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Resource Allocation for Multiuser Cooperative OFDM Networks: Who Helps Whom and How to Cooperate Zhu Han∗ , Thanongsak Himsoon+ , W. Pam Siriwongpairat+ , and K. J. Ray Liu ∗ Department of Electrical and Computer Engineering University of Houston, Houston, TX 77204 + Meteor Communications Corporation, Kent, WA 98032 Department of Electrical and Computer Engineering and the Institute of Systems Research University of Maryland, College Park, MD 20742 Abstract— Cooperative transmissions have been shown in wireless networks. The power control constantly ad- to be able to greatly improve system performance by justs the transmit power so as to maintain the received exploring the broadcasting nature of wireless channels link quality, while improving the system performance. and cooperation among users. We focus, in this paper, on leveraging cooperation for resource allocation among In some applications such as wireless sensor networks users such that the network performance can be improved. in which each user is powered by batteries, optimizing Two important questions are answered: who should help the power management can greatly extend the network whom among the distributively located users, and how lifetime. Much work has been done for resource al- many resources the users should use for cooperation location for multiuser wireless networks. In [1], the to improve the performance. To answer these questions, a power-optimization, subcarrier-allocation, and relay- authors gave an overview of radio resource management selection problem is formulated over a multiuser OFDM for wireless networks. In [2] and [3], a closed-loop network, which is applicable to systems such as wireless power control framework was proposed and proved to LAN networks. In the multiuser OFDM network, cooper- converge to a unique optimal point. In [4], a second ation among different users is conducted by assigning the order power control algorithm was proposed to improve subcarriers of the helping users to relay a certain part of the helped users’ data, while maintaining the desired rates the convergence speed. In [5], rate adaption and power of both helping users and helped users by means of power control were combined to increase system throughput. In control and rate adaption. In this way, the bandwidth efﬁ- [6] and [7], power control was combined with antenna ciency of the multiuser OFDM system with cooperation is array processing to improve the network performance. the same as that of the non-cooperative OFDM system. The In [8], an algorithm considering time, space, and mul- formulated optimization problem is an assignment problem for subcarrier usage and corresponding bit loading as tiuser diversity was proposed for enhancing the system well as power control. We provide an approximate closed- efﬁciency. form solution for a two-user two-subcarrier case. Then, a Recently, cooperative transmissions have gained much suboptimal heuristic algorithm for multiple-user multiple- attention as an emerging transmit strategy for future subcarrier case is proposed and implemented in the base wireless networks. The cooperative transmissions efﬁ- station to solve the formulated N P hard problem. From the simulation results, the proposed scheme achieves up ciently take advantage of the broadcasting nature of wire- to 50% overall power saving for the two-user system and less networks, as well as exploit the inherent spatial and 19% ∼ 54% overall power saving for the multiuser case multiuser diversities. By exploring space diversity and with random locations, compared to the current multiuser multiuser diversity, the cooperative transmission scheme OFDM system without cooperative diversity. The proposed and implementation algorithms were proposed in [9], scheme is also compared to a much more complicated OFDMA system. assuming full channel state information at the cooperat- ing nodes that utilize beamforming. In [10], transmission protocols of the cooperative transmissions were classi- I. I NTRODUCTION ﬁed into different approaches and their performance was analyzed in terms of outage probabilities. The work in Resource allocation such as power control has long [11] analyzed more complicated transmitter cooperative been regarded as an effective way to dynamically combat schemes involving dirty paper coding. The authors in channel ﬂuctuations and reduce co-channel interference [12] provided rigorous analysis on symbol error rate Part of this work was presented in IEEE Wireless Communications and optimum power allocation for the multi-node decode and Networking Conference, 2005 and forward protocol. Energy-efﬁcient transmission was considered for broadcast networks in [13]. In [14], can maintain the desired rates of both helping users and oversampling was combined with the intrinsic properties helped users. By doing this, the power of the helped users of orthogonal frequency division multiplexing (OFDM) is greatly reduced, because some of its subcarriers have symbols, in the context of maximal ratio combining cooperative diversity and consequent power-reduction (MRC) and amplify and forward, so that this rate loss offered by the helping users. On the other hand, the of cooperative transmissions can be overcome. In [15], power of the helping users is slightly increased, because the authors evaluated cooperative-diversity performances the helping users not only have to relay the information when the best relay is chosen according to the average of the helped users, but also utilize the remaining sub- SNR, and the outage probability of relay selection based carriers to transmit their own data by higher modulation on the instantaneous SNR. In [16], the authors proposed and power. By careful design, the overall system power a distributed relay selection scheme that requires limited can be reduced. For such a shared subcarriers cooperative network knowledge and is based on instantaneous SNRs. scheme, there is no extra stage purely dedicated for relay. These ideas are also working their way into standards; In this way, the bandwidth efﬁciency of the multiuser e.g., the IEEE 802.16 (WiMAX) standards body for OFDM systems with cooperation is the same as that future broadband wireless access has established the of the non-cooperative OFDM systems such as the 802.16j Relay Task Group to incorporate cooperative IEEE 802.11a/g standard [20]. The optimization for the relaying mechanisms into this technology. system resource allocation is performed by modifying Most of the existing cooperative transmission works the OFDM subcarrier assignment for cooperation and concentrate on improving the one-to-one link quality, the corresponding bit loading as well as power control. while most resource allocation works in the literature have not considered cooperation transmission