A Distributed Power Allocation Algorithm with Inter ... - Semantic Scholar

A Distributed Power Allocation Algorithm with Inter-cell Interference Coordination for Multi-cell OFDMA Systems Gangming Lv, Shihua Zhu, and Hui Hui Department of Information and Communication Engineering, Xi’an Jiaotong University, Xi’an, 710049, P.R. China. Email:{gmlv, szhu, huihui}@mail.xjtu.edu.cn Abstract—Inter-cell interference is a serious problem in multicell OFDMA systems. In this paper, an iterative interference price based power allocation algorithm with inter-cell interference coordination is proposed for downlink multi-cell OFDMA systems. In the algorithm, each base station announces a price for interference and performs power allocation based on the price information received from other base stations with the objective of maximizing the net utility of the local cell. Through price information updating and transmit power reallocation in every iteration, the algorithm can intelligently coordinate the intercell interference in the system. Besides, the algorithm can be implemented distributedly with price information shared among neighboring cells. Considering the heavy feedback overhead from the users to the base stations, a simplified algorithm which considers only the major interference is also proposed. Simulation results show that when the inter-cell interference is serious, the proposed algorithms can effectively increase the transmission rate compared with iterative water-filling.

I. I NTRODUCTION Due to its strong potential in dealing with impairments and uncertainties of wireless channels, orthogonal frequency division multiple access (OFDMA) has been accepted as one of the most promising multiple access techniques for future wireless communications. Besides, its multi-carrier nature provides a high degree of flexibility for adaptive resource allocation. In the last decade, many works have been done on intelligent resource allocation in OFDMA systems so as to fully take advantages of OFDMA. However, most of the prior studies have focused on the single cell scenario (see in [1]–[3] and references therein), and can not be applied directly into multicell systems due to inter-cell interference (ICI). ICI is an inherent problem in cellular OFDMA systems. With ICI, any change of resource allocation in a specific cell will affect the performance of neighboring cells. One possible way to reduce the influence of ICI is to use a large frequency reuse factor. However this may result in low spectrum efficiency. In [4] and [5], fractional frequency reuse is proposed to increase the spectrum efficiency while reducing the ICI. However, user group division in the algorithm reduces the freedom of subcarrier allocation. In [6], interference range detection is introduced into multi-cell channel allocation where no ICI is allowed in each user’s interference range. Aiming at maximizing the total transmission rate, a centralized multi-cell resource allocation is studied in [7] where ICI is not allowed only when it cannot bring any positive contribution to the total

system throughput. However, the result of the algorithm relies heavily on the allocation order of neighboring cells and is not unique. In [8] and [9], game theory is applied into multicell resource allocation for OFDMA networks with minimum transmission rate request. However, since no cooperation is assumed, they may not be optimal. Although cost of ICI is considered in the objective utility functions in [9], the constant interference price assumption is not reasonable for practical systems. In [10], ICI is coordinated more effectively by using adaptive interference prices in ad hoc networks but no subchannel allocation is considered therein. In this paper, we study the power allocation problem for downlink multi-cell OFDMA systems with full frequency reuse (FFR). Our objective is to maximize the total utility which is defined as the weighted sum rate of all users in the system. Through analysis of the necessary conditions of the optimum solution, we develop an iterative interference price based power allocation algorithm. In our algorithm, any allocated transmit power which may cause inter-cell interference will be charged with a certain price. The interference price is obtained from neighboring cells according to how serious the ICI may depress their performances. By dynamically updating the interference price in every iteration, the proposed algorithm can intelligently coordinate the ICI among neighboring cells. Furthermore, a simplified algorithm is also proposed to reduce the heavy feedback load. Simulation results show that the proposed algorithms can effectively reduce the negative influence of ICI when ICI is serious. The organization of this paper is as follows. Section II outlines the system model, followed by the formulation of the resource allocation problem. Then, a semi-distributed heuristic multi-cell resource allocation algorithm is deduced and simplified in Section III. In section IV, the proposed algorithms are simulated and compared with existing known approaches. Finally, we conclude in section V. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider the downlink of an M -cell S-subcarrier OFDMA system. Let B and S respectively denote the cell set and subcarrier set, let Um denote the user set in cell m. We assume all cells sharing a total bandwidth B. Therefore, inter-cell interference exists among neighboring cells. Let Am denote the set of neighboring cells that may cause interference

