Alternating Optimization for Capacity Region of Gaussian MIMO Broadcast Channels with Per-antenna Power Constraint

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1 Alternatng Optmzaton for Capacty Regon of Gaussan MIMO Broadcast Channels wth Per-antenna Power Constrant Thuy M. Pham, Ronan Farrell, and Le-Nam Tran Department of Electronc Engneerng, Maynooth Unversty, Maynooth, Co. Kldare, Ireland Emal: {mnhthuy.pham, ronan.farrell, arxv: v2 [cs.it] 17 May 2017 Abstract Ths paper characterzes the capacty regon of Gaussan MIMO broadcast channels (BCs) wth per-antenna power constrant (PAPC). Whle the capacty regon of MIMO BCs wth a sum power constrant (SPC) was extensvely studed, that under PAPC has receved less attenton. A reason s that effcent solutons for ths problem are hard to fnd. The goal of ths paper s to devse an effcent algorthm for determnng the capacty regon of Gaussan MIMO BCs PAPC, whch s scalable to the problem sze. To ths end, we frst transform the weghted sum capacty maxmzaton problem, whch s nherently nonconvex wth the nput covarance matrces, nto a convex formulaton n the dual multple access channel by mnmax dualty. Then we derve a computatonally effcent algorthm combnng the concept of alternatng optmzaton and successve convex approxmaton. The proposed algorthm acheves much lower complexty compared to an exstng nterorpont method. Moreover, numercal results demonstrate that the proposed algorthm converges very fast under varous scenaros. Index Terms MIMO, mnmax dualty, drty paper codng, alternatng optmzaton, successve convex optmzaton. I. INTRODUCTION Snce ts nventon n the md-90s [1], [2], multple-nput multple-output (MIMO) technology has been adopted n all modern moble wreless networs. From a system desgn perspectve, one of the most fundamental problems s to compute the capacty of the system of nterest. For a sngle user MIMO (SU-MIMO) channel, poneer studes proved that the capacty can be acheved by Gaussan nput sgnalng [1], [2]. For multuser MIMO scenaros, the semnal wor of [3] showed that drty-paper codng (DPC) n fact acheves the entre capacty regon of Gaussan MIMO broadcast channels (BCs). The partcular case of the sum capacty of MIMO BCs was studed n several poneer studes ncludng [3] [7], The capacty of MIMO systems s nvestgated along wth a certan type of constrant on the nput covarance matrces. In ths regard, we remar that all papers mentoned above assume a sum power constrant (SPC), and ths usually leads to effcently computatonal algorthms. For the sum-capacty computaton, Vswanathan et al. [8] appled a steepest descend method, whle Yu [9] proposed a dual decomposton-based algorthm. In ths lne of research, Jndal et al. presented a sum power teratve water-fllng algorthm by explotng the MAC- BC dualty. The entre capacty regon of MIMO BCs wth a SPC was characterzed n [10], [11], usng conjugate gradent projecton (CGP)- and pre-condtoned gradent projectonbased approaches. Whle SPC has been wdely consdered, t s less appealng n realty due to the fact that each antenna s usually equpped wth a dfferent power amplfer, whch has ts own power budget. Despte ts practcal and fundamental mportance, the research on effcent methods for computng the capacty regon of Gaussan MIMO BCs has been qute lmted. For SU-MIMO, ths problem was solved n [12], [13] by the socalled mode-droppng algorthm. In [3], t was shown that DPC stll acheves the full capacty regon of the MIMO BC under PAPC. However, fndng the DPC regon wth PAPC s more numercally dffcult than wth a SPC. In fact, no closed-form desgn has been reported for the computaton of the capacty regon of the MIMO BC PAPC. To the best of our nowledge, the only attempt to characterze the entre capacty regon of the MIMO BC PAPC was made n [14]. Specfcally, the authors n [14] establshed a modfed dualty between the MAC and BC and transformed the nput optmzaton problem n the BC nto a mnmax optmzaton problem n the dual MAC. Then resultng program s solved by a standard barrer nteror-pont routne. Thus, as a common property for the class of