yet. Due to the limited of radio resources and increasing demand Furthermore, we develop a suboptimal algorithm im- of variety of services, it is important to consider resource plemented in the base station to solve the proposed allocation to fully exploit the cooperative transmission N P hard problem. We analyze the situation which paradigm. In this paper, we consider resource allo- users should be helped or helping, i.e., we answer the cation among multiple users to optimize the system question of “who helps whom”. In addition, we optimize performance by taking into consideration the cooperative how many OFDM subcarriers and how much power transmission strategy. In multiuser wireless networks, should be used for helping others, i.e., we answer the there are many open resource allocation questions for question of “how to cooperate”. An approximate closed- cooperative transmissions. The most important ones are form solution for the two-user two-subcarrier case is who should help whom among the distributively located derived. Then, a heuristic algorithm for the multiple- users (i.e. “who helps whom” and relay selection), user multiple-subcarrier case is constructed. Performance and how many resources (like power and subcarrier) comparison with a bound using OFDM multiple access the users should utilize for cooperation to improve the (OFDMA) system is also studied. From the simulation system performance (“how to cooperate”). In [17], the results, the proposed scheme can save up to 50% of basic problem was formulated for 2-user case and some the overall transmit power for the two-user system and preliminary heuristics were constructed. In this paper, we 19% ∼ 54% overall power saving for the multiple-user aim at answering these two major questions in multiple- case with random locations, compared with the current user case and in a more rigorous way. OFDM systems without cooperative diversity. To answer the questions, we consider the power- control, subcarrier-allocation, and relay-selection prob- lem that seeks to minimize the system power over a The rest of the paper is organized as follows: In multiuser OFDM network [18]- [19], which is a key Section II, we give the multiuser OFDM system model element in 4G cellular networks, wireless metropolitan and provide the traditional non-cooperative transmission area network (WMAN), and wireless local area networks solution using the waterﬁlling method. We construct (WLAN). In most of current OFDM systems, each time the cooperative transmission over multiuser OFDM net- only one user transmits at a time as in a TDD system, works and formulate the cooperative optimization as an and users are scheduled for transmission over different assignment problem in Section III. We provide algo- times. The cooperation can be implemented by assigning rithms to solve the problem in Section IV. Simulation some subcarriers of the helping users to relay parts of the results are provided in Section V. Finally, Section VI helped users’ data, while power control and rate adaption concludes the paper. 2 II. M ULTIUSER OFDM S YSTEM M ODEL AND where y + = max(y, 0) and µi is the water level obtained N ON - COOPERATIVE S OLUTION by bisection search of the following expression We consider an uplink multiuser OFDM system1 . N n (µi − Ii )+ Suppose that there are N subcarriers and K users in W log2 1 + n = Ti . (5) Ii the network. The system is time-multiplexed to serve n=1 all users and each time only one user transmits. We represent Ti as the transmission rate of the ith user, and III. P ROBLEM F ORMULATION FOR C OOPERATIVE n the rate is divided into N subcarriers. We denote ri as R ESOURCE A LLOCATION th th the transmission rate of the i user at the n subcarrier, Note that, the solution in (4) is based on the assump- and Pin represents the corresponding transmit power. tion that all users do not cooperate with each other. Due Using adaption techniques such as adaptive modulation, to the broadcasting nature of wireless communication we have [21] systems, not only the base station (BS) but also the other Pin Gn users can hear the transmitted data. If the other users n i ri = W log2 1 + , (1) can cooperate and help the transmission, cooperative σ2 Λ diversity can be exploited and the system performance where W is the subcarrier bandwidth, Λ is a constant can be signiﬁcantly improved. In this section, we em- for the capacity gap, Gn is the subcarrier gain, and σ 2 i ploy cooperative transmission over the OFDM network is the thermal noise plus interference power. Without loss and then formulate the cooperative resource allocation of generality, we assume that the noise plus interference problem. power is stable and the same for all subcarriers and all users. We also assume the channels are constant over A. Cooperative Transmissions over OFDM Networks each power control interval. The goal of this paper is to minimize the overall In the current OFDM system such as in the IEEE power consumption, under the sum rate constraint over 802.11a/g standard, the media access control (MAC) all subcarriers for each user. If there is no cooperation layer provides two different wireless access mechanisms among users, the overall power minimization problem is for wireless medium sharing, namely, the distributed the same as minimizing each user’s power independently. coordination function (DCF) and point coordination We deﬁne Pi = [Pi1 , . . . , PiN ] as a power assignment function (PCF). The DCF achieves automatic medium vector. With the bit loading in (1), the ith user’s power sharing among users using carrier sense multiple access minimization problem can be expressed as: with collision avoidance (CSMA/CA) and request to N send/clear to send (RTS/CTS). The PCF is a more cen- min Pin , (2) tralized control mechanism. In both mechanisms, time Pi n=1 division multiple access (TDMA) technology is utilized N for all users to share the channels. Similarly in this paper, n we assume at each time only one user occupies all the s.t. ri = Ti . n=1 bandwidth. This is also optimal for TDMA over the single cell case due to the severe interference. The above constrained optimization can be solved by the Most of current systems use ARQ with the traditional waterﬁlling method [21]. By representing ACK/NACK mechanism to ensure the successful packet n Λσ 2 deliver. In cooperative transmission, the reason to use Ii = , (3) Gni relay instead of ACK/NACK is due to the wireless the optimal solution of the waterﬁlling method is given channel. If the source-destination wireless link is not by good, there is possibility that ACK/NACK messages may not be able to successfully transmit. Moreover, due to + Pin n Pin = (µi − Ii ) n and ri = W log2 1 + n (4) the bursty nature of wireless channel, if the source- Ii destination channel is not good now, in the immediate 1 Since the relay can improve the performance only if it is located future time, the channel might not recover yet. How- close to the source-destination link, we only need to consider the ever, since the broadcast nature of wireless channel, the resource allocation within a relatively small region, i.e. within one cell. relay can also hear the transmission from the source to For the inter-cell interference that comes from the far away sources, we destination. Due to the spatial diversity, the relay can consider the case in which the inter-cell interference has been either considered as the stable noise-plus-interference or has been solved by have better channel condition and can help the source to frequency reuse schemes. transmit to the destination. 