CONSIDERING the limited budget of transmit power and

Transcription

1 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY Liited-Feedback-Based Adaptive Power Allocation and ubcarrier Pairing for OFDM DF elay Networks With Diversity Yong Liu and Wen Chen, enior Meber, IEEE Abstract A liited-feedback-based dynaic resource allocation algorith is proposed for a relay cooperative network with orthogonal frequency-division-ultiplexing OFDM odulation. A counication odel where one source node counicates with one destination node assisted by one half-duplex decode-and forward DF relay is considered in this paper. We first consider the selective DF schee, in which soe relay subcarriers will keep idle if they do not have the advantage of forwarding the received sybols. Furtherore, we consider the enhanced DF schee where the idle subcarriers are used to transit new essages at the source. We ai to axiize the syste s instantaneous rate by jointly optiizing power allocation and subcarrier pairing on each subcarrier based on the Lloyd algorith. Both su and individual power constraints are considered. The joint optiization turns out to be a ixed integer prograing proble. We then transfor it into a convex optiization by continuous relaxation and achieve the solution in the dual doain. The perforance of the proposed joint resource allocation algorith is verified by siulations. We find that the proposed schee outperfors the existing ethods in various channel conditions. We also observe that only a few feedback bits can achieve ost of the perforance gain of the perfect channel-state-inforation CI-based resource allocation algorith at different levels of signal-to-noise ratio N. Index Ters Decode and forward DF, liited feedback, Lloyd algorith, orthogonal frequency-division ultiplexing OFDM, power allocation, subcarrier pairing. I. INTODUCTION CONIDEING the liited budget of transit power and hardware coplexity, cooperative relaying has recently attracted a lot of research interest, which is eployed to exploit spatial diversity, cobat wireless channel fading, and extend coverage without antenna arrays ], ]. For exaple, IEEE 80.6 currently integrates relays for ultihop counications 3]. Two ain relay strategies have been adopted in such scenarios: aplify and forward AF and decode and forward DF. Manuscript received August 3, 0; revised March 6, 0 and April 8, 0; accepted April 9, 0. Date of publication May 8, 0; date of current version July 0, 0. This work was supported by the National 973 Project under Grant 0CB3606, by the National cience Foundation of China under Grant and Grant , by the National 973 Project under Grant 009CB84900, and by the National Key Laboratory Project under Grant IN-0. The review of this paper was coordinated by Prof. M. D. Yacoub. The authors are with the Departent of Electronic Engineering, hanghai Jiao Tong University, hanghai 00030, China, and also with the tate Key Laboratory for Integrated ervice Networks, Xidian University, Xi an 7007, China e-ail: yongliu98@sjtu.edu.cn; wenchen@sjtu.edu.cn. Digital Object Identifier 0.09/TVT The AF relay aplifies and retransits the received signal without decoding, whereas the latter reencodes the received signal before retransission. Orthogonal frequency-division ultiplexing OFDM is a technique to itigate frequency selectivity and intersybol interference with its inherent robustness against frequencyselective fading 4]. Because of its potential for high spectral efficiency, OFDM-based relaying offers a ore proising perspective in iproving syste perforance. Power allocation is always critical in wireless networks due to the liited budget of transit power. It has been widely discussed in the context of both single-carrier and ulticarrier relaying channels 5] ]. We have proposed liitedfeedback-based power allocation algoriths for a single-carrier relaying channel and a ulticarrier-based relaying odel in 5] and 6], respectively. In 7], Ahed et al. propose a power control algorith for AF relaying with liited feedback. Then, they study the rate and power control to iprove the throughput gain of DF in 8]. On the other hand, power allocation for OFDM-based relaying is also extensively studied. The authors in 9] investigate the power allocation for an OFDM-based AF relaying by separately optiizing the source and relay powers. In 0], the sae authors propose a power allocation schee for ultiple-input ultiple-output MIMO OFDM relay syste in the sae way. In ], Ying et al. work on a siilar proble but for DF relaying OFDM systes. Ma et al. introduce power loading algoriths to iniize the transit power for OFDMbased AF and selective DF odes with respect to various power constraint conditions in ]. Due to the independent fading on each subcarrier in each hop, subcarrier pairing is eployed in OFDM power allocation to further iprove syste perforance 3] 7]. Most works in the literature focus only on relay odels without diversity. A sorted subcarrier pairing schee is proposed in 3]. The authors deterine the pairing sequence by ordering the source relay subcarriers and the relay destination D subcarriers, respectively, according to the channel gains. The authors in 4] prove that the sorted pairing ethod is optial for both DF and AF relaying without the source destination D link. In 5], channel pairing, channel user assignent, and power allocation are jointly optiized in a ultiple-access syste by a polynoial-tie algorith based on continuous relaxation and dual iniization. Wang and Wu 6] propose a joint subcarrier pairing and power allocation algorith for an OFDM two-hop relay syste with separate power constraints /$ IEEE

