SOCP Approaches to Joint Subcarrier Allocation and Precoder Design for Downlink OFDMA Systems

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1 SOCP Approaches to Joint Subcarrier Allocation and Precoder Design for Downlink OFDMA Systems Dan Nguyen, Le-Nam Tran, Pekka Pirinen, and Matti Latva-aho Centre for Wireless Communications and Dept. Commun. Eng., University of Oulu, Oulu, Finland Abstract We study the joint subcarrier allocation and precoder design (JSAPD) problem to the sum rate of downlink orthogonal frequency division multiple access (OFDMA) systems under a sum power constraint. Naturally, this problem belongs to a class of combinatorial optimization problems which are difficult to solve in general. Based on the concept of big-m formulation, and by exploiting its specific structure, we can transform the JSAPD problem into a mixed integer second order cone program (MI-SOCP), which then offers two advantages. Firstly, when the number of subcarriers/users is small, the design problem can be solved to global optimum in reasonable time by dedicated solvers. Secondly, when the number of subcarriers/users is large, near-optimal solutions of the JSAPD problem can be found by considering the continuous convex relaxation of the MI-SOCP. Numerical experiments are carried out to demonstrate the improved performance of the proposed designs compared to known solutions. I. INTRODUCTION Orthogonal frequency division multiple access (OFDMA) and multiple input multiple output (MIMO) techniques are vital solutions to satisfying explosive need for high speed data rate services. In fact, these techniques are the core components in many advanced wireless communications standards such as LTE [1] and WIMAX [2]. The main benefits of OFDMA are the capabilities of not only overcoming frequency selective fading caused by time dispersive channels, but also allowing simultaneous transmissions from multiple users over the disparate subset of subcarriers [3], [4]. On the other hand, MIMO communications takes advantage of multipath transmission brought by multiple antennas at transceivers to achieve spatial multiplexing gain and diversity gain [5], [6]. Thus, the combination of these two technologies certainly results in multifold improvements in link reliability as well as system capacity. In order to fully exploit frequency and multiuser diversities, radio resources such as subcarrier and power should be appropriately allocated to users since fading conditions of a channel vary from one user to another. In the literature, radio resource allocation (RRA) problems in OFDMA systems have been intensively studied under some performance criteria, e.g., power minimization [7], [8] and sum rate maximization [3], This research work has been funded from the Academy of Finland under grant agreement n project Juliet. [4], [9], [10], etc. These problems are generally formulated as combinatorial optimization ones, which are usually intractable and difficult to solve. Hence, many heuristic RRA algorithms have been proposed. For example, the study in [11] presented two heuristic subcarrier algorithms, called maximum-eigencriterion and product-criterion, but only efficient in low and high signal to noise ratio (SNR) regions, respectively. In [4], the authors proposed a two step RRA method, in which the first step is subcarrier assignment where each subcarrier is exclusively given to one user with the highest channel gain, and the second step is the transmit power allocation. Alternatively, the works in [3], [8] [10] employed convexification techniques and optimization tools, e.g., the dual decomposition, to find the solulions of the RRA problem. In this paper, we propose efficient algorithms for the joint subcarrier allocation and precoder design (JSAPD) problem aiming at maximizing the sum rate of an OFDMA wireless communications system. In particular, we first formulate the JSAPD problem of interest as a generic mixed integer convex programing (MICP) based on the big-m concept [12], usually employed in mixed integer programming. By exploiting its specific structure, we then arrive at a mixed integer second order cone program (MI-SOCP), which exhibits two advantages. Firstly, when the problem size, i.e., the number of subcarriers and/or users, is small, the MI-SOCP can be solved optimally in acceptable time using some powerfull optimization solvers such as CPLEX [13], MOSEK [14] or GUROBI [15]. Secondly, to deal with problems of larger size and take advantages of many state-of-the-art SOCP solvers, the resulting MI-SOCP can be easily reduced to SOCP by allowing the binary variables to take any value on the interval [0,1]. To evaluate the effectiveness of our proposed algorithms, we conduct numerical experiments under an outdoor to indoor scenario following IMT-2000 recommendations [16]. The numerical results demonstrate that the solutions of the relaxed SOCP based scheme are close to optimal ones achieved by the MI-SOCP method, and better than known solutions, reported in [11]. The rest of the paper is organized as follows. Section II presents the system model and formulates the considered sum rate problem. The proposed methods are detailed in Section III, followed by simulation results in Section IV. Finally, 1271

2 conclusions are drawn in Section V. Notation: Standard notations are used in this paper. Bold lower and upper case letters represent vectors and matrices, respectively; H H and H T are Hermitian and standard transpose of H, respectively; tr(h) and H are the trace and determinant of H, respectively; H 0 means that H is a positive semidefinite matrix; rank(h) is the rank of H. II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model We consider a single hop multiuser MIMO OFDMA wireless system in which a serving access point (AP) installed with L antennas simultaneously communicates with K users in a downlink channel, each equipped with M k antennas. Let B denote the total system bandwidth which is equally divided into N mutually orthogonal subcarriers, each with a bandwidth W = B/N. Throughout the paper, we assume that guard interval is large enough so that inter-symbol interference (ISI) and inter-carrier interference (ICI) are ignored, and channel state information (CSI) is perfectly known at the both ends of the system. For OFDMA systems, since each subcarrier is allocated to only one user at a time, the received signal of user k on subcarrier n is written as y k,n = H k,n s k,n +n k,n (1) where s k,n represents the L 1 vector of OFDM symbol transmitted by the AP to user k on the subcarrier n; H k,n is the M k L complex channel matrix and n k,n is a background noise vector, each element assumed to be zero mean additive white Gaussian (AWGN) with variance 2. Let E([s k,n s H k,n ]) = Q k,n for k = 1,...,K and n = 1,...,N, the sum rate of user k on the subcarrier n is achieved by R k,n = log 2 I+ Γ 2 H k,n Q k,n H H k,n (2) where Γ is a signal to noise ratio gap related to practical modulation and coding scheme. For this issue, the interested reader is referred to [4] for more details. B. Problem Formulation As we know different users undergo different channel gains on each subcarrier. Therefore, how to assign subcarriers and design precoders for users is an important task in OFDMA systems since this greatly impacts on the performance of each user as well as the total system. Herein, we aim at jointly designing the subcarrier allocation and precoders to the sum rate of the downlink OFDMA wireless system. For the purpose of problem formulation, let us define c k,n to be a binary variable that reflexes the selection status of user k over subcarrier n. Precisely, c k,n = 1 if subcarrier n is allocated to user k, and c k,n = 0, otherwise. The selection problem imposes that if c k,n = 0, then tr(q k,n ) = 0. With the introduction of binary variables c k,n, we can formulate the JSAPD problem as Q k,n 0,c k,n K N R k,n K N tr(q k,n ) P max, tr(q k,n ) c k,n P max, k, n, c k,n {0,1}, k, n, K c k,n = 1, n k=1 (3a) (3b) (3c) (3d) (3e) where the constraint in (3b) is the sum power constraint at the AP with a maximum transmit powerp max, the ones in (3d) and (3e) ensure that each subcarrier is shared by only one user. In (3c), we have used the concept of big-m formulation where tr(q k,n ) should be limited by its bound (possibly maximum value) when c k,n = 1. From (3b), it is clear that tr(q k,n ) is bounded by P max, and therefore leading to (3c). We can see that the objective in (3a) is concave with respect to Q k,n, and all the constraints, except (3d), are convex. In fact, the problem (3) is a mixed integer convex programming (MICP) which is hard to solve in general. A global solution to MICPs can be found using branch-and-bound methods where lower and upper bounds are computed based on the convex part of the problem. Hence, the difficulty in solving MICPs will significantly reduced if we can transform the convex part of (3) into a popular standard convex program such as linear program, second order cone program (SOCP). In the following section, we will show how to reformulate (3) into a mixed integer SOCP (MI-SOCP) to take advantages of efficient current MI-SOCP solvers. III. SOCP BASED METHODS To begin with, we denote a compact singular value decomposition ofh k,n, i.e.,h k,n = U k,n Σ k,n Vk,n H whereu k,n and V k,n are unitary matrices and Σ k,n is a diagonal matrix with positive singular values {δ k,n,1,...,δ k,n,rk,n } in which = rank(h H k,n H k,n). Then, letting Q k,n = Vk,n H Q k,nv k,n and noting that tr( Q k,n ) = tr(vk,n H Q k,nv k,n ) = tr(q k,n V k,n Vk,n H ) = tr(q k,n), we can equivalently express (3) as Q k,n 0,c k,n K N log 2 I+ Γ K N tr( Q k,n ) P max, Σ 2 k,n Q k,n tr( Q k,n ) c k,n P max, k, n, (3d) (3e). (4a) (4b) (4c) (4d) The key to arriving at an MI-SOCP relies on the observation that Q k,n must be diagonal to be an optimal solution. This conclusion is due to the Hadamard s inequality, i.e., if X = [x ij ] is a positive semidefinite matrix with dimension 1272