3 User 1's User 2's User K's … User j data data data data v= 1 ... N 1 ... N 1 ... ... 1 ... N Time 1 u= j 1 1 o e/ e P w r at R 2 ... ... e Time 1 m Ti Ti m N e 1 BS 1 Subcarrier 1 1 0 i ... ... Time 2 N 1 User i data 1 ... User j data ... wr e ... ... Po e/ at R ... ... 1 Subcarrier 1 ... ... Time K Fig. 1. OFDM Cooperative Transmission Network N 1 In OFDM networks, each user has ﬂexibility to assign Fig. 2. Assignment Matrix A Example: User 2’s Subcarrier 1 Helps User 1’s Subcarrier N the transmission over different subcarriers. This ﬂexi- bility gives the possibility of cooperation among users. In this paper, we consider cooperative OFDM system over TDMA by using this ﬂexibility. At each time, still answer the question of “how to cooperate”. Moreover, only one user transmits with positive power. However, because of the users’ different locations and channel this user can select the number of subcarriers for its conditions, some users are more effective to help others’ own data while keeping the same rate by using adaptive transmissions. Hence, it is essential to ﬁnd the optimal modulation and power control. At the same time, this cooperative groups, i.e., to answer the question of “who user can determine the number of subcarriers for relaying helps whom”. parts of others’ data. Notice that, unlike most of current We will answer the above two questions by ﬁrst deﬁn- cooperative transmission schemes in which extra relay ing an assignment matrix AKN ×KN , given in Figure 2, stages are necessary, the overall bandwidth efﬁciency for whose element Au,v ∈ {0, 1} where u = 1, · · · , KN relay and self-transmission of the proposed system is and v = 1, · · · , KN . The value of Au,v represents the same as the current OFDM system such as IEEE the subcarrier indexes of both helping user and helped 802.11a/g standard. To fully understand the proposed user. For notation convenience, we denote (i, n) = (i − scheme, an example is given as follows. 1)N + n. We use (i, n) to represent the helping user In Figure 1, user i relays user j’s data to the BS. i at subcarrier n, and (j, n ) as the helped user j at At time one, user j transmits data, while all other users subcarrier n . The value of each element of A has the including the BS can listen. In the next time period, user following interpretation: i transmits its own data, while at the same time user i 1) A(i,n),(i,n) = 1 means the ith user transmits its can help transmit user j’s data if user i’s location is close own data at the nth subcarrier to the BS. to the BS and the channel is good. Speciﬁcally, user i 2) A(i,n),(j,n ) = 1, for i = j, means the ith user at can relay some parts of user j’s data in some of the N the nth subcarrier relays the data for the j th user subcarriers, so as to reduce user j’s transmit power. In at the n th subcarrier . doing so, user i has to transmit its own data in the rest Since each subcarrier contains only data from one user of the available subcarriers2 . Consequently the power for KN at a time, we have v=1 Au,v = 1, ∀u = 1, . . . , KN . user i is increased to maintain its own data transmission. Note that, in case of A = IKN ×KN , the solutions of However, the overall system power can be reduced. From the proposed scheme are the same as those obtained the system optimization point of view, the overall power from the traditional non-cooperative method in Section of both user i and user j can be minimized by selecting II. We also show an example in Figure 2 where user the proper number of subcarriers for cooperation, i.e., to 2 uses its subcarrier 1 to relay the data for user 1 at 2 In practice, this can be implemented by adaptive modulation and the N th subcarrier, i.e. A(2,1),(1,N ) = 1. As shown in by sending zero over the relaying subcarriers. Figure 2, each set of N rows represents data transmitted 4 at a speciﬁc time and each set of N columns represents where without loss of generality ω2 ∼ N (0, σ 2 ) and whose data are being transmitted at that time. Rj,i Xi,BS = (9) B. Problem Formulation for Resource Allocation over |Rj,i | Cooperative Transmissions is the transmitted signal from user i to the BS that is In this subsection, we formulate the resource alloca- normalized to have unit energy. tion optimization with cooperative transmissions as an Substituting (7) into (9), then we can rewrite (8) as assignment problem. We deﬁne PK×N = [P1 , . . . , PN ] Ri,BS = as the power allocation matrix whose components are all nonnegative, and GKN ×KN as the channel gain matrix Pin G(i,n)(i,n) ( Pjn G(j,n )(i,n) Xj,i + ω1 ) whose elements obey the following rules: + ω2 . (10) 1) G(j,n ),(i,n) , for i = j, denotes the channel gain Pjn G(j,n )(i,n) + σ2 from the j th user at the n -th subcarrier to the ith Using (10), the relayed SNR for the n -th subcarrier user at the nth subcarrier. of the j th user, which is helped by the nth subcarrier of 2) G(i,n),(i,n) represents the channel gain from the the ith user, is given by: ith user at the nth subcarrier to the BS. 3) In order to prevent A(i,n),(i,n ) = 1, for n = n j,n A(i,n),(j,n ) Pin Pjn G(i,n)(i,n) G(j,n )(i,n) (each user will not relay its own data), we deﬁne Γr = . (11) G(i,n),(i,n ) = 0, ∀n = n , and ∀i. σ 2 Pin G(i,n)(i,n) + Pjn G(j,n )(i,n) + σ2 In this paper, we use the ampliﬁed-and-forward (AF) Here A(i,n),(j,n ) has a value of 0 or 1, depending on the cooperative protocol [10] which is simple to be im- helping and helped relation. Therefore, by (6) and (11), plemented in relays and can be more mathematically we have the rate at the output of MRC as tractable. Other cooperative protocols such as decode- and-forward can be employed in a similar way. The n Γj,n + Γj,n d r rj = W log2 1 + . (12) helping user helps the helped user by relaying the data on Λ the selected subcarriers using AF. The receiver at the BS If user i helps with subcarrier n, user i transmit no combines together the directly received signal from the information for its own data with this subcarrier, i.e., helped user and the relayed signal from the helping user, n ri = 0. The overall occupied bandwidth for user j’s n using MRC. In what