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

to cell m. Furthermore, we define pm = [pm k,s ]Km ×S as the power allocation strategy of BS m, where pm k,s denotes the transmit power on subcarrier s for user k and Km denotes the number of users in cell m. For each BS, we assume that the total transmission power is constrained to be less than pmax . In addition, we denote by p = [p1 , . . . , pM ] the network power allocation strategy and pm − the reduced network power allocation strategy by removing pm from p. Similar notations will be used for other quantities. m be the channel gain between user k and BS m on Let gk,s subcarrier s. Given the power allocation strategy p, the signalto-interference-plus-noise ratio (SINR) of user k in cell m on subcarrier s can be written as m (p) γk,s

m pm k,s gk,s = m m m n0 + Ik,s (p− ) + gk,s

m∗ M∗ μ1∗ 1,1 , . . . , μk,s , . . . , μKM ,S such that for ∀m ∈ B

wim

i∈Um −λ∗m

j∈Um ,j=k

pm j,s

,

(1)

m ∈Am

m ∈Am i∈Um

(5)

k∈Um s∈S m∗ μm∗ k,s pk,s = 0, ∀k

(6)

j∈Um

s∈S

log2 (1 +

s∈S

m (p) γk,s

max

Γ

).

(2)

ωkm rkm (p)

(3)

m∈B k∈Um

s.t.

m m m 2 ∂f (γi,s ∂f (γi,s ) ) (γi,s ) =− = m . m m m ∂Ii,s ∂γi,s pi,s gi,s

(7)

m Here ηi,s represents user i’s marginal increase in transmission rate by decreasing one unit interference on subcarrier s. With (7), we can easily derive that

m m m ∂f (γi,s (p)) (p)) ∂Ii,s ∂f (γi,s m m = m = −ηi,s (p)gi,s . m ∂pm ∂p ∂I k,s k,s i,s

For simplicity, we further define m m δsm ,m (p) = ωim ηi,s (p)gi,s ,

(8)

i∈Um

and

δsm (p) =

δsm ,m (p).

(9)

m ∈Am

where, Γ = − ln(5BER)/1.5 is the bit error rate (BER) gap. Our objective is to find the optimal power allocation strategy p to maximize the network utility which is defined as the weighted sum of data rate of all users in the system. Mathematically, the problem is given as follows

(4)

∈ Um , ∀s ∈ S.

m (p) ηi,s

m m gk,s ps is

power of BS m on subcarrier s. Assuming M-ary QAM is used, then the achievable data rate of user k per symbol is [11]

m ∂f (γi,s (p∗ )) ∂pm k,s

We define

the total interference to user k on subcarrier s generated by the = pm neighboring cells and pm s j,s is the total transmit

m f (γk,s (p)) =

m where, n0 is the noise power, Ik,s (pm −) =

wim

+ μm∗ k,s = 0, ∀k ∈ Um , ∀s ∈ S; pm∗ λ∗m ( k,s − pmax ) = 0;

rkm (p) =

m ∂f (γi,s (p∗ )) + m ∂pk,s

pm k,s ≤ pmax , ∀m ∈ B,

k∈Um s∈S

0 ≤ pm k,s , ∀m ∈ B, k ∈ Um , s ∈ S.

