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Communications and Network, 2013, 5, 303-307 http://dx.doi.org/10.4236/cn.2013.53B2056 Published Online September 2013 (http://www.scirp.org/journal/cn) Copyright © 2013 SciRes. CN Resource Allocation for OFDMA-MIMO Relay Systems with Proportional Fairness Constraints Cuiru Zhao, Youming Li, Bin Chen, Zhao Wang, Jiongtao Wang Institute of Communication Technology, Ningbo University, Ningbo 315211, China Email: liyouming@nbu.edu.cn, pengpengyuagt@126.com Received July, 2013 ABSTRACT In this paper, we study resource allocation problem in orthogonal frequency division multiple access multiple-input multiple-output (OFDMA-MIMO) relay systems and formulate the optimal instantaneous resource allocation problem including subcarrier assignment, relay selection and power allocation to maximize system capacity. Based on the as- sumption that the availability of perfect channel state information (CSI) is known at the resource allocation controller, we propose a new resource allocation algorithm which can guarantee proportional fairness among users. In the proposed algorithm, a two-step suboptimal method is taken into account. Firstly, we assume equal power allocation for each user to linearize the problem and propose the subcarrier assignment and relay selection scheme based on equivalent channel gain. Secondly, we derive the closed-form expressions for power allocation through relaxing the proportional fairness constraints. Numerical simulations show that the proposed algorithm performs well in terms of satisfying proportional fairness among users in strict sense and achieving improvement on system total capacity. Keywords: Relay Network; OFDMA; MIMO; Proportional Fairness; Resource Allocation 1. Introduction The Orthogonal Frequency Division Multiple Access (OFDMA) is regarded as a leading candidate for the fourth generation (4G) wireless communication system because of its high spectral efficiency, flexible resource allocation and inherent robustness against frequency- selective fading. Furthermore, Multiple-Input Multiple- Output (MIMO) system has been extensively studied in recent years. Since the multiple antenna technology pro- vides extra spatial degrees of freedom, an OFDMA- MIMO system can improve transmission reliability and capacity without the need of increasing power or band- width. On the other hand, due to its potential benefits of enlarging the coverage of communication systems, in- creasing the capacity and enhancing the link reliability, cooperative relaying has attracted significant attention of many researchers. Recently, there have been many research efforts on improving the system throughput by resource allocation in OFDMA-MIMO relay system. Resource allocations of Amplify-and-Forward (AF) and Decode-and-Forward (DF) MIMO-OFDM relay systems are proposed in [1,2] respectively. But, both [1] and [2] only focus on a single user scenario. A number of results have been published on resource allocation for multi-user MIMO-OFDMA relay systems [3-5]. The optimal instantaneous resource allocation problem, including path selection, power allocation and sub-channel scheduling, is formulated in [3]. Optimal and suboptimal resource allocation algorithms for weighted sum rate maximization are presented in [4]. In [5], heterogeneous data rate requirements for delay sensi- tive and non-delay sensitive users are taken into consid- eration; the authors derive a distributed iterative resource allocation and scheduling algorithm with closed-form power and subcarrier allocation via employing dual de- composition. However, most existing research resource allocation in OFDMA-MIMO relay systems focus on maximizing the system capacity, fairness among multiple users has not received much attention. In most practical communica- tion system, the different business types of users have different rate, are endowed with different resource allo- cation priority. Therefore, to study the resource alloca- tion problem with proportional fairness constraint is es- sential for OFDMA-MIMO relay systems. In this paper, we investigate subcarrier assignment, relay selection and power allocation problem with pro- portional