follows, we will derive ri in (1) 1 information is thus 2W . So there is no factor of 2 in (12). that incorporates cooperative transmissions. Notice that Each helping user (like user i) utilizes less bandwidth for helping users can select multiple subcarriers to assist the its own transmission while using the rest of its available helped users. Suppose the ith user at the nth subcarrier bandwidth (like subcarrier n) for helping others. There helps the j th user at the n th subcarrier. We express the is no stage purely dedicated to the cooperative trans- SNR that results from the direct transmission from the mission. This fact leads to the same average bandwidth j th user at the n th subcarrier to the BS by efﬁciency as the non-cooperative case. A(j,n ),(j,n ) Pjn G(j,n ),(j,n ) In this paper, we determine the assignment matrix A Γj,n = d . (6) with combinatorial components, and the corresponding σ2 Here A(j,n ),(j,n ) has a value of 0 or 1, depending on if power allocation matrix P with nonnegative real com- user j transmits its own information on subcarrier n . ponents, for an objective to minimize the overall power Next, we consider the SNR at the BS that results from and satisfy all the constraints. The optimization problem user i relaying user j’s data to the BS. By assuming that can be formulated as: Xj,i is the transmitted signal from user j to user i, the K N received signal at user i is min Pin (13) A,P i=1 n=1 Rj,i = Pjn G(j,n )(i,n) Xj,i + ω1 , (7) N n Transmission Rate: n=1 ri (A, P) = Ti , ∀i; where ω1 ∼ N (0, σ 2 ) and σ 2 is the noise plus inter- KN s.t. Assignment: v=1 Au,v = 1, ∀u = 1, . . . , KN, ference variance. The noise values at different users are and Au,v ∈ {0, 1}, ∀u, v, assumed to be the same. User i ampliﬁes Rj,i and relays N n max Power Constraint: n=1 Pi ≤ Pi , ∀i, it to the BS in which the received signal is where Pimax is the maximal power constraint due to Ri,BS = Pin G(i,n)(i,n) Xi,BS + ω2 , (8) the hardware limitation. Here the subscript i represents 5 users which can be helped users, helping users, or Note that the problem in (13) can be viewed as a non-cooperative users. Notice that A indicates both generalized assignment problem, which is an N P hard “who help whom” and the number of subcarriers for problem [22]. When A is ﬁxed, the problem in (13) can cooperation, and P illustrates the level of power for be viewed as a nonlinear continuous optimization over n cooperation. From (6), (11) and (12), ri is a function of P. So we divide the problem into two subproblems in the both assignment matrix A and power allocation matrix next section. The ﬁrst subproblem is to ﬁnd the optimal P. The optimal choice of A also depends on the channel P with a ﬁxed A. Then in the second subproblem, we conditions from the helped users to the helping users as try to ﬁnd A that generates the optimal solution by using well as those from the helping users to the BS. In case of the results of the ﬁrst subproblem. A = IKN ×KN , the problem in (13) reduces to the non- cooperative problem in (2) and the waterﬁlling method IV. P ROPOSED R ESOURCE A LLOCATION S CHEME can be used to ﬁnd the optimal solution. OVER C OOPERATIVE T RANSMISSIONS It is worth mentioning that the problem formulation In this Section, we ﬁrst provide an analytical approx- in (13) is from a network point of view based on the imation of optimum power allocation with ﬁxed A for assumptions that all the users will follow the protocol a two-user two-subcarrier case. This analysis provides and there is no greedy or malicious user. In [27], the some insight for the formulated problem. Second, we behaviors and incentives of greedy users are studied for prove the unique optimality of (13) with ﬁxed A. Then, cooperative transmissions. based on the insight obtained from the two-user two- Next, we study the maximum transmit power con- subcarrier case, we develop a greedy suboptimal algo- straint in (13). To cope with this constraint, one common rithm to optimize A and solve the problem in (13) for the approach is to add the barrier functions which can be multiple-user case. Finally, a performance comparison written as: with OFDMA for multi-user multi-subcarrier case is also N n maxN provided. if 0, n=1 Pi ≤ Pi , Ii ( Pin , Pimax ) = ∞, otherwise. A. Analytical Approximation for Two-User Two- n=1 (14) Subcarrier System With Fixed A We can modify the objective function in (13) as In what follows, we will show the analytical evaluation K N K N of the optimum power allocation for the cooperative min Pin + Ii ( Pin , Pimax ), (15) transmission system with two users, as shown in Figure A,P i=1 n=1 i=1 n=1 1. Since the users’ locations are random and mostly The optimization goal is equal to the original goal in asymmetric, it is effective for one user to help the other (13) if each user’s power is less than or equal to the instead of both users helping each other3 . Without loss maximum power limit. Otherwise, the optimization goal of generality, we consider the case in which user 2 will achieve inﬁnity. One good approximation for Ii is helps user 1. For simplicity of the exposition, we assume the log function which is widely utilized in nonlinear nu- that an OFDM modulator for each user utilizes two merical optimization. By adding these barrier functions, subcarriers (N = 2), and we consider the case that user the power constraint in (13) can be removed. In addition, 2 allocates subcarrier 1 to relay the data of user 1 at for the single cell case, the users located close to the BS subcarrier 24 . Based on the system under consideration, have better channels and need lower transmitted power. we know that the power of user 2 at subcarrier 2 is These users can help other faraway users with higher not used for relay transmission. Therefore, the optimum transmitted power. The cooperation transmission reduces power allocation can be determined by the helped users’ power a lot and increases the helping 2 users’ power slightly. The underlying reason for this min P2 , (16) unequal power reduction and power increase is because 2 s.t. r2 = T2 , the propagation loss factor in the wireless networks is usually greater than 1. Since the helping users’ power Thus, we have 2 is usually much smaller than the helped users’ power, if G(2,2)(2,2) P2 T2 log2 1 + = . (17) there exists a feasible non-cooperative solution in (13), Λσ 2 W the cooperative solutions still satisfy the maximal power 3 Nevertheless, the two users helping each other case has been studied constraint. This fact will also be shown in the simulation in [23]. results. For mathematical simplicity, we do not consider 4 This can also be generalized to the N -subcarrier case with helping the maximal power constraint in the following analysis. percentage equal to 50% by using similar techniques. 