second order optmzaton methods, the algorthm proposed n [14] has computatonal complexty that does not favor large-scale antenna systems whch are envsoned n next wreless communcatons generatons. In ths paper, we determne the capacty regon of MIMO BCs under PAPC. In partcular, the problem of nterest s also nown as weghted sum rate maxmzaton (WSRMax) for MIMO BCs. As mentoned earler, the capacty regon can be acheved by DPC, but the resultng WSRMax problem s nonconvex. As a standard step, we apply the mnmax dualty presented n [14] to transform the WSRMax problem nto a mnmax program. However, unle [14] whch solves the resultng mnmax problem by a barrer nterorpont method, we tae advantage of the problem specfcs to propose an effcent algorthm that blends the concept of alternatng optmzaton (AO) and successve convex approxmaton (SCA). Especally, n each teraton of the proposed algorthm, closed-form expressons based on conjugate gradent projecton (CGP) method are derved. As a result, the complexty of the proposed algorthm s much lower than that of the barrer method n [14], and scales lnearly wth the

2 number of users n the system, mang t partcularly sutable for large-scale networs. Numercal experments are carred out to demonstrate that the proposed algorthm can converge rapdly, especally for networs of hgh sgnal-to-nose rato, and the number of teratons requred for convergence s qute nsenstve to the number of users. The remander of the paper s organzed as follows. The system model s descrbed n Secton II followed by the proposed algorthm n Secton III. Secton IV provdes the complexty analyss of the proposed algorthm whle Secton V presents the numercal results. Fnally, we conclude the paper n Secton VI. Notaton: Standard notatons are used n ths paper. Bold lower and upper case letters represent vectors and matrces, respectvely. I defnes an dentty matrx, of whch the sze can be easly nferred from the context; C M N denotes the space of M N complex matrces; H and H T are Hermtan and normal transpose of H, respectvely; H,j s the (, j)th entry of H; H s the determnant of H; ran(h) stands for the ran of H; dag(x) denotes the dagonal matrx wth dagonal elements beng x. II. SYSTEM MODEL Consder a K-user MIMO BC where the base staton and each user have N and M antennas, respectvely. The channel matrx for user s denoted by H. Let s be the composte sgnal that combnes the data for all users n the downln. Then, the receved sgnal at user s expressed as y = H s + z (1) where z s the Gaussan nose wth dstrbuton CN (0, I M ). When DPC s appled to acheve the capacty regon, for a gven user, the nterference caused by users j < s completely canceled wthout affectng the optmalty. As a result, the WSRMax under PAPC s formulated as maxmze {S 0} w log I+H =1 SH [S ], P, I+H 1 =1 SH where S s the nput covarance matrx for the th user, P s the power constrant on antenna, and w s the weghtng factor assgned to user. Wthout loss of optmalty, we assume that 0 < w 1 w 2... w K n the followng. Snce (2) s a nonconvex problem, solvng t drectly s not a good opton. However, we can explot the MAC-BC dualty to transform (2) nto mnmax program n the dual MAC whch can be solved effcently by the novel AO as presented n the next secton. III. PROPOSED SOLUTION (2) and w = 1 Applyng the modfed MAC-BC dualty ntroduced n [15], we can equvalently rewrte (2) as the followng mnmax optmzaton problem mn max Q 0 { S 0} log Q + = H S H w K log Q tr( S ) = P, tr(qp) = P, Q : dagonal where = w w 1 0, P N =1 P ; { S } and Q are nput covarance and nose covarance matrces n the dual MAC, respectvely. As shown n [15], the objectve n (3) s convex wth Q 0 and concave wth { S 0}. Thus, there exsts a saddle pont for (3), whch also solves (2). The mnmax formulaton n (3) also suggests a way to fnd { S } and Q by AO. However, a pure AO algorthm s not guaranteed to converge. In fact, a counterexample was already gven n [16]. In what follows, we propose an teratve algorthm based on combnng AO and SCA, of whch convergence can be proved. Let { S n } be the soluton to the