Optimal Power Allocation for Two-Way Decode-and-Forward OFDM Relay Networks

Optimal Po wer Allocat ion for T wo-W ay Decode-and-F orward OFDM Rela y Networks Fei He ∗ , Y in Sun ∗ , Xiang Chen, Limin Xiao, Shidong Zhou State Ke y Labo ratory on Micro wa ve and Digital Communicatio ns Tsinghua National Laborato ry for Infor mation Science and T echnology Departmen t of E lectronic Engineer ing, Tsinghua University , Beijing 1000 84, China Email: h ef08@mails.tsingh ua.edu.cn , sunyin02@gma il.com Abstract —This paper presents a no vel two-way decode-and- fo rward (DF) relay strategy for Orthogonal Frequency Division Multipl exing (OFDM) relay netwo rks. This DF relay strategy em- ploys multi- subcarrier joint chann el coding to lev erage frequency selectiv e fading, and thus can achiev e a higher data rate than the con ventional p er -subcarrier DF relay strategies. W e fu rther propose a low-complexity , optimal power allocation strategy to maximize the data rate of the proposed rela y strategy . S imulation results suggest that our strategy obtains a su bstantial gain ove r the per -sub carrier DF r elay strategies, and also out perf orms the amplify-and-forwa rd (AF) r elay strategy in a wide signal- to-noise-ratio (SNR) region. I . I N T R O D U C T I O N In rec ent years, re laying has emerged as a powerful tech- nique to improve the coverage and th roughp ut of wireless networks. Comp ared with the tradition al one-way relay ing, two-way relay ing provid es better spec tral efficiency , where two terminal nod es employs an in termediate relay no de to exchange infor mation simultaneou sly [1], [2]. Orthogo nal Frequency Division Mu ltiplexing (OFDM) is an essential broadb and transmission techniqu e to improve the spectral ef ficiency of wireless networks. A comb ination of OFDM and relay ing tec hