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IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013

Proportional Resource Allocation with Subcarrier Grouping in OFDM Wireless Systems Zhanyang Ren, Shanzhi Chen, Bo Hu, and Weiguo Ma

Abstract—In practical orthogonal frequency division multiplexing (OFDM) wireless systems, such as long term evolution (LTE), the subcarriers assigned to one user in one scheduling determination always use the same modulation and coding schemes (MCS). Taking this implementation issue into consideration, multiuser resource allocation with subcarrier grouping is investigated in this paper. The resource allocation is modeled as an optimization problem, in which a set of proportional rate constraints ensures system fairness. Firstly, a preliminary subcarrier assignment is developed, and then, an optimal power allocation scheme is carried out to ensure the proportional rate constraints precisely. Furthermore, an entire iterative algorithm is proposed to utilize subcarrier as effectively as possible. Numerical results demonstrate that the proposed algorithm achieves sum-rate capacity which is close to optimum and distributes resource among users flexibly as well.

In this paper, the MCS constraint mentioned above is referred to as subcarrier grouping, and the insight into resource allocation for OFDM wireless systems is provided. Sum-rate capacity is to be maximized and, for the sake of fairness issue, proportional rate constraints [2] are considered to ensure users reach data rates corresponding to the predefined proportion. At first, subcarriers are assigned with equal power distribution, and then the proportional rates are ensured precisely through a bisection-based optimal power allocation scheme. Moreover, in order to make utilization of frequency resource more effective, an entire iterative algorithm is carried out. Numerical results are finally given to verify the performance of the proposed algorithm.

Index Terms—Resource allocation, subcarrier grouping, sumrate capacity, proportional rate constraints.

II. S YSTEM M ODEL

I. I NTRODUCTION RTHOGONAL frequency division multiplexing (OFDM), as one of the most promising technologies adopted in state-of-the-art wireless systems, provides a high performance physical layer. OFDM wireless systems benefit from efficient and flexible utilization of radio resource since rate and power can be adjusted on a per-subcarrier basis. Extensive research work has been focusing on resource allocation for OFDM wireless systems in the downlink direction [1]–[5]. However, the algorithms in these literatures are all developed without considering implementation issue. In practical OFDM wireless systems, such as long term evolution (LTE), for the purpose of signaling cost reduction and device simplicity, the subcarriers assigned to one user always use the same modulation and coding schemes (MCS). To the best of our knowledge, there is few published work related to this area. Liang et al. in [6] introduced blockwise resource allocation problem, and simply assumed that the subcarriers grouped into a block have the same channel gain. This scheme ignores frequency-selectivity and thus neglects frequency diversity gain. Reference [7] and reference [8] focused on resource allocation in LTE cellular networks and tried to maximize the system throughput without fairness consideration.

O

Manuscript received November 30, 2012. The associate editor coordinating the review of this letter and approving it for publication was D. Popescu. Z. Ren and B. Hu are with the State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, China (e-mail: [email protected]). S. Chen and W. Ma are with the State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunications Technology (CATT), Beijing, China. This work is financially supported by the National High-Technology Program of China (863) (Grant No.2011AA01A101), and International S&T Cooperation: Joint Reaserch project (Gran No.2010DFB13020). Digital Object Identifier 10.1109/LCOMM.2013.031913.122706

In this paper, a downlink multiuser scenario in OFDM systems is considered and, perfect instantaneous channel information is assumed to be available at both the base station and the user side. Based on these channel state information, the resource allocation algorithm assigns a total of N subcarriers to K users, with total transmit power constraint P total . The objective of this paper is to allocate the subcarriers and power to achieve sum-rate capacity maximization. A set of proportional rate constraints {α1 , ..., αK } is also considered to address system fairness issue. According to the subcarrier grouping constraint, all subcarriers assigned to one user should use the same MCS. Since the same predetermined bit-error rate (BER) performance level should be achieved for each subcarrier in a group, these grouped subcarriers are required to have the same signal-tonoise-ratio (SNR) condition. Mathematically, using the Shannon capacity formula for Gaussian channel, the resource allocation optimization problem is formulated as K bits 1 sec log2 (1 + pn,k gn,k ) maximize N Hz k=1n∈Dk

subject to

K

pn,k ≤ Ptotal

k=1n∈Dk

pn,k ≥ 0 for all k, n D1 ∪ D2 ... ∪ DK ⊆ {1, ..., N } Di ∩ Dj = φ for i = j and i, j ∈ {1, ..., K} r1 : r2 : ... : rK = α1 : α2 ... : αK ,

(1)

where Dk is the set of subcarriers assigned to user k, and pn,k is the amount of power allocated to user k on subcarrier n. For simplicity, we let g n,k denote the channel-gain-to-noise-ratio B (CNR) h2n,k /N0 N , where hn,k and N 0 are channel gain and

c 2013 IEEE 1089-7798/13$31.00

REN et al.: PROPORTIONAL RESOURCE ALLOCATION WITH SUBCARRIER GROUPING IN OFDM WIRELESS SYSTEMS

TABLE I P RELIMINARY S UBCARRIER A SSIGNMENT

TABLE II POWER ALLOCATION

Initialization: set λlow , λhigh as (14) and (15), and tolerance 1: Repeat calculate λnew = (λlow + λhigh ) /2; If f (λnew ) > Ptotal update λhigh ← λnew ; Else update λlow ← λnew ; End If Until | f (λnew ) − Ptotal |≤ 2: calculate Pk with λnew based on (9) and (10)

Initialization: set D = {1, ..., N }, and Dk = φ for k ∈ {1, ..., K} 1: For user k ∈ {1, ..., K} j = argmaxn {gn,k | n ∈ D}; update Dk ← Dk + {j} and D ← D − {j}; End For 2: While D = φ k = argmini {ri /αi | i ∈ {1, ..., K}}; j = argmaxn {gn,k | n ∈ D}; calculate rk based on (6); If rk > 0 update Dk ← Dk + {j} and D ← D − {j}; update rk based on (5); Else break; End If End While

noise power spectral density of AWGN, respectively. B/N is the subcarrier bandwidth with system bandwidth B divided by subcarrier number N . The fourth constraint shows that each subcarrier can only be assigned to one user, and the last equality presents the proportional data rate requirements, where rk is expressed as 1 log2 (1 + pn,k gn,k ) , (2) rk = N n∈Dk

with pn,k gn,k = pm,k gm,k for m, n ∈ Dk

(3)

indicating the subcarrier grouping constraint aforementioned.

III. P RELIMINARY S UBCARRIER A SSIGNMENT AND P OWER A LLOCATION S CHEME A. Preliminary Subcarrier Assignment For user k, we intuitively find that if the subcarrier set Dk and the user-level power Pk are fixed, the subcarrierlevel power allocation can be determined directly from the subcarrier grouping constraint (3) as follows: pn,k =

gn,k

P k

1 i∈Dk gi,k

for n ∈ Dk .

Therefore, user rate rk can be calculated as Sk Pk rk = log2 1 + , 1 N n∈Dk g

869

this user, we can avoid the suboptimal bad-subcarrier-taking event caused in (5) to a certain extent. As a result, the simple solution (5) can be used instead, which only needs a direct calculation. Hence, we are motivated to develop a similar subcarrier assignment scheme with that of [2], and equal power distribution across all subcarriers Ptotal /N is also assumed. The algorithm is shown in detail in Table I. In the first step each user chooses its best subcarrier, and then in the second step, we ensure that the user whose proportional rate is minimum can get the assignment opportunity. The best subcarrier for the chosen user would be assigned if the updated rate can be enhanced. This choose-and-update policy is repeated until there is no subcarrier left or the chosen user cannot get further rate increment. Calculation of the rate increment rk is given as follows: ⎛ ⎞ + 1) P (Sk + 1) (S k total ⎠ log2 ⎝1 + Δrk = 1 1 N N + n∈Dk gn,k gj,k Sk Ptotal Sk , (6) − log2 1 + 1 N N n∈Dk gn,k where j denotes the index of the subcarrier to be assigned for user k. B. Optimal Power Allocation Scheme

(4)

Since the subcarrier-level power allocation can be directly determined by the user-level power based on (4), the original optimization problem (1) with the determined subcarrier assignment turns into a user-level power allocation as follows:

(5)

n,k

where S k denotes the cardinality of Dk . Notice that this rk ∼ Pk approach is by no means optimal, since all subcarriers are assumed to obtain power allocation. In order to achieve maximum data rate, the relatively deepfading subcarriers may be dropped, as waterfilling method does. However, the optimization problem is non-convex and much more complex. Moreover, in the greedy-based subcarrier assignment proposed later, expensive computation would happen. Nevertheless, in a multiuser scenario, if the best subcarrier is always assigned to the user during subcarrier assignment, i.e., the assigned subcarrier has the highest channel gain for

K Sk log2 (1 + Pk Hk ) N

maximize

k=1 K

subject to

Pk ≤ Ptotal , Pk ≥ 0 for all k

k=1

r1 : r2 : ... : rK = α1 : α2 ... : αK , (7)

1 can be regarded as the equivalent where Hk = 1/ n∈Dk gn,k CNR for user k, and rk is expressed as (5). For proportional rate constraints, we introduce an intermediate variable λ, which is defined as r1 r2 rK = = ... = = λ. (8) α1 α2 αK

870

IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013

K , and Hence, sum-rate capacity is reformed as λ α k k=1 objective of (7) is converted into seeking the maximum λ. On the other hand, power requirement for user k can be expressed based on (5) as 2βk − 1 , (9) Pk = Hk where βk =

λN αk . Sk

TABLE III ENTIRE ALGORITHM

Initialization: set D = {1, ..., N } 1: execute Preliminary Subcarrier Assignment and Power Allocation to get initial subcarrier set Dk and user power level Pk , for k ∈ {1, ..., K} 2: Repeat For user k ∈ {1, ..., K} j = argmaxn {gn,k | n ∈ D − ∪k Dk }; calculate rk base on (16) If rk > 0 update Dk ← Dk + {j}; End If End For update Pk via Power Allocation; Until D = ∪k Dk or Dk doesn’t change 3: calculate Pn,k with final Pk based on (4)

(10)

Hence, with the following definition f (λ) =

K 2

λN αk Sk

Hk

k=1

−1

,

(11)

the optimization problem (7) can be rewritten as follows: maximize

λ

subject to

f (λ) ≤ Ptotal .

10

8 Sum−rate Capacity (bits/s/Hz)

Obviously, f (λ) is a monotonously increasing function of λ. Therefore, (12) is equivalent to seeking λ as follows: f (λ) = Ptotal .

(13)

This is a one-dimensional root-finding problem, for which the bisection method can be used to find the optimal solution. To speed up the searching process, we set the initial point as

/K) λlow = min rk (Ptotal , for k ∈ {1, ..., K} (14) αk

/K) λhigh = max rk (Ptotal , for k ∈ {1, ..., K} , (15) αk where rk (Ptotal /K) denotes rate of user k with power level of Ptotal /K. The bisection-based power allocation scheme is described in Table II: after the optimal λ found by the bisection method, user power is calculated by (9) and (10). IV. E NTIRE I TERATIVE A LGORITHM In the preliminary subcarrier assignment scheme, when the chosen user experiences severe deep-fading on all the left subcarriers, its data rate may decrease with further assignment. In this case, the algorithm would stop, and both the subcarriers and power would be left. Intuitively, it is can be explained by the fact that the current power level prevents the user from utilizing more bandwidth resource, and makes this situation happen. Notice that, this power level results from the equal power distribution we assumed in the preliminary subcarrier assignment. On the other hand, the proposed bisection-based power allocation surely may enhance power level for some users. Hence, further subcarrier assignment is possible. In this context, an entire iterative algorithm is proposed in Table III. As shown in the table, firstly, the preliminary subcarrier assignment and the power allocation scheme are applied to get initial subcarrier set and user power level respectively. Secondly, users are checked to decide if additional subcarriers can be assigned to increase user rate. If the data rates can be further enhanced for some users, the corresponding subcarriers would be assigned and the power allocation would be executed again to ensure the proportionality. The second step is repeated until there is no extra subcarrier can be taken, or no subcarrier left in the system. Finally, the subcarrier-level power allocation

MaxThr Optimal Proposed PSA

9

(12)

7 6 5 4 3 2 1 0

Fig. 1.

0

5

10

15 20 Average SNR (dB)

25

30

Sum-rate Capacity versus Average SNR.

is directly obtained via (4). Suppose that the temporary power level for user k is Pk during the iteration, the rate increment calculation in the algorithm is given as follows: Sk + 1 Pk log2 1 + Δrk = 1 1 N n∈Dk gn,k + gj,k Pk Sk . (16) − log2 1 + 1 N n∈Dk g n,k

To explain why this iterative algorithm can increase sum-rate capacity, we give a simple analysis below. Proof: Without loss of generality, consider the ith iteration, and suppose that user k can be assigned an additional subcarrier. Hence, with the updated subcarrier set Dk , data rate of user k increases and is thus proportional maximum. From the perspective of power consumption, it means that if we keep the current proportional rate level, user k would need less power. Therefore, through the (i + 1)th iteration, the marginal power of user k will be reallocated among all the users to reach a higher proportional rate level, and thus the sum-rate capacity is increased. V. S IMULATIONS A. Simulation Environment In simulations, we assume that each user’s subcarrier signal undergoes identical Rayleigh fading independently. The

REN et al.: PROPORTIONAL RESOURCE ALLOCATION WITH SUBCARRIER GROUPING IN OFDM WIRELESS SYSTEMS

871

Average User Rate (bits/s/Hz)

5

4

3

2

User1 (PSA) User2 (PSA) User1 (Proposed) User2 (Proposed) User1 (Optimal) User2 (Optimal) User1 (MaxThr) User2 (MaxThr)

1

0

0

5

10

15

20

25

30

20

25

30

Average SNR (dB) Instantaneous User Rate (bits/s/Hz)

7 6 5 4 3

User1 (PSA) User2 (PSA) User1 (Proposed) User2 (Proposed) User1 (Optimal) User2 (Optimal) User1 (MaxThr) User2 (MaxThr)

2 1 0

0

5

10

15

Average SNR (dB)

Fig. 2.

Average & Instantaneous User Rate versus Average SNR.

2 = 1, average channel gain is normalized such that E gk,n for all k and n. We define average SNR as Ptotal /N0 B, and adopt Monte Carlo simulation method to obtain the average data from 1,000 channel realizations. As the comparison schemes, following methods are considered: 1) Optimal: Optimal solution for the problem discussed in this paper, which is obtained through full enumeration. It is adopted as the upper bound of the proposed algorithm. 2) MaxThr: The method in [8] to maximize throughput without fairness consideration. In simulations it is directly applied in subcarrier level instead of resource block level. 3) PSA: The preliminary subcarrier assignment scheme. It achieves a coarse tradeoff between capacity and fairness. For all methods, proportional rate constraint is set to 1:1, and 10 subcarriers and 2 users are set to reduce the computational time. B. Simulation Results Figure 1 depicts the sum-rate capacity, as a function of average SNR. As shown in the figure, the MaxThr method reaches the highest capacity. This is reasonable, since in the MaxThr method no fairness issue is considered. It can also be seen that as average SNR increases, the proposed algorithm achieves capacity close to the optimum. Though in the low SNR region, the proposed scheme and the PSA scheme reach a similar level, whereas in the high SNR region, the proposed scheme strictly outperforms the latter one.

User data rates as expressed in (5) are shown in Figure 2. User rate averaged over 1,000 channel realizations and instantaneous user rate sampled from the 500th channel realization are both given to consider long-term and instantaneous behavior respectively. It is clear that the proportional rate constraints are ensured precisely in the proposed algorithm. On the other hand, it should be noted that, the reason why the average rates between the 2 users are similar in the MaxThr method is that the long-term channel conditions are similar in simulations. While for instantaneous rate, both the MaxThr method and the PSA scheme have significant discrepancy between the 2 users, and the case is even worse in the MaxThr method, since fairness issue is not taken into account. VI. C ONCLUSION From the implementation point of view, proportional resource allocation with subcarrier grouping for OFDM systems is investigated in this paper. Firstly, the preliminary subcarrier assignment and simple bisection-based power allocation scheme are proposed, and then, the entire iterative algorithm is developed to utilize bandwidth resource effectively. Compared with the throughput maximization method in [8], the proposed scheme distributes resource among users more flexibly with proportional rates being ensured precisely. Moreover, the proposed scheme outperforms the simple scheme where only PSA is used, and as SNR increases, the proposed scheme can also achieve sum-rate capacity close to the optimum. R EFERENCES [1] C. 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. 1747–1758, 1999. [2] Z. Shen, J. Andrews, and B. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, 2005. [3] H. Yin and H. Liu, “An efficient multiuser loading algorithm for OFDMbased broadband wireless systems,” in Proc. 2000 IEEE Global Telecommunications Conference, vol. 1, pp. 103–107. [4] D. Kivanc, G. Li, and H. Liu, “Computationally efficient bandwidth allocation and power control for OFDMA,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1150–1158, 2003. [5] G. Song and Y. Li, “Cross-layer optimization for OFDM wireless networks—part I: theoretical framework,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 614–624, 2005. [6] L. Xiaowen and Z. Jinkang, “An adaptive subcarrier allocation algorithm for multiuser OFDM system,” in Proc. 2003 IEEE Vehicular Technology Conference – Fall, vol. 3, pp. 1502–1506. [7] R. Kwan, C. Leung, and J. Zhang, “Resource allocation in an LTE cellular communication system,” in Proc. 2009 IEEE International Conference on Communications, pp. 1–5. [8] J. Fan, Q. Yin, G. Li, B. Peng, and X. Zhu, “Adaptive block-level resource allocation in OFDMA networks,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3966–3972, 2011.

IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013

Proportional Resource Allocation with Subcarrier Grouping in OFDM Wireless Systems Zhanyang Ren, Shanzhi Chen, Bo Hu, and Weiguo Ma

Abstract—In practical orthogonal frequency division multiplexing (OFDM) wireless systems, such as long term evolution (LTE), the subcarriers assigned to one user in one scheduling determination always use the same modulation and coding schemes (MCS). Taking this implementation issue into consideration, multiuser resource allocation with subcarrier grouping is investigated in this paper. The resource allocation is modeled as an optimization problem, in which a set of proportional rate constraints ensures system fairness. Firstly, a preliminary subcarrier assignment is developed, and then, an optimal power allocation scheme is carried out to ensure the proportional rate constraints precisely. Furthermore, an entire iterative algorithm is proposed to utilize subcarrier as effectively as possible. Numerical results demonstrate that the proposed algorithm achieves sum-rate capacity which is close to optimum and distributes resource among users flexibly as well.

In this paper, the MCS constraint mentioned above is referred to as subcarrier grouping, and the insight into resource allocation for OFDM wireless systems is provided. Sum-rate capacity is to be maximized and, for the sake of fairness issue, proportional rate constraints [2] are considered to ensure users reach data rates corresponding to the predefined proportion. At first, subcarriers are assigned with equal power distribution, and then the proportional rates are ensured precisely through a bisection-based optimal power allocation scheme. Moreover, in order to make utilization of frequency resource more effective, an entire iterative algorithm is carried out. Numerical results are finally given to verify the performance of the proposed algorithm.

Index Terms—Resource allocation, subcarrier grouping, sumrate capacity, proportional rate constraints.

II. S YSTEM M ODEL

I. I NTRODUCTION RTHOGONAL frequency division multiplexing (OFDM), as one of the most promising technologies adopted in state-of-the-art wireless systems, provides a high performance physical layer. OFDM wireless systems benefit from efficient and flexible utilization of radio resource since rate and power can be adjusted on a per-subcarrier basis. Extensive research work has been focusing on resource allocation for OFDM wireless systems in the downlink direction [1]–[5]. However, the algorithms in these literatures are all developed without considering implementation issue. In practical OFDM wireless systems, such as long term evolution (LTE), for the purpose of signaling cost reduction and device simplicity, the subcarriers assigned to one user always use the same modulation and coding schemes (MCS). To the best of our knowledge, there is few published work related to this area. Liang et al. in [6] introduced blockwise resource allocation problem, and simply assumed that the subcarriers grouped into a block have the same channel gain. This scheme ignores frequency-selectivity and thus neglects frequency diversity gain. Reference [7] and reference [8] focused on resource allocation in LTE cellular networks and tried to maximize the system throughput without fairness consideration.

O

Manuscript received November 30, 2012. The associate editor coordinating the review of this letter and approving it for publication was D. Popescu. Z. Ren and B. Hu are with the State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, China (e-mail: [email protected]). S. Chen and W. Ma are with the State Key Laboratory of Wireless Mobile Communications, China Academy of Telecommunications Technology (CATT), Beijing, China. This work is financially supported by the National High-Technology Program of China (863) (Grant No.2011AA01A101), and International S&T Cooperation: Joint Reaserch project (Gran No.2010DFB13020). Digital Object Identifier 10.1109/LCOMM.2013.031913.122706

In this paper, a downlink multiuser scenario in OFDM systems is considered and, perfect instantaneous channel information is assumed to be available at both the base station and the user side. Based on these channel state information, the resource allocation algorithm assigns a total of N subcarriers to K users, with total transmit power constraint P total . The objective of this paper is to allocate the subcarriers and power to achieve sum-rate capacity maximization. A set of proportional rate constraints {α1 , ..., αK } is also considered to address system fairness issue. According to the subcarrier grouping constraint, all subcarriers assigned to one user should use the same MCS. Since the same predetermined bit-error rate (BER) performance level should be achieved for each subcarrier in a group, these grouped subcarriers are required to have the same signal-tonoise-ratio (SNR) condition. Mathematically, using the Shannon capacity formula for Gaussian channel, the resource allocation optimization problem is formulated as K bits 1 sec log2 (1 + pn,k gn,k ) maximize N Hz k=1n∈Dk

subject to

K

pn,k ≤ Ptotal

k=1n∈Dk

pn,k ≥ 0 for all k, n D1 ∪ D2 ... ∪ DK ⊆ {1, ..., N } Di ∩ Dj = φ for i = j and i, j ∈ {1, ..., K} r1 : r2 : ... : rK = α1 : α2 ... : αK ,

