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Dynamic Power and Sub-carrier Allocation for OFDMA-based Wireless Multicast Systems Juan Liu∗ , Wei Chen† , Member, IEEE, Zhigang Cao† , Senior Member, IEEE and K. B. Letaief‡ , Fellow, IEEE E-mail: ∗ [email protected] † {wchen, czg-dee}@tsinghua.edu.cn ‡ [email protected]

Abstract—Dynamic resource allocation is a key technique that can significantly improve the performance of next generation wireless systems under guaranteed QoS to users. Most of the current resource allocation algorithms are, however, limited to unicast traffics. In practice, how to efficiently allocate various resources in multicast wireless systems is not known. In this paper, we shall study dynamic resource allocation for OFDMAbased single-cell multicast systems. Specifically, we shall formulate an optimization problem to maximize the system throughput given a set of available resources (power and sub-carriers). The optimal resource allocation solution is proposed along with a lowcomplexity algorithm. In two extreme cases, namely, low and high SNR regimes, the low-complexity allocation algorithm is further simplified. Numerical results will show that the system throughput is significantly improved by using our proposed algorithms.

I. I NTRODUCTION The next-generation wireless networks are expected to provide broadband multimedia services such as voice, web browsing, video conference etc. with diverse Quality of Service (QoS) requirements [1]. Multicast over wireless networks is an important and challenging goal oriented to many multimedia applications such as audio/video clips, mobile TV, interactive game [2]. There are two key traffics, namely, unicast traffics and multicast traffics, in wireless multimedia communications. Current studies mainly focus on unicast traffics. In particular, dynamic resource allocation has been identified as one of the most efficient techniques to achieve better QoS and higher system spectral efficiency in unicast wireless networks. Furthermore, more attention is paid to the unicast OFDM systems. Orthogonal Frequency Division Multiplexing (OFDM) is regarded as one of the promising techniques for future broadband wireless networks due to its ability to provide very high data rates in the multi-path fading environment [3]. Orthogonal Frequency Division Multiple Access (OFDMA) is a multiuser version of the popular OFDM scheme and it is referred to as multiuser OFDM sometimes. In multiuser OFDM systems, dynamic resource allocation always exploit multiuser diversity gain to improve system performance [4] [5]. The resource allocation in multiuser OFDM systems is cast into two types of optimization problems: 1) to minimize the overall transmit power with constraints on data rates or Bit Error Rates (BER) [5]; and 2) to maximize the system throughput with the total transmission power constraint [4] [6]. Various QoS and fairness constraints are also This work is supported in part by NSFC under No. 60472027 and NSFC/RGC joint funding under Grants No. 60618001 and N HKUST622/06.

included in the optimization problems [6] [7]. In general, these optimization problems are non-convex and computationallyintensive. To overcome this, low-complexity algorithms were developed to obtain suboptimal performance with low computational complexity. In [6], the adaptive subcarrier and power allocation problem was solved in two-stages: 1) suboptimal subcarrier allocation with coarse proportional fairness; and 2) optimal power allocation with precise proportional fairness. To the best of our knowledge, most dynamic resource allocation algorithms, however, only consider unicast multiuser OFDM systems. In wireless networks, many multimedia applications adapt to the multicast transmission from the Base Station (BS) to a group of users. These targeted users consist of a multicast group and they receive the data packets of the same traffic flow. The simultaneously achievable transmission rates to these users were investigated in [8] [9]. Recently scientific researches of multicast transmission in the wireless networks are paid more attention. For example, proportional fair scheduling algorithms were developed to deal with multiple multicast groups in each time slot in cellular data networks [10]. In this paper, we shall propose dynamic subcarrier and power allocation algorithms for OFDMA-based single-cell wireless multicast systems. In the proposed algorithms, the subcarriers and powers are dynamically allocated to the multicast groups. Our aim is to maximize the system throughput given the total power constraint. Assume that there are multiple multicast groups in a cell and each multicast group may contain a different number of users. The users included in the same multicast group are called co-group users and these can be located in different places in the cell. Superposition coding is not applied here due to the hardware limitation. The BS shall transmit data packets to each multicast group at one rate. If this data rate is higher than the maximum rate that a user’s mobile device can handle, the device is incapable of decoding any data. For simplicity, data packets of each traffic flow will be transmitted to the targeted users with the minimum one of their individual data rate so that all the users can receive and decode the data packets. The dynamic subcarrier and power allocation problem in this paper is formulated into a constrained optimization problem. The optimal allocation is obtained by solving a relaxed optimization problem that is convex. However, it will be shown that the computational complexity is huge for a multicast OFDM system with hundreds of subcarriers and users. Therefore, we shall propose some low-complexity subcarrier and power algorithms. Numerical results will show that the