followng maxmzaton problem n the nth teraton maxmze { S 0} log Q n + = H S H tr( S ) = P. The above maneuver s nothng but a standard routne of optmzng { S } when Q s held fxed. Problem (4) can be solved by off-the-shelf nteror-pont convex solvers but the complexty s not affordable for large-scale systems. In our numercal experments, all of nown (free and commercal) solvers fal to solve (4) on a relatvely powerful destop PC for N 100, regardless of the number of users. That s, nterorpont methods are not an effcent approach to solvng (4) for massve MIMO technques whch are lely to be adopted n 5G systems. To overcome ths shortcomng, we now present an effcent method to solve (4) based on the CGP framewor. To proceed, let S = { S S 0, tr( S ) = P } be the feasble set of (4). The man operaton of a CGP method s to project a gven { S } onto S. Our motvaton s that the projecton of { S } onto S can be reduced to a projecton of a resultng vector onto a canoncal smplex, whch can be computed effcently. The projecton of { S } onto the feasble set S s formulated as Ṡ S 2 F {Ṡ 0} (5) K tr(ṡ) = P. Let U D U = S be the EVD of S, where U s untary and D s dagonal. Then we can wrte Ṡ = U Ḋ U for some Ḋ 0. Snce U s untary, t holds that tr(ṡ) = tr(ḋ) and that Ṡ S F = Ḋ D F. That s to say, (5) s equvalent to {Ḋ 0} Ḋ D 2 F tr(ḋ) = P. (3) (4) (6)

3 It s easy to see that Ḋ must be dagonal to the objectve of (6). Next let d = dag(ḋ), d = dag( D ), d = [ d T 1, d T 2,..., d T K ]T, and d = [ d T 1, d T 2,..., d T K ]T. Then (6) can be reduced to 1 2 d d 2 2 d 0 (7) 1 M d = P where M = 1 M. It s now clear that (7) s the projecton onto a canoncal smplex and effcent algorthms (smlar to water-fllng algorthms) can be found n [17]. The complete descrpton of the proposed CGP method for solvng (4) s provded n Algorthm 1. We note that smlar approaches were also presented n [10], [11]. Algorthm 1: The proposed CGP algorthm for solvng (4). Input: P, ǫ > 0 1 Intalzaton: τ = 1 + ǫ, m = 0, { S 0 } S. 2 whle (τ > ǫ) do 3 Calculate the conjugate gradent G m. 4 Choose an approprate postve scalar s m and create S m = S m + sm Gm. 5 Project S m onto S to obtan Ṡm. 6 Choose approprate step sze α m and set S m+1 = S m + αm (Ṡm S m ). 7 τ = tr( f( S m ) ( S m+1 S m )). 8 m := m end Output: S as the optmal soluton to (4). Another man step of a CGP method s the computaton of the conjugated gradent of the objectve, as requred n lne 3 of Algorthm 1. The conjugate gradent, denoted as G m, can be calculated as follows. Frst, we compute the gradent of the objectve n (4) as ) 1 f( S ) = H j (Q + H S H H. (8) j=1 =j Then the conjugate gradent drecton s gven by G m = f( S m ) + β m Gm 1 (9) where the parameter β m s the Fletcher choce of deflecton [18] { 0 m = 0 β m = f( S m (10) ) 2 m 1 tr(( G m 1 ) f( S m 1 )) For the step sze n lne 6 of Algorthm 1, we perform an Armjo lne search [19] to determne approprate value. For the mportance case of the sum capacty of the MIMO BC, we note that more effcent approaches to solvng (4) do exst. In fact, n ths case (4) becomes maxmze { S 0} log Q n + =1 H S H tr( S ) = P. (11) We remar that (11) s equvalent to the problem of fndng the sum-capacty of a MAC wth a SPC for whch the sum power teratve water-fllng proposed n [20] or the dual decomposton method n [9] have been shown to be partcularly computatonally effcent. We now turn our attenton to the problem of fndng Q n+1. If a pure AO method s followed, we arrve at the optmzaton problem below: Q 0 log Q + = H S H w K log Q tr(qp) = P (12) However, as mentoned n [14] and also observed n [16], the convergence of such a nave AO method s not guaranteed. The novelty of our proposed AO algorthm s that, nstead of optmzng the orgnal objectve n (12) whch can lead to fluctuatons, we opt to an upper bound of the objectve n (12). Ths s n lght of the SCA prncple, and wll lead to a monotonc convergence as shown n the Appendx. To ths end, by nvong the concavty of logdet functon, we have the followng nequalty log Q + = H S n H log Φ n + tr( Φ n ( Q Q n )) (13) where Φ n = Q