPC SIC where the th nk s the th one to be encoded decoded, the coupng matrx s gven by Φ,k = 0, k and Φ,k = 1, k >. For genera B-MAC networks, certan combnatons of DPC and SIC are aso aowed as dscussed n [4]. We gve the achevabe rate. Note that Φ, = 0 by defnton. The nterference-pus-nose covarance of nk s Ω = W + Φ,k H,k Σ k H,k, 2 where Σ k s the covarance matrx of x k. We denote a the covarance matrces as Σ 1: = Σ 1,..., Σ. Then the achevabe rate of nk s gven by [4] I Σ 1:, Φ = og I + H, Σ H, Ω 1. 3 The paper consders the weghted sum-rate mzaton probem under mutpe near constrants wth a fxed Φ: WSRM_MC: Σ 1: s.t. w I Σ 1:, Φ 4 Σ 0,, Σ Ŵ n, δ n, n = 1,..., N, where w > 0 s the weght for nk ; Ŵ n, s are postve semdefnte matrces and δ n 0, n. We focus on a fxed coupng matrx Φ. The optmzaton of the coupng matrx Φ, or equvaenty, the optmzaton of the encodng order n DPC and decodng order n SIC s dscussed n [4]. The structure of the optma souton rees on a rate duaty between the forward and reverse nks of a MIMO B-MAC network [4], whch s revewed as foows. et [H,k ], Σ Ŵ P T, [W ], 5 denote a network where the channe matrces are [H,k ]; the nput covarance matrces must satsfy a snge near constrant Σ Ŵ P T ; and the covarance matrx of the nose at the recever of nk s W. Then the dua network or reverse nks s defned as [ ] ] H k,, ˆΣ W P T, [Ŵ, 6 whereˆdenote the correspondng terms n the reverse nks. The coupng matrx for the reverse nks s the transpose of that for the forward nks. The nterference-pus-nose covarance matrx of reverse nk s ˆΩ = Ŵ + Φ k, H ˆΣ k, k H k,, 7 and the rate of reverse nk s Î ˆΣ1:, Φ T = og I + H ˆΣ, H ˆΩ 1,. The key of the rate duaty s a covarance transformaton from the forward nput covarance matrces to the reverse ones. The detas of the covarance transformaton can be found n [4], [7]. We restate the rate duaty n [4] as foows. Theorem 1: For any Σ 1: satsfyng Σ Ŵ P T and achevng a rate pont r n the network 5, ts covarance transformaton ˆΣ 1: acheves a rate pont ˆr r n the dua network 6 under the near constrant ˆΣ W = Σ Ŵ P T. Therefore, the set of a achevabe rates n the forward and reverse nks are the same. III. POITE WATER-FIING UNDER MUTIPE INEAR CONSTRAINTS In ths secton, we prove that the optma souton of probem 4 has a pote water-fng structure from the KKT condtons. The agrange functon of probem 4 s µ 1:N, Θ 1:, Σ 1: = + Σ Θ + w og I + H, Σ H, Ω 1 8 N µ n δ n n=1 Σ Ŵ n, where the agrange mutpers µ 1:N = [µ n ] n=1,...,n R N 1 + are assocated wth the mutpe near constrants; and Θ 1: = 2379

3 Θ 1,..., Θ wth Θ C T T are the matrx agrange mutpers assocated wth the postve semdefnteness constrants on Σ s. et Σ 1: be an optma souton for probem 4. Then t must satsfy the KKT condtons [7],.e., there exst optma agrange mutpers µ n 0, n = 1,..., N and Θ 0,, = 1,...,, such that Σ Σ µ 1:N, Θ 1:, Σ 1: = 0; 9 µ n δ n Σ Ŵ n, = 0; 10 for a = 1,..., and n = 1,..., N. Defne Ŵ = N n=1 µ nŵn,. It foows from 10 that Σ 1: aso satsfes the near constrant Σ Ŵ = N n=1 µ nδ n. By the rate duaty n [4], the nose covarance matrx of reverse nk s gven by Ŵ. It was shown n Theorem 3 of [7] that ˆΣ 1: = ˆΣ1,..., ˆΣ defned beow s the covarance transformaton of Σ 1: : 1 ˆΣ = w Ω 1 Ω + H, Σ H,,. 