3 n n, X n x ii and the equality holds if and only if X is a diagonal matrix. In this way, we can write Q k,n = diag{q k,n,1,...,q k,n,rk,n } and transform (4) into q k,n,i 0,c k,n r K N k,n ( log 2 1+ Γ K N q k,n,i P max, q k,n,i c k,n P max, k, n, (3d) (3e). δ 2 k,n,i q k,n,i ) (5a) (5b) (5c) (5d) Now to proceed further, we observe that log 2 x with x > 0 is monotonically increasing, and thus the problem in (5) is equivalent to r K N k,n ( 1+ Γ ) c k,n σ 2 δk,n,i 2 q k,n,i (6a) k,n (3d) (3e) (5b) (5c) (6b) which can be boiled down to r K N k,n c k,n q k,n,i 0, k, n, i t k,n,i (6c) (7a) 1+ Γ 2 δk,n,iq 2 k,n,i t k,n,i, k, n, i, (7b) (3d) (3e) (5b) (5c) (7c) q k,n,i 0, k, n, i (7d) by using the epigraph form of (6). We note that since all constraints (7b)-(7d) are linear with mixed binary and continuous variables, and the objective function of (7a) can be transformed to a set of SOC constraints (see [17] and also [18]). Thus, the resulting problem in (7) is a mixed integer second order cone program (MI-SOCP) which can be optimally solved using existing powerful solvers such as CPEX, GUROBI or MOSEK. In particular, these solvers can efficiently solve MI- SOCP problems of small size. In our numerical experiments, when the number of subcarriers N is set to 8 and the number of users K is equal to 4, the MI-SOCP solvers mentioned above can solve (7) within a few seconds. This scenario, albeit of small sizes, is interesting from a practical perspective. For example, the AP is serving several users and left with a small number of subcarriers. If there are a few incoming users, the AP can solve the MI-SOCP in (7) to allocate the unoccupied subcarriers to the new users. That is, the AP does not necessarily solve the MI-SOCP for all number of subcarriers and users. For cases where the number of available subcarriers is large, e.g., up to 64, 512 or 1024, solving the JSAPD problem using MI-SOCP solvers is not a practical option and it merely serves as a theoretical benchmark for comparison purpose. Thus, we have to resort to suboptimal solutions. For this purpose, we first relax the binary variables in constraints (5c) and (5d) to the continuous ones, i.e., c k,n is allowed to take a value on the interval [0,1]. Let c k,n denote the continuous relaxed variable of c k,n for ease of description. Then, the continuous relaxation of (7) is q k,n,i 0, c k,n K r N k,n 1+ Γ 2 t k,n,i (8a) δ 2 k,n,i q k,n,i t k,n,i, k, n, i, (8b) K N q k,n,i P max, q k,n,i c k,n P max, k, n, c k,n [0,1], k, n, K c k,n = 1, n. k=1 (8c) (8d) (8e) (8f) Thanks to recent advances in mathematical methods for solving SOCPs, the problem in (8) can be solved very efficiently even for a large number of subcarriers and users. After obtaining the optimal solutions of qk,n,i and c k,n of (8), we still need to determine the subcarrier allocation for the model in (5). We notice that since one subcarrier is exclusively assigned to one user in an OFDMA system, it is reasonable to select the user with a value of c k,n closest to 1 for a particular subcarrier n. Hence, we propose the following mapping rule between c k,n and c k,n c k,n = 1 if k = argmax c i,n n = 1,...,N 0 otherwise. Once c k,n is determined, the remaining problem is to compute precoders{q k,n } of the original problem in (3), which can be easily found through solving (5) with the known {c k,n }. To be specific, let K = {k c k,n = 1} and N = {n c k,n = 1}, then the problem (5) reduces to q k,n,i 0 k K,n N k K,n N i r k,n ( log 2 1+ Γ q k,n,i P max. (9) ) δk,n,iq 2 k,n,i (10a) (10b) We have omitted (5c) since it is automatically satisfied due to (10b) as mentioned previously. Now the problem in (10) is a standard power allocation problem with a total power constraint which can be solved in closed-form expressions via the water-filling algorithm as [ ] qk,n,i = 1 λ σ2 k,n δk,n,i 2 (11)