Here, δsm ,m (p) represents cell m ’s marginal increase in utility per unite power decreasing in BS m on subcarrier s, i.e., by decreasing one unit transmit power on subcarrier s in BS m, the total utility in cell m can increase by δsm ,m . Then, condition (4) can be rewritten as m ∂f (γi,s (p∗ )) wim − δsm (p∗ ) − λ∗m + μm∗ (10) k,s = 0. m ∂pk,s i∈Um

m For ∀m ∈ B, suppose that pm − and δs are fixed, then conditions (5), (6) and (10) are the necessary and sufficient optimality conditions for the following local optimization problem : m m ωkm f (γk,s (pm , pm δsm (pm , pm max − )) − − )ps s∈S k∈Um

s.t. III. S EMI -D ISTRIBUTED P OWER A LLOCATION S CHEME The objective function of problem (3) may not be concave in p because of ICI, and thus cannot be solved directly. However, it is easy to verify that the objective function is continuous and differentiable in p, thus any local optimum of this problem must satisfy the Karush-Kuhn-Tucker (KKT) necessary conditions [12]. Theorem 1 (KKT Conditions): for any local optimum p∗ = (p1∗ , p2∗ , . . . , pM ∗ ) of problem (3). There exist unique nonnegative Lagrange multipliers λ∗1 , . . . , λ∗M and

s∈S

pm k,s

k∈U (m) s∈S 0 ≤ pm k,s , ∀k

≤

pm max ;

(11)

∈ Um , ∀s ∈ S.

In the objective function of problem (11), the first part is the income, which is the total utility of cell m obtained by the power allocation strategy pm . The second part can be viewed as the cost, which is the utility loss of neighboring cells caused by the ICI generated by cell m, with δsm the total price charged by neighboring cells. Therefore, problem (11) equals to maximize the surplus of cell m. The analysis above motivates us to optimize the system utility through maximizing the net utility in each cell with


price information exchanged among neighboring cells. In the following, we firstly discuss the solution of the local problem (11). For the optimum solution of problem (11) we have the following theorem. Theorem 2 (Intra-cell exclusive subcarrier allocation): The optimum p∗m of problem (11) satisfies: for ∀s ∈ S and ks ∈ m∗ m∗ and pm∗ Um , if pm∗ ks ,s > 0, then pks ,s = ps k,s = 0, ∀k = ks ∈ Um , where ks is chosen as follows, ks = arg max(ωkm f ( k∈Um

m pm∗ s gk,s m (pm ) )). n0 + Ik,s −

(12)

Proof: Without loss of generality, denote the total transmit power on subcarrier s as pm∗ s , then problem (11) can be decoupled into several parallel optimization problems for different subcarriers. On subcarrier s, the optimization problem is: m pm k,s gk,s ωkm f ( 2 max m (pm ) + (pm∗ − pm )g m ) σ + Ik,s s − k,s k,s k∈Um

s.t.

−δsm pm∗ s m∗ m pm k,s ≤ ps andpk,s ≥ 0, ∀k ∈ Um .

(13)

k∈Um

Since δsm pm∗ is a constant, (13) is equivalent to the optimizas tion problem (5.12) in [13]. Following a similar argument we can prove the theorem. Theorem 2 indicates that for certain subcarrier, only one user which can produce the highest utility with the assigned power is allowed to transmit on it. According to Theorem 2 and KKT conditions (5),(6) and (10), we can futher derive the optimum transmit power for subcarrier s ∈ S as the following, + m m m + I (p )) Γ(n ω 0 − k ,s ks s − = pm∗ , pm∗ s ks ,s = (λm + δsm ) ln 2 gkms ,s (14) where λm is chosen to satisfy the total transmit power constraint. Obviously, (14) is a revised water filling problem where the water filling level is adjusted by the interference price δsm on each subcarrier. (12) and (14) give us the optimum solution for the local power allocation problem (11). Based on the two equations, we design an power allocation algorithm for the local cell optimization, which is shown in Algorithm 1. It is worth noting that since user selection and power allocation may mutually impact on each other, we assume equal power allocation on all subcarriers during user selection. After user is selected for each subcarrier, transmit power is reallocated according to (14). Algorithm 1 gives the local power allocation method for maximizing the net utility of each cell given that the interference price is known and fixed. Based on our previous observation that system utility can be optimized by maximizing the net utility in each cell, we propose a distributed interference price based power allocation (IPPA) algorithm for downlink OFDMA systems. The algorithm works iteratively, where in each iteration, each BS performs power allocation using algorithm 1 based on the interference price it received in the

