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IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 5, MAY 2010

411

User Selection and Resource Allocation Algorithm with Fairness in MISO-OFDMA Vasileios D. Papoutsis, Ioannis G. Fraimis, and Stavros A. Kotsopoulos

Abstract—The problem of user selection and resource allocation for the downlink of wireless systems operating over a frequency-selective channel is investigated. It is assumed that the Base Station (BS) uses many antennas, whereas a single antenna is available to each user. A suboptimal, but efficient algorithm is devised that is based on Zero Forcing (ZF) beamforming and on spatial correlation. The algorithm maximizes the sum of the users’ data rates subject to constraints on total available power and proportional fairness among users’ data rates. The main feature of it is the proportionality among users’ data rates which is guaranteed as it is shown by simulations.

between the BS and user 𝑘 in subcarrier 𝑛. Thus, for each subcarrier 𝑛, the baseband equivalent model for the system can be written as y𝑛 = H𝑛 x𝑛 + z𝑛 ,

(1)

FDMA is a scheme that helps exploit multiuser diversity in frequency-selective channels [1] while MIMO technology offers significant increase in the data throughput [2] without additional bandwidth or transmit power requirements. Zf technique has been proposed to remove co-channel interuser interference [3]. By combining OFDMA with MIMO transmission, wireless systems can offer larger system capacities and improved reliability. In general, in order to transmit on the boundary of the capacity region, the BS needs to transmit to multiple users simultaneously in each subchannel employing Dirty Paper Coding (DPC) [2]. However, DPC has large implementation complexity. In [4]-[6], user selection and beamforming algorithms, that are based on ZF [3], are proposed in order to maximize the system capacity without guaranteeing any kind of fairness among the users’ data rates. In [7], a kind of fairness is supported. In this paper, a user selection and resource allocation algorithm for multiuser downlink MISO-OFDMA is developed that incorporates fairness by imposing proportional constraints [1] among the users’ data rates. The beamforming scheme of [4] [6] is applied in each subcarrier but the user selection procedure takes fairness into account.

where H𝑛 = [h1,𝑛 h2,𝑛 . . . h𝐾,𝑛 ]𝑇 is a 𝐾 × 𝑇 matrix with complex entries, x𝑛 is the 𝑇 × 1 transmitted signal vector in subcarrier 𝑛, y𝑛 is a 𝐾 × 1 vector containing the received signal of each user, and z𝑛 is a 𝐾 × 1 noise vector. The practically important case where 𝐾 ≥ 𝑇 is considered. Hence, rank(H𝑛 ) = min(𝑇, 𝐾) = 𝑇 . The total transmitted power is 𝑃𝑡𝑜𝑡 and equal power is allocated to each subcarrier. ∗ 𝑡𝑜𝑡 , where C𝑛 = 𝔼[x𝑛 (x𝑛 ) ] is the Hence, trace[C𝑛 ] ≤ 𝑃𝑁 covariance matrix of the transmitted signal x𝑛 . Using only transmit beamforming, the following model 1 2 𝑇 𝑇 𝑤𝑘,𝑛 . . . 𝑤𝑘,𝑛 ] be the is obtained. Let w𝑘,𝑛 = [𝑤𝑘,𝑛 𝑇 × 1 beamforming weight vector for user 𝑘 in subcarrier 𝑛. Then, the baseband model (1) can be written as y𝑛 = H𝑛 W𝑛 D𝑛 s𝑛 + z𝑛 , where W𝑛 = [w1,𝑛 w2,𝑛 . . . w𝐾,𝑛 ] is the 𝑇 × 𝐾 beamforming weight matrix, s𝑛 is a 𝐾 × 1 vector containing the transmitting signals, and D𝑛 = √ √ √ diag( 𝑝1,𝑛 , 𝑝2,𝑛 , . . . , 𝑝𝐾,𝑛 ) is the power distribution to subcarrier 𝑛 among the 𝐾 users. Using ZF beamforming it is possible to encode users individually, and with smaller complexity compared to DPC [2]. If 𝐾 ≤ 𝑇 and rank(H𝑛 ) = 𝐾, the ZF beamforming matrix is W𝑛 = H∗𝑛 (H𝑛 H∗𝑛 )−1 . However, if 𝐾 > 𝑇 , it is not possible to use it because H𝑛 H∗𝑛 is singular. In that case it is necessary to select 𝑡 ≤ 𝑇 out of 𝐾 users in each subcarrier. Hence, there are 𝐼 possible combinations of users transmitting in the same subcarrier denoted as 𝐴𝑖 , where 𝐴𝑖 ⊂ {1, 2, . . . , 𝐾}, 0 < ∣𝐴𝑖 ∣ ≤ ( 𝑇 , ∣𝐴 ) 𝑖 ∣ denotes the cardinality of set 𝐴𝑖 , and ∑𝑇 𝐾 𝐼 = 𝑙=1 . 𝑙 Let a set of users 𝐴𝑖 = {𝑠1 , . . . , 𝑠𝑡 }, in each subcarrier, such that H𝑛 (𝐴𝑖 ) = [h𝑠1 ,𝑛 h𝑠2 ,𝑛 . . . h𝑠𝑡 ,𝑛 ]𝑇 . When ZF is used, the data rate of user 𝑘 ∈ 𝐴𝑖 in subcarrier 𝑛 is [2]