(1)

where Dk is the set of subcarriers assigned to user k, and pn,k is the amount of power allocated to user k on subcarrier n. For simplicity, we let g n,k denote the channel-gain-to-noise-ratio B (CNR) h2n,k /N0 N , where hn,k and N 0 are channel gain and

c 2013 IEEE 1089-7798/13$31.00

REN et al.: PROPORTIONAL RESOURCE ALLOCATION WITH SUBCARRIER GROUPING IN OFDM WIRELESS SYSTEMS

TABLE I P RELIMINARY S UBCARRIER A SSIGNMENT

TABLE II POWER ALLOCATION

Initialization: set λlow , λhigh as (14) and (15), and tolerance 1: Repeat calculate λnew = (λlow + λhigh ) /2; If f (λnew ) > Ptotal update λhigh ← λnew ; Else update λlow ← λnew ; End If Until | f (λnew ) − Ptotal |≤ 2: calculate Pk with λnew based on (9) and (10)

Initialization: set D = {1, ..., N }, and Dk = φ for k ∈ {1, ..., K} 1: For user k ∈ {1, ..., K} j = argmaxn {gn,k | n ∈ D}; update Dk ← Dk + {j} and D ← D − {j}; End For 2: While D = φ k = argmini {ri /αi | i ∈ {1, ..., K}}; j = argmaxn {gn,k | n ∈ D}; calculate rk based on (6); If rk > 0 update Dk ← Dk + {j} and D ← D − {j}; update rk based on (5); Else break; End If End While

noise power spectral density of AWGN, respectively. B/N is the subcarrier bandwidth with system bandwidth B divided by subcarrier number N . The fourth constraint shows that each subcarrier can only be assigned to one user, and the last equality presents the proportional data rate requirements, where rk is expressed as 1 log2 (1 + pn,k gn,k ) , (2) rk = N n∈Dk

with pn,k gn,k = pm,k gm,k for m, n ∈ Dk

(3)

indicating the subcarrier grouping constraint aforementioned.

III. P RELIMINARY S UBCARRIER A SSIGNMENT AND P OWER A LLOCATION S CHEME A. Preliminary Subcarrier Assignment For user k, we intuitively find that if the subcarrier set Dk and the user-level power Pk are fixed, the subcarrierlevel power allocation can be determined directly from the subcarrier grouping constraint (3) as follows: pn,k =

gn,k

P k

1 i∈Dk gi,k

for n ∈ Dk .

Therefore, user rate rk can be calculated as Sk Pk rk = log2 1 + , 1 N n∈Dk g

869

this user, we can avoid the suboptimal bad-subcarrier-taking event caused in (5) to a certain extent. As a result, the simple solution (5) can be used instead, which only needs a direct calculation. Hence, we are motivated to develop a similar subcarrier assignment scheme with that of [2], and equal power distribution across all subcarriers Ptotal /N is also assumed. The algorithm is shown in detail in Table I. In the first step each user chooses its best subcarrier, and then in the second step, we ensure that the user whose proportional rate is minimum can get the assignment opportunity. The best subcarrier for the chosen user would be assigned if the updated rate can be enhanced. This choose-and-update policy is repeated until there is no subcarrier left or the chosen user cannot get further rate increment. Calculation of the rate increment rk is given as follows: ⎛ ⎞ + 1) P (Sk + 1) (S k total ⎠ log2 ⎝1 + Δrk = 1 1 N N + n∈Dk gn,k gj,k Sk Ptotal Sk , (6) − log2 1 + 1 N N n∈Dk gn,k where j denotes the index of the subcarrier to be assigned for user k. B. Optimal Power Allocation Scheme

(4)

Since the subcarrier-level power allocation can be directly determined by the user-level power based on (4), the original optimization problem (1) with the determined subcarrier assignment turns into a user-level power allocation as follows:

(5)

n,k

where S k denotes the cardinality of Dk . Notice that this rk ∼ Pk approach is by no means optimal, since all subcarriers are assumed to obtain power allocation. In order to achieve maximum data rate, the relatively deepfading subcarriers may be dropped, as waterfilling method does. However, the optimization problem is non-convex and much more complex. Moreover, in the greedy-based subcarrier assignment proposed later, expensive computation would happen. Nevertheless, in a multiuser scenario, if the best subcarrier is always assigned to the user during subcarrier assignment, i.e., the assigned subcarrier has the highest channel gain for

K Sk log2 (1 + Pk Hk ) N

maximize

k=1 K

subject to

Pk ≤ Ptotal , Pk ≥ 0 for all k

k=1

r1 : r2 : ... : rK = α1 : α2 ... : αK , (7)

1 can be regarded as the equivalent where Hk = 1/ n∈Dk gn,k CNR for user k, and rk is expressed as (5). For proportional rate constraints, we introduce an intermediate variable λ, which is defined as r1 r2 rK = = ... = = λ. (8) α1 α2 αK

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IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 5, MAY 2013

K , and Hence, sum-rate capacity is reformed as λ α k k=1 objective of (7) is converted into seeking the maximum λ. On the other hand, power requirement for user k can be expressed based on (5) as 2βk − 1 , (9) Pk = Hk where βk =

λN αk . Sk

TABLE III ENTIRE ALGORITHM

Initialization: set D = {1, ..., N } 1: execute Preliminary Subcarrier Assignment and Power Allocation to get initial subcarrier set Dk and user power level Pk , for k ∈ {1, ..., K} 2: Repeat For user k ∈ {1, ..., K} j = argmaxn {gn,k | n ∈ D − ∪k Dk }; calculate rk base on (16) If rk > 0 update Dk ← Dk + {j}; End If End For update Pk via Power Allocation; Until D = ∪k Dk or Dk doesn’t change 3: calculate Pn,k with final Pk based on (4)

(10)

Hence, with the following definition f (λ) =

K 2

λN αk Sk

Hk

k=1

−1

,

(11)

the optimization problem (7) can be rewritten as follows: maximize

λ

subject to

f (λ) ≤ Ptotal .

10

8 Sum−rate Capacity (bits/s/Hz)

Obviously, f (λ) is a monotonously increasing function of λ. Therefore, (12) is equivalent to seeking λ as follows: f (λ) = Ptotal .

(13)

This is a one-dimensional root-finding problem, for which the bisection method can be used to find the optimal solution. To speed up the searching process, we set the initial point as

/K) λlow = min rk (Ptotal , for k ∈ {1, ..., K} (14) αk

/K) λhigh = max rk (Ptotal , for k ∈ {1, ..., K} , (15) αk where rk (Ptotal /K) denotes rate of user k with power level of Ptotal /K. The bisection-based power allocation scheme is described in Table II: after the optimal λ found by the bisection method, user power is calculated by (9) and (10). IV. E NTIRE I TERATIVE A LGORITHM In the preliminary subcarrier assignment scheme, when the chosen user experiences severe deep-fading on all the left subcarriers, its data rate may decrease with further assignment. In this case, the algorithm would stop, and both the subcarriers and power would be left. Intuitively, it is can be explained by the fact that the current power level prevents the user from utilizing more bandwidth resource, and makes this situation happen. Notice that, this power level results from the equal power distribution we assumed in the preliminary subcarrier assignment. On the other hand, the proposed bisection-based power allocation surely may enhance power level for some users. Hence, further subcarrier assignment is possible. In this context, an entire iterative algorithm is proposed in Table III. As shown in the table, firstly, the preliminary subcarrier assignment and the power allocation scheme are applied to get initial subcarrier set and user power level respectively. Secondly, users are checked to decide if additional subcarriers can be assigned to increase user rate. If the data rates can be further enhanced for some users, the corresponding subcarriers would be assigned and the power allocation would be executed again to ensure the proportionality. The second step is repeated until there is no extra subcarrier can be taken, or no subcarrier left in the system. Finally, the subcarrier-level power allocation

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is directly obtained via (4). Suppose that the temporary power level for user k is Pk during the iteration, the rate increment calculation in the algorithm is given as follows: Sk + 1 Pk log2 1 + Δrk = 1 1 N n∈Dk gn,k + gj,k Pk Sk . (16) − log2 1 + 1 N n∈Dk g n,k