fairness constraint for OFDMA-MIMO relay assisted cellular system, and formulate the problem as a joint optimization problem. Since the problem is a NP-hard combination optimization problem with non- linear constraints, we use a two-step suboptimal method to solve it. Firstly, we assume equal power allocation for C. R. ZHAO ET AL. Copyright © 2013 SciRes. CN 304 each user to linearize the problem and propose the sub- carrier assignment and relay selection scheme based on equivalent channel gain. Secondly, we derive the closed- form expressions for power allocation through relaxing the proportional fairness constraints. Simulation results show that the proposed algorithm performs well in terms of satisfying proportional date rate constrains in strict sense and achieving improvement on system total rate. The rest of this paper is organized as follows. The OFDMA-MIMO relay assisted cellular communication system model and optimization problem formulation is described in Section Ⅱ. The proposed resource alloca- tion solution is presented in Section Ⅲ. In Section Ⅳ, simulations results are given and discussed. Finally, conclusions are drawn in section Ⅴ. 2. System Model We consider a OFDMA-MIMO relay assisted single cel- lular communication system as shown in Figure 1. In the adopted system, the overall bandwidth is B which is di- vided into N orthogonal subcarriers. A base station (BS) is located at the center, while K relay stations (RS) at the inner boundary, operating in decode-and-forward (DF) mode. M mobile stations (MS) are separated between the inner and the outer boundaries. The BS and the RSs are equipped with , B SRS NN antennas, respectively. While, each MS is equipped with only one antenna. In order to guarantee all users can receive signals from the base sta- tion, the system operates in a time division duplex (TDD) mode, where transmission takes place in two time slots. In the first time slot, each RS receives and decodes the signal from BS, then re-enc odes and forwards the signal to the relevant MSs during the second time slot. For sim- plicity, we assume that the subcarriers used in the first time slot and the second time slot are the same. It is fur- ther assumed that each subcarrier can be assigned to only link in each slot [6]. Let ,,, , , B Skn kmn HH represent the channel gain matrix of the link from BS to relay k on the subcarrier n and the link from relay k to user m on the subcarrier n, respec- tively. And the corresponding transmission power is ,,, , , B Skn kmn pp. ,, ,,,, 1,1 1,21, ,, ,,,, 2,12,22, ,, ,, ,,,, ,1 ,2, ,, ,,,, , ,1,11,21, BS BS RSRSRSBS RS BS k nBSk nBS k n N BS k nBSk nBS k n N BS k n BS k nBSk nBS k n NN NN kmn kmnkmn kmn N hh h hh h H hh h Hhh h The singular value decomposition of ,, B Skn H and ,,kmn H are given by ' ,,,, ,,,, 1 N H ii i BS knBS k nBSknBS kn i Hu ,,,,,, ,, H kmnkmnkmn kmn Hu where ' ,, (1, 2,...) i BS kniN and ,,kmn are the singular value of the matrix,, B Skn H, ,,kmn H, respectively. ', B SRS NNN is the rank of the matrix ,, ,, H B Skn BSkn HH. The instantaneous rate between BS and relay k on the subcarrier n can be written: ,,2,, ,, log 1 2 B SknBSkn BSkn B Rpg N Similarly, the instantaneous rate between relay k and user m on the subcarrier n in the second time slot is given by ,,2,, ,, log 1 2 kmnkmnkmn B Rpg N The achievable rate of user m assisted by the relay k on the subcarrier n is the minimum rate of ,, B Skn R and ,,kmn R[7] ,,,,, , min , n B SkmBSkn kmn RRR (1) where 2 1 ,, ,, 0/ BS k n BS k n gNB N and 2 ,, ,, 0/ kmn kmn gNB N are the main characteristic sub-channel gain of the link from BS to relay k on the subcarrier n and the link from relay k to user m on the subcarrier n, respectively. is the SNR gap related to BER, ln(5)/1.5BER and 0 N denotes the power spectral density of the Gaussian white noise. Define ,, 0,1 kmn as the joint user selection, path selection and subcarrier allocation indicator. ,, 1 kmn if and only if the subcarrier n is assigned to the user m and relayed by relay k. Thus the achievable data rate of the user m is ,, ,, 11 KN n mkmnBSkm kn RR . Therefore, the adaptive optimal resource allocation problem subject to total transmit power constraint and guaranteeing propor- tional fairness among users is expressed as follows MS BS 2 RS Figure 1. Single cellular OFDMA-MIMO relay system model. C. R. ZHAO ET AL. Copyright © 2013 SciRes. CN 305 ,, ,, 111 ,, ,, 11 ,, 11 ,, , 11 12 12 max : 1: {0,1} 2: 1 3: 4: 5 :::... ::: ... : KMN n kmnBSkm kmn kmn KM kmn km KN BS k nT kn MN kmn kT mn M M R subject to A A ApP ApP ARRR (2) where 1 A and 2 A emphasize that every subcarrier can be allocated at most one link in each slot. 3 A and 4 A are the total transmit power constraints for BS and RS, respectively. 