Algorithm 1: Intra-Cell Power Allocation Algorithm for cell m ∈ B pmax m Initialization: pm k,s = 0, ps = S , ∀s ∈ S, k ∈ Um ; for s = 1 : S do m∗ m ps gk,s ks ← arg max(ωkm f ( n0 +I m (pm ) )); k∈Um

k,s

−

end for Perform power allocation according to (14); Algorithm 2: IPPA Algorithm Executed by Each BS m ∈ B m = 0, δsm ,m = 0, δsm,m = Initialization: pm k,s = 0, δs 0, ∀s ∈ S, k ∈ Um , m ∈ Am ; repeat Find the optimum Pm∗ for cell m using Algorithm 1; Transmit data for each user in cell m with the allocated power Pm∗ ; for m ∈ Am do Update the prices charged to cell m , δsm,m (s = 1, . . . , S), according to (8); m,m , . . . , δSm,m ) to BS Send price message Δm m = (δ1 m ; end for Once a message arrived, update the total price charged by other cells, δsm , according to (9); until no user is active in the cell;

last iteration. Then, interference price information is updated and exchanged among neighboring BSs. By making charges for ICIs with prices that can closely reflect their influences, the algorithm can intelligently coordinate the interferences in the system. The detail process of the IPPA algorithm is given in Algorithm 2. Since the local power allocation strategy in each cell pm (m = 1, . . . , M ) satisfies the KKT conditions assuming fixed power allocation in neighboring cells, the converge point of the proposed algorithm p will then satisfy the KKT conditions also. This means that p is at least one of the local optimum of problem (3). Especially, if the objective function of problem (3) is concave, then p is actually the global optimum. However, if the objective function is not concave, the converge point of the proposed method is not necessarily the optimal solution. As there is no effective way other than exhaustive search to find the optimum solution, the proposed algorithm provides an applicable method to find a suboptimal solution. In the IPPA algorithm, each BS decides the power allocation of its own cell and do not need to know the transmit power and channel state information (CSI) in other cells. However, it needs price information from neighboring cells. According to (8), in order to obtain the price information, each user has to feedback the CSI of its serving BS and the BSs which have interference on it. This will cause a heavy feedback load for each user. However, in practical cases, the downlink performance loss caused by ICI is mainly impacted by one or two dominant interfering BSs. This motivates us to reduce the feedback by considering only the major interferences. Redefine


IPPA: γ = 110dB, α=3

Iterative WF: γ = 110dB, α=3

3

1.75

Total power (w)

Total power (w)

2.5 2 1.5 1 0.5 0 0

Fig. 1.

10

20 30 Iterations

Fig. 2.

Simulation environment

40

50

1.7

1.65

1.6

1.55

0

10

20 30 Iterations

40

50

Convergence of the transmit power γ = 110dB, α=3

105

as

δsm ,m (p) =

m m ωim ηi,s (p)gi,s 1(i, m),

(15)