6 2 2 1 2 Hence, the optimum power allocation P2 is given by To ﬁnd the optimum power P1 and P2 in terms of ν1 , Λσ 2 we apply the Lagrange multiplier method to (23). After 2 P2 = 2T2 /W − 1 . (18) some manipulations, we ﬁnd that the optimum power is G(2,2)(2,2) the solution of a quadratic equation: 1 For the relay transmission link, r2 = 0 since user 2 uses subcarrier 1 to help user 1. So the bandwidth 1 2 Y 2 G(1,2)(1,2) + G(1,2)(2,1) G(2,1)(2,1) Y (P2 − P1 ) + 2 1 efﬁciency is the same as the noncooperative scheme. The G(1,2)(2,1) G(2,1)(2,1) P1 P2 (G(1,2)(2,1) − G(2,1)(2,1) ) = 0, (25) optimum power allocation can be obtained by solving the following optimization problem: where 1 2 1 2 1 min P1 + P1 + P2 , (19) Y = G(1,2)(2,1) P1 + G(2,1)(2,1) P2 . (26) P 1 2 s.t. r1 + r1 = T1 . By substituting (26) into (25), we have the quadratic form: Lagrange multiplier method can be applied to obtain an analytical solution. However, it is difﬁcult, if possible, to 2 2 1 1 A(P1 )2 + BP1 P2 + C(P2 )2 = 0, (27) get a closed-form solution by directly applying Lagrange Where A, B, and C are functions of channel gains only. multiplier technique to (19). In the sequel, we provide Assuming that the channel gains are known, we can an alternative approach that allows us to obtain a closed- 2 express the transmit power P1 in terms of the relay form solution for the optimization problem in (19). 1 power P2 as First, we use a tight approximation on the SNR 2 1 P1 = ηP2 . (28) expression in (11) as proved in [24] 2 1 G(1,2)(2,1) G(2,1)(2,1) P1 P2 By substituting (28) into (25) and solving the quadratic Γ1,2 ≈ r 2 1 . (20) equation for η, if G(1,2)(2,1) = G(2,1)(2,1) , we obtain σ2 G(1,2)(2,1) P1 + G(2,1)(2,1) P2 We divide W in both sides of the rate constraint in (19) −G(2,1)(2,1) G(1,2)(1,2) + G(2,1)(2,1) G(1,2)(2,1) G(2,1)(2,1) − G(1,2)( and denote ν1 = T1 . Then we use the approximation in W η= (20) to the constraint as: G(1,2)(2,1) G(2,1)(2,1) − G(1,2)(1,2) (29) 1 2 2 1 G(1,1)(1,1) P1 G(1,2)(1,2) P1 If (1,2)(2,1) G(2,1)(2,1) P1 P2, then we solve for P1 and GG 2 log2 1 + + log2 1 + + (1,2)(2,1) = G(2,1)(2,1) = ν1 , Λσ 2 Λσ 2 2 1 P 1 from the P1 + G(2,1)(2,1) P2 Λσ 22 G(1,2)(2,1)optimization problem in (23) by applying 1 ν1 the 2Lagrange multiplier method in a similar way. After ν1 (21) some manipulations, we can ﬁnd that 1 with ν1 corresponding to the ﬁrst logarithmic term 1 1 1 1 2 −(2G(1,2)(1,2) P2 + G(1,2)(2,1) )P2 + [(2G(1,2)(1,2) P2 + G(1,2)(2,1) that involves only P1 , and ν1 relating to the second η = 2 1 1 2(G(1,2)(1,2) − G(1,2)(2,1) )(P2 )2 logarithmic term that contains both P1 and P2 . Both 1 2 (30) ν1 and ν1 have non-negative values. Therefore, we can separate (19) into two subproblems: where Ω 4(G(1,2)(1,2) − G(1,2)(2,1) )[(G(1,2)(1,2) + 1 G(1,2)(2,1) )P2 − G(1,2)(2,1) ](P2 )3 . 1 1 min P1 , (22) If η is not a positive real number or not a number 1 G(1,1)(1,1) P1 1 caused by dividing by zero in (29) and (30), because of s.t. log2 1 + = ν1 polynomial nature of (27), the optimal solution of (23) Λσ 2 1 happens on the boundary, i.e. P2 = 0. So under this and 2 1 condition, user 1 and user 2 do not cooperate with each min P1 + P2 (23) P12 ≥0,P 1 ≥0 2 other. Otherwise if η is a positive real number, using 2 (28) and (29) to solve the optimization problem in (23), 2 1 G(1,2)(1,2) P1 G(1,2)(2,1) G(2,1)(2,1) P1 P2 can express 2 optimum P 2 and P 1 in terms of ν 2 we the s.t. log2 1 + + 2 1 = ν1 . 1 2 1 Λσ 2 Λσ 2 G(1,2)(2,1) P1 + G(2,1)(2,1) P2 as 1 1 2 1 The optimum power P1 in (22) can be easily obtained P2 = 2ν1 − 1 , (31) 1 ηβ1 in terms of ν1 as and 1 Λσ 2 1 ν1 1 2 P1 = 2 −1 . (24) 2 P1 = 2ν1 − 1 , (32) G(1,1)(1,1) β1 7 where Proof: All users are divided into two groups. The G(1,2)(1,2) G(1,2)(2,1) G(2,1)(2,1) ﬁrst group of users does not cooperate with others, i.e., β1 = + . {i : A(i,n),(j,n ) = 0, ∀i = j or n = n }. Therefore, the Λσ 2 Λσ 2 ηG(1,2)(2,1) + G(2,1)(2,1) (33) problem can be considered in the same way as the single 1 2 user case and the waterﬁlling method can be used to ﬁnd From (24), (31), (32), and the relation ν1 = ν1 − ν1 , the 1 2 1 the only local optimum which is the global optimum for total power P1 + P1 + P2 that satisﬁes the constraint in 2 this kind of users. (19) can be expressed in terms of ν1 , i.e., In the second group, users cooperate with each other, 1 2 1 Λσ 2 2 P1 + P1 + P2 = 2(ν1 −ν1 ) − 1 (34) i.e., {i : ∃A(i,n),(j,n ) = 1, ∀i = j or n = n }. For G(1,1)(1,1) a ﬁxed A, the optimization problem for this group of 1 2 1 2 users can be expressed as + 2ν1 −1 + 2ν1 − 1 2 f (ν1 ). β1 ηβ1 K N 2 Finally, we can ﬁnd the optimum value of ν1 by solving min Pjn (39) the unconstrained optimization problem P j=1 n =1 2 min f (ν1 ), (35) N n n =1 rj = Tj , ∀j. 2 ν1 s.t. which results in We express the constraint by the use of (12) as: 2 Λσ 2 ηβ1 ν1 = 0.5 ν1 + 0.5 log2 . (36) N G(1,1)(1,1) (η + β1 ) Γj,n + Γj,n d r Tj 1+ = 2W . (40) 1 Consequently, ν1 can be found from (21) as Λ n =1 1 Λσ 2 ηβ1 Since each user can be helped by at most one other ν1 = 0.5 ν1 − 0.5 log2 . (37) G(1,1)(1,1) (η + β1 ) user, each product for each subcarrier n in (40) is not 1 2 coupled with each other. Suppose the cooperative rate for Substituting the obtained ν1 and ν1 into (24), (31) Γj,n +Γj,n 1 2 and (32) gives the optimum power P1 , P1 , and P2 , 1 user j and subcarrier n is 2τ , i.e., 1+ d Λ r = respectively. 2τ . For a ﬁxed A and using (6) and (11), all power From the above derivations, we know that in order to components result in a quadratic form of Pjn , because minimize the overall power, the ratio η in (28) (i.e. the of the uncoupling fact. If the two roots of the polynomial helped user’s transmit power over the helping user’s re- function have opposite signs, the problem in (39) has lay power) is only determined by the channel conditions a unique positive solution for power P. Suppose each between the two users and those to the BS. Then for cooperative subcarrier has the rate larger than W if τ > each helped user, it balances the power allocation used 1, and the helping user i helps by subcarrier n. From (6) between the subcarriers that do not get help from others and approximation of (11), we can rewrite (40) as and the subcarriers whose information is relayed by others, so as to minimize the overall power. We will show G(j,n )(j,n ) (G(j,n )(j,n ) + G(i,n)(i,n) )(Pjn )2 (41) by computer simulation that the above analysis with the (n ) 2 +C Pj + Λσ G(i,n)(i,n) Pin (1 − τ ) = 0. SNR approximation in (20) provides the solutions that are very close to the optimal solutions. where C is a constant of channel gain and Pin . It is obvious that in order to obtain unique positive Pjn , τ B. Power Minimization Optimality for Multiple-user should be greater than 1. As a result, the global optimum Multiple-Subcarrier System with Fixed A can always be achieved. In this subsection, we assume that the assignment ma- The above theorem proves that there is a unique trix A is known and ﬁxed. We show the characteristics optimal solution for the case with ﬁxed A under the high of the solution by the following theorem. rate assumption. So with the ﬁxed A, any nonlinear or Theorem 1: For a ﬁxed A, as long as each cooperative convex optimization methods [25], [26] can be employed n subcarrier has the rate rj in (12) that is larger than W , to solve (39). In the case of low rates, the optimization there is only one local optimum which is also the global problem will have multiple local optima. Under this optimum for (13). In other words, the corresponding condition, there are generally two possible approaches. SNR satisﬁes The ﬁrst one is simulated annealing, and the other is to Γj,n + Γj,n > Λ. d r (38) ﬁnd good heuristic initialization. 