II. S YSTEM M ODEL AND P ROBLEM F ORMULATION

𝑟𝑘,𝑖,𝑛 = log2 (𝜇𝑛 𝑐𝑘,𝑛 (𝐴𝑖 )),

Consider an OFDMA downlink transmission with 𝑁 subcarriers, 𝑇 antennas at the BS and 𝐾 active users, each equipped with a single receive antenna. Also, let 𝐵 be the overall available bandwidth, and h𝑘,𝑛 = [ℎ1𝑘,𝑛 . . . ℎ𝑇𝑘,𝑛 ]𝑇 be the 𝑇 × 1 baseband equivalent gain vector of the channel

where 𝑐𝑘,𝑛 (𝐴𝑖 ) = {[(H𝑛 (𝐴𝑖 )H𝑛 (𝐴𝑖 )∗ )−1 ]𝑘,𝑘 }−1 , and 𝜇 is obtained by solving the water-filling equation [4] [ ] ∑𝑛 𝑃𝑡𝑜𝑡 1 𝜇 − = 𝑛 𝑘∈𝐴𝑖 𝑐𝑘,𝑛 (𝐴𝑖 ) + 𝑁 . The power loading then ] [ yields 𝑝𝑘,𝑖,𝑛 = 𝑐𝑘,𝑛 (𝐴𝑖 ) 𝜇𝑛 − 𝑐𝑘,𝑛1(𝐴𝑖 ) , ∀𝑘 ∈ 𝐴𝑖 . + By applying the conclusions above, the linear beamforming optimization problem can be formulated as

Index Terms—MISO, OFDMA, resource allocation, ZeroForcing, proportional fairness.

O

I. I NTRODUCTION

Manuscript received January 10, 2010. The associate editor coordinating the review of this letter and approving it for publication was H. Shafiee. The authors are with the Wireless Telecommunications Laboratory, Department of Electrical and Computer Engineering, University of Patras, Greece (e-mail: {vpapoutsis, ifraimhs, kotsop}@ece.upatras.gr). Digital Object Identifier 10.1109/LCOMM.2010.05.100044

max

𝜌𝑘,𝑖,𝑛 ,𝑝𝑘,𝑖,𝑛

c 2010 IEEE 1089-7798/10$25.00 ⃝

𝐾 𝑁 𝐼 𝐵 ∑∑∑ 𝜌𝑘,𝑖,𝑛 𝑟𝑘,𝑖,𝑛 𝑁 𝑛=1 𝑖=1 𝑘=1

(2)

(3)

IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 5, MAY 2010

𝑅𝑘 =

𝑁 𝐼 𝐵 ∑∑ 𝜌𝑘,𝑖,𝑛 𝑟𝑘,𝑖,𝑛 𝑁 𝑛=1 𝑖=1

(4)

and {𝛾𝑘 }𝐾 𝑘=1 are the proportional data rate constraints. The problem above (3) is an NP-hard combinatorial optimization problem with non-linear constraints. The optimal solution can be obtained by exhaustive search of all possible user assignment sets in all subcarriers but the complexity is given by 𝐼 𝑁 , which is extremely complicated even for moderate 𝐾, 𝑁 . In the following, a suboptimal user selection and resource allocation algorithm is proposed. III. T HE P ROPOSED R ESOURCE A LLOCATION A LGORITHM The proposed algorithm is based on ZF beamforming and on spatial correlation [6] between different users, denoting 𝜂𝑙,𝑚 =