i∈Um

where, 1(i, m) = 1 if the interference from BS m is the major interference to user i in cell m and 1(i, m) = 0 otherwise. By replacing (8) with (15) in Algorithm 2, a simplified algorithm can be formulated. In this simplified algorithm, each user only has to track and feedback its channel state with the BS it is connected to and the BSs which have major interference on it. This dose not add any extra feedback, because same CSI is needed for non-blind handover [14]. IV. SIMULATION RESULTS Performances of the IPPA algorithm and the simplified IPPA algorithm which consider only the major interference (IPPA-MI) are illustrated in this section through simulations. Specifically, in the IPPA-MI algorithm only two major interferences are considered for each user. Besides, two other FFR based algorithms: iterative water-filling (WF) algorithm and RNC algorithm proposed in [7] are also simulated for comparison. In our simulations, hexagon cells with a base station located at the center of each cell are assumed. The propagation model assumes the operation in a suburban environment. The path loss (in dB) at distance d (in meter) from BS is P L(d) = 100 + 10α log(d/1000), where α is the path loss factor. Frequency selective fading is generated using COST207-TU model [15]. Time scale is divided into frames and channel state is assumed to be static during one frame. Each frame is further divided into slots and power allocation is performed at the beginning of every slot. In our simulations, 50 slots are assumed in each frame with every slot containing 20 OFDM symbols. Other parameters are assumed as follows: cell radius is 1km; the bandwidth B = 3.2M Hz; the number of subcarriers S = 32; the maximum power constraint is /S ; the BER gap Γ = 1; the weight normalized as γ pmax n0 m for each user ωk = 1, with which the utility is actually the transmission rate. The simulation scenario is shown in Fig.1, where 37 neighboring cells are considered in the system. In each cell, 15 mobile users are assumed and randomly located around the BS with same distances. Specifically, in the center cell the distance between users and the BS is denoted by d, while in other cells

Total transmission rate (w)

δsm ,m

100 95 90 85 80 75

IPPA Iterative WF 0

10

20

30

40

50

Iterations

Fig. 3.

Average transmission rate with respect to iterations

the distance is fixed to be 700 meters. To mitigate the edge effect, we only measure the performances of the central 7 cells which are marked by shadow. Fig. 2 shows the transmit power on a certain subcarrier s with respect to iterations for the 7 cells. In the figure, each curve corresponds to the total transmit power on subcarrier s for one cell. We can see that both the proposed IPPA algorithm and the iterative WF algorithm can converge quickly. However, the results are quite different. Especially, in the IPPA algorithm, the transmit power of one cell converges to 0 which corresponds to the case where our IPPA algorithm intelligently choose not to reuse the subcarrier when the ICI is too severe. From the corresponding performance demonstrated in Fig. 3, we can see that our IPPA algorithm outperforms iterative WF algorithm obviously in terms of transmission rate. This is due to the fact that by making proper charges for the interferences, ICIs between adjacent cells can be adaptively avoided or coordinated. Besides, the average transmission rate of the IPPA algorithm increases constantly with the iteration number until it converges, which indicates that every iteration is effective in improving the power allocation. Fig. 4 shows the average transmission rate of the four algorithms under different power constraint. Here, the path loss factor α = 3 and the distance d = 700 meters. For comparison, we also simulate the frequency reuse strategy with the reuse factor (RF) being 3 where no interference exists between neighbouring cells. We can see that the four FFR based algorithms outperform the algorithm with RF = 3 remarkably. This may indicate that full frequency reuse is more efficient for OFDMA networks. Besides, our IPPA algorithm


70

1.05

1

12 11 10 9 8 7

60 6

50

0.95

102

104

106

108 110 γ (dB)

112

114

116

IPPA α=2 IPPA−MI α=2 RNC α=2 IPPA α=3 IPPA−MI α=3 RNC α=3 IPPA α=4 IPPA−MI α=4 RNC α=4

25

20

15

10

5

0 100

Fig. 5.

102

104

106

108 110 γ (dB)

500 d (m)

1000

(a) Frequency reuse factor

Average transmission rate per cell under different power constraint

30

0

112

114

116

5

0

200

400 600 d (m)

800

1000

118

35

Normalized performance gain G (%)

Normalized performance gain G(%)

80

IPPA IPPA−MI RNC

13

1.1

90

40 100

Fig. 4.