8 C. Finding Suboptimal A for Multiple-user Multiple- we go back to determine the helping and helped user Subcarrier Case pair again and continue the iteration. Notice that there In previous sections, we analyze the optimization might be more than one subcarrier of a helping user for problem with a ﬁxed A. In this subsection, we develop relaying the information of the helped users. If the user a suboptimal solution to ﬁnd A, which represents “who with the minimal power cannot help others anymore, helps whom” and how many subcarriers should be used the random pair of helping user and helped user is for cooperation (i.e. “how to cooperate”). Because of formed to further explore the possibility that the power the combinatorial nature, the problem in (39) is also an can be reduced. If the power cannot be reduced, this N P hard problem [25], [26], since any element of A random pair of users is not applicable and the original has a value of either 0 or 1 and the search dimension A and P are restored. Otherwise, the new A and its of A is 2KN ×KN . For any speciﬁc A, we calculate the corresponding P are updated. This process stops when transmit power vector P(A), and then select the one that the power cannot be further reduced for a period of generates the minimal overall power. The problem can iterations. The resulting A is the bandwidth assignment be formulated as for the helping users to the helped users, which is the K N answer to the questions of “who helps whom” and “how min n Pi (A) (42) many subcarriers to cooperate”. The resulting matrix P A i=1 n=1 represents power allocation. So ”how to cooperate” is also answered. Notice that in each iteration, the sum rate KN for each user is kept unchanged. The detailed algorithm s.t. Au,v = 1, ∀u = 1, . . . , KN, and Au,v ∈ {0, 1}, ∀u, v. is shown as follows: v=1 However, the complexity is high especially when a large Multiuser Suboptimal Algorithm for A number of subcarriers are utilized, and when there are a Initialization: A = IKN ×KN , and calculate (39) to substantial number of users in the OFDM network. This obtain P. prohibits the full search method in practice. Iteration: Next, we propose a suboptimal greedy algorithm to 1) select user i with the minimal power and user j ﬁnd the assignment matrix A. The basic idea is to let with maximal power under the condition: the user with the least transmit power (likely close to the BS) to help the user with the most transmit power • user i has at least one subcarrier for transmit- (likely far away from the BS). Consequently, the user ting its own data. with the least power has to increase its power to help the user with the most power in order to reduce the overall 2) Hypotheses: system power. Then we determine the users’ selection • Only the subcarrier that transmits the user’s again, and repeat the above steps. The iteration stops own data is eligible for the hypotheses. when no overall power can be reduced. The suboptimal • If user i’s subcarrier n helps user j’s subcarrier greedy algorithm is implemented in the base station. n: Initially, A is assigned as an identity matrix, i.e., the Set [A](i−1)∗N +n,(i−1)∗N +n = 0 and initial scheme is the non-cooperative scheme. Second, [A](i−1)∗N +n,(j−1)∗N +n = 1; we sort according to users’ transmit power. We select • Solve (39) for P. the users with the maximal and minimal transmit power Among all hypotheses, ﬁnd the maximal power as helped user and helping user, respectively, according reduction. to the following condition: Each user has at least one subcarrier to send its own data. The above condition 3) If all hypotheses are not effective for power reduc- makes sure that helping user’s data must be transmitted. tion, random helping and helped pairs are formed Among N subcarriers of the helping user, we make to see if the power can be further reduced or not. N hypotheses that the nth subcarrier is assigned to 4) Update A, go to step 1. End if no power reduction assist the helped user and the remaining subcarriers are for a period of iterations, return A and P. unchanged. The overall power for this hypothesis is Since the system power is non-increasing in each obtained by solving (39) with some numerical method. iteration 5 and is lower bounded by the full search result, From all these hypotheses, the algorithm selects the one that maximally reduces the overall power and keeps the 5 The results from the random disturbance are applicable only when rest N − 1 subcarriers the same in each iteration. Then there is power reduction. 9 the iteration always converges. Note that the complexity αi is no more than 2. αi is equal to 2 only if all the for each iteration of the proposed algorithm is O(N 3 ) subcarriers of the helped user get helped by others. The and the algorithm is suboptimal because of the greedy problem formulation for the performance comparison is local search. But the local optimum problem is alleviated to minimize the overall power under the constraints of by the random disturbance in step 3 of the algorithm. total bandwidth and individual rates: Moreover, the helping users cannot be helped again in K the future. As we have mentioned in the two-user case, min αi N Pi (43) due to the asymmetry of channel conditions for different 0<αi ≤2 i=1 users, it is not effective for users to help each other or form helping loop. So in the proposed algorithm, we K implicitly exclude the helping loop and users cannot be i=1 αi = K; helping users and helped users at the same time. s.t. ¯ Pi Gi αi N W log2 1 + Λσ 2 = Ti , The channel information required to perform the al- gorithm can be obtained in the following way: Since the ¯ where Gi is the average channel gain 6 over the band- users take turns to transmit information to the BS, the width occupied by user i. The objective function is the channel information from the source to the destination overall transmitted power since here Pi is the power and from the relay to the destination can be easily per bandwidth W , the ﬁrst constraint is the overall estimated. When one user transmits