Resource Allocation for Uplink Multiuser OFDM Relay ... - CiteSeerX

Resource Allocation for Uplink Multiuser OFDM Relay Networks with Fairness Constraints Hyundoo Jeong∗ , Jae Hong Lee∗ , and Hanbyul Seo† ∗ School

of Electrical Engineering and INMC Seoul National University Seoul 151-744, Korea † Advanced Wireless Communication Group Mobile Communication Technology Research Lab. LG Electronics, Inc. E-mail : [email protected] Abstract—In this paper, a new subcarrier allocation algorithm is proposed for an uplink multiuser OFDM relay network. To achieve fairness for a source and relay, we consider the minimum rate requirement for a source and the power constraint on a relay, respectively. An optimization problem with fairness constraints is formulated and the subcarrier allocation criterion is obtained by using convex optimization. In the proposed algorithm, if all sources meet the minimum rate requirement, a subcarrier is allocated to a source-relay pair which has the largest instantaneous rate to maximize the sum of the achievable rate of a source. Otherwise, a subcarrier is allocated to a source-relay pair which has the smallest achievable rate to improve fairness for a source. Simulation results show that the proposed algorithm improves fairness and spectral efficiency for a source. Index terms — Relay networks, OFDM, resource allocation, fairness.

I. I NTRODUCTION A cooperative relay network has become an emerging research subject to obtain spatial diversity without a multiple input multiple output (MIMO) antenna array [1]-[2]. In a cooperative relay network, a source transmits information to the destination using one or multiple relays. Each terminal with a single antenna forms virtual MIMO by sharing their antennas and obtains spatial diversity. Orthogonal frequency division multiplexing (OFDM) systems make reliable and high data-rate wireless communications feasible. In OFDM systems, a high rate data stream is split up into a number of lower rate streams, which are transmitted simultaneously over a number of orthogonal subcarriers. It is well known that OFDM systems achieve a higher data-rate by adaptive resource allocation [3]-[7]. Adaptive resource allocation is that subcarrier, bit, and power are allocated to a user based on channel state information of each subcarrier. An achievable rate and a diversity gain are increased by adaptive resource allocation for an uplink OFDM relay network [8], [9]. Most of the previous works consider a single source-destination pair. It is not easy to apply adaptive resource allocation for a single source-destination pair to an uplink multiuser OFDM relay network which consists

of multiple sources, multiple relays, and a destination. In [10], a subcarrier allocation algorithm was investigated for an uplink multiuser OFDM relay network. The same number of subcarriers is allocated to each relay to achieve fairness for a relay. When each relay uses same transmit power, fairness for a relay is achieved in terms of power consumption. To maximize the sum of the achievable rate, a subcarrier is allocated to the source in good channel condition and not to the source in bad channel condition. As a result, the achievable rate of the source in bad channel condition is reduced. By a subcarrier allocation algorithm in [10], the sum of the achievable rate of a source is maximized while achieving fairness for a relay. However, fairness for a source does not achieved. In this paper, we propose a new subcarrier allocation algorithm for an uplink multiuser OFDM relay network. We formulate an optimization problem with fairness constraints on a source and relay, and obtain a subcarrier allocation criterion by using convex optimization. To improve fairness for a source, a subcarrier is allocated to a source to meet the minimum rate requirement. The equal power constraint on each relay is imposed to achieve fairness for a relay. We adopt equal power allocation for each source and relay to reduce computational complexity. In [5]-[7], it is shown that equal power allocation achieves similar performance of the waterfilling power allocation which is known to be optimal but equal power allocation has less computational complexity. This paper is organized as follows. Section II provides a system model and formulates an optimization problem. Section III proposes a new subcarrier allocation algorithm. Section IV provides simulation results for the outage probability and spectral efficiency. Conclusions are given in Section V. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model Consider an uplink multiuser OFDM relay network which consists of K sources, M relays, and a single destination as shown in Fig. 1. Let K, M, and N denote the sets of the source, relay, and subcarrier, respectively. Assume that K sources are uniformly distributed and M relays are fixed in

978-1-4244-2517-4/09/$20.00 ©2009 IEEE Authorized licensed use limited to: Seoul National University. Downloaded on July 15,2010 at 01:21:13 UTC from IEEE Xplore. Restrictions apply.