IPPA IPPA−MI RNC Iterative WF Frequency reuse factor

Average transmission rate per cell (bits/symbol)

100

14

1.15

IPPA IPPA−MI RNC Iterative WF RF=3

110

α=3,γ=110dB

α=3,γ=110dB

α=3 120

118

Normalized performance gain with different path loss factor

and the simplified IPPA-MI algorithm achieve comparable performances with the central RNC algorithm and outperform the iterative WF algorithm obviously. Actually, as shown in the figure, when γ exceeds 112dB, the transmission rate of the iterative WF algorithm can hardly increase any more. This is because that when transmit power is high enough, transmission rate is mainly limited by the ICI but not the noise. In such situation, no improvement can be achieved by only increasing the transmit power. However, through ICI coordination, our algorithms can avoid this limitation. We can also see from the figure that the performance difference between IPPA and IPPA-MI is quite small in this setting. Fig.5 further illustrates the performance gain of our proposed algorithms under different path loss factor assumptions. In the figure, we define the performance gain G = (G1 − G2)/G2 where G1 is the average transmission rate of the interested algorithm and G2 is the average transmission rate of the iterative WF algorithm. We can see that the

Fig. 6.

(b) Normalized performance gain

Performances with respect to user distance

performance gain of our algorithms is evident. Besides, it is shown that the performance gain increases with transmit power and decreases with path loss factor. This is because that with larger transmit power or smaller path loss factor, ICI is more serious, consequently, the performance gain obtained through ICI coordination becomes more remarkable. Compared with RNC algorithm, the performance gain achieved by our algorithms are smaller when α = 2. However for α ≥ 3, which is always practical in most wireless environments, our algorithm can achieve comparable or even better performance. Another point worth noting is that the performance loss of IPPA-MI, caused by ignoring some non-dominant ICI, decreases with path loss factor. Specifically, when α ≥ 3, the performance loss is negligible. Therefore, the simplified IPPA-MI algorithm is applicable in practice. Fig. 6 shows the average frequency reuse factor and performance gain of the proposed algorithms with respect to different distances between the users and the BS in the center cell, d, through 100 independent simulations. We use the same performance gain definition as in Fig. 5. As shown in Fig. 6(a), the effective frequency reuse factor keeps almost static when d ≤ 400m and increases quickly when d ≥ 400m. This indicates that the proposed algorithm can adaptively change the frequency reuse strategy according to the ICI level. Compared with fixed frequency reuse strategy, the proposed algorithm is more flexible. Besides, according to the corresponding performance gain shown in Fig. 6(b), we can see that the dynamic frequency reuse of the proposed algorithms are effective in improving the system transmission rate. V. C ONCLUSIONS Power allocation for downlink multi-cell OFDMA systems with full frequency reuse is studied in this paper. Motivated by the necessary conditions for the optimum power allocation, we derive a distributed interference price based power allocation algorithm. In the algorithm, charges are made for ICIs with certain prices. Through dynamic interference price updating


and sharing among neighboring cells, the proposed algorithm can intelligently coordinate the ICI between adjacent BSs by optimizing the resource allocation in each cell. Besides, the algorithm can be implemented distributedly which is attractive to the future flat network. Considering the heavy feedback load required for each user in order to derive the accurate interference price, we also propose a simplified algorithm which considers only the major interference. Simulation results show that both algorithms can effectively coordinating the ICIs and further increase the throughput. However, fairness needs to be considered in future studies. ACKNOWLEDGEMENT This work is supported by the National Natural Science Foundation of China under Grant 60872028. R EFERENCES [1] C. Y. Wong, R. S. Cheng, K. B. Letaief, 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, 1999. [2] Y. J. Zhang and K. B. Letaief, “Multiuser adaptive subcarrier-and-bit allocation with adaptive cell selection for ofdm systems,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1566–1575, 2004. [3] G. Song and Y. Li, “Utility-based resource allocation and scheduling in ofdm-based wireless broadband networks,” IEEE Commun. Magazine, vol. 43, no. 12, pp. 127–134, 2005. [4] L. Haipeng, Z. Xin, and Y. Dacheng, “A novel frequency reuse scheme for multi-cell ofdma systems,” in VTC-2007 Fall, IEEE , pp. 347–351, 2007. [5] R1050507, Ed., Soft frequency reuse scheme for UTRAN LTE, Athens, Greece, http://www.3gpp.org, May 2005.

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