11 where Ω s the nterference-pus-nose covarance matrx of forward nk resuted from Σ 1:. Then the condton Σ = 0 n 9 can be rewrtten as ˆΩ = w H, Ω + H, Σ H, 1 H, + Θ, 12 where ˆΩ = Ŵ + k Φ k,h k, ˆΣ k H k, s the nterferencepus-nose covarance matrx of reverse nk. In [7], ˆΩ was shown to be non-snguar wth probabty one. Assumng ˆΩ s non-snguar and defne H 1/2 = Ω H ˆΩ 1/2, and Q = ˆΩ 1/2 Σ ˆΩ1/2 respectvey be the equvaent channe and the equvaent nput covarance matrx for nk. Then Equaton 12 can be rewrtten as I = w H I + H 1 Q H H + ˆΩ 1/2 Θ ˆΩ 1/2, 13 whch s the KKT condton of a snge-user optmzaton probem. Snce the souton of 13 s unque and s gven by the water-fng over H wth w as the water-fng eve [2], the foowng theorem s proved. Theorem 2: For each, perform the thn SVD on the Ω 1/2 equvaent channe as H H ˆΩ 1/2, = F G, where F C R N, G C T N, R N N ++, and N = Rank H,. If Σ 1: satsfes the KKT condtons n 9 wth the optma agrange mutpers µ n, n = 1,..., N, t must have a pote water-fng structure,.e., the equvaent nput covarance matrx Q s a water-fng over H as foows Q ˆΩ1/2 Σ ˆΩ1/2 = G D G 14 D = w I 2 +,. Furthermore, the covarance transformaton ˆΣ 1: = ˆΣ1,..., ˆΣ n 11 possesses the pote water-fng structure,.e., for each reverse nk, the reverse equvaent nput covarance matrx ˆQ Ω 1/2 ˆΣ Ω 1/2 s a water-fng over H 1/2 = ˆΩ H, Ω 1/2 as ˆQ = F D F. IV. AGORITHM In ths secton, we expot the pote water-fng to sove probem 4 for a sub-cass of B-MAC networks named the nterference tree ee networks. ee networks defned n [4] s a natura extenson of MAC and BC. Defnton 1: A B-MAC network wth a fxed coupng matrx s caed an Interference ee ee Network f after nterference canceaton, the nks can be ndexed such that any nk s not nterfered by the nks wth smaer ndces. An ee network s dfferent from a network whose channe gans has a tree topoogy. For exampe, a BC MAC whch has tree topoogy s not an ee network f DPC SIC s not empoyed, whe a BC MAC empoyng DPC SIC s an ee network. Wthout oss of generaty, we consder ee networks where the th nk s not nterfered by the frst 1 nks,.e., the coupng matrx satsfes Φ,k = 0, k and Φ,k = 1, k >. Even for ee networks, probem 4 s n genera nonconvex. Nevertheess, the agrange duaty method [10] can be apped to obtan a suboptma souton. Defne g Σ 1:, µ 1:N w og I + H, Σ H, Ω 1 Σ Ŵ, where Ŵ = N n=1 µ nŵn,. The dua functon of probem 4 s gven by g µ 1:N = Σ 0, N g Σ 1:, µ 1:N + µ n δ n 15 Hence, the dua probem can be defned as n=1 mn g µ 1:N 16 µ n 0, n To sove the mzaton probem n 15, we desgn a pote water-fng based agorthm whch monotoncay ncreases g unt t converges to a statonary pont. We frst defne some notatons. For any {1,..., }, fxng Σ j, j = + 1,..., for the ast nks, the frst nks form a sub-network [H,k ] k,,...,, [ ] Σ Ŵ = PT, W,, where W = W + j=+1 Φ,jH,j Σ j H,j s the covarance matrx of the equvaent coored nose of nk ; PT = Σ Ŵ. The reverse nks of the sub-network s [ H k, ]k,,...,, ˆΣ W ] = PT, [Ŵ.,..., 18 et ˆΣ1: be the covarance transformaton of Σ 1:. Then by emma 9 n [4], ˆΣ1: = ˆΣ1,..., ˆΣ s aso the covarance transformaton of Σ 1: = Σ 1,..., Σ, apped to the sub-network 17. Denote I Σ 1:, Φ and Î ˆΣ 1:, Φ T 2380

4 the forward and reverse nk rates of the th nk of the sub-network 17 acheved by Σ 1: and ˆΣ 1: respectvey. Defne g Σ 1:, µ 1:N w I Σ 1:, Φ Σ Ŵ and ĝ ˆΣ 1:, µ 1:N w Î ˆΣ1:, Φ T ˆΣ W. The foowng 3-step agorthm ncreases g Σ 1:, µ 1:N for any {1,..., } wthout affectng the vaue of g Σ 1:, µ 1:N, >. Step 1: Cacuate ˆΣ 1: 1 by the covarance transformaton n [4], [7] apped to the th sub-network. By Theorem 1, we have ˆΣ ĝ 1:, µ 1:N g Σ 1:, µ 1:N = g Σ 1:, µ 1:N. Step 2: Improve ĝ by sovng the foowng snge-user optmzaton probem: ˆΣ 0 w og I + H ˆΣ, H, ˆΩ 1 ˆΣ W 19 where ˆΩ = Ŵ + 1 Φ k,h ˆΣ k, k H k, and W = W + j=+1 Φ,jH,j Σ j H,j = Ω. Perform the thn SVD Ω 1/2 H, ˆΩ 1/2 = F G. Appyng Theorem 2 to a snge user network, the optma souton s unquey gven by the foowng pote water-fng souton ˆΣ = Ω 1/2 F w I 2 + F Ω 1/2. 