1

Since each terminal cannot transmit and receive simultaneously, sources transmit information to the destination over two orthogonal time slots. In the first time slot, each source transmits its own information to the relays and the destination simultaneously. In the the second time slot, each relay transmits the received information to the destination using either amplify-and-forward (AF) or decode-and-forward (DF) [2].

F1,1 n

1

1

F1,M n

F2,1 n

2

H1

n

H 2

n

G1

n

F2, M n

B. Problem Formulation If the subcarrier n is allocated to the source k and the relay m, the instantaneous rate of the source k with the AF relay is given by [2]

H K

(n)

Rk,m = ⎛

n

n

GM

1 2

FK ,1 n

K

M

FK , M n

Fig. 1.

System model for an uplink multiuser OFDM relay network.

log2 ⎝1 +

(n)

Hk (n)

(n)

=

T t=1

(t)

hk e−j2π(n−1)(t−1)/N . (n)

(1)

Let Hk , Fk,m , and Gm denote the channel coefficient of the subcarrier n between the source k to the destination, the source k to the relay m, and the relay m to the destination, respectively. Assume that the bandwidth of each subcarrier is much smaller than the coherence bandwidth of the wireless channel so that the wireless channel appears frequency-flat on every subcarrier. Assume that the cyclic guard interval of an OFDM symbol is longer than the delay spread of the wireless channel so that there is no intersymbol interference (ISI), and is much shorter than an OFDM symbol so that it has negligible influence on the achievable rate of each source. Assume that perfect channel state information is available at the destination. Assume that the channel coefficient of each subcarrier is constant over one OFDM frame, which consists of several OFDM symbols so that centralized resource allocation is feasible.

N0

⎞

+

(n) 2 (n) 2 (n) (n) Fk,m |Gm | pr,m p s,k ⎠ N0 (n) 2 (n) (n) 2 (n) Fk,m ps,k +Gm pr,m +N0

(2) and the instantaneous rate of the source k with the DF relay is given by [2]

2 (n)

Rk,m = a cell. Assume that each terminal is equipped with a single antenna and cannot transmit and receive simultaneously. The wireless channel is assumed to be frequency-selective (t) (t) so that it is modeled as a T -tap filter [14]. Let hk , fk,m , and (t) gm denote the t-th time-domain channel coefficient between the source k to the destination, the source k to the relay m, and the relay m to the destination, respectively. The t-th time-domain channel coefficient is assumed to be a Rayleigh random variable. The channel coefficient of the subcarrier n is obtained by the discrete Fourier transform (DFT). For example, the channel coefficient of the subcarrier n between the source k to the destination is given by

(n) 2 (n) Hk ps,k

1 2

min

log2 log2

1+ 1+

(n) (n) Fk,m ps,k

,

N0 (n) 2 (n) Hk ps,k N0

+

|G(n) m |

2 (n) pr,m

N0

(3) where N0 is the single-sided power spectral density of AWGN, (n) (n) and ps,k and pr,m are power allocated to the source k and the relay m on the subcarrier n, respectively. Assume that N0 is equal for all subcarriers in a wireless network, and the source k and the relay m have their maximum transmit power Ps,k and Pr,m , respectively. The subcarrier assignment indicator variable is denoted by (n) ρk,m . If the subcarrier n is allocated to the source k and the (n) relay m, ρk,m is equal to 1. Otherwise, it is equal to 0. Assume that the subcarrier is allocated to only one source and one relay so that there is no interference between sources. Assume that the information of the source on the subcarrier is forwarded by the assigned relay on the same subcarrier. Then, the achievable rate of the source k is given by Rk =

N M m=1 n=1

(n)

(n)

ρk,m Rk,m .

(4)

The sum of the achievable rate over all sources is denoted by R and given by K R= Rk . (5) k=1

In order to maximize the sum of the achievable rate of a source, a subcarrier is allocated to the source in good channel condition. For a source in good channel condition, the achievable rate is enhanced but that of the source in bad channel condition is reduced significantly. When the achievable rate of the source is much smaller than the others, fairness for a source is not achieved. If each source meets the

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minimum rate requirement which is a target achievable rate, fairness for a source is improved. Since the maximum transmit power for each relay is limited to Pr,m , fairness for a relay is achieved. The optimization problem with fairness constraints is formulated as R∗ = max

N M K k=1 m=1 n=1

subject to:

(n)

(n)

ρk,m Rk,m

(n) ρk,m ∈ {0, 1} , ∀k, m, n M K (n) ρk,m = 1, ∀n k=1 m=1 N (n) ps,k ≤ Ps,k , ∀k n=1 N p(n) r,m ≤ Pr,m , ∀m n=1 N M (n) (n) ρk,m Rk,m ≥ Rmin , ∀k m=1 n=1 (n) ps,k ≥ 0, ∀k, n p(n) r,m ≥ 0, ∀m, n.