n + = H S n H, Φ n (Φ n ) 1. Thus, usng the above upper bound, Q n+1 s found to be the optmal soluton to the followng problem ( ) K Q 0 w K tr Φ n Q log Q (14) tr(qp) = P. Snce Q n (14) s dagonal,.e., Q = dag(q), we can rewrte (14) as N q>0 =1 φn q log q N =1 P (15) q = P, ] where φ n = [ w K Φ n,,.e., φn s the th dagonal element of w K Φ n. By settng the dervatve of Lagrangan functon of (15) to zero, we obtan 1 q = φn + γp (16) where γ 0 s the soluton of the equaton N P φn + γp = P. (17) =1 Denote g(γ) = N K P =1 P. It s easy to see that φn +γp g(γ) s decreasng wth γ. When γ = 0, snce q 1 we have N φn P P q = P. (18) =1 φn =1 Therefore, (17) always has a unque soluton, whch can found effcently, e.g., by the bsecton or Newton method. The proposed algorthm based on AO s summarzed n Algorthm 2. N

4 The man pont of Algorthm 2 s to elmnate the possble png-pong effect of the obtaned objectve by the use of the nequalty n (13). The convergence proof of Algorthm 2 s provded n the Appendx. Note that the error tolerance τ of Algorthm 2 s only computed for n 1. Algorthm 2: Proposed algorthm for the computaton of the capacty regon of a MIMO BC based on AO. Input: Q := Q 0 dagonal matrx of postve elements, ǫ > 0 1 Intalzaton: Set n := 0 and τ = 1 + ǫ. 2 whle (τ > ǫ) do 3 Solve (4) and denote the optmal soluton by { S n } 4 For n 1 compute τ = f DPC (Q n, { S n }) f DPC (Q n 1, { S n 1 }), where f DPC ( ) denotes the objectve n (3). 5 For each, compute Φ n = (Q n + = H S n H ) 1. 6 Solve (14) to fnd Q n+1. 7 n := n end Output: Use the obtaned { S n }K and the BC-MAC transformaton n [5] to fnd the optmal soluton to (2). IV. COMPLEXITY ANALYSIS In ths secton, we analyze the complexty estmaton of the proposed algorthm n terms of the number of flops. The results of flop countng for typcal matrx operatons such as EVD, matrx nverson are taen from [21] and [22]. Moreover,we treat every complex operaton as 6 real flops as consdered n [22], [23]. In the complexty analyss presented below, we only consder the man operatons of hgh complexty n the overall complexty. A. Complexty of Algorthm 2 Algorthm 2 has two ey procedures, one for fndng { S } and one for fndng Q. Of all steps n Algorthm 1, the computaton of the conjugate gradent drecton gven n (8) has the largest computaton cost. Smlarly, the complexty for fndng Q s mostly due to the computaton of Φ n defned n (13). Therefore, the per-teraton complexty of Algorthm 2 s O(KN 3 ). That s, the complexty of the proposed algorthm ncreases lnearly wth the number of users. B. Complexty of the Barrer Interor-Pont Method n [14] The method presented n [14] s based on solvng the KKT condton for (3). To fnd the gradent w.r.t each nput covarance matrx for the logdet functon, the complexty s 6N 3. Then, to fnd a Newton drecton, ths algorthm stll needs to solve a lnear system of M(M + 1)/2 unnowns, whch requres 6( KM(M+1) 2 + 3N) 3 flops. 1 In total, the perteraton complexty s O(K 2 N 3 + K 3 M 6 ). 1 Here we assume M = M for all to smplfy the notaton. TABLE I PER-ITERATION COMPLEXITY COMPARISON Algorthms Per-teraton complexty Algorthm 2 O(KN 3 ) Barrer nteror-pont method [14] O(K 2 N 3 + K 3 M 6 ) The complexty comparson s summarzed n Table I. We remar that the complexty of Algorthm 2 ncreases lnearly wth the number of users K, whch was also acheved n [20] for the sum-capacty wth SPC. Thus, Algorthm 2 scales much better than the barrer nteror-pont method n [14] wth the number of users. V. NUMERICAL RESULTS Ths secton provdes numercal results to verfy the proposed algorthm. If not otherwse mentoned, the number of transmt antennas at the BS s set to N = 5 and the number of receve antennas at each user s set to M = 2 for all.. The total power P s vared from a low value to a hgh one to nvestgate the convergence rate of the proposed algorthm.the power constrant for each transmt antenna s set equally. The ntal value Q 0 n the proposed algorthm s set to the dentty matrx. An error tolerance of ǫ = 10 6 s selected as the stoppng crteron for the proposed algorthm. In the frst