∣(h𝑙,𝑛 )∗ h𝑚,𝑛 ∣ , 0 ≤ 𝜂𝑙,𝑚 ≤ 1, ∥ h𝑙,𝑛 ∥∥ h𝑚,𝑛 ∥

(5)

the spatial correlation between users 𝑙, 𝑚 in subcarrier 𝑛. The larger the 𝜂𝑙,𝑚 is, the more power is required to eliminate the interference between users 𝑙, 𝑚, and the less sum data rate is achieved. The proposed algorithm comprises the following steps 𝐾 ∙ Set 𝒮 = {1 . . . 𝑁 }, 𝑅𝑘 = 0, {𝛾𝑘 }𝑘=1 , ∀𝑘 = 1, . . . , 𝐾, 𝜌𝑘,𝑖,𝑛 = 0, ∀𝑘 = 1, . . . , 𝐾, 𝑖 = 1, 2, . . . , 𝐼, and 𝑛 ∈ 𝒮. ∙ While ∣𝒮∣ ∕= ∅: – Set 𝒰 = {1, . . . , 𝐾}, ∣𝐴𝑖 ∣ = ∅. 𝑅𝑖 𝑘 – Find user 𝑘 satisfying 𝑅 𝛾𝑘 ≤ 𝛾𝑖 ∀𝑖, 1 ≤ 𝑖 ≤ 𝐾. – Find subcarrier 𝑛 = arg max𝑗∈𝒮 ∥ h𝑘,𝑗 ∥. – Set 𝑡 = 1, 𝜌𝑘,𝑖,𝑛 = 1, 𝐴𝑖 (𝑡) = {𝑘}, 𝒰 = 𝒰 − {𝑘}, and compute 𝑅𝑘 , according to (2), (4). 𝐴𝑖 (𝑡) means the allocation result of the 𝑡 step. – For 𝑡 = 2 to 𝑇 : ∗ ∀𝑙 ∈ 𝐴𝑖 (𝑡 − 1) and 𝑚 ∈ 𝒰, compute 𝜂𝑙,𝑚 (5). ∗ Compute the average correlation between already − 1) and ∀𝑚 ∈ 𝒰, according selected users 𝐴𝑖 (𝑡 ∑ 𝜂𝑙,𝑚 𝑖 (𝑡−1) . to equation 𝐶 𝑚 = 𝑙∈𝐴 ∣𝐴𝑖 (𝑡−1)∣ ∗ Form the group, 𝒬, of candidates that contains the 𝐿 users with the lowest values of 𝐶 𝑚 , 𝑚 ∈ 𝒰. ∗ Find a user, 𝑠𝑡 ∈ 𝒬, such that ∑ ∑ 𝑟𝑘,𝑖,𝑛 > 𝑟𝑘,𝑖,𝑛 and 𝑘∈𝐴𝑖 (𝑡−1)∪{𝑠𝑡 }

𝑘∈𝐴𝑖 (𝑡−1)

   𝑅𝑠𝑡 + 𝑟𝑠𝑡 ,𝑖,𝑛 𝑅𝑘   − ≤ 𝐷, ∀𝑘 ∈ 𝐴𝑖 (𝑡 − 1)  𝛾𝑠𝑡 𝛾𝑘  where D is a system parameter that indicates the relation between proportional fairness among the users’ data rates and sum of the users’ data rates. ∗ If user 𝑠𝑡 is found, set 𝜌𝑠𝑡 ,𝑖,𝑛 = 1, 𝐴𝑖 (𝑡) = 𝐴𝑖 (𝑡− 1) ∪ 𝑠𝑡 , and 𝒰 = 𝒰 − {𝑠𝑡 }.

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0.08 0.1 0.12 0.14 System parameter D

0.16

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0.025 Fairness pointer

subject to 𝜌𝑘,𝑖,𝑛 ∈ {0, 1}, ∀𝑘, 𝑖, 𝑛, 𝑝𝑘,𝑖,𝑛 ≥ 0, ∀𝑘, 𝑖, 𝑛, ∑𝐾 ∑𝐾 𝑃𝑡𝑜𝑡 𝑘=1 𝑝𝑘,𝑖,𝑛 ≤ 𝑘=1 𝜌𝑘,𝑖,𝑛 ≤ 𝑇 , ∀𝑛, 𝑖, 𝑅1 : 𝑁 , ∀𝑛, 𝑖, 𝑅2 : . . . : 𝑅𝐾 = 𝛾1 : 𝛾2 : . . . : 𝛾𝐾 . 𝜌𝑘,𝑖,𝑛 is the subcarrier allocation indicator such that 𝜌𝑘,𝑖,𝑛 = 1 if user 𝑘 ∈ 𝐴𝑖 , and 𝐴𝑖 is selected in subcarrier 𝑛; otherwise 𝜌𝑘,𝑖,𝑛 = 0, for 𝑘 = 1, 2, . . . 𝐾, 𝑖 = 1, 2, . . . 𝐼, and 𝑛 = 1, 2, . . . 𝑁 . The total data rate for user 𝑘, denoted as 𝑅𝑘 , is defined as

Sum of the users’ data rates (bits/s/Hz)

412

0.02 0.015 0.01 0.005 0.02

Fig. 1.

Choice of system parameter.

∗ Compute 𝑅𝑘 , ∀𝑘 ∈ 𝐴𝑖 (𝑡) , according to (2), (4). – Set 𝒮 = 𝒮 − {𝑛}. 𝑘 The algorithm finds user 𝑘 by calculating 𝑅 𝛾𝑘 , ∀𝑘 = 1, . . . , 𝐾 after each allocation. During the first iteration, when all users have zero data rates any user can be chosen. Then, 𝑛 is chosen that maximizes the data rate of user 𝑘 if that user were to transmit alone in that subcarrier. Additional users are admitted to the subcarrier based on two criteria: 1) similar to [4] [6], the sum data rate in the subcarrier should increase, and 2) the newly admitted user 𝑠𝑡 can achieve “fair” sum data rate compared to the sum data rate of the other users that have already been admitted to the subcarrier, according to system parameter 𝐷. The size of the set 𝒬 is set heuristically equal to 𝐿 = min{𝒰, 𝑇 }, because it was shown to lead to good performance in most simulated cases.

IV. S IMULATION R ESULTS The proposed algorithm is compared with the algorithms proposed in [4],[5],[7], Round Robin (RR) algorithm, and Maximal Ratio Combining (MRC) transmission, only to the user with the strongest channel. In RR algorithm, each user is given a fair share of the channel resource regardless of the channel state and 𝑇 users are selected in each subcarrier. Both equal power (EQ) allocation and water-filling (WF) power allocation over the parallel subchannels are considered. In all simulations presented in this section, the frequencyselective channel consists of six independent Rayleigh multipath components. As in [1], an exponentially decaying power delay profile is assumed, the ratio of the energy of the 𝑙th tap to the first tap being equal to 𝑒−2𝑙 . A maximum delay spread of 5𝜇s and maximum doppler of 30Hz is assumed. The channel information is sampled every 0.5ms to update the proposed algorithm, 𝑇 = 4, 𝑁 = 64, 𝑆𝑁 𝑅 = 15, and the number of channel realizations is equal to 1000. For each channel realization, 100 time samples are used for each user and the available users are been assigned a set of proportional constants {𝛾𝑘 }𝐾 𝑘=1 which follow the probability

PAPOUTSIS et al.: USER SELECTION AND RESOURCE ALLOCATION ALGORITHM WITH FAIRNESS IN MISO-OFDMA

Alg. in [5] RR−EQ RR−WF MRC Alg. in [4] Alg. in [7] Prop. Alg.

Sum of the users’ data rates (bit/s/Hz)

22 20 18 16 14 12 10 8 6

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Fig. 2.

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Sum of the users’ data rates vs number of users.

Fairness index (Fp)

0.9 0.8 0.7 0.6 Alg. in [5] RR−EQ RR−WF MRC Prop. alg. Alg. in [7] Alg. in [4]

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Fig. 3.