(6)

The Lagrangian for the relaxed optimization problem is given by K M M K N N (n) (n) (n) ρk,m Rk,m + λn ρk,m − 1 L= − k=1 m=1 n=1 n=1 K Nk=1 m=1 K N (n) (n) + ηk ps,k − Ps,k − αk,n ps,k n=1 n=1 k=1 k=1M N M N (n) (n) + νm pr,m − Pr,m − βm,n pr,m m=1 n=1 m=1 n=1 K M N (n) (n) + μk Rmin − ρk,m Rk,m

(7a) (7b)

(7c)

−

k=1 M K

k=1 m=1 n=1

(7f)

∂L

(n)

γk,m,n ρk,m

(n)

(n) ∂ρk,m

= λn − (1 + μk ) Rk,m − γk,m,n .

(n)

γk,m,n = λn − (1 + μk ) Rk,m .

(10)

Due to the non-negativity of the Lagrange multiplier, λn becomes (n) (11) λn ≥ (1 + μk ) Rk,m . (n)

If ρk,m is positive, the subcarrier n is allocated to the source (n) k and the relay m. The necessary condition for positive ρk,m is obtained by the KKT condition and complementary slackness condition. The KKT condition is given by (n)

γk,m,n ρk,m = 0. III. R ELAXED O PTIMIZATION P ROBLEM AND P ROPOSED S UBCARRIER A LLOCATION A LGORITHM In this section, we obtain the subcarrier allocation criterion and propose a new subcarrier allocation algorithm for an uplink multiuser OFDM relay network.

(9)

By the Karush-Kuhn-Tucker (KKT) conditions, the partial derivative of the Lagrangian is equal to zero at the optimal solution. Hence, from (9), γk,m,n becomes

(7g)

Constraints (7a) and (7b) represent that a subcarrier is exclusively allocated to only one source and one relay to avoid interference between sources. Constraints (7c) and (7d) are the maximum transmit power for each source and relay, respectively. Constraint (7e) is the minimum rate requirement for a source. Assume that each source has the same minimum rate requirement. However, each source meets the different minimum rate requirement by a slight modification of a new subcarrier allocation algorithm.

m=1 n=1

(8) where λn , ηk , νm , μk , αk,n , βm,n , and γk,m,n are the Lagrange multipliers. The partial derivative of the Lagrangian with respective to (n) the optimization variable ρk,m is given by

(7d) (7e)

N

(12)

The complementary slackness condition is given by (n)

ρk,m > 0, if γk,m,n = 0 or

(13)

(n)

ρk,m = 0, if γk,m,n > 0.

(14) (n)

A. Relaxed Optimization Problem and KKT Conditions The optimization problem in (6) contains both continuous and integer variables. It is a combinatorial optimization problem which has excessive computational complexity to obtain a global optimal solution. To make the problem tractable, relax the constraint on the subcarrier assignment indicator variable. After the constraint relaxation, the subcarrier assignment indicator variable takes a real number within [0, 1]. It is shown that the relaxed optimization problem provides an upper bound of the original optimization problem [3].

The subcarrier assignment indicator variable ρk,m is positive only when γk,m,n is equal to zero and an equality holds in (11). Therefore, to maximize the sum of the achievable rate and meet the minimum rate requirement for a source, the subcarrier n is allocated to the source k ∗ and the relay m∗ such that (n)

(k ∗ , m∗ ) = arg max (1 + μk ) Rk,m k,m

(15)

where μk is the Lagrange multiplier for the minimum rate requirement for the source k. By using (15), to obtain the optimal solution for subcarrier allocation, it is needed to know the exact value of μk . In [3],


the Lagrange multiplier is obtained by an iterative searching algorithm which has excessive computational complexity [4]. A substitute value of μk is obtained by the KKT condition and complementary slackness condition. The KKT condition is given by N M (n) (n) (16) ρk,m Rk,m = 0. μk Rmin − m=1 n=1

The complementary slackness condition is given by μk = 0, if

N M m=1 n=1

or μk > 0, if

M

N

m=1 n=1

(n)

(n)

ρk,m Rk,m > Rmin (n)

(n)

ρk,m Rk,m = Rmin .