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system throughput is greatly improved by using our proposed algorithms. Though fairness is an important issue for multicast systems [10], it is out of the scope of this paper. The rest of this paper is organized as follows. Section II presents the system model and formulates the optimization problem. The optimal resource allocation algorithm is described in Section III. Section IV studies the low-complexity resource allocation algorithms. In Section V and Section VI, the simulation results and conclusions are presented, respectively. II. S YSTEM M ODEL The block diagram of the considered multicast OFDM system is shown in Fig. 1. The BS, which is assumed to have perfect Channel State Information (CSI) of all the users, will carry out the power and subcarrier allocation in a centralized manner. In this model, G downlink traffic flows are transmitted to K users on M subcarriers. Without loss of generality, we assume that each user receives only one traffic flow each time. Clearly, these users are included in G multicast groups. Let Ki (i = 1, · · · , G) denote the user set of the ith multicast group corresponding to the ith traffic flow. Thus, the user set denoted by K = K1 ∪ K2 · · · ∪ KG contains all the users in the system. In this paper, we consider a general framework consisting of mere multicast groups as each unicast group can be represented by a multicast group with one user. The symbol |Kg |1 represents the number of users included in the gth multicast group. If |Kg | > 1, the gth group is a multicast group. Otherwise, it is a unicast group. Obviously, the conventional allocation scheme for unicast systems can be viewed as special case of our methodology. In this paper, each subcarrier is assumed to have an equal bandwidth denoted by B0 = B/M , where B is the total system bandwidth. By considering the limitations in hardware, superposition coding is not applied in this paper. Every user in a multicast group is assumed to receive the desirable data packets with the same data rate. Therefore, it must be no more than the minimum one of the users’ data rates on the specified subcarriers. As it is well known that the maximum data rate of user k receiving the gth traffic flow on the mth subcarrier can be represented by |hk,m |2 pm /B, ∀k ∈ Kg (1) ck,m = B0 log2 1 + N0 B0 where N0 represents the single-sided power spectral density of the white noise, |hk,m | is the channel coefficient of user k of the gth group on the mth subcarrier, pm denote the amount of power transmitted on the mth subcarrier. Then, the data rate of the gth traffic flow carried on the mth subcarrier may be equal to the minimal one in (1) that can be denoted as rg,m = mink∈Kg (ck,m ). The equivalent channel gain of the gth group on the mth subcarrier is exactly the channel gain of the user with the worst CSI denoted by αg,m = mink∈Kg |hk,m |2 . It can be regarded as the individual channel gain of all the users in the gth group on the mth subcarrier as they are assumed to 1 Throughout

this paper, |X | denotes the cardinality of a set X .

Dynamic Subcarrierand-power Allocation

Subcarrier Selector and AGC

OFDM Transceiver

OFDM Transceiver

Base Station

User k receiver

Traffic flow 1 Traffic flow 2

Data packets of traffic flow g

Traffic flow G

Fig. 1.