2 560 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 and find the solution by separately considering the subcarrier pairing and the power allocation. In 7], optial subcarrier assignent and power allocation schees for the ultiuser ultirelay odel is investigated, and the optial subcarrier and power allocation policy in a quasi-closed for is obtained. esource allocation utilizing channel state inforation CI can yield significant perforance iproveent 5], 6], 8], 9]. Treendous innovation that realizes instantaneous channel adaptation is to use feedback whose history ay trace back to hannon 0]. It is proved that with perfect CI at the source, the error and capacity perforance are significantly better than that without CI 9], ]. oe research has been carried out to achieve the perforance gain based on liited feedback, since perfect CI at the source is always ipractical. One can either send back a quantized CI or quantized power allocation vectors 7] or the index of the best vector in a power allocation codebook shared by all nodes ], 3]. These works are ostly studied in point-to-point MIMO and OFDM systes. Only a few works exist on OFDM relay networks. The power allocation issue for a single OFDM AF relay network with liited feedback is investigated in 4]. They construct the codebook based on the Lloyd algorith. iilarly, Zhang et al. introduce the sae idea into the DF odel in 5]. In view of the lack of joint optiization of power allocation and subcarrier pairing for OFDM relay systes with diversity based on liited feedback, we ai to solve this proble in this paper. This work is developed based on our previous works 5], 6]. We present a liited-feedback-based joint power allocation and subcarrier pairing for a selective OFDM DF relay network under different levels of quantized CI feedback. In our feedback schee, we construct a codebook based on an iterative Lloyd algorith with a odified distortion easure. The joint optiization proble is forulated as a ixed integer prograing proble that is hard to solve. We transfor it into a convex proble by continuous relaxation 6], 7] and solve it in the dual-doain instead. In our siulation, we observe that the duality gap virtually turns out to be zero when the nuber of subcarriers is reasonably large, which is consistent with that observed in 7] and 9]. We then relax the constraint that only the relay can transit in the relaying phase. When the relay does not transit on soe subcarriers, we eploy the enhanced DF that allows the source to transit new essages on these idle subcarriers. Then, we extend the joint optiization proble for selective DF to that for enhanced DF under both su power constraint and individual power constraints. It is shown that the extra direct-link transission leads to a rearkable rate enhanceent in the siulation. In addition, soe existing schees such as the conventional unifor power allocation without subcarrier pairing UPA w/o P, the optial power allocation without subcarrier pairing OPA w/o P, and the unifor power allocation with subcarrier pairing UPA with P are copared with the proposed algorith. iulation results deonstrate that the proposed algorith outperfors the existing ones. We also find that a negligible perforance loss can be achieved with just a few feedback bits at different levels of signal-to-noise ratio N. Fig.. yste block diagra of OFDM cooperative diversity odel with liited feedback. The reainder of this paper is organized as follows. The syste odel is introduced in ection II. In ection III, we solve the joint optiization proble for selective DF relay networks and propose a liited-feedback-based resource allocation algorith. In ection IV, we solve the optiization proble for enhanced DF relay networks and then consider the joint optiization under individual power constraints. iulations are perfored in ection V to verify the perforance of the proposed algorith. Finally, the conclusions are drawn in ection VI. II. YTEM MODEL The scenario of a three-node DF diversity odel is considered, where one source counicates with one destination assisted by one half-duplex relay, as shown in Fig.. The channel on each hop is divided into N subcarriers. Counication takes place in two phases. The source broadcasts its signal in the listening phase, whereas the relay and destination listen. The relay decodes and forwards in the relaying phase. It is assued that each subcarrier in the listening phase is paired with one subcarrier in the relaying phase. Therefore, the nuber of subcarrier pairs is N. We utilize P, n to denote the subcarrier in the listening phase pairing with the subcarrier n in the relaying phase. For subcarrier pair P, n, it ight not be the actual pair participating in counication. If P, n actually participates in counication, it is said to be selected. We denote h D, h, and hn D as channel coefficients of the th subcarrier of D and and the nth subcarrier of D, respectively. For a potential P, n, the source transits sybol s over subcarrier with power in the listening phase, and the received signals at the relay and destination are respectively given by y r = h s + z r y d = h Ds + z d where z d CN0,σ d and z r CN0,σr are the additive noises at the relay and the destination, respectively. In the relaying phase, the relay transits the reencoded signal ŝ with power on the nth subcarrier, and the received signal at the destination is y dn = hn Dŝ + z dn where z dn CNµ, σ d is the additive noise at the destination in the relaying phase.