4 where λ is a water level chosen so that r k,n [ ] 1 λ σ2 k,n δk,n,i 2 = P max (12) k K,n N and [x] + = max(0,x). Finally, precoders of users are determined by Q k,n = V k,n diag{q k,n,1,...,q k,n, }V H k,n, k, n (13) due to Q k,n = V H k,n Q k,nv k,n as defined above. In summary, the proposed SOCP based algorithm is outlined as follows Algorithm 1 The SOCP based algorithm 1: Solve the relaxed SOCP problem in (8) to find optimal solutions of { c k,n } and {q k,n,i }. 2: Determine subcarrier allocation {c k,n } based on { c k,n } for the considered original problem using (9). 3: Find power allocation using (11) and (12). 4: Calculate precoders for users using (13). We have presented a joint design for subcarrier allocation and precoding for downlink OFDMA systems. It is worth mentioning that our proposed algorithms also apply to the uplink channel by slightly modifying the sum power constraint in (3b) as N n=1 tr(q k,n) P k,max, k = 1,...,K, i.e., peruser power constraints. + IV. NUMERICAL RESULTS This section numerically evaluates the proposed SOCP based designs. Particularly, we compare the proposed algorithms with the heuristic schemes introduced in [11] which consist of two sequential steps: subcarrier allocation and precoder design. In the first step (i.e, subcarrier allocation), assigning a specific subcarrier n to user k is implemented by maximum eigenvalue criterion (MEC) which is claimed to be the optimal subcarrrier assignment scheme in low SNR region, and product of eigenvalues criterion (PEC) which is suitable for high SNR region. Specifically, in MEC scheme, user k with the highest eigenvalue of H H k,n H k,n is chosen for transmission on subcarrier n while, in PEC method, subcarrier n is allocated to user k with the highest product of eigenvalues of H H k,n H k,n. In the second step, precoders are determined through (11), (12) and (13). We consider an exemplary scenario for an outdoor to indoor pedestrian environment including one AP with L = 2 antennas serving K = 2 users, each equipped with M k = 2 antennas. The distances between the users and AP are set to 50 m. The basic simulation parameters are listed in Table I. The small scale fading of the channel link from a transmit antenna to a receive one is modeled as multipath Rayleigh fading with the power delay profile, taken from IMT-2000 specifications [16] (see Table I). For the large scale fading, all channel links experience the outdoor to indoor pedestrian path loss model in [16] as PL = 40log 10 d km +30log 10 f MHz +49 where f MHz is Relative sum rate (%) System parameters Carrier frequency System bandwidth Sampling frequency Relative path delay Relative average path power Thermal noise Receiver noise figure TABLE I SIMULATION PARAMETERS 2000 MHz 5 MHz 5.6 MHz [0; 200; 800; 1200; 2300; 3700] ns [0; 0.9; 4.9; 8.0; 7.8; 23.9] db 174 dbm/hz 9 db MEC method PEC method Proposed SOCP method Maximum power at an access point, P max (dbm) Fig. 1. Relative sum rate (%) versus maximum transmit power at an access point, P max (dbm). In this setup, the numbers of users and subcarriers are set to K = 4 and N = 8, respectively. carrier frequency in MHz and d km is the distance in kilometers between the AP and a specific user. In Fig. 1, we plot the relative sum rates of the considered schemes (the proposed relaxed SOCP, MEC, and PEC) over the MI-SOCP in percentage versus the maximum transmit power of the AP. Each point in the curve is calculated as (A B) 100 /B where A is the sum rate of the relaxed SOCP/MEC/PEC and B is the one of the optimal design obtained by solving MI-SOCP in (7). In this simulation setup, the number of users is set to K = 4 and the system bandwidth is divided into N = 8 orthogonal subcarriers. For our methods, the MOSEK solver is used to solve the MI-SOCP in (7) and the continuous relaxation in (8). It is clearly observed that the SOCP based scheme achieves the sum rate close to the optimal one of the MI-SOCP for a wide range of trasmit power at the AP. In contrast, the MEC algorithm is only effective in the low SNR region whereas the PEC is in the high SNR region. Particularly, the sum rates of the relaxed SOCP and MEC methods reach the optimal value as P max < 0 dbm while at P max 12 dbm the relaxed SOCP yields a slightly smaller sum rate than the PEC (it is proved in [11] that the PEC algorithm is optimal in high SNR regime). The reason is that our proposed method is jointly designed subcarrier allocation and precoding scheme for the users instead of the two-stage design of the PEC and MEC. Fig. 2 compares the sum rate of the relaxed SOCP design with those of the PEC and MEC schemes as a function of 1274

