Maximizing Sum Rate and Minimizing MSE on Multiuser Downlink: Optimality, Fast Algorithms and Equivalence via Max-min SIR

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1 1 Maximizing Sum Rate and Minimizing MSE on Mutiuser Downink: Optimaity, Fast Agorithms and Equivaence via Max-min SIR Chee Wei Tan 1,2, Mung Chiang 2 and R. Srikant 3 1 Caifornia Institute of Technoogy, CA USA 2 Princeton University, NJ USA 3 University of Iinois at Urbana-Champaign, I USA Abstract Maximizing the minimum weighted SIR, minimizing the weighted sum MSE and maximizing the weighted sum rate in a mutiuser downink system are three important performance objectives in joint transceiver and power optimization, where a the users have a tota power constraint. We show that, through connections with the noninear Perron-Frobenius theory, jointy optimizing power and beamformers in the max-min weighted SIR probem can be soved optimay in a distributed fashion. Then, connecting these three performance objectives through the arithmetic-geometric mean inequaity and nonnegative matrix theory, we sove the weighted sum MSE minimization and weighted sum rate maximization in the ow to moderate interference regimes using fast agorithms. I. INTRODUCTION In this paper, we focus on the downink transmission, where the transmitter at the base station) is equipped with an antenna array and each user has a singe receive antenna. Fu channe information is avaiabe at the transmitter to adapt the beamformers to interference. A the users share the same bandwidth and meet a tota power constraint. We consider a joint optimization of power and transmit beamformer for the min-max weighted) mean squared error MSE) probem or, equivaenty, the max-min weighted) signa-to-interference ratio SIR) probem. This probem is chaenging to sove, because the transmit beamformers are couped across users, making them hard to optimize in a distributed fashion. Whie previous agorithms in the iterature require centraized computation of the eigenvaue and eigenvector of an extended couping matrix, we propose a fast distributed agorithm that computes the optima power and transmit beamformer in the max-min weighted SIR probem with geometric convergence rate. This is achieved by appying the noninear Perron-Frobenius theory in [1], [2], [3] and the upink-downink duaity in [4], [5], [6], [7], [8], [9], wherein the upink acts as an intermediate mechanism to optimize transmit beamformers in the downink. We next keep the beamformers fixed and study the nonconvex probems of, 1) minimizing the weighted sum MSE between the transmitted and estimated symbos, and, 2) maximizing the weighted sum rate. The max-min SIR probem is shown to be a specia case of these two probems. Previous work in the iterature, see e.g., [10], ony sove these two Fig. 1. Overview of the connection soid ines) between the three optimization probems in the paper: i) Weighted sum MSE minimization in 18), ii) weighted sum rate maximization in 28), and iii) max-min weighted SIR in 5). The upper haf of the dotted ine considers power contro ony, whie the ower haf considers both power contro and beamforming. nonconvex probems suboptimay. We deveop fast agorithms independent of stepsize) to sove these two nonconvex probems optimay under ow to medium interference conditions. We everage the standard interference function approach in [11] to show that our agorithms converge under synchronous and asynchronous updates. Proofs can be found in [12]. We refer the readers to Figure 1 for an overview of the connection between the three main optimization probems in the paper. The foowing notations are used. Bodface uppercase etters denote matrices, bodface owercase etters denote coumn vectors, itaics denote scaars, and u v B F) denotes componentwise inequaity between vectors u and v matrices B and F). We et By) denote the th eement of By. The Perron-Frobenius eigenvaue of a nonnegative matrix F is denoted as ρf), and the Perron right) and eft eigenvectors of F associated with ρf) are denoted by xf) and yf), respectivey. The super-scripts ) and ) denote transpose and compex conjugate transpose respectivey. We et e denote the th unit coordinate vector and I denote the identity matrix. et x y denote x y = [x 1 y 1,...,x y ].

2 2 II. SYSTEM MODE We consider a singe ce mutiuser system with N antennas at the base station and decentraized users, each equipped with a singe receive antenna, operating in a frequency-fat fading channe. The downink channe can be modeed as a vector Gaussian broadcast channe: y = h x + z, = 1,...,, 1) where y C 1 1 is the received signa of the th user, h C N 1 is the channe matrix between the base station and the th user, x C N 1 is the transmitted signa vector, and z s are the i.i.d. additive compex Gaussian noise vectors with variance n /2 on each of its rea and imaginary components. We assume that the mutiuser system adopts a inear transmission and reception strategy. In transmit beamforming, the base station transmits a signa x in the form of x = =1 d ŵ, where ŵ C N 1 is the transmit beamformer that carries the information signa d of the th user. We assume a tota power constraint at the transmit antennas, i.e., E[x x] = P. From 1), the received ) signa for the th user can be expressed as y = h ŵ d + ) j h ŵj d j + z. Next, we write ŵ = p u, where p is the downink transmit power and u is the normaized transmit beamformer, i.e., u u = 1, of the th user. Now, the received SIR of the th user in the downink transmission can be given in terms of p and U = [u 1...u ]: SIR p,u) = p h u 2 j p j h u j 2 + n. 2) We define the matrix G with entries G j = h u j 2 in the downink transmission. In terms of the beamforming matrix U, we aso define the cross channe interference) matrix FU) with entries: and vu) = F j U) = { 0, if = j G j U) G U), if j 3) ) n1 G 11 U), n 2 G 22 U),..., n. 4) G U) For brevity, we omit the dependency on U when we fix the beamformers. III. MAX-MIN WEIGHTED SIR MAXIMIZATION In this section, we first consider optimizing ony power before we consider a joint optimization between power and transmit beamformers. et β be a positive vector, where the th entry is assigned by the network to the th ink to refect some ong-term priority). We first consider the foowing maxmin weighted SIR probem: maximize min SIR p) 1 p P, p 0, variabes: p. Note that 5) is equivaent to the min-max weighted MSE probem: min p max /1 + SIR p)) 1 p P. 5) Next, et us define the matrix B = F + 1/ P)v1. 6) By expoiting a connection between the noninear Perron- Frobenius theory in [1], [2] and the agebraic structure of 5), we can give a cosed form soution to 5). 1 emma 1: The optima objective and soution of 5) is given by 1/ρdiagβ)B) and P/1 xdiagβ)b))xdiagβ)b) respectivey. The foowing agorithm computes the optima power of 5) given in emma 1. We et k index discrete time sots. Agorithm 1 Max-min weighted SIR): 1) Update power pk + 1): ) p k) SIR pk)). 7) 2) Normaize pk + 1): pk + 1) pk + 1) P/1 pk + 1). 8) Coroary 1: Starting from any initia point p0), pk) in Agorithm 1 converges geometricay fast to the optima soution of 5), P/1 xdiagβ)b))xdiagβ)b). Remark 1: Interestingy, 7) in Agorithm 1 is simpy the Distributed Power Contro DPC) agorithm in [13], where the th user has a virtua SIR threshod of in both the virtua) upink and downink transmission. However, due to 8), the standard interference function approach in [11] cannot be used to prove the convergence of Agorithm 1. Next, we consider the joint optimization of power and transmit beamformer in the foowing max-min weighted SIR probem: SIR p,u) maximize min β =1 p P, p 0, u u = 1, variabes: U = [u 1...u ], p. We first review the notion of upink-downink duaity. The duaity theory states that, under a same tota power constraint and additive white noise for a users, the achievabe SIR region for a downink transmission with joint transmit beamforming and power contro optimization is equivaent to that of an upink transmission with joint receive beamforming and power contro optimization. Further, the optima receive beamforming vectors in the upink is aso the optima transmit beamforming vectors in the downink. Since joint power contro and beamforming optimization in the upink does not have the beamformer couping difficuty associated with the downink hence easier to sove), the dua) upink probem can be first used to obtain the optima transmit beamformers in the downink. The optima downink transmit power is then computed by keeping the transmit beamformers fixed. In the 1 A cosed-form soution to 5) was first obtained in [8] using a nonnegative increased dimension) matrix totay different from B. As such, the agorithmic soution to 5) in [8] is different and is mainy centraized. On the other hand, our soution expoits the DPC agorithm in [13] and is distributed. 9)