3 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF 56 Let λ = h /σr, λ n D = hn D /σd, and λ D = h D /σd denote the noralized channel gains, respectively. Depending on whether the relay is helpful, each subcarrier pair P, n ay work in either the relaying ode or the idle ode in a selective DF relay 30]. For an P, n, therelay forwards the essage ŝ on subcarrier n in the relaying phase when it works in the relaying ode; while in the idle ode, the relay does not forward = 0, and s is transitted to the destination by the D link in the listening phase only. Then, the end-to-end rate achieved by P, n during the two phases is given by 3, shown at the botto of the page. A criterion to decide the working ode of P, n, that is, using relay is advantageous when in λ,λ n D} >λ D 4 in selective DF ode is presented in ] and 30]. Otherwise, the relay keeps idle on subcarrier n in the relaying phase for ŝ. III. LIMITED-FEEDBACK-BAED OPTIMAL EOUCE ALLOCATION In this section, we analyze the joint optiization of power allocation and subcarrier pairing for selective DF based on liited feedback. The optiization proble is forulated first and then solved in the dual doain. A. Optiization Proble Forulation Let = + for P, n. We first consider the rate in the relaying ode. Then, the su rate is axiized when log + that is λ =log + λ D+ λn D 5 + λ = + λ D + λn D. 6 Together with = +, we obtain = λ n D λ +λn D λ D = λ λ D λ +λn D λ D. When the syste works in the idle ode, we can easily get P = 8 = 0. Denote as the equivalent channel gain given by = λ λ n D 7 λ, relaying ode +λn D λ D λ,d, idle ode. 9 By now, we can unify the rate as =log +. 0 We define a subcarrier pairing paraeter 0, } that takes if P, n is selected, and 0 otherwise. Then, the su rate optiization proble can be forulated as ax P,t} s.t. C : C3 : P t, C : 0 = n, C4 : = = n=, n where P t is the transit power budget, and t and P are two N N atrices with the, nth entry and, respectively. C3 and C4 correspond to the pairing constraint that each subcarrier in listening phase only pairs with one subcarrier n in the relaying phase. ince it is a ixed integer prograing proble that is difficult to solve, we relax the integer constraint of 0, } as 0, ], n, as in 6] and 34]. Denote = as the actual power consued on P, n. Then, the optiization proble becoes ax,t} log + s.t. C5 : 0, n C6 : P t λ C7 : 0, n, and C3 C4 where = NN is an N N atrix. Obviously, the preceding objective function is concave with respect to, t. In the following, we will eploy the dual ethod 7], 8] to solve this optiization proble. In 7], the authors have shown that under a so-called tiesharing condition, the duality gap of the optiization proble is always zero, regardless of the convexity of the objective function. Further, the authors show that the tie-sharing condition is always satisfied for practical ultiuser spectru optiization probles in ulticarrier systes when the nuber of frequency carriers goes to infinity. This suggests that we can solve the proble by the dual ethod 8], which will provide an upper bound for the original proble. More iportantly, the ethod can guarantee being integer valued. = log + λ D, idle ode in log + λ D + λn D, log + λ }, relaying ode 3

4 56 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 B. olution by the Dual Method Dualizing the constraints C4 and C6, we obtain the generated Lagrange function as L, t,α,β = λ log + t + α P t + β 3 = n= where α 0 and β =β,β,...,β N 0 are dual variables. Then, the dual objective function and the dual proble are, respectively gα, β =axls, t,α,β, s.t. C3, C5, C6 4,t} in gα, β, s.t. α 0, β 0. 5 α,β} ince a dual function is always optiized by first optiizing soe variables and then optiizing the reaining ones 8]. We first optiize with the assuption that α and β are given. Taking the partial differentiation of L with respect to,wehave that is L = + α = α = 0. 7 Together with the constraint 0, we obtain the optial solution = α ] + 8 where x] + =ax0,x}. We find that is associated with the subcarrier pairing paraeter. To find the optial solution for, we first substitute 8 into 3 to obtain the updated Lagrange function Lp, t, α,β = log + α = + α P t + = β n= ] + α ] + T + αp t + β = 9 where T = log + α ] + α α ] + β. 0 ince both T and αp t + N = β are independent of, we obtain the optial t for any n as t = : =argax=,...,n T 0 : otherwise. uppose that there is a subcarrier corresponding to two different n. Then, it is conflict to the constraint C4, which is ebedded in the Lagrangian. Therefore, N = t = n. ince both and t include the dual variables α and β, we have to find values α and β that iniize gα, β. Given and t i in the ith iteration, the optial values i of dual variables can iteratively be achieved by the subgradient ethod 35] α i+ = α i a i β i+ P t N = N n= i = β i b i N n= ti, =,...,N in which i is the iteration nuber, and a i and b i are step sizes designed properly. Within each iteration, the subcarrier pairing paraeter and power allocation vectors can respectively be updated by 8 and with the updated α and β. Then, the algorith to find the optial resource allocation vectors can be designed as in Algorith. Algorith The Optial esource Allocation Algorith tep : et i =, and initialize α i, β i, ε, and ax iter, tep : If i <ax iter, a i = b i = 0.0/ i, tep 3: Copute t i by using α = α i and β = β i, tep 4: Copute i by 8 using α = α i and = t i, tep 5: Copute α i+, β i+ by using α = α i, β = β i, = i, and = t i, tep 6: If α i+ α i / α i+ <ε and β i+ β i / β i <ε, exit and output α = α i+, β = β i+, = i, and t = t i ; otherwise, set i = i + and go to tep. Denote the optial values of the original proble, the relaxed proble, and the relaxed dual proble 4 as o, r, and d, respectively. It is obviously d r o.