smulaton, we plot the convergence of the proposed algorthm for dfferent value of the total transmt power for a set of randomly generated channel realzatons. As can be seen, Fg. 1 shows that the convergence of the proposed algorthm s strctly monotonc as proved n the Appendx. An nterestng observaton s that the number of teratons decreases when the total power ncreases. As mentoned n the complexty analyss, the proposed algorthm has a desrable property,.e., the per-teraton complexty ncreases lnearly wth the number of users K. Ths s an attractve property n the system of a large number of users whch s usually the case for a massve MIMO setup. However, t s dffcult, f not mpossble, to analyze the convergence rate of the proposed algorthm wth K by analytcal expressons. Instead, we study the convergence property of the proposed algorthm wth K by numercal experments. For the purpose, we plot n Fg. 2 the average number of teratons requred by the proposed algorthm to converge. As can be seen, the number of teratons for the proposed algorthm to converge s relatvely nsenstve to K. It s worth notng that for smlar setups, the nteror-pont method n [14] requres at least 60 teratons to converge, whle the proposed algorthm only taes 4 teratons even n the case of 50 users. Ths promsng characterstc of the proposed algorthm maes t sutable for studyng the capacty regon of massve MIMO systems where the number of transmt antennas and/or the number of users can be very large. Ths pont wll be further elaborated n the next numercal experment. Tang the advantage that the proposed algorthm has low complexty, n the last numercal experment we characterze the capacty regon of a massve MIMO system wth PAPC. In

5 Dualty gap P = 0 db P = 10 db P = 20 db R2 4 2 ZF, PAPC DPC, PAPC DPC, SPC Average number of teratons Iteraton Fg. 1. Convergence behavor of the proposed algorthm wth K = P = 0 db P = 20 db Number of users Fg. 2. Average number of teratons versus number of users. partcular, we also consder achevable rate regon of the wellnown ZF scheme [24], [25]. The purpose s to understand the performance of ZF (whch s thought to be sub-optmal) n comparson wth the capacty achevng codng scheme under some realstc channel models. To ths end we consder a smple urban scenaro usng WINNER II B1 channel model [26], where a base staton, equpped wth N = 128 antennas, s located at the center of the cell and sngle-antenna recevers are dstrbuted randomly. The total power at the BS s P = 46 dbm and each antenna s equal power constrant,.e., P = P/N for = 1, 2,..., 128. As can be seen clearly n Fg. 3, there s a remarable gap between the achevable rate regon of ZF and the capacty regon, especally when the number of users ncreases. Ths bascally mples that ZF s stll far from optmal for a practcal number of transmt antennas. Our observaton opens research opportuntes n the future to stre the balance between optmal performance by DPC and low-complexty by ZF. VI. CONCLUSIONS In ths paper, we have consdered the problem of computng the capacty regon of Gaussan MIMO BCs PAPC. R R 1 (a) Number of users K = 2 ZF, PAPC DPC, PAPC DPC, SPC R 1 (b) Number of users K = 8 Fg. 3. Comparson of capacty regon of a massve MIMO setup wth N = 128 and M = 1 for all. For the case K = 8 users, the capacty regon s projected on the frst two users. Towards ths end, the problem of WSRMax wth PAPC has been solved by a low-complexty algorthm. We have frst converted the noncovex problem of MIMO BC nto an equvalent mnmax problem n the correspondng dual MAC. Then a novel AO algothm has been proposed to solve the resultng mmmax program n combnaton wth the successve convex optmzaton prncple. In partcular, all the computaton n the proposed algorthm s based on closed-form expressons. In addton, the smulaton results have demonstrated a fast and stable convergence of the proposed algorthm, even for large-scale settngs. ACKNOWLEDGEMENTS Ths wor was supported by a research grant from Scence Foundaton Ireland and s co-funded by the European Regonal Development Fund under Grant 13/RC/2077.