To explain why this iterative algorithm can increase sum-rate capacity, we give a simple analysis below. Proof: Without loss of generality, consider the ith iteration, and suppose that user k can be assigned an additional subcarrier. Hence, with the updated subcarrier set Dk , data rate of user k increases and is thus proportional maximum. From the perspective of power consumption, it means that if we keep the current proportional rate level, user k would need less power. Therefore, through the (i + 1)th iteration, the marginal power of user k will be reallocated among all the users to reach a higher proportional rate level, and thus the sum-rate capacity is increased. V. S IMULATIONS A. Simulation Environment In simulations, we assume that each user’s subcarrier signal undergoes identical Rayleigh fading independently. The

REN et al.: PROPORTIONAL RESOURCE ALLOCATION WITH SUBCARRIER GROUPING IN OFDM WIRELESS SYSTEMS

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2 = 1, average channel gain is normalized such that E gk,n for all k and n. We define average SNR as Ptotal /N0 B, and adopt Monte Carlo simulation method to obtain the average data from 1,000 channel realizations. As the comparison schemes, following methods are considered: 1) Optimal: Optimal solution for the problem discussed in this paper, which is obtained through full enumeration. It is adopted as the upper bound of the proposed algorithm. 2) MaxThr: The method in [8] to maximize throughput without fairness consideration. In simulations it is directly applied in subcarrier level instead of resource block level. 3) PSA: The preliminary subcarrier assignment scheme. It achieves a coarse tradeoff between capacity and fairness. For all methods, proportional rate constraint is set to 1:1, and 10 subcarriers and 2 users are set to reduce the computational time. B. Simulation Results Figure 1 depicts the sum-rate capacity, as a function of average SNR. As shown in the figure, the MaxThr method reaches the highest capacity. This is reasonable, since in the MaxThr method no fairness issue is considered. It can also be seen that as average SNR increases, the proposed algorithm achieves capacity close to the optimum. Though in the low SNR region, the proposed scheme and the PSA scheme reach a similar level, whereas in the high SNR region, the proposed scheme strictly outperforms the latter one.

User data rates as expressed in (5) are shown in Figure 2. User rate averaged over 1,000 channel realizations and instantaneous user rate sampled from the 500th channel realization are both given to consider long-term and instantaneous behavior respectively. It is clear that the proportional rate constraints are ensured precisely in the proposed algorithm. On the other hand, it should be noted that, the reason why the average rates between the 2 users are similar in the MaxThr method is that the long-term channel conditions are similar in simulations. While for instantaneous rate, both the MaxThr method and the PSA scheme have significant discrepancy between the 2 users, and the case is even worse in the MaxThr method, since fairness issue is not taken into account. VI. C ONCLUSION From the implementation point of view, proportional resource allocation with subcarrier grouping for OFDM systems is investigated in this paper. Firstly, the preliminary subcarrier assignment and simple bisection-based power allocation scheme are proposed, and then, the entire iterative algorithm is developed to utilize bandwidth resource effectively. Compared with the throughput maximization method in [8], the proposed scheme distributes resource among users more flexibly with proportional rates being ensured precisely. Moreover, the proposed scheme outperforms the simple scheme where only PSA is used, and as SNR increases, the proposed scheme can also achieve sum-rate capacity close to the optimum. R EFERENCES [1] C. 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. 1747–1758, 1999. [2] Z. Shen, J. Andrews, and B. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2726–2737, 2005. [3] H. Yin and H. Liu, “An efficient multiuser loading algorithm for OFDMbased broadband wireless systems,” in Proc. 2000 IEEE Global Telecommunications Conference, vol. 1, pp. 103–107. [4] D. Kivanc, G. Li, and H. Liu, “Computationally efficient bandwidth allocation and power control for OFDMA,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1150–1158, 2003. [5] G. Song and Y. Li, “Cross-layer optimization for OFDM wireless networks—part I: theoretical framework,” IEEE Trans. Wireless Commun., vol. 4, no. 2, pp. 614–624, 2005. [6] L. Xiaowen and Z. Jinkang, “An adaptive subcarrier allocation algorithm for multiuser OFDM system,” in Proc. 2003 IEEE Vehicular Technology Conference – Fall, vol. 3, pp. 1502–1506. [7] R. Kwan, C. Leung, and J. Zhang, “Resource allocation in an LTE cellular communication system,” in Proc. 2009 IEEE International Conference on Communications, pp. 1–5. [8] J. Fan, Q. Yin, G. Li, B. Peng, and X. Zhu, “Adaptive block-level resource allocation in OFDMA networks,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3966–3972, 2011.