5 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF 563 Because the optial t achieved by solving 4 and 5 satisfy C3, C4, and 0, }, d is also the dual optiu value for proble. In our siulation, we find that the duality gap is asyptotically zero when the nuber of subcarriers is reasonably large. Based on the analysis and siulations of 7] and 9], as well as our paper, it can be concluded that d. = r. = o for ost of the practical cases. If the subcarrier nuber is N, then the total nuber of all possible pairing configurations is N!. The coplexity of coputing the achieved rate is N for a given subcarrier pairing schee. Thus, the coplexity of exhaustive search is ON N!, which is prohibitively high. However, within each iteration of Algorith, the coplexity of the proposed algorith is doinated by the coputation of 9, which is ON in ters of logarith and ultiplication operations. The coplexity of coputing the optial power allocation and the su rate is ON. Therefore, the total coplexity for Algorith is OkN, where k is the nuber of iterations. It is obvious that the coplexity is tractable. In 7] and 9] it is shown that the duality gap of the optiization proble is always zero when the tie sharing condition is satisfied, regardless of the convexity of the objective function. They also showed that the tie-sharing condition will be satisfied if the optial value of the optiization proble is a concave function of the constraints. In our case, the optial subcarrier pairing ay vary as the power constraint changes. Therefore, the axiu su rate as a function of the su power constraint ay have discrete changes in the slope at the transition points where the optial subcarrier pairing schee changes. The sudden jup in the slope ight ake the optiization nonconcave with the su power. However, 9] also indicates analytically and through siulations that the concavity will be asyptotically satisfied as the nuber of subcarriers becoes large. This is because the aount of discrete slope change tends to decrease with ore subcarriers since the bandwidth affected by each change becoes narrower. Therefore, the curve is expected to be ore concave as the nuber of subcarriers increases. However, 9] as well as we cannot rigorously prove this in atheatics. In our siulations, we found that the concavity is ostly satisfied when the nuber of subcarriers is reasonably large, which is consistent with that observed in 9]. For exaple, when N =, we have observed that only about 0.8% of the possible channel conditions will result in the nonconcavity, and when N=4, the probability turns to be 0.%, and the su rate is alost always concave in the su power constraint when N = 8. Therefore, it can be concluded that the duality gap is virtually zero for ost of the practical OFDM cases. This will also be verified by our siulation results. C. Lloyd Algorith-Based Codebook Design If perfect CI can be achieved at the source and relay, the resource allocation vectors can be siply deterined by Algorith. However, as stated earlier, due to liited resource of feedback link, full knowledge of the CI available at the transit sides is difficult in OFDM systes. To solve this proble, we propose a liited feedback algorith for power allocation and subcarrier pairing in this section. In this algorith, the destination, which is assued to have full forward CI, selects a resource allocation vector fro an elaborately designed codebook upon receiving the current CI and transits its index to the source and relay through a liited nuber of feedback bits. This technique eploys a codebook of quantized power allocation and subcarrier pairing designed offline and equipped on the source, relay, and destination. The codebook construction for liited-bit feedback can be linked to a vector quantization proble. We use the Lloyd algorith 3] to search for good resource allocation codebooks based on su rate criterion. To design the liited-feedback-based codebook, we have to construct and iteratively use the nearest neighbor rule and the centroid condition, which play crucial roles in the Lloyd algorith. In the proposed algorith, the centroid condition is designed to select the optial codeword with axiu syste rate in a given region, whereas the task of the nearest neighbor rule is to deterine the region in which the vectors are closest to the optial codeword of this region. Notice that the optiization of finding the regions and optial resource allocation schee is equivalent to designing a vector quantizer with a odified distortion easure 3]. Taking the optial rate perforance as the design criterion, we use the error distance function to easure the average distortion. Using the centroid condition and the nearest neighbor rule iteratively, the error distance will decrease. uppose that the destination has perfect CI h =h D, h, h D, where h D =h D,...,hN D, h =h,...,hn, and h D =h D,...,hN D, respectively, denote the CIs of D,, and D at a particular period. Given b bits of feedback, the space defined by all possible sets of h is quantized into B = b regions. In the sequel, we set codeword as c =,,, n =,...,N} and denote c h as the end-to-end su rate of a given channel condition h and codeword c. Then c h = log +. 3 We first randoly generate the training channel condition set H = h l,l =,...,M} with M B. Then, we can easily obtain the training code set T = ch h H}, in which ch denotes the optial code achieved by Algorith for a given h H. The objective of the Lloyd-algorith-based codebook design is to randoly choose a codebook C = c, c,...,c B } of size B fro the training code set T and refine it. The error distance function is defined as } DC =E h H ch h ax c k h 0 k B 4 where E } is the expectation of a rando variable. Using this distortion function, the codebook design algorith can be suarized as in Algorith.

6 564 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 Algorith The Lloyd Algorith Based Codebook Design tep : tep : et j =, ε>0, randoly generate the training code set and select the initial codebook tep : C j = c j, cj,...,cj B } fro T, then calculate DC j by 4; tep 3: tep : Cluster the set of possible channel realization vectors H into B quantization regions with the kth region tep 4: Q j k, k =,...,B, denoted as } Q j k h = c jk h c jl h l,,...,b} tep 5: tep 3: Using 3, generate a new codebook C j+ with the kth codeword c j+ k defined as c j+ k =argax E c T h Q j c h,k=,,...,b k tep 6: tep 4: Calculate the average distortion DC j+ by 4; tep 7: tep 5: If DC j+ <DC j +ε for soe sall ε, stop iteration and set the optial C = C j+ ; otherwise set j = j + and go back to tep. While the offline design of codebook sees to be coputationally coplex and tie consuing, the real-tie feed back process is quite siple. D. Feedback chee Upon receiving the instantaneous CI h, the destination searches over all the codewords in the designed codebook of size B and selects the qth codeword provided with axiu su rate, i.e., q =argax q c q h. Afterward, the destination sends back the index q to both the source and the relay through a noiseless feedback link. ince the source and relay have been equipped with the sae codebook copies, upon receiving q, the source transits with power and the relay with power indexed by q. IV. OPTIMAL EOUCE ALLOCATION FO ENHANCED DF MODE Depending on whether the relay is helpful, each subcarrier pairing ay work in either the relaying ode or the idle ode. For a subcarrier pair working in the idle ode, the idle subcarrier in the relaying phase is not utilized. We further allow the source to transit extra essages on those idle subcarriers in the relaying phase, which is called enhanced DF ode in this paper. A. Forulation of the Optiization Proble iilarly, the achieved rate of the enhanced DF ode is given at the botto of the page.,,,, and P respectively denote the source power in the listening phase, the source power in the relaying phase, and the relay power in the relaying phase. Because the condition to activate the relay depends not only on the channel gains but also on the power allocation, we define an indicator ρ 0, } to show the status of P, n at the relay, i.e., the relay is used for P, n if ρ =, otherwise, it is not used. Let the equivalent channel gain = λ λn D /λ + λn D λ D, and let =, +. Then, the optiization proble based on the su rate of the enhanced DF ode can be forulated as ax P,t,ρ} s.t. D : D : ρ log + λ + ρ log +, λ D } + ρ log +, λn D = = n, = n= ρ, +, } + ρ P t, D3 :,,P,,P 0 n 5 where p =,P,,P, 3 NN, t = NN, and ρ =ρ NN. iilarly, we ake a continuous relaxation to the optiization proble and obtain a standard convex proble. Moreover, we respectively denote ρ, and, = ρ, = ρ,, =, as the actual power consuption at the source and the relay in = log +, λ D +log in log +, λ +, λn D, log + }, idle ode }, λ D + λn D, relaying ode

7 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF 565 the two phases. Then, the relaxed optiization proble is forulated as ax s,t,ρ} ρ log + ρ + ρ log + λ D, ρ + ρ log + λ n } D, ρ s.t. D4 : 0 n, D5 : ρ 0 n D6 : +, +, = P t D7 :,,,, 0 n, and D. 6 B. Dual olution of the elaxed Proble Dualizing the constraints D and D6, we obtain the Lagrangian Ls, t,ρ,α, β = ρ log + + ρ log ρ + λ D, ρ + ρ log + λ n } D, ρ + α P t,i + + β n n= i= = 7 where α and β n are dual variables, as before. Then, the dual objective function and the dual proble can, respectively, be expressed as gα, β = ax Ls, t, ρ, α, β, s.t. D, D4 D6 8 s,t,ρ} gα, β, s.t. α 0, β 0. 9 in α,β} Taking derivatives of L with respect to,,, and,, we obtain the optial solutions = ρ α ] + 30, = ρ α ] + λ 3 D, = ρ α ] + λ n. 3 D Denote as the rate contribution of P, n to the Lagrangian in the relaying ode and I in the idle ode. Then, we have = log + α ] + α α ] + 33 I = log + λ D α ] + λ D α α ] + λ D + log + λ n D α ] + λ n D α α ] + λ n. 34 D Easily, we obtain the optial indictor as ρ =, when > I 0, otherwise. 35 Denoting =ρ + ρ I β n for briefness, we obtain the optial subcarrier pairing paraeter as t =, =argax =,...,N 0, otherwise n. 36 We siilarly update the Lagrange ultipliers α and β by subgradient ethod as α i+ = α i a i β i+ P t N = N n= = β i b i N n= ti +, +, }, =,...,N. 37 With the updated α and β in each iteration, we can update the subcarrier pairing t, the power allocation vectors,,,,, and the indicator ρ by Algorith. Notice that the iteration procedure in Algorith should be odified in soe places. For exaple, before coputing t i in Algorith, we have to figure ou, I and ρ by 54, 34, and 55, respectively. iilarly, we can use Algorith to design a codebook for liited feedback. C. esource Allocation Under Individual Power Constraints In this section, we investigate the resource allocation under individual power constraints for the source and the relay. For the individual power constraints, the su powers at the source

8 566 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 and the relay have separate constraints, which can be expressed as = P, n= P 38 where P and P denote the source power constraint and the relay power constraint, respectively. For a given subcarrier pairing P, n, the ode selection criterion 30], 3] is expressed as elaying ode : λ λ D Then, we can siilarly obtain a Lagrangian L = + µ P = + ρ λ } + µ P λ D + λ n D. 39 n= λ n D 40 where is the P set of the relaying ode. The Lagrange coefficients µ,µ 0 are chosen such that the individual power constraints are satisfied. The Lagrange ultiplier ρ 0 corresponds to the ode selection criterion. For alost all of the subcarrier pairs belonging to relaying ode, the authors in 3] conclude that the selection criterion will be satisfied when λ = λ D + λ n D 4 with a possible exception pair satisfying γd n /γ D = λ /λ. However, usually, there will be at ost one subcarrier pair in this set. Fortunately, we find that the exception P, n has the sae contribution and cost to the Lagrangian in the odel, which are respectively /logλ D /µ and µ /µ /λ D, no atter if it is classified into relaying ode or idle ode. Therefore, we assign it to relaying ode thereafter. For relaying ode, 4 iplies =λ n D /λ λ D P. Then, P and will be zero or positive siultaneously. Thus, in the relaying ode, we can first allocate the total power of P, n and then obtain the corresponding and.let λ = n D λ +λn D λ D 4 = λ λ D λ +λn D λ D in the idle ode. Denote the equivalent channel gain of P, n by λ λn D = λ, relaying ode +λn D λ D λ D, idle ode. 44 We can also use a unified rate expression to deonstrate the original optiization as = log + λ. 45 The unified rate helps siplifying the optiization in the sae way. Let = ρ log + + ρ ρ log + λ D ρ +log + 3 λ n ] } D 46 ρ where is the su power of P, n in the relaying ode, which can be obtained fro 4. and 3 are respectively the powers used by the direct link of P, n in the listening and relaying phases. Then, the su rate optiization is forulated as ax,,t,ρ} s.t. E : = n, E : = = n= E3 : ρ 0, }, E4 : 0, } E5 : η P E6 : η P, E7 : j 0 j. 47 Let P=,, 3 NN3, t= NN and ρ =ρ NN. Denote η λ n,d = λ, relaying ode +λn D λ D 48, idle ode λ η = λ D λ, relaying ode +λn D λ D 49 0, idle ode. We dualize the constraints E, E5, E6, and 4. Then, the generated Lagrange function is in the relaying ode, and P = = 0 43 LP, t,λ,λ, β = + β n n= =

9 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF µ P + µ P η η 3 50 where µ 0, µ 0, and β =β,β,...,β N 0 are dual variables. The dual objective function is gµ,µ, β = ax LP, t,µ,µ, β, ρ P,t,ρ} s.t. E, E7, E8, E9 5 and the dual proble is in µ,µ,β} gµ,µ, β, s.t. µ 0,µ 0. 5 We take derivatives of L with respect to,, and 3 and obtain ] + = ρ µ η + µ η = ρ ] + µ λ D 3 = ρ ] + µ λ n. 53 D Denote and I as the rate contributions of P, n to the Lagrange function in relaying and idle odes, respectively. Then = log + P µ η I = µ P + P log P + +λ D P 3 µ η P +log + λ n D ] P 3 P 54 where =/µ η +µ η /λ ] +, = /µ /λ D ]+, and P 3 =/µ /λ n D ]+. We easily obtain ρ =, when > I 0, otherwise ubstituting 53 into 50, we obtain. 55 LP, t,µ,µ, β, ρ = T + K µ,µ,β 56 where T = ρ + ρ N β n, and K µ,µ,β = µ P + µ P + N n= β n. Both T and K µ,µ,β are independent of. Therefore, the optial t is obtained as t = : =argax=,...,n T 0 : n otherwise n. 57 µ, µ, and β n that iniize gµ,µ, β are achieved by the subgradient ethod µ i+ = µ i ai P N N µ i+ η = µ i bi P N = i i N n= η 3i i β n i+ = β n i c i N = ti,n=,...,n. 58 By now, we have obtained the optial ode selection vector, subcarrier pairing vector t, and power allocation vector,, 3 for the given dual variables, respectively. We can siilarly update the subcarrier pairing and power allocation vectors as in Algorith with soe slight odifications. The Lloyd algorith can be eployed again to design the codebook. V. IMULATION EULT We present soe siulations to deonstrate the perforance of the proposed algoriths in this section. The channels of the subcarriers are independent identically distributed i.i.d. subject to ayleigh fading, with a large-scale fading path loss exponent.5. The channel coefficients are assued to be constant within two phases and varying independently fro one period to another. We assue equal noise power at relay and destination nodes, i.e., σr = σd. In the siulations, quadrature phase-shift keying odulation is adopted, and the step sizes a i and b i for the subgradient ethod are set to be 0.0/ i, where i is the iteration index. The size of the CI set in Algorith is 0 4, which is far ore than the quantized regions. everal existing schees are copared with the proposed algorith in ters of su rate. These existing schees include the following: i UPA w/o P: The essages transitted on subcarrier at the source node will be retransitted on subcarrier at the relay node; the power is allocated equally at the source and relay subcarriers. ii OPA w/o P: The essages transitted on the subcarrier at the source node will be retransitted on the subcarrier at the relay node; the power allocation is perfored according to waterfilling at the source and the relay subcarriers. iii UPA with P: The essages transitted on the subcarrier at the source node will be retransitted on subcarrier n at the relay node by subcarrier pairing; the power is allocated equally at the source and relay subcarriers. Then, the perforance of the proposed algorith with different feedback bits is deonstrated. In addition, the perforance gap between the enhanced DF and the selective DF odes versus the subcarrier nuber is also revealed in our siulations. A. ate Coparison for Different chees chees and 3 are copared with the proposed algorith with perfect CI and liited feedback schee in Fig.. The upper curve denotes the proposed joint power allocation and