Block diagram for multicast OFDM systems

have the same data rate in this paper. Hence, αg,m pm /B, ∀k ∈ Kg rg,m = rk,m = B0 log2 1 + N0 B0

(2)

Define the aggregate data rate of the gth flow on the mth subcarrier as the summation of all the data rates in (2). Thus, it is given by Rg,m = k∈Kg rk,m = |Kg | · rg,m . Definition 1: The subcarrier allocation indicator is denoted by ρg,m . If the mth subcarrier is allocated to the gth flow, then ρg,m = 1. Otherwise, ρg,m = 0. The throughput of the gth flow can be defined as the summation of Rg,m over all the subcarriers represented by M Rg = m=1 ρg,m Rg,m . Let us define the set Ωg as the set containing all the subcarriers allocated toflow g. The throughput Rg can also be written as Rg = m∈Ωg Rg,m . Define the system throughput C as the summation of the G throughputs of all the flows, then, C = g=1 Rg . Our objective is to maximize the system throughput C by jointly optimizing the subcarrier allocation ρg,m and power allocation pm . The optimization problem is formulated as follows M G αg,m pm |Kg |B0 ρg,m log2 1 + ρg,m , pm B N0 B0 g=1 m=1 max

M

(3.a)

pm ≤ Pmax , pm ≥ 0

(3.b)

ρg,m = 1, m = 1, · · · , M

(3.c)

m=1 G g=1

ρg,m ∈ {0, 1}

(3.d)

Constraint (3.b) corresponds to the transmission power limitation. Constraint (3.c) means that all the subcarriers should be allocated. Constraint (3.d) shows that no subcarrier can be shared by multiple groups. III. O PTIMAL R ESOURCE A LLOCATION In this section, the Optimal Subcarrier and Power Allocation (OSPA) algorithm is derived and is studied. Clearly, (3) is a NP-hard problem. As a result, exhaustive searching algorithms can hardly be applied in practice. To solve this problem, we shall convert it into a convex optimization problem by relaxing


the integer constraint for the variables ρg,m in problem (3). As a result, we get min

ρg,m , pm

s.t.

−             

M G

αg,m pm |Kg | ρg,m log2 1 + M N0 B0 g=1 m=1 M m=1 G g=1

> ρg1 ,m Rg1 ,m +

pm ≤ Pmax , pm ≥ 0

ρg,m ≥ 0

f (ρg,m , pm ) = ρg,m log2 (1 + Hg,m pm )

The gradient of f (ρg,m , pm ) is

f (ρg,m , pm ) =

(5)

log2 (1 + Hg,m pm )

H

f (ρg,m , pm ) =

0 µg (pm ) log2

µg (pm ) log2 ρg,m (µg (pm ))2 − log2

(8)

ρgi ,m Rgi ,m

As a result, the total system throughput can be improved further. This contradicts the assumption that x is an optimal solution to (4). Theorem 1: x∗2 in Lemma 2 is exactly the optimal solution of (3) denoted by x∗1 . Proof: Denote the feasible set of (3) and (4) as S1 and S2 , respectively. The optimal value of (3) and (4) are denoted as f1∗ and f2∗ . It can be seen that S1 is a subset of S2 represented by S1 ⊆ S2 . We must have f1∗ f2∗ . From Lemma 2, we have x∗2 ∈ S1 . This in turn results in f2∗ f1∗ . Therefore, f2∗ and f1∗ are equal. Correspondingly x∗2 is exactly the optimal solution of (3) denoted by x∗1 . The global optimum solution of (4) is obtained by using the Reduced Gradient (RG) [11] method. However, this is still computationally complex, especially when there are hundreds of users and subcarriers in the system. To solve this problem, we propose low-complexity allocation algorithms as described in the next section.

(6)

Hg,m ρg,m log2 1+Hg,m pm

g,m By substituting 1+Hg,m pm with µg (pm ), then the Hessian matrix of f (ρg,m , pm ) is given by

2

L i=2

(4)

ρg,m = 1, m = 1, · · · , M

αg,m N0 B0 .