5 A denotes the proportional data rate constraint. 3. Proposed Resource Allocation Algorithm It is difficult to solve the optimization problem in (2), because it includes both integer and continuous variables. In this section, we use a two-step suboptimal algorithm to reduce the computational complexity. In step one, the subcarrier allocation and relay selection under equal power allocation are discussed. In step two, the problem of power allocation is solved. 3.1. Subcarrier Allocation and Relay Selection Scheme In order to approaches the maximum rate in (1), the fol- lowing equation should be satisfied ,,,,, ,, , B Skn BSknkmnkmn pg pg So we obtain ,, ,, ,, ,, B SknBSkn kmn kmn pg pg (3) and the equivalent channel gain of link [8], ,,, , ,, ,,, , BSkn kmn n BS k m B Skn kmn gg ggg Thus (1) can be rewritten as ,,2,, ,, 2,,,, log 1 2 log 1 2 n B SkmBSkn BSkn kmnkmn B Rpg N Bpg N (4) The subcarrier allocation and relay selection algorithm is described as follows A) Determine the minimum number of subcarriers as- signed for each user 1 /,1 M mm m m NN mM B) Implement the subcarrier allocation and relay se- lection for each user a) Initialization {1,2 ,,},{1,2 ,,} , {1,2 ,,},0 , KN T Mm K N P MRp N b) While 0 m N 1) Find m satisfying arg min/ M mm mR 2) For the found m, find n* and * ksatisfying ** ** ,, , (, )argmax NK n B Skm nk nk g 3) Let ** * ,, 1, / NN kmn n , update * * ,, n B Skm R according to (3). C) The remaining subcarrier allocation a) While 0 N , for 1n to N and n doesn’t be used, find the user * m and relay * k satisfying ** ** ,, , (,) argmax MK n B Skm mk mk g b) Let ** ,, 1, / NN kmnn and update ** ,, n B Sk m R according to (3). 3.2. Proposed Power Allocation Scheme After the subcarrier allocation and relay selection, the next is to assign the available power on subcarriers. The optimization problem can be reformed as 2,,,, 1 ,, 11 1212 maxlog (1) 2 : 1: 2::: ...:::... : k K BS knBSkn knC KN BS k nT kn K K BpH N subject to BpP BRRR NNN (5) Formulate Lagrangian problem as following 00 0 0 0 ,, 2,,,, 11 ,, 11 2,,,, 1 2,,,, (,,) log (1) 2 () [log(1 ) 2 log (1)] 2 k k BS knk KN BS knBSkn kn KN BS kn kn K kBSknBSkn knC kk k BS k nBS kn nC k Lp BpH N pP BpH N NBpH NN C. R. ZHAO ET AL. Copyright © 2013 SciRes. CN 306 Then differentiate ,, ,, B Skn k Lp with respect to 0 ,, ,, ,, ,, ,, ,, ,, ,, ,, 12ln2 12ln2 0 BS k nk BS kn BS k n BS k nBS kn kBS kn kkBS knBS k n Lp p BH pH N NBH NpH N (6) From (6) we have ' '' ,, ,, ,, ,, ,, ,, 11 BS k nBS kn B Skn BSkn BS k nBS k n HH pH pH (7) Then equation (7) can be reformed as ,, ,,1 1 11 BS k nBS kn pp HH BS,k, BS,k, +- (8) The rate of link k 2,,1 2,, ,,1 1 log log 2k kkBSk BSkn nC BS k B RNp H NH According to the last constraint in (5), we can obtain: 00 0 0 2,,1 ,,1 2,,1 ,,1 1 log 1 log kBSk k BS k kBSk k BS k Np w H Np w H (9) where 2,, log k kBSkn nC wH , 0 k is a reference relay. From (9), we can derive 0 ,,1, ,1 B Skk BSkk papb (10) where 00 0 0 ,,1 ,,1 1 2, kkk k kk NwN w NN k kk B Sk BSk a ab HH Therefore, 0 1 ,,1 1 K Tkkk k BS kK kk k PNbe p Na (11) According to (8) and (11), we have 0 ,,, ,1 ,,1 ,, ,, ,, ,, ,, 11 BS knkBS kk B Sk BSkn BS k nBS k n kmn kmn papb HH pg pg (12) 4. Numerical Results and Analysis 4.1. Simulation Parameters In this section, we consider a single cellular OFDMA- MIMO relay communication system with a BS located in the center, RSs equally distributed at the inner boundary and MSs separated between the inner and the outer boundaries. Both BS and RS are equipped with multiple antennas, each MS is equipped with one antenna. We adopt the channel for simulation consists of six-path Rayleigh fading and the maximum doppler shift is 30Hz. The total bandwidth is set to be 1MHz. Assume that Gaussian white noise single PSD is 8 10 and BER is 3 10 . 