10 568 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 Fig.. yste su rate versus N for the proposed enhanced DF relay schees and the existing schees, where w/o denotes without. Fig. 4. u rate versus the nuber of subcarriers for the enhanced DF and selective DF with fixed feedback bit of. These curves are obtained with d = 0.4. Fig. 3. yste su rate versus N for the proposed enhanced DF relay schee with different levels of feedback bits. The upper curve denotes the perfect CI case, and the lowest curve denotes the schee without feedback, where the power is uniforly allocated. Other curves deonstrate the effect of different feedback bits on su rate. subcarrier pairing for the enhanced DF schee with perfect CI. The second upper curve denotes the proposed joint power allocation and subcarrier pairing for the enhanced DF schee with -bit feedback. The other three curves denote the existing schees and 3, respectively. We can observe that, only with -bit feedback, the proposed joint power allocation and subcarrier pairing for the enhanced DF relay outperfors the existing schees and 3 greatly. Therefore, we can conclude that the joint power allocation and subcarrier pairing ake valuable contribution to syste su rate. B. ate Coparison for Different Feedback Bits The joint power allocation and subcarrier pairing for the enhanced DF relay with different feedback bits are copared in Fig. 3. We can find that only a few feedback bits are enough to Fig. 5. u rate versus the nuber of subcarriers and the perforance gap between the schees with odified idle and selective relaying odes. We assue that syste operates with fixed feedback bit level of. The curves are obtained with d = 0.8. achieve ost of the perforance gains of the perfect feedback. For exaple, with 4 bits of feedback at a rate of.5 in Fig. 3, there is only a -.7-dB gap to the perfect CI case, and we also notice that further increasing the feedback bits brings degressive iproveent, which iplies that the feedback bits as well as the codebook size in the odel are not necessarily too large. C. ate Versus Different ubcarrier Nubers Under u and Individual Power Constraints With the su and individual power constraints, the su rates versus the nuber of subcarriers for the selective DF and the enhanced DF relaying odes are illustrated in Figs. 4 and 5. We

11 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF 569 Fig. 6. u rate versus average N for d = 0.5, 0.5, 0.75, with feedback bits of. Fig. 7. u rate versus the relay location for the enhanced DF and selective DF with -bit feedback when N = 4. consider the cases that subcarrier nuber N =, 4, 8, 6 with fixed feedback bit level of. For the sake of fairness, we assue P + P = P t. In addition, as for the case with individual power constraints, we assue P =3P. The constraints are set with the practical consideration that the relay often plays the role of assisting the transission between the source and the destination. Moreover, if ore power is assigned to the relay node, the achievable rate will be liited since soe of the relay power will not be used. Assuing P =3/4P t will ake the coparison with the su power constraints uch fairer. In addition, we assue that the relay locates in a line between the source and the destination, and the D distance is one unit. Denote d 0 <d< as the distance. Thus, the D distance is d. Fig. 4 is obtained with d = 0.4, whereas d = 0.8 in Fig. 5. We find that the enhanced DF ode always outperfors the selective DF ode and the schees without subcarrier pairing, especially when the channel condition of D is relatively poor. We can also observe that the bigger the subcarrier nuber is, the bigger the perforance gap between the two odes is. As for the cases under different constraints, the perforance of the su power constraint is better than that of the individual power constraints, which is due to the ore flexibility of power allocation between source and relay under the su power constraint. In addition, we consider the duality gaps in the two figures. The siulation results exactly coincide with our analysis in ection III. We find that the dual solutions approxiate to the optial values of 47 in our siulation. The duality gaps turn out to be nearly 0 when the nuber of subcarriers is reasonably large, which is consistent with the prediction in 9]. D. Effect of elay Location On ate To exploit the syste rate versus N for different relay locations, we siulate the rate versus N by setting d = 0.5, d = 0.5, and d = 0.75, respectively. Fig. 6 deonstrates the effect of relay location to syste su rate at different Ns with a fixed feedback bit level of. We can find that the enhanced DF ode always outperfors the selective DF ode and the OPA w/o P in any kind of d, and the channel condition of plays a ore iportant role than the channel condition of D in general. In addition, we find that the gap between the syste su rates achieved by the enhanced DF and the selective DF is larger when d 0.5 is larger, whereas the perforance gap between the enhanced DF and the selective DF is tiny when d = 0.5. To exploit the effect of relay location to the syste perforance, we siulate the su rate versus relay location d in Fig. 7. The figure is obtained with the fixed feedback bit and subcarrier nuber N = 4. We find that the rate reaches axiu at about d = We also observe that coparing with the proposed schee with -bit feedback, the perforance loss of the schee UPA with P is not very big at this location, which iplies that if none of or D channels is very poor, or there is no great difference between the channel conditions of and D, and the schee UPA with P can provide acceptable perforance with N = 4. However, if at least one of these channel conditions is very poor, we would be better to dynaically allocate the power and subcarrier resources, since the proposed algorith can achieve rearkable perforance gain. In addition, the perforance loss of the schee UPA with P increases with the nuber of subcarriers due to frequency diversity and ore flexibility in pairing of large N. Figs.8 and 9 are obtained with N = 3 and 64, respectively. We observe that the perforance gains of the proposed algorith are uch ore rearkable. The rearkable perforance gain results fro the uch ore pairing degree provided by the big subcarrier nuber. There is another general trend that can be observed fro the two figures. The rate gap between the enhanced DF and the selective DF is larger when d 0.5 is larger. The perforance iproveent of the enhanced DF is due to the extra direct-link transission in the second phase, since the relay has a high possibility of being idle when the or D channel is poor because of the large d 0.5.

12 570 IEEE TANACTION ON VEHICULA TECHNOLOGY, VOL. 6, NO. 6, JULY 0 Fig. 8. u rate versus the relay location for the enhanced DF and selective DF with -bit feedback when N = 3. Fig. 9. u rate versus the relay location for the enhanced DF and selective DF with -bit feedback when N = 64. VI. CONCLUION In this paper, we have discussed a liited-feedback-based joint power allocation and subcarrier pairing algorith for OFDM DF relay networks with diversity. When the relay does not forward the received sybols on soe subcarriers, we further allow the source node to transit new essages on these idle subcarriers. Both su power constraint and individual power constraints for the source and relay nodes are considered. ince the forulated optiization is a ixed integer prograing proble, we transfor it into a convex proble by continuous relaxation and then solve it in the dual doain. iulations show that the proposed algoriths can achieve considerable rate gain with tractable coplexities. It outperfors several existing schees under various channel conditions. The contribution of the extra direct-link transission is also clearly deonstrated in the siulation. In addition, we notice that a negligible perforance loss can be achieved with just a few feedback bits at different levels of N values. EFEENCE ] J. N. Lanean, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: Efficient protocols and outage behavior, IEEE Trans. Inf. Theory, vol. 50, no., pp , Dec ] A. Host Madsen and J. Zhang, Capacity bounds and power allocation in wireless relay channel, IEEE Trans. Inf. Theory, vol. 5, no. 6, pp , Jun ] C. Hoyann, K. Klagges, and M. chinnenberg, Multihop counication in relay enhanced IEEE 80.6 networks, in Proc. IEEE PIMC, ep. 006, pp. 4. 4] N.. Van and P. ajee, OFDM for Wireless Multiedia Counications. Boston, MA: Artech House, ] Y. Liu and W. Chen, Power allocation for the fading relay channel with liited feedback, in Proc. 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13 LIU AND CHEN: LIMITED-FEEDBACK-BAED POWE ALLOCATION AND UBCAIE PAIING FO OFDM DF 57 8]. Boyd and L. Vandenberghe, Convex Optiization. Cabridge, U.K.: Cabridge Univ. Press, Aug ] K. eong, M. Mohseni, and J. Cioffi, Optial resource allocation for OFDMA downlink systes, in Proc. IEEE IIT, Jul. 006, pp ] V. Luc, L. Jeroe, O. Onur, and Z. Abdellatif, ate-optiized power allocation for DF-relayed OFDM transission under su and individual power constraints, EUAIP J. Wireless Coun. Network., vol. 009, p. 6, Feb ] J. Louveaux,. Duran, and L. Vandendorpe, Efficient algorith for optial power allocation in OFDM transission with relaying, in Proc. IEEE Int. Conf. Acoust., peech, ignal Process., Mar. 008, pp ] A. Gersho and. M. Gray, Vector Quantization and ignal Copression. Boston, MA: Kluwer, ] J. G. Proakis, Digital Counications, 4th ed. New York: McGraw- Hill, ] L. M. C. Hoo, B. Halder, J. Tellado, and J. M. Cioffi, Multiuser transit optiization for ulticarrier broadcast channels: Asyptotic FDMA capacity region and algoriths, IEEE Trans. Coun., vol. 5, no. 6, pp , Jun ]. Boyd, L. Xiao, and A. Mutapcic, ubgradient ethods, in Lecture Notes of EE39O. tanford, CA: tanford Univ. Press, Oct Yong Liu received the B.. and M.. degrees in electronic engineering fro Taiyuan University of Technology, Taiyuan, China, in 004 and 007, respectively. He is currently working toward the Ph.D. degree in electronic engineering with the Departent of Electric Engineering, hanghai Jiao Tong University, hanghai, China. He is also with the tate Key Laboratory for IN, Xidian University, Xi an, China. His current research interests lie in the area of wireless counication, relay cooperation, and orthogonal frequencydivision ultiplexing systes. Wen Chen M 03 M received B.. and M.. degrees fro Wuhan University, Wuhan, China, in 990 and 993, respectively and the Ph.D. degree fro the University of Electro-Counications, Tokyo, Japan, in 999. He was a esearcher with the Japan ociety for the Prootion of ciences fro 999 to 00. In 00, he joined the University of Alberta, Edonton, AB, Canada, starting as a Postdoctoral Fellow with the Inforation esearch Laboratory and continuing as a esearch Associate with the Departent of Electrical and Coputer Engineering. ince 006, he has been a Full Professor with the Departent of Electronic Engineering, hanghai Jiao Tong University, hanghai, China, where he is also the Director of the Institute for ignal Processing and ystes. He is also with the tate Key Laboratory for Integrated ervice Networks, Xidian University, Xi an, China. He has published ore than 00 papers in IEEE journals and conferences. His interests cover network coding, cooperative counications, cognitive radio, and utiple-input ultipleoutput orthogonal frequency-division ultiplexing systes. Dr. Chen received the Ariyaa Meorial esearch Prize in 997 and the PIM Post-Doctoral Fellowship in 00. He received the honors of New Century Excellent cholar in China in 006 and Pujiang Excellent cholar in hanghai in 007. He was elected to the Vice General ecretary of hanghai Institute of Electronics in 008. He is in the editorial board of the International Journal of Wireless Counications and Networking and serves on the Journal of Counications, Journal of Coputers, Journal of Networks, and EUAIP Journal on Wireless Counications and Networking as lead Guest Editor. He is the Technical Progra Coittee Chair for IEEE- ICCC008 and IEEE-ICCT0 and the General Conference Chair for IEEE- ICI009, IEEE-WCNI00, and IEEE-WiMob0.