Rm = Rg1 ,m ≥ ρg1 ,m Rg1 ,m + (1 − ρg1 ,m ) max Rgi ,m i=2,··· ,L

where ρg,m ∈ [0, 1] are real numbers for all g and m. Lemma 1: The optimization problem (4) is convex with at least an optimal solution. Proof: Now, we prove (4) is a convex optimization problem. The constraints consist of linear equations and inequalities, so the feasible set is convex. The convexity of the objective function is shown as follows. First, we consider

where Hg,m = calculated as

in the following that

(7)

As ρg,m and µg (pm ) are nonnegative, the Hessian matrix of f (ρg,m , pm ) is negative semi-definite. Hence, f (ρg,m , pm ) is concave and −f (ρg,m , pm ) is convex. So the objective function is a summation of a set of convex functions. Therefore, problem (4) is convex. As a result, there always exist at least one optimal solution. Lemma 2: There always exists an optimal solution x∗2 ∗ (ρg,m and p∗m ) for (4) which satisfies ρ∗g,m ∈ {1, 0}. Proof: If all the subcarrier allocation indicators ρg,m of an optimal solution x to (4) belong to the binary set {1, 0}, Lemma 2 is proved by setting x∗2 = x. Otherwise, there must have multiple ρgi ,m ∈ (0, 1) (i = 1, · · · , L) satisfying equation (3.c). This means that the mth subcarrier is shared by multiple flows (g1 , · · · , gL ). If the throughput of all the flows is exactly the same on the mth subcarrier, randomly assigning this subcarrier to one of the flows makes no difference to the total system throughput. Thus, we obtain another optimal solution x∗2 that satisfies Lemma 2. Otherwise, the throughput of some flows must be greater than that of others. Without loss of generality, the throughput of the g1 th flow on this subcarrier is assumed to be the largest one. Therefore, allocating the mth subcarrier exclusively to the g1 th flow will result in an improvement of the throughput on this subcarrier. It is proved

IV. L OW-C OMPLEXITY R ESOURCE A LLOCATION A LGORITHMS In this section, three low-complexity allocation algorithms based on heuristic methods are proposed. The algorithms are divided into two steps: Dynamic subcarrier allocation based on equal power allocation is performed first followed by optimal power allocation on the pre-determined best subcarrier allocation. A. Step 1 In this subsection, we first propose a dynamic subcarrier allocation algorithm. Two subcarrier allocation algorithms are then developed under extreme cases. In the proposed algorithms, the total power is equally allocated to the subcarriers so as to simplify the allocation mechanism. In this context, the aggregate data rate of the gth flow on the mth subcarrier can be represented by αg,m Pmax Rg,m = |Kg | log2 1 + N0 B0 M (9) |Kg | αg,m Pmax = log2 1 + N0 B0 M Let γ denote the Signal-to-Noise Ratio (SNR) product term in (9) is expanded as: ηg,m = (1 + γαg,m )

Pmax /M N0 B0 .

The

|Kg |

= 1 + |Kg |γαg,m +, · · · , +(γαg,m )

|Kg |

(10)

A dynamic subcarrier allocation algorithm is described as follows. We shall refer to it as Equal Power-based Subcarrier


18

Allocation (EPSA). For the mth subcarrier, EPSA will compare Rg,m for all the flows (g = 1, · · · , G) and allocate the subcarrier to the flow with the maximal aggregate data rate. In practice, EPSA will compare ηg,m instead of Rg,m as the comparison result is equivalent.

spectrum efficiency (bit/s/Hz)

16

Algorithm 1 EPSA 1: Initialization 2: Set Ωg = φ and ρg,m = 0 for ∀ g, m 3: for m = 1 to M do ∗ 4: gm = argg max ηg,m ∗ = Ωg ∗ ∪ { m } 5: Ωgm m ∗ ,m = 1, 6: ρgm ρg=ρg∗ ,m=0 m 7: end for

12 10 8 6 4 2 0 0

Fig. 2.