20 et ˆΣ 1: = ˆΣ1,..., ˆΣ 1, ˆΣ. Then we have ĝ ˆΣ 1:, µ 1:N ĝ ˆΣ 1:, µ 1:N and the equaty hods f and ony f ˆΣ satsfes the pote water-fng structure n 20. Snce the nterference reaton s reversed n the reverse nks, the reverse nk causes no nterference to the frst 1 reverse nks and thus ˆΣ ĝ 1:, µ 1:N ˆΣ ĝ 1:, µ 1:N must hod. Step 3: Improve g by the covarance transformaton from ˆΣ 1: to Σ 1: = Σ 1,..., Σ for the sub-network. By Theorem 1, we have g Σ 1:, µ 1:N ˆΣ ĝ 1: 1:N, µ. et Σ 1: = Σ 1,, Σ, Σ +1,, Σ. Combnng the above resuts and the fact that the frst nks cause no nterference to a other nks n the orgna network, we must have g Σ 1:, µ 1:N g Σ 1:, µ 1:N. Then the agorthm to sove the probem n 15 s summarzed n Tabe I and s referred to as Agorthm PF. The foowng theorem hods for Agorthm PF. Theorem 3: Agorthm PF monotoncay ncreases g Σ 1:, µ 1:N unt t converges to a statonary pont of the probem n 15. The proof s sketched beow. Snce g Σ 1:, µ 1:N s upper bounded, Agorthm PF must converge to a fxed pont. At the fxed pont, f Σ 1: does not satsfy the KKT condtons, then t can be shown that for some reverse nk, ˆΣ does not satsfy the pote water-fng structure n 20 and thus g Σ 1:, µ 1:N can be strcty ncreased by Agorthm PF, whch contradcts wth the assumpton of fxed pont. Tabe I AGORITHM PF FOR THE MAXIMIZATION PROBEM IN 15 Intaze Σ 1: such that Σ 0,. Whe not converge do For = 1 : 1. Cacuate ˆΣ 1: 1 by the covarance transformaton of Σ 1:. 2. Obtan ˆΣ by pote water-fng as n Cacuate Σ 1: by the covarance transformaton of ˆΣ 1:. 4. Update Σ 1: as Σ 1: = Σ 1,..., Σ, Σ +1,..., Σ. End End Tabe II AGORITHM P2 SOVING WSRM_MC FOR ITREE NETWORKS Intaze µ 1:N such that µ n > 0, n. 1. Sovng for Σ 1: n probem 15 wth fxed µ 1:N usng Ag. PF. 2. Update µ 1:N usng 21 or Return to step 1 unt convergence. After sovng the mzaton probem n 15, a subgradent method can be adopted to sove the dua probem n 16. et Σ 1: denote the souton of Agorthm PF. If Σ 1: s the goba [ optmum of the probem n 15, t can be shown that δ n Σ Ŵ n, s a subgradent of ]n=1,...,n g µ 1:N at pont µ 1:N [8]. Then µ 1:N s updated by the subgradent method as [ µ n = µ n t δ n Σ Ŵ n, ] +, 21 where t denotes the step sze of the subgradent agorthm. For convenence, the update n 21 s caed the subgradent update even f Σ 1: s not the goba optmum. We aso propose an aternatve update of µ n as µ n = µ n Σ Ŵ n, /δ n. 22 Smuatons show that the update n 22 usuay works better than the subgradent update. The overa agorthm to sove probem 4 s summarzed n tabe II and s referred to as Agorthm P2. The foowng theorem can be proved usng convex optmzaton theory [10] and the rate duaty.. Theorem 4: In an ee network, f ether w I Σ 1:, Φ or w Î ˆΣ 1:, Φ T s concave for fxed Φ, Agorthm P2 wth the subgradent update converges to the optma souton of probem 4. For genera cases, smuatons show that Agorthm P2 st converges to a statonary pont of probem 4. V. NUMERICA RESUTS In ths secton, we compare wth the state-of-the-art and demonstrate the superor convergence speed of the proposed agorthm for a 3-user MISO BC. A smuaton parameters are set the same as those n Fg. 2 of [9] for comparson wth the nfeasbe start Newton agorthm n [9]. Defne the term N n=1 Ŵn, Σ δ n as the constrant error for convergence comparson. The sum-rate and constrant error 2381

5 Sum rate bts/channe use Constrant error Agorthm P2 wth update 22 Agorthm P2 wth subgradent update Infeasbe start Netwon Number of teratons Sum rate error Constrant error Agorthm P2 wth update 22 Agorthm P2 wth subgradent update Infeasbe start Netwon Number of teratons Fgure 1. Convergence comparson for a 3-user MISO BC Fgure 2. Accuracy comparson for a 3-user MISO BC versus the number of teratons are potted n Fg. 1. In Fg. 2, we compare the asymptotc convergence speed, or accuracy, n terms of sum-rate error and constrant error n ogarthmc scae. The proposed agorthm converges much faster and are severa orders of magntude more accurate than the