3 3 case where the noise is different for each user, a virtua upink transmission assuming that a users have the same noise, i.e., n = 1 for a ) is constructed as an intermediary step to compute the optima transmit beamforming vector. et the virtua upink power be given by q. Now, suppose there exists positive vaues γ optima max-min weighted SIR) and q for a such that the virtua upink SIR ˆ satisfies SIR ˆ q h p,u) = u 2 j q j h j u γ 10) for a. Since SIR ˆ in 10) ony depends on the beamforming vector u, the receive beamforming optimization, with the power fixed at q, is soved by u = arg min u j G j U) G U) q 1 j + G U), 11) whose soution is the inear minimum mean squared error MMSE) receiver given by optima up to a scaing factor): ) + u = j q jh j h j + I h, where ) + denotes pseudoinversion. Using this MMSE receiver, the SIR constraint in 10) is aways met with equaity, i.e., SIR ˆ p,u) = γ. By the upink-downink duaity, the MMSE receiver is aso the optima transmit beamformer in the downink max-min weighted SIR probem given by 9). Now, we are ready to use Agorithm 1 to sove the joint power contro and beamforming probem in 9) in a fast and distributed fashion. The foowing agorithm computes the optima power and transmit beamformer in 9): geometricay fast to p = xbu )) unique up to a scaing constant). Remark 2: Note that 12) and 15) of Agorithm 2 use the DPC agorithm in [13], where the th user has a virtua SIR threshod of in both the virtua) upink and downink transmission. In the case where n s are equa for a, q in 12) is the exact upink transmit power, and ony computing 1 qk + 1) in 13) requires a goba coordination at the base station. Compared to previous centraized soution in [4], [8], our soution has ess compexity and provabe geometric convergence rate. IV. WEIGHTED SUM MSE MINIMIZATION In this section, we study minimizing the weighted sum of the MSE s of individua data streams under a sum power constraint. We assume that a the receivers use the MMSE fiter for estimating the received symbos of a users. The weighted sum MSE at the output of the MMSE receiver is given by [14]: 1 w 1 + SIR p), 17) =1 where w is some positive weight assigned by the network to the th ink to refect some ong-term priority). Without oss of generaity, we assume that w is a probabiity vector. The weighted sum MSE minimization probem is given by =1 w 1 1+SIR p) =1 p P, p 0, 18) Agorithm 2 Max-min SIR Power Contro & Beamforming): We denote the optima power vector to 18) by p. We can 1) Update virtua) upink power qk + 1): rewrite 18) as P ) =1 w P k G kp k +n k q k + 1) = q SIR ˆ k). 12) kp k +n qk),uk)) =1 p P, p 0, 19) 2) Normaize qk + 1): qk + 1) qk + 1) P/1 qk + 1). 13) It can be shown that the tota power constraint in 18) and 19) are tight at optimaity, which we expoit to transform 19) in the variabes p into another optimization probem that can 3) Update transmit beamforming matrix Uk) = be used to sove 19) optimay. To proceed further, we need [u 1 k)...u k)]: to introduce the notion of quasi-invertibiity of a nonnegative u k) = + matrix in [15], which wi be usefu in soving 19) optimay. q j k)h j h j + I h. 14) Definition 1 Quasi-invertibiity): A square nonnegative j matrix B is a quasi-inverse of a square nonnegative matrix B if B B = B B = BB. Furthermore, I B) 1 = I + B 4) Update downink power pk + 1): [15]. ) Using B in 6), we next study the existence of B, which can p k). 15) SIR pk),uk)) interestingy be associated with the SNR regime. In the case where the tota maximum power is very arge, i.e., P 5) Normaize pk + 1): high SNR regime) or when interference off-diagonas of F) pk + 1) pk + 1) P/1 is very arge, it is deduced in the foowing that B does not pk + 1). 16) exist. emma 2: B does not exist when B = F, where F j > 0 Theorem 1: et the optima power and beamforming matrix in 9) be p and U respectivey. Then, starting from any initia point q0) and p0), pk) in Agorithm 2 converges for a, j and j. However, when F = 0 no interference) such that B = v1 / P or when P is sufficienty sma ow SNR regime)