Next we consider two extreme scenarios, namely, low and high SNR regimes. In (10), γ is close to zero for very low SNR so that the higher-order terms can be ignored. In this context, (10) can be represented by L = 1 + γ|Kg |αg,m ηg,m = 1 + ηg,m

(11)

We shall refer to EPSA for Low SNR as LEPSA. In this L in (11) instead of ηg,m in (10). In high case, we compare ηg,m SNR, when γ is large enough compared with the expectation |K | of αg,m , the highest-order term (γαg,m ) g dominates the product. For high SNR, (10) can also be represented by H = 1 + (γαg,m ) ηg,m = 1 + ηg,m

14

OSPA unicast LcSPA unicast OSPA unicast-multicast LcSPA unicast-multicast OSPA multicast LcSPA multicast

|Kg |

(12)

We shall refer to EPSA for High SNR as HEPSA. In this H in (12) is compared. If |Kg | = 1 for all the algorithm, ηg,m flows, the multicast OFDM systems in this paper regress to the conventional unicast OFDM systems. In this case, ηg,m equals to 1 + γαg,m for (10)-(12). Therefore, these three algorithms are equivalent for the unicast multiuser systems. Their computational complexity is very low as they are carried out through linear operations. B. Step 2 Given ρ∗g,m from Step 1, the optimization problem (3)-(4) is transformed as M ∗ | αg∗ ,m pm |Kgm log2 1 + m max pm M N0 B0 m=1 (13) M pm ≤ Pmax , pm ≥ 0 s.t. m=1

The solution to the power allocation problem (13) can be found by using the Lagrange multiplier technique. We define the Lagrangian function as M M ∗ | αg∗ ,m pm |Kgm log2 1 + m +λ( pm − Pmax ) L= M N0 B0 m=1 m=1 (14)

5

10 15 average SNR (dB)

20

Performance comparison between OSPA and LcSPA.

where λ is a Lagrange multiplier and the solution of problem (13) can be obtained by solving ∂L/∂pm = 0. Consequently, the transmission power for each subcarrier should satisfy ∗ | ∗ ,m |Kgm αgm ∂L = + λ = 0, ∀m. ∗ ,m pm ∂pm M log 2 N0 B0 + αgm

(15)

The amount of power pm allocated to subcarrier m can be represented by ∗ | |Kgm N0 B − , 0 (16) pm = max λ0 αg,m The power allocation scheme is similar to the conventional water-filling rule [12] except that the water-level in (16) may M = be different for the subcarriers. From ∂L m=1 pm − ∂λ = 0, the optimal power allocation in (13) satisfies P max M p = P . λ in (16) is determined by substituting max 0 m=1 m all the pm into the constraint equation. The problem (13) is also a convex optimization problem and this can be proved in the same way as that in Section III. We can adopt the RG method to obtain the optimal power allocation as well. By combing the above two steps, three low-complexity subcarrier and power allocation algorithms are developed. We shall refer to them as LcSPA, L-LcSPA and H-LcSPA, respectively. For any of the three algorithms, the overall complexity is very low compared to that of OSPA due to the linear operation based subcarrier allocation. V. N UMERICAL R ESULTS In this section, some simulation results are presented to demonstrate the potential of the proposed algorithms. Here, the sub-channels are assumed to undergo i.i.d. flat Rayleigh fading and the average channel gain, E(|hk,m |2 ), is assumed to be one. The noise power of every subcarrier is assumed to be one for simplicity. A. Comparison of OSPA and LcSPA We consider an OFDM system with 10 subcarriers and 5 users among which 3 ones desire to receive the same contents. Fig. 2 shows the performance comparison between OSPA and LcSPA for the average SNR ranging from 0dB to


60

40

LcSPA unicast L-LcSPA unicast H-LcSPA unicast LcSPA multicast L-LcSPA multicast H-LcSPA multicast

40 spectrum efficiency (bit/s/Hz)

spectrum efficiency (bit/s/Hz)

50

45

30

20

35 30 25

EPSA unicast LcSPA unicast FSPA unicast EPSA multicast LcSPA multicast FSPA multicast

20 15 10

10 5

0 0

Fig. 3.

5

10 15 average SNR (dB)

0 0

20

Performance comparison between L-LcSPA/H-LcSPA and LcSPA.