nfeasbe start Newton agorthm. It s aso observed that Agorthm P2 wth update 22 outperforms that wth the subgradent update. Fnay, we brefy compare the compexty of the agorthms. The man computaton compexty of Agorthm P2 es n the SVD and matrx nverse operatons n the pote waterfng and n the cacuaton of the MMSE receve vectors for covarance transformaton. Both operatons are performed over the matrces whose dmensons are equa to the number of antennas at each node. We use the number of SVD and matrx nverse operatons to measure the compexty of Agorthm P2. The compexty order per teraton of Agorthm P2 s O 2, whch s ower than the two-oop agorthm n [8], where the nteror pont method used to sove the nner oop probem has a compexty order of O 3. The nfeasbe start Newton agorthm n [9] ony works for MISO BC. In each teraton, the order of the number of T T matrx mutpcatons s O + N 2, where T s the number of transmt antennas. Each teraton aso requres a + N N + 1 matrx nverse operaton, whose compexty s cubc wth respect to. Note that for MISO case, the channe matrces reduce to vectors, and the SVD n the pote water-fng reduces to vector normazaton. Therefore, the compexty of the nfeasbe start Newton agorthm s hgher than that of Agorthm P2. VI. CONCUSION We extend the pote water-fng resuts from a snge near constrant to mutpe near constrants for the generazed MIMO nterference networks, named MIMO B-MAC networks, and use t to desgn hghy effcent agorthms for weghted sum-rate mzaton under mutpe near constrants n ee networks, whch s a natura extenson of MAC and BC and cas for more nformaton theoretca study. The agorthm has ower compexty, much faster convergence speed and hgher accuracy, than the state-of-the-art [8], [9]. Furthermore, as shown n [7], the proposed agorthm can be easy generazed to sove the non-convex optmzaton of genera MIMO B-MAC networks wth nterference oops. ACKNOWEDGMENT The authors woud ke to thank Guseppe Care and Hoon Huh for the hep on the comparson between the nfeasbe start Newton agorthm [9] and the proposed agorthm. REFERENCES [1] W. Yu, W. Rhee, S. Boyd, and J. Coff, Iteratve water-fng for Gaussan vector mutpe-access channes, IEEE ans. Info. Theory, vo. 50, no. 1, pp , [2] N. Jnda, W. Rhee, S. Vshwanath, S. A. Jafar, and A. Godsmth, Sum power teratve water-fng for mut-antenna Gaussan broadcast channes, IEEE ans. Info. Theory, vo. 51, no. 4, pp , Apr [3] O. Popescu, D. C. Popescu, and C. Rose, Smutaneous water fng n mutuay nterferng systems, IEEE ans. Wreess Commu., vo. 6, no. 3, pp , Mar [4] A. u, Y. u, H. Xang, and W. uo, Duaty, pote water-fng, and optmzaton for MIMO B-MAC nterference networks and ee networks, submtted to IEEE ans. Info. Theory, Apr. 2010; revsed Oct [Onne]. Avaabe: [5], MIMO B-MAC nterference network optmzaton under rate constrants by pote water-fng and duaty, IEEE ans. Sgna Processng, vo. 59, no. 1, pp , Jan [Onne]. Avaabe: [6] V. Cadambe and S. Jafar, Interference agnment and the degrees of freedom of wreess X networks, IEEE ansactons on Informaton Theory, vo. 55, no. 9, pp , sept [7] A. u, Y. u, H. Xang, and W. uo, Technca report: Pote water-fng for weghted sum-rate mzaton n MIMO B-MAC networks under mutpe near constrants, Pekng Unversty and Unversty of Coorado at Bouder Jont Technca Report, Dec [Onne]. Avaabe: ue/pubcatons/ndex.htm [8]. Zhang, R. Zhang, Y. ang, Y. Xn, and H. V. Poor, On gaussan MIMO BC-MAC duaty wth mutpe transmt covarance constrants, submtted to IEEE ans. on Informaton Theory, Sept [9] H. Huh, H. C. Papadopouos, and G. Care, Mutuser MISO transmtter optmzaton for nterce nterference mtgaton, IEEE ansactons on Sgna Processng, vo. 58, no. 8, pp , Aug [10] S. Boyd and. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press,