6 APPENDIX For notatonal convenence, we denote Q {Q Q : dagonal, Q 0, tr(qp) = P }, ˆ = w K. Snce the functon log Q + = H S H s jontly concave wth Q and { S }, the followng nequalty holds ˆ log Q + H S H = ˆ log Q n + H S n H + + = } {{ } Φ n ( ) ˆ tr Φ n (Q Qn ) ˆ K = H ( S S n )) (19) for all Q Q and { S } S. The above nequalty comes from the frst order approxmaton of log Q+ = H S H around the pont (Q n, { S n }). Substtute Q := Qn+1 and S := nto the above equalty, we have S n+1 ˆ log Q n+1 + H n+1 S H = ˆ (log Φ n + tr(φ n (Qn+1 Q n )) + = H ( S n+1 S n )) ). (20) Snce { S n } = arg max log Q n + = H S H, { S } S the optmalty condton results n ˆ K = H ( S S n )) 0 (21) for all { S } S. For { S } = { S n+1 } the above nequalty s equal to ˆ whch leads to K = H ( S n+1 S n )) 0 (22) ˆ log Q n+1 + H n+1 S H = ˆ (log Φ n + tr( Φ n (Qn+1 Q n ) )). (23) Subtract both sdes of the above nequalty by log Q n+1 results n ˆ log Q n+1 + H n+1 S H log Q n+1 = } {{ } f DPC (Q n+1,{ S n+1 }) ˆ (log Φ n +tr( Φ n (Qn+1 Q n ) )) log Q n+1. Snce Q n+1 solves (14) t holds that (24) ˆ (log Φ n + tr( Φ n (Qn+1 Q n ) )) log Q n+1 ˆ (log Φ n + tr( Φ n (Q Qn ) )) log Q (25) for all Q Q. For the specal case Q := Q n, the nequalty above s reduced to ˆ (log Φ n + tr ( Φ n (Qn+1 Q n ) )) log Q n+1 ˆ log Φ n log Q n. (26) } {{ } f DPC (Q n,{ S n }) Combnng (24) and (26) results n f DPC (Q n, { S n }) f DPC (Q n+1, { S n+1 }). It s easy to see that {f DPC (Q n, { S n })} s bounded above, and thus {f DPC (Q n, { S n })} s convergent. We also note that (13) s strct f Q Q n. Consequently, the sequence {f DPC (Q n, { S n })} s strctly decreasng unless t s convergent. Therefore, the contnuty of f DPC ( ) and the compactness of S and Q mply lm f DPC (Q n, { S n n }) = f DPC (Q, { S }). REFERENCES [1] E. Telatar, Capacty of mult-antenna Gaussan channels, Eur. Trans. Telecommun, vol. 10, pp , Nov [2] G. J. Foschn and M. J. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Pers.Commun, vol. 6, pp , Mar [3] H. Wengarten, Y. Stenberg, and S. S. Shama, The capacty regon of the Gaussan multple-nput multple-output broadcast channel, IEEE Trans. Inf. Theory, vol. 52, no. 9, pp , Sep [4] G. Care and S. Shama, On the achevable throughput of a multantenna Gaussan broadcast channel, IEEE Trans. Inf. Theory, vol. 49, no. 7, pp , Jul [5] S. Vshwanath, N. Jndal, and A. Goldsmth, Dualty, achevable rates and sum-rate capacty of Gaussan MIMO broadcast channels, IEEE Trans. Inf. Theory, vol. 49, pp , Oct [6] P. Vswanath and D. Tse, Sum capacty of the vector Gaussan broadcast channel and upln-downln dualty, IEEE Trans. Inf. Theory, vol. 49, no. 8, pp , Aug [7] W. Yu and J. Coff, Sum capacty of Gaussan vector broadcast channels, IEEE Trans. Inf. Theory, vol. 50, no. 9, pp , Sep [8] H. Vswanathan, S. Venatesan, and H. Huang, Downln capacty evaluaton of cellular networs wth nown-nterference cancellaton, IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp , Jun

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