to the BS, the bandwidth, and the second constraint is the required rate. possible relays of this user can estimate the channels The problem in (43) is different from the problem in (from sources to relays). All this information can be sent (13) for two reasons. First, the bandwidth assignment to the BS for the optimization. Since the average number is relaxed to continuous functions αi instead of the of users is small and number of subcarriers are around 48 discrete function A 7 . So the solution to (43) is an upper for WLAN, the amount of information is limited. The BS bound for OFDMA. Second, the received SNR of the needs to send control bits to the potential helping/helped helped user is obtained by the assumption that all the users to indicate relay-selection, power-optimization, and bandwidth is occupied by the helped user directly. The subcarrier-allocation. This requires some overheard for purpose is to have a fair comparison from the bandwidth signaling. Here we also assume the mobility of the users point of view. Moreover, this simpliﬁes the analysis but is low, so that the channel conditions are stable for removes the multipath diversity provided by cooperative sufﬁciently long time and the frequency and overhead transmission. Since each users’ transmission can be to update the channel information are also low. allocated to not only the different time slots but also dif- ferent frequency in (43), the multiuser, time, frequency D. Performance Comparison diversity of OFDMA can be achieved. The problem in For the multiuser case, it is very difﬁcult to obtain (43) can be solved by some numerical methods [25], the analytical closed-form solution of (13), because of [26]. The solutions of (43) give us some insights on the non-convexity and integer nature of the N P hard the performance comparisons of the proposed suboptimal problem. In order to better understand the performance solution, which are shown in the next Section. of the proposed suboptimal solution, we also compare the performances with OFDMA. Since the helped user’s information is relayed by the helping user in some V. S IMULATION R ESULTS AND A NALYSIS subcarriers, it can be viewed as the helped user occupied We perform computer simulations for multiuser more bandwidth (subcarriers) by the help of the helping OFDM systems to evaluate the system performance and users. The basic idea is to view these helping subcarriers answer the questions like “who helps whom” and “how as a part of the bandwidth of the helped users. Suppose to cooperate”. In what follows, the simulation results for the helping user i helps the helped user j with N the two-user system are presented in the ﬁrst part, and subcarriers. So the bandwidth occupied by the helped those for the multiuser system are shown in the second user is W N (1 + N ) and the bandwidth for the helping N part. user is W N (1 − N ). We deﬁne each user to have a N bandwidth of αi N W , where αi ∈ (0, 2]. Note that αi is 6 Notice that ﬂat fading is assumed for OFDMA system to reduce the greater than 0 since any user must have some bandwidth analysis complexity, since OFDMA resource allocation in frequency to transmit its own data. Since we only consider the case selective fading is still an open issue in the literature. in which a user can only get helped by one other user, 7 This is typical relaxation in the OFDMA literature [18]. 10 Comparison of Three Methods for Overall Power Cooperation Region for 50% Helping 30 water filing user 1 helps user 2 user 2 helps user 1 User 1 help User 2 analysis : user 1 helps user 2 20 analysis : user 2 helps user 1 10 Overall Power (dBm) 10 Waterfilling Area User 2 Location Y 0 User 2 help User 1 0 −10 −10 −20 −30 −30 −20 −10 0 10 20 30 User 2 Location X −20 −30 −20 −10 0 10 20 30 User 2 Location Fig. 4. Cooperative Region For Two-User System with User 1 Located at (10m, 0m) Fig. 3. Comparison of Overall Power of Three Different Schemes and Analytical Results when user 2 is located far away from the BS, even in the opposite direction to user 1 such as (-30m,0), the 1- A. Two-User System H-2 scheme can reduce the overall transmit power. This We set simulations of the ﬁrst part as follows: There can be explained by the same reason as above. In the are a total of K = 2 users in the OFDM network. A BS extreme case, when user 2 is located very far away from is located at coordinate (0,0), user 1 is ﬁxed at coordinate the BS compared with the location of user 1 to the BS, (10m,0), and user 2 is randomly located within the range both user 1 and the BS can be considered as multiple of [-30m 30m] in both x-axis and y-axis. The propaga- sinks for user 2’s signal. This provides so called “virtual tion loss factor is set to 3. The noise plus interference multiple antenna diversity”. In addition, the analytical level is σ 2 = −60dBm and we select the capacity gap results obtained by derivation closely match the optimal as Λ = 1. An OFDM modulator for each user utilizes results by the numerical algorithm. N = 32 subcarriers and each subcarrier occupying a In Figure 4, we show the region where different bandwidth of W = 1. Without loss of generality, we schemes should be applied based on user 2’s location. assume ﬂat fading in simulations so that the results such With the same simulation setup as in the previous case, as regions can be clearly related to distances and can be we can see that when user 2’s location is close to compared with OFDMA results obtained from Section the BS or lies in between user 1 and the BS, the 2- IV-D. The system under frequency selective fading can H-1 scheme is preferred. When user 2 is located far be studied in a similar way. The performance can be away from the BS, the 1-H-2 scheme produces minimal further improved by exploring frequency diversity. overall power. When user 2’s location is in between the In Figure 3, we show a comparison of the overall above two cases, the waterﬁlling scheme is the optimum power in dB of the waterﬁlling scheme, user 1 helps choice. This ﬁgure presents the answers to the question user 2 (1-H-2) scheme, user 2 helps user 1 (2-H-1) of “who helps whom”. It is worth mentioning that both scheme, and the analyzed results from Section IV-B. In x axis and y axis represent users’ channel conditions this simulation, user 2 moves from location (-30m,0) to by using locations, so that the answer of “who helps (30m,0). The transmission rate for each user is Ti = whom” can be clearly illustrated. If channel effects such 2N W and half of the subcarriers are used for helping as shadowing and fading are considered, both x axis and the other. We observe that when user 2 is located close to y axis can be channel conditions and are not related to the BS, the 2-H-1 scheme can reduce the overall power the distances. up to 50%. The reason is that user 1 can use user 2 as In Figure 5, we