5 Downlink sum rate (Mbits/s) MEC method PEC method Proposed SOCP method Number of users, K Fig. 2. Downlink sum rate (Mbits/s) versus a number of users. The maximum transmit power of the AP is fixed at 6 dbm, and the number of subcarrier is equal to 64. number of users, K. In this setting, we fix the maximum transmit power atp max = 6 dbm and the number of subcarriers at N = 64. For this scenario, it takes very long time to solve (7) by any MI-SOCP solver, and thus we do not report the optimal sum rate in this setting. As can be seen in Fig. 2, the performances of all methods increase with the number of users, meaning that all methods can exploit multiuser diversity. However, our SOCP based design is superior to the other two ones. In particularly, the sum rates of the relaxed SOCP and MEC methods are equal and better than that of the PEC one at K = 2. At K = 12, the relaxed design and PEC scheme nearly produce the same performance and they all have a better sum rate than the MEC design. Note that the different behaviors of the MEC and PEC methods are due to heuristic assumptions of the sum rate function in low and high SNR regimes, respectively. V. CONCLUSIONS In this paper, we have developed efficient radio resource management schemes which jointly consider subcarrier allocation and precoding design to the sum rate of a downlink OFDMA wireless system a sum power constraint at a serving access point. Specifically, based on the notion of big-m formulation and singular value decomposition, we have transformed the JSAPD problem into an MI-SOCP which can be solved efficiently by several dedicated solvers when the problem size is not so large. For problem of large dimensions, we have proposed a convex approximation for the MI-SOCP of interest by relaxing the binary variables to be continuous ones on the interval [0,1]. Our relaxation method is different from existing heuristic schemes which attempt to approximate the sum rate in a specific SNR regime. The numerical results demonstrate that our proposed SOCP based design is superior to the MEC and PEC algorithms, and its performance is close to the one of the proposed MI-SOCP design, which is the optimal one. REFERENCES [1] 3GPP Technical Specification Group Radio Access Network, Evolved Universal Terrestrial Radio Access (E-UTRA): Physical channels and Modulation, 3GPP Std. TS v9.1.0, [2] IEEE Broadband Wireless Access Working Group, IEEE m System Description Document (SDD), Dec [Online]. Available: [3] G. Fang, Y. Sun, J. Zhou, J. Shi, Z. Li, and E. Dutkiewicz, Subcarrier allocation for OFDMA wireless channels using lagrangian relaxation methods, in Proc. IEEE GLOBECOM 06, 2006, pp [4] J. Jang and K.-B. Lee, Transmit power adaptation for multiuser OFDM systems, IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp , [5] L.-N. Tran, M. Bengtsson, and B. Ottersten, Iterative precoder design and user scheduling for block-diagonalized systems, IEEE Trans. Signal Process., vol. 60, no. 7, pp , Jul [6] L.-N. Tran and E.-K. Hong, Multiuser diversity for successive zeroforcing dirty paper coding: Greedy scheduling algorithms and asymptotic performance analysis, IEEE Trans. Signal Process., vol. 58, no. 6, pp , Jun [7] J. M. Choi, J. S. Kwak, H. S. Kim, and J. H. Lee, Adaptive subcarrier, bit, and power allocation algorithm for MIMO-OFDMA system, in Proc. IEEE VTC 2004-Spring, vol. 3, 2004, pp [8] C. Y. Wong, R. Cheng, K. Lataief, and R. Murch, Multiuser OFDM with adaptive subcarrier, bit, and power allocation, IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp , [9] M. Awad, V. Mahinthan, M. Mehrjoo, X. Shen, and J. W. Mark, A dualdecomposition-based resource allocation for OFDMA networks with imperfect CSI, IEEE Trans. Veh. Technol., vol. 59, no. 5, pp , [10] W. Rhee and J. Cioffi, Increase in capacity of multiuser OFDM system using dynamic subchannel allocation, in Proc. IEEE VTC 2000-Spring, vol. 2, 2000, pp vol.2. [11] H. Li and H. Liu, On the optimality of OFDMA MIMO channels, in Proc. Asilomar Conf. Signals Syst. Comput. (ASILOMAR), 2006, pp [12] J. Lee and S. Leyffer, Mixed Integer Nonlinear Programming, ser. The IMA Volumes in Mathematics and its Applications. Springer New York, 2011, vol [13] IBM ILOG CPLEX Optimizer Version l v12.4. [Online]. Available: [14] MOSEK Optimization Software Version 7.0. [Online]. Available: [15] Gurobi Optimizer Version 5.1. Houston, Texas: Gurobi Optimization, Mar., April [Online]. Available: [16] ITU document, Rec. ITU-R M Guidelines for Evaluation of Radio Transmission Technologies for IMT-2000, 1997, (question ITU- R 39/8). [17] F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, vol. 95, pp. 3 51, [18] L.-N. Tran, M. F. Hanif, A. Tölli, and M. Juntti, Fast converging algorithm for weighted sum rate maximization in multicell MISO downlink, IEEE Signal Process. Lett., vol. 19, no. 12, pp , Dec