Algorithm 1: Proposed subcarrier allocation algorithm Step 1: Initialization Set K = {1, 2, . . . , K}, M = {1, 2, . . . , M }, N = (n) {1, 2, . . . , N }, and ρk,m = 0, ∀k, m, n; Step 2: Initial subcarrier allocation for k = 1 : K do (n) (m∗ , n∗ ) = arg max Rk,m , m ∈ M, n ∈ N ; m,n

(n∗ )

ρk,m∗ = 1, N = N − {n∗ }; (17)

(18)

From (17) and (18), if the source k does not meet the minimum rate requirement, μk is positive. Otherwise, μk becomes 0. Then, we substitute μk as the difference between the minimum rate requirement and the achievable rate of the source k. A substitute value of μk is denoted ζk and is given by (19) ζk = Rmin − Rk . B. Proposed Subcarrier Allocation Algorithm In a new subcarrier allocation algorithm, the source and relay allocate equal power over assigned subcarriers. It is shown that, for an uplink system, the sum of the achievable rate is maximized by allocation subcarriers based on both the channel coefficient and transmit power of the subcarrier [7]. In the proposed algorithm, a subcarrier is allocated to the source and relay based on the instantaneous rate which is obtained by both the channel coefficient and the transmit power of the subcarrier. The performance of a new subcarrier allocation algorithm is compared with that of the greedy and static subcarrier allocation algorithm using equal power allocation. The greedy subcarrier allocation algorithm is the solution of the problem 1 in [10]. In the greedy subcarrier allocation algorithm, a subcarrier is allocated to a source-relay pair which has the largest instantaneous rate so that the sum of the achievable rate of a source is maximized. However, the achievable rate is reduced significantly for the source in bad channel condition. The static subcarrier allocation algorithm is a modified scheme of OFDM-TDMA in [3]. In the static subcarrier allocation algorithm, a source and the nearest relay use all subcarriers for a predetermined time slot exclusively. We propose a new subcarrier allocation algorithm for an uplink multiuser OFDM relay network, which is maximizing the sum of the achievable rate of a source and improving fairness for a source and relay. The proposed subcarrier allocation algorithm consists of three steps. In the first step, all sets and variables are initialized. In the second step, one subcarrierrelay pair which has the largest instantaneous rate is allocated to one source until each source has one subcarrier-relay pair. In the third step, remaining subcarriers are allocated to the

update Rk ; Step 3: Remaining subcarrier allocation while N = ∅ do k ∗ = arg max ζk , k ∈ K; k

if ζk∗ > 0 then (n) (n) calculate ps,k∗ , pr,m , m ∈ M, n ∈ N ; (n) (m∗ , n∗ ) = arg max Rk∗ ,m , m ∈ M, n ∈ N ; m,n

(n∗ )

ρk∗ ,m∗ = 1, N = N − {n∗ }; else n∗ = rand (N ); ∗ (n∗ ) (n ) calculate ps,k , pr,m , k ∈ K, m ∈ M; (n∗ ) (k ∗ , m∗ ) = arg max Rk,m , k ∈ K, m ∈ M; k,m

(n∗ )

ρk∗ ,m∗ = 1, N = N − {n∗ }; update Rk ;

source and relay to maximize the sum of the achievable rate or meet the minimum rate requirement for a source. Subcarrier allocation depends on the sign of ζk . If ζk is positive, the source k does not meet the minimum rate requirement so that the subcarrier is allocated to the source k-relay pair to meet the minimum rate requirement for the source k. Otherwise, the randomly selected subcarrier is allocated to the source-relay pair which has the largest instantaneous rate to maximize the sum of the achievable rate. The third step is repeated until all subcarriers are allocated. IV. S IMULATION R ESULTS Suppose that the base station is located at the center of a circular cell with the radius of 1000 meters. Suppose that sources are uniformly distributed in the cell and relays are fixed symmetrically on the circle with the radius of 500 meters. Similar to the worst possible average signal to noise ratio (WSNR) [6], the received SNR is defined as the average received SNR of the source on the middle of the cell radius. Assume that all sources and relays have the same initial power. The number of subcarriers is 128 and the total bandwidth is 5 MHz. Suppose that the channel of ITU pedestrian B model is adopted [11]. Suppose that path the loss exponent is 3.5 for a relay to the destination channel and is 4 for both a source to a relay and a source to the destination channel.