4.2. Results and Analysis In our simulation, a heuristic algorithm, PAARS+EquPo and a static algorithm, Static+ EquPo are compared with the proposed algorithm. PAARS+EquPo means the subcarriers allocation and relay selection according to the proposed scheme, equal power allocation for each subcarrier, which is adopted in [6]. Static+ EquPo means each user is assigned fixed subcarriers and the power equally allocated to subcarriers. In the following, the performance of proposed algorithm is evaluated from two aspects: user fairness and system capacity. To evaluate user fairness, we first define the normal- ized capacity The normalized capacity ＝ m M m im R M R Figure 2 and Figure 3 depict the normalized capacity of each user with different subcarrier numbers, respec- tively. It is shown that, the normalized capacity of pro- posed algorithm is close to the original normalized fair- ness constraint. PAARS+EquPo algorithm can achieve better fairness among users when the number of subcar- riers is large. However, Static+ EquPo algorithm can’t obtain proportional fairness among users. Figure 4 shows the performance of system capacity with respect to the number of users. As shown in this figure, the capacity increases as the number of users in the system increases. This is due to multiuser diversity gain, which is more prominent in systems with larger number of users. Furthermore, the capacity using PAARS + EquPo algorithm is higher than that adopting the proposed algorithm, because in the proposed algo- rithm, we consider proportional fairness among users in subcarrier allocation, relay selection scheme, and power allocation scheme. The proposed algorithm outperforms Static+EquPo algorithm, because the proposed algorithm is an adaptive resource allocation algorithm which can adapt to different channel information. C. R. ZHAO ET AL. Copyright © 2013 SciRes. CN 307 12345678910 1112 0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 User i ndex Normalized capacity PF PSARS+EquPo Propos ed Stat ic+E qupo Figure 2. Normalized capacity versus user index after 50 iterations, the number of subcarriers is 1024. 1 234 5 6 78910 11 12 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 User index Normalized capacity PF PSARS+EquPo Propos ed St at i c + EquP o Figure 3. Normalized capacity versus user index after 50 iterations, the number of subcarriers is 2048. 234567891011 12 4.1 4.12 4.14 4.16 4.18 4.2 4.22 4.24 4.26 4.28 Number of us ers System capacity(Mbit/s/Hz) Proposed al gori t hm PSARA+EquPo St at i c + E quP o Figure 4. System capacity versus number of users after 50 iterations. 5. Conclusions In this paper, we propose a new resource allocation algo- rithm for OFDMA-MIMO relay assisted single cellular communication system, which consider proportional fairness among users. The performance of the proposed algorithm is compared with other algorithms and simula- tion results demonstrate that he proposed algorithm per- forms well in terms of satisfying proportional fairness among users in strict sense and achieving improvement on system total capacity. 6. Acknowledgements This work was supported in part by the National Science Foundation of China (61071119), the Ningbo Natural Science Foundation (2012A610017), and the Innovation Team of Ningbo (2011B81002). REFERENCES [1] I. Hammerström and A. Wittneben, “Power Allocation Schemes for Amplify-and-forward MIMO-OFDM Relay Links,” IEEE Transactions on Wireless Communication, Vol. 6, No. 8, 2007, pp. 2798-2802. [2] T. Li, H. Yu and S. Liu, “Adaptive Power Allocation for Decode-and-forward MIMO-OFDM Relay System,” Proceedings of the 5th International Conference on Wire- less Communications, Beijing, 24-26 September 2009, pp. 1-4. [3] L. J. Zu, Y. S. Ji, L.P. Wang, L. Zhong, F. Q. Liu, P. Wang and J. Xu, “Joint Optimization in Multi-user MIMO-OFDMA Relay-enhanced Cellular Networks,” IEEE Wireless Communication and Networking Confer- ence, Cancunications, 28-31 March 2011, pp. 13-18. [4] Y. B. Lin, W. H. Wu and Y. T. 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