24dB when the specific contents are delivered to the 3 users by unicast/multicast transmission, or multicast transmission to 2 users and unicast transmission to the left one. The very slight gap between OSPA and LcSPA implies that LcSPA is able to achieve a near optimal solution. It also shows that the spectrum efficiency is improved about 2bit/s/Hz with multicast transmission to one more user for SNR around 15dB. B. Comparison of L-LcSPA/H-LcSPA and LcSPA Then we consider an OFDM system with 256 subcarriers and 32 users among which 16 users are attempt to receive the same contents. Fig. 3 shows the performance comparison between L-LcSPA /H-LcSPA and LcSPA for an average SNR ranging from 0dB to 24dB when the specific contents are delivered to the 16 users by unicast or multicast transmission. It is observed that the performance of L-LcSPA and H-LcSPA is near to that of LcSPA for very small SNR and for SNR values larger than 21dB respectively with multicast transmission applied. It is also observed that these three algorithms are equivalent in the case of unicast transmission. C. Comparison of EPSA and LcSPA Fig. 4 shows the performance comparison of FSPA, EPSA and LcSPA for an average SNR ranging from 0dB to 21dB. The simulation parameters are exactly the same as the last simulation. As is seen, the system performance is significantly improved by LcSPA compared to EPSA when multicast transmission is applied. However, there almost has no gap between the curves for LcSPA and EPSA in the case of unicast transmission. This implies that the optimal power allocation on each subcarrier should satisfy Eqn. (16) rather than the conventional water-filling rule. The Fixed Subcarrier and optimal Power Allocation (FSPA) algorithm first assigns almost an equal number of subcarriers to the multicast groups, then employs the optimal power allocation scheme described in Section IV-B. Compared to the performance with FSPA, more performance improvement is obtained in the multicast OFDM system than the unicast OFDM system by using our proposed algorithms (LcSPA/EPSA). This implies that dynamic resource allocation may play a more important role in the multicast OFDM systems.

Fig. 4.

5

10 average SNR (dB)

15

20

Performance comparison of FSPA, EPSA and LcSPA.

VI. C ONCLUSION This paper considered dynamic subcarrier and power allocation for single-cell multicast OFDM systems. We formulated an optimization problem to maximize the system throughput with a total transmission power constraint. OSPA is obtained by solving this optimization problem. Moreover, low-complexity resource allocation algorithms (LcSPA/LLcSPA/H-LcSPA) are proposed to reduce the computational complexity. Simulation results have shown that LcSPA can perform almost as well as OSPA. However, in contrast to OSPA, LcSPA has a much lower computational complexity. Compared with LcSPA, its modified algorithms (L-LcSPA/HLcSPA) perform well in the low and high SNR regimes. It is also proved that the system throughput is significantly improved by using our proposed algorithms. R EFERENCES [1] S. Y. Hui and K. H. Yeung, “Challenges in the migration to 4g mobile systems,” IEEE Commun. Mag., vol. 41, pp. 54–56, December 2003. [2] U. Varshney, “Multicast over wireless networks,” Communications of the ACM, vol. 45, pp. 31–37, Dec. 2002. [3] L. J. Cimini and N. R. Sollenberger, “OFDM with diversity and coding for advanced cellular internet services,” in Proc. Globecom’1997. IEEE, Nov. 1997, pp. 305–309. [4] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser ofdm systems,” IEEE J. Select. Areas Commun., vol. 21, no. 2, February 2003. [5] C. Y. Wong, R. S. Cheng, K. B. Letaief, et al., “Multiuser ofdm with adaptive subcarrier, bit, and power allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747–1758, Oct. 1999. [6] 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. [7] Y. J. Zhang and K. B. Letaief, “Cross-layer adaptive resource management for wireless packet networks with ofdm signaling,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3244–3254, November 2006. [8] T. M. Cover, “Broadcast channels,” IEEE Trans. Inform. Theory, vol. IT-18, no. 1, pp. 2–14, Jan. 1972. [9] L. Li and A. Goldsmith, “Capacity and optimal resource allocation for fading broadcast channels. i. ergodic capacity,” IEEE Trans. Info. Theory, vol. 47, no. 3, pp. 1083–1102, 2001. [10] H. Won, H. Cai, et al., “Multicast scheduling in cellular data networks,” in Proc. Infocom 2007. IEEE, May 2007, pp. 1172–1180. [11] D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. Massachusetts: Addison-Wesley, 1983. [12] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.