answer the question of how the users a relay node to transmit user 1’s data such that user 1’s should cooperate with each other. The simulation setup power can be reduced. Since user 2 is close to the BS, is the same as in the previous case except that the even with only half of the subcarriers to carry its own number of helped subcarriers is not ﬁxed. Instead, we data, the increase of power for user 2 is still smaller ﬁnd the optimal percentage of subcarriers that are used than the power reduction for user 1. On the other hand, for helping others. We show the helping percentage as a 11 Helping Percentage vs. Rate and Location 100 Optimal Helping Percentage For Different Location of User 2 Helping Percentage 50 80 60 0 40 Helping Percentage 20 −50 0 −20 −100 −40 0 −60 2 −30 30 −30 −20 4 −20 20 −10 10 6 −10 0 0 0 8 10 User 2 Location X Rate (NW) User 2 Location 10 −10 User 2 Location Y 20 10 30 20 −20 30 −30 Fig. 7. Helping Percentage versus User’s Rate Fig. 5. Cooperative Percentage For Two-User System Helping Range vs. Rate and User 2 Location 30 to (30m, 0). As we can see from Figure 6, the helping regions are changed with increase of the transmission 20 User 1 help User 2 rate. The waterﬁlling regions increase to maxima around the rate equal to 3.4N W , then the waterﬁlling regions 10 Waterfilling are shrunk until all regions are occupied by 1-H-2 or User 2 Location 2-H-1 regions. This is because when the power grows 0 User 2 help User 1 exponentially with increase of rate, the helping with little percentage can reduce a lot of power in the proposed −10 OFDM cooperative network. In Figure 7, however, we Waterfilling observe that the helping percentages are reduced as the −20 User 1 help User 2 users’ rates keep increasing. This is because the users need more subcarriers to send their own data, so the −30 number of subcarriers to help others is reduced. Note 0.2 1 2 3 4 5 6 7 8 9 10 Rate (NW) also from the ﬁgure that when the rate is high, the original waterﬁlling area becomes the 1-H-2 or 2-H-1 Fig. 6. Cooperative Region versus User’s Rate area. But the percentage of helping is very small. function of different user 2’s locations. For pure notation B. Multiuser System purpose, when the 2-H-1 scheme is used, the percentage We consider, in this subsection, the performance of is illustrated to be positive; when the waterﬁlling scheme the proposed scheme for the multiuser scenario. In the is chosen, the percentage is zero; when the 1-H-2 scheme simulations, all distributed users are located within a is applied, the percentage is shown to be negative. By circle of radius 50m. The BS is located at the center doing this, with the same percentages, we can distinguish of the circle, i.e. (0m, 0m) and the closest distance from who helps whom. From Figure 5, the closer user 2 is a user to the BS is 1m. The other settings are the same to the BS, the larger percentage user 2 will help user as those for the two-user case. 1. This observation follows the fact that user 2’s own In Figure 8, we show simulation results for a 4- transmission virtually costs nothing in terms of power user case. Over different random locations, we select usage. On the other hand, when user 2 is far away from a typical snapshot of user 1, 2, 3, and 4 with location the BS, user 1 helps user 2 more and more. Therefore, the (3.2331m, 7.2071m), (1.7768m, -7.8635m), (35.7636m, ﬁgure gives us insight as to how the level of cooperation 5.8430m), and (-32.7265m, 21.5281m), respectively. We and partner assignments affect the system performance. show the power changes in the proposed cooperative Figure 6 and Figure 7 show the effects of user’s scheme versus the iteration number. Because user 1 and transmission rate Ti on who should help whom and how user 2 are close to the BS, they help user 3 and user users should cooperate. We modify Ti from 0.5N W to 4’s transmission to reduce their power. The subcarriers 10N W for all users. User 2 is located from (-30m,0) of user 1 and user 2 are assigned to help user 3 and user 12 Power Reduction vs. Iteration Power vs. Rate −10 Average User 1 −20 User 2 User 3 User 4 −10 −30 Power (dB) Power (dB) −40 −20 Waterfilling, No. of User=4 −50 Cooperation, No. of User =4 Waterfilling, No. of User=8 Cooperation, No. of User=8 −60 OFDMA, No. of User=4 OFDMA, No. of User=8 −70 −30 0 10 20 30 40 50 60 1 2 3 4 5 6 Iteration Rate Fig. 8. Power Convergence Fig. 10. Overall Power versus Each User’s Rate in the Unit of N W Power vs. Number of Users TABLE I E FFECT OF S HADOW FADING ON P OWER S AVING (P ERCENTAGE OF P OWER R EDUCTION USING C OOPERATION OVER THE P OWER OF 20 N ONCOOPERATION ) Power (dBm) Shadow Fading Standard Deviation Rate=2NW Rate=4NW 1dB 23.5% 37.3% 3dB 41.0% 53.5% 5dB 59.5% 78.9% 10 Waterfilling Rate=2NW Cooperation Rate=2NW Waterfilling Rate=4NW the waterﬁlling scheme will produce the minimal overall Cooperation Rate=4NW power. Under this occasion, the proposed cooperative 2 3 4 5 6 Number of Users 7 8 9 10 scheme has the same performance as the non-cooperative scheme. Fig. 9. Overall Power versus Number of Users In Figure 10, we show the overall system power versus the rate requirements with 4-user and 8-user cases. Obviously the overall power increases almost exponen- 4. Consequently, the remaining numbers of subcarriers tially with the rate requirement. This corresponds to the for user 1 and user 2 are reduced, so that their transmit relation between rate and power given in (1) and (12). power is increased. But the overall system power is The proposed cooperative scheme can reduce 37% to reduced. Notice that after iteration 37, user 4 still has 54% of overall power under 4-user condition and 8-user the largest power, but no user can further help him/her, condition. The proposed cooperative scheme can reduce since to help user 4 will increase the power of the helping more power when the rate is high. This is because the users more than the power reduction of user 4. From power reduction is large if the user is helped by others iteration 50, user 2 tries to help user 3 which has the when the rate is higher, even though the percentages second largest power. This is due to the third step of the of subcarriers that other users would like to help are algorithm, in which the random pair of users is formed. reduced according to Figure 7. In Figure 9, we show the system overall power in dB In Table I, we investigate the shadow fading effect versus the number of users with different rate require- on the proposed scheme. Speciﬁcally, we study the ments. We can see that the proposed cooperative scheme lognormal shadow fading model with different variance can save up to 19% of overall power for rate equal to in dB. As shown in the table, the proposed scheme can 2N W and 42% for rate equal to 4N W , respectively. 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