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Fairness index vs number of users.

mass function 𝑝 𝛾𝑘

available to [4] [5] that only target sum data rate maximization. On the other hand, more users put more constraints to the proposed algorithm, because new users need to share the same resources. In addition, sum data rate of the proposed algorithm is significantly enhanced over both RR-WF and RR-EQ algorithm. MRC algorithm is the lower bound of the proposed algorithm as in MRC each subcarrier is allocated to only one user. Sum data rate of [7] is degraded compared with [4] [5] and enhanced over the other algorithms, because it imposes a kind of fairness between users’ data rates. In Fig. 3, fairness index 𝐹𝑝 is a modified version of the one introduced in [1], and is defined as ∑𝐾 𝑘 2 ( 𝑘=1 𝑅 𝛾𝑘 ) , 𝐹𝑝 = ∑𝐾 𝑅 𝐾 𝑘=1 ( 𝛾𝑘𝑘 )2 where 𝐹𝑝 is a real number in the interval (0, 1] with the maximum value of 1 for the case that the achieved data rate proportions among the users are the same as the predetermined set {𝛾𝑘 }𝐾 𝑘=1 . Employing [4] [5], no guarantees are provided for the fairness between user data rates and as the number of users increases, fairness index degrades. RR-WF, RR-EQ, and [7] experience almost the same 𝐹𝑝 regardless of the number of users, because these algorithms achieve approximately equal data rates among users. The proposed algorithm and MRC distribute the sum data rate very well among users, very close to the defined ideal data rate constraints which is the main point of this paper. However, MRC does not exploit the 𝑇 = 4 degrees of freedom that are available in each subcarrier.

1

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⎧ ⎨ 1 with probability 0.5 = 2 with probability 0.3 ⎩ 4 with probability 0.2

In Fig. 1, the performance of the proposed algorithm is shown for different values of system parameter 𝐷, when 𝐾 = 16. Fairness pointer indicates the maximum difference between users’ data rates and respective fairness constraints, namely max𝑘=1,...,𝐾 (𝑅𝑘 − 𝛾𝑘 ). It is shown that as the system parameter becomes larger, the sum of the users’ data rates becomes larger too, but the fairness criterion is more relaxed. Thus, the system parameter indicates a tradeoff between sum of the users’ data rates and accomplishing fairness between users’ data rates. Figs. 2, 3, are shown for 𝐷 = 0.1, which is chosen heuristically to ensure a reasonable trade off between sum of the users’ data rates and accomplishing fairness between users’ data rates. In Fig. 2, the number of users varies from 4-16 in increment of 2, while in Fig. 3, 𝐾 = 16. In Fig. 2, it can be seen the reasonable price being paid in order to guarantee fairness by using the proposed algorithm. As the number of users increases, the difference in sum data rates increases because additional multiuser diversity is

V. C ONCLUSION A fairness-aware user selection and resource allocation algorithm, which is based both on ZF beamforming and on spatial correlation, for the MISO downlink over frequencyselective channels was introduced. The main goal was to achieve proportional fairness among users’ data rates despite the loss with respect to the unconstrained case where the only target is the maximization of the sum data rate. Simulation results provide proofs about these statements. R EFERENCES [1] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726-2737, Nov. 2005. [2] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian MIMO broadcast channel,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3936-3964, Sep. 2006. [3] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461-471, Feb. 2004. [4] G. Dimic and N. D. Sidiropoulos, “On downlink beamforming with greedy user selection: performance analysis and a simple new algorithm,” IEEE Trans. Signal Process., vol. 53, no. 10, pp. 3857-3868, Oct. 2005. [5] P. W. C. Chan and R. S. Cheng, “Capacity maximization for zero-forcing MIMO-OFDMA downlink systems with multiuser diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1880-1889, May 2007. [6] S. Karachontzitis and D. Toumpakaris, “Efficient and low-complexity user selection for the multiuser MISO downlink,” IEEE Personal, Indoor and Mobile Radio Commun. Symposium, Tokyo, Japan, Sep. 2009. [7] S. Kai, W. Ying, C. Zi-xiong, and Z. Ping, “Fairness based resource allocation for multiuser MISO-OFDMA systems with beamforming,” J. China Univ. of Posts and Telec., vol. 16, no. 1, pp. 38-43, Feb. 2009.