4 4 such that B v1 / P, then B aways exists, as shown by the foowing emma. emma 3: For any nonnegative vector v, B = 1/1 + 1 v)v1 when B = v1. In a numerica exampe for a ten-user IEEE b network, we experiment with the maximum power constraint of 33mW and 1W the argest possibe vaue aowed in IEEE b). Averaging over 10, 000 random channe coefficient instances, the percentage of instances where B exists is 99% and 65% corresponding to the maximum power constraint of 33mW and 1W, respectivey. For the rest of the paper, we focus on the case when B exists. We next sove 19) in the foowing. et us define z = I + B)p. 20) Note that G z is the tota received desired and interfering) signa power pus the additive white noise at the th receiver. Then, we can rewrite 19) in terms of z as =1 w Bz) z z Bz), = 1,...,, 21) variabes: z, where the constraints in 21) are due to the nonnegativity of p, since, using Definition 1, p = I+B) 1 z = I B)z 0. The foowing resut provides a condition under which the optima soution to 21), z, can be transformed to yied the optima soution to 19) or equivaenty 18). Theorem 2: The optima soution to 18) is given by p = I+B) 1 z, where z is the optima soution to 21), if B is the quasi-inverse of a nonnegative matrix B, where ρ B) < 1. emma 4: If B exists, then B has the spectra radius ρ B) = ρb) 1 + ρb), 22) with the corresponding eft and Perron eigenvectors of B. In the foowing, we derive usefu ower bounds to 18), investigate specia cases, and finay characterize the exact soution to 18). A. A ower Bound to Weighted Sum MSE Minimization By expoiting the eigenspace of B and the Friedand-Karin inequaity in nonnegative matrix theory [16] see [12] for its extension), the foowing resut gives a ower bound on the weighted sum MSE probem in 18). Theorem 3: If B exists, =1 w SIR p) /ρB) ) w xb) yb) 23) for a feasibe p in 18). Equaity is achieved if and ony if w = xb) yb). Thus, SIR p ) = 1/ρB))1. In this case, p = xb) soves 18). Interestingy, Theorem 3 shows that soving 5) with β = 1 can be seen as an approximation method to soving 18) suboptimay, but with an approximation guarantee. In particuar, by taking the ogarithm of the objective function of 18), can be interpreted as the approximation ratio of Agorithm 1 with β = 1 in soving 18). w xb) yb) Remark 3: Based on Theorem 3, we obtain a connection between the min-max MSE probem and the weighted sum MSE optimization. Suppose we consider min-max MSE: min p max 1/1 + SIR p)). Then, by the max-min characterization of ρ B): max min z>0 Bz) z = min z>0 max Bz) z = ρ B), 24) the optima objective of min-max MSE is simpy ρ B) = 1/1 + 1/ρB)). It immediatey foows from Theorem 3 that 5) with β = 1 yieds the equivaent power aocation as minmax MSE and the optima sum MSE with w = xb) yb). B. Exact Soution to Weighted Sum MSE Minimization The existence of B aows us to deineate cases of 18) that can be soved optimay from the genera probem. The foowing resut gives the exact cosed-form soution to 18), which motivates a fast agorithm Agorithm 3 beow) to compute the optima soution. Theorem 4: If B exists, then the optima soution to 18) is given by p = I B)z 0, where z is given by z = w j B j z j j w j B j /z j 25) for a and satisfies 1 z 1 Bz = P. Now, 25) in Theorem 4 can be written in the form of z = Iz), where I is a homogeneous function. We wi everage the standard interference function resuts in [11] to propose the foowing step size free) agorithm that computes z in Theorem 4, and impicity, the optima transmit power of 18). Agorithm 3 Sum MSE Minimization): 1) Initiaize an arbitrariy sma ǫ > 0. 2) Update auxiiary variabe zk + 1): z k + 1) = w B j j z j k) j w j B + ǫ. 26) j /z j k) 3) Update pk + 1): SIR pk)) 1 + SIR pk)) z k + 1). 27) 4) Normaize pk+1): pk+1) pk+1) P/1 pk+ 1)). The foowing theorem shows that Agorithm 3 converges to the optima soution z and p in Theorem 4. Theorem 5: If B exists, for arbitrariy sma ǫ > 0, Agorithm 3 converges to the unique fixed point z and p in Theorem 4 from any initia point z0) under synchronous and asynchronous updates. Remark 4: Agorithm 3 requires a compexity of O 3 ) to compute B. Step 2 of Agorithm 3 can be impemented by distributed message passing. Transforming from zk + 1) to pk + 1) in 27) is performed ocay by each user, and the normaization at Step 4 is performed at the base station.

5 5 Remark 5: The convergence of Agorithm 3 everages the standard interference function approach in [11] by introducing ǫ in 26) of Agorithm 3. V. WEIGHTED SUM RATE MAXIMIZATION In this section, we consider the weighted sum rate of a users as a performance metric to be optimized. We then quantify the connection of the weighted sum rate maximization and the weighted sum MSE minimization. A. Exact Soution to Weighted Sum Rate Maximization The weighted sum rate maximization probem is given by maximize =1 w og1 + SIR p)) =1 p P, p 0, 28) We can rewrite 28) to be equivaent to =1 P P k G kp k +1 k G kp k +1 ) w =1 p P, p 0, 29) Simiar to Section IV, if B is the quasi-inverse of B, we can rewrite 29) as ) w Bz) =1 z z Bz), = 1,...,, 30) variabes: z, where z is given by 20). Simiar to Theorem 2, the foowing Theorem 6 gives the condition under which 28) is soved optimay. Theorem 6: If B exists, the optima soution to 28) is given by p = I+B) 1 z, where z is the optima soution of 30). Next, we connect the three optimization probems given in 28), 18) and 5) with β = 1. Remark 6: Appying the arithmetic-geometric mean inequaity to connect 21) and 30), we deduce that the weighted sum rate maximization has the same optima power as the weighted sum MSE minimization when w = xb) yb). Theorem 6 is used to give the foowing soution of 28). Theorem 7: If B exists, then the optima soution to 28) is given by p = I B)z 0, where z is given by z w = j w j B 31) j / Bz ) j for a and satisfies 1 z 1 Bz = P. As in the previous, 31) in Theorem 7 can be expressed as z = Iz), where I is a homogeneous function. Using the standard interference function approach in [11], the foowing agorithm computes the optima soution of 28). Agorithm 4 Sum Rate Maximization): 1) Initiaize an arbitrariy sma ǫ > 0. 2) Update auxiiary variabe zk + 1): w z k + 1) = j w j B + ǫ. 32) j / Bzk)) j 3) Update pk + 1): SIR pk)) 1 + SIR pk)) z k + 1). 33) 4) Normaize pk+1): pk+1) pk+1) P/1 pk+ 1)). The foowing resut shows that Agorithm 4 converges to the optima soution z and p in Theorem 7. Theorem 8: If B exists, for arbitrariy sma ǫ > 0, Agorithm 4 converges to the unique fixed point z and p in Theorem 7 from any initia point z0) under synchronous and asynchronous updates. ACKNOWEDGEMENT We acknowedge hepfu discussions with Steven ow at Catech and Kevin Tang at Corne. This research has been supported in part by ONR grant N and NSF CNS REFERENCES [1] V. D. Bonde,. Ninove, and P. Van Dooren. An affine eigenvaue probem on the nonnegative orthant. inear Agebra and its Appications, 404:69 84, [2] U. Krause. Concave Perron-Frobenius theory and appications. Noninear anaysis, ): , [3] C. W. Tan, M. Chiang, and R. Srikant. Fast agorithms and performance bounds for sum rate maximization in wireess networks. Proc. of IEEE Infocom, [4] M. Schubert and H. Boche. Soution of the mutiuser downink beamforming probem with individua SINR constraints. IEEE Trans. on Vehicuar Technoogy, 531):18 28, [5] F. Rashid-Farrokhi, K. J. R. iu, and. Tassiuas. Transmit beamforming and power contro for ceuar wireess systems. IEEE Journa on Seected Areas in Communications, 168): , [6] P. Viswanath and D. N. C. Tse. Sum capacity of the vector Gaussian broadcast channe and upink-downink duaity. IEEE Trans. on Information Theory, 498): , [7] W. Yu. Upink-downink duaity via minimax duaity. IEEE Trans. on Information Theory, 522): , [8] W. Yang and G. Xu. Optima downink power assignment for smart antenna systems. Proc. of IEEE ICASSP, [9] M. Chiang, P. Hande, T. an, and C. W. Tan. Power contro in wireess ceuar networks. Foundations and Trends in Networking, 24): , [10] M. Codreanu, A. Toi, M. Juntti, and M. atva-aho. Joint design of Tx-Rx beamformers in MIMO downink channe. IEEE Trans. on Signa Processing, 559): , [11] R. D. Yates. A framework for upink power contro in ceuar radio systems. IEEE Journa on Seected Areas in Communications, 137): , [12] C. W. Tan. Nonconvex power contro in mutiuser communication systems. Ph.D. Thesis, Princeton University, November [13] G. J. Foschini and Z. Mijanic. A simpe distributed autonomous power contro agorithm and its convergence. IEEE Trans. on Vehicuar Technoogy, 424): , [14] P. Viswanath, V. Anantharam, and D. N. C. Tse. Optima sequences, power contro, and user capacity of synchronous CDMA systems with inear MMSE mutiuser receivers. IEEE Trans. on Information Theory, 456): , [15] Y. K. Wong. Some mathematica concepts for inear economic modes. Chapter in Economic Activity Anaysis, Edited by O. Morgenstern, John Wiey & Sons, Inc.: , [16] S. Friedand and S. Karin. Some inequaities for the spectra radius of non-negative matrices and appications. Duke Mathematica Journa, 423): , 1975.