Fig. 2. Outage probability of the proposed subcarrier allocation algorithm. K=8, M =4, and Rmin =0.625 Mbits/s.

A. Outage Probability Fig. 2 shows the outage probability of the proposed algorithm for AF and DF for K=8, M =4, and Rmin = 0.625 Mbits/s. An outage event occurs when at least one source does not meet the minimum rate requirement. It is shown that the proposed algorithm achieves much smaller outage probability than the greedy and static algorithm. It is also shown that, at the outage probability of 10−3 , the proposed algorithm achieves the SNR gain of 12 dB and 14 dB over the static algorithm for AF and DF, respectively. B. Spectral Efficiency Fig. 3 shows spectral efficiency of the proposed algorithm for AF and DF for K=8, M =4, and Rmin = 0.625 Mbits/s. It is shown that, at SNR of 20 dB, the proposed algorithm achieves the higher spectral efficiency by 2.1 bps/Hz and 1.8 bps/Hz than the static algorithm for AF and DF, respectively. It is also shown that, at SNR of 20 dB, the proposed algorithm achieves the lower spectral efficiency by 0.3 bps/Hz and 0.15 bps/Hz than the greedy algorithm for AF and DF, respectively. However, the proposed algorithm achieves significant SNR gain in the outage probability but negligible spectral efficiency loss over the greedy algorithm. V. C ONCLUSION In this paper, we propose a new subcarrier allocation algorithm for an uplink multiuser OFDM relay network. In the proposed algorithm, a subcarrier is allocated to a source and relay to maximize the sum of the achievable rate while meeting the minimum rate requirement for a source. The performance of the proposed algorithm is compared with that of the greedy and static algorithm. The proposed algorithm improves fairness and spectral efficiency for a source while achieving fairness for a relay.

Fig. 3. Spectral efficiency of the proposed subcarrier allocation algorithm. K=8, M =4, and Rmin =0.625 Mbits/s.

ACKNOWLEDGMENT The authors would like to thank Harin Jeong and Dr. Jung Min Choi for helpful comments. This work was supported by LG Electronics. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang,“User cooperation diversityPart I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927-1938, Nov. 2003. [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [3] C. Y. Wong, R. S. Cheng, K. B. Letief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Commun., vol. 17, no. 10, pp. 1747-1758, Oct. 1999. [4] K. B. Letaief and Y. J. Zhang, “Dynamic multiuser resource allocation and adaptation for wireless systems,” IEEE Trans. Wireless Commun., vol. 13, no. 4, pp. 38-47, Aug. 2006. [5] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 2, pp. 171-178, Feb. 2003. [6] W. Rhee and J. M. Cioffi, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. IEEE Veh. Tech. Conf. (VTC) 2000-Spring, Tokyo, Japan, May 2000, pp. 1085-1089. [7] K. Kim, Y. Han, and S. L. Kim, “Joint subcarrier and power allocation in uplink OFDMA systems,” IEEE Commun. Letters, vol. 8, no. 6, pp. 526-528, June 2005. [8] I. Hammerstrom and A. Wittneben, “On the optimal power allocation for nonregenerative OFDM relay links,” in Proc. IEEE ICC 2006 , Istanbul, Turkey, June 2006, pp. 4463-4468. [9] B. Gui, L. Dai, and L. J. Cimini, “Selective relaying in cooperative OFDM systems: Two-hop random network,” in Proc. IEEE WCNC 2008 , Las Vegas, NV, March 2008, pp. 574-578 [10] G. Li and H. Liu, “Resource allocation for OFDMA relay networks with fairness constraints,” IEEE J. Sel. Areas Commun., vol. 24, no. 11, pp. 2061-2069, Nov. 2006. [11] WiMAX Forum, “Mobile WiMAX-Part I: A technical overview and performance evaluation,” WiMAX Forum White Paper, June 2006. [12] R. Nee and R. Prasad, OFDM Wireless Multimedia Communications. Artech House, 2000. [13] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [14] J. G. Proakis, Digital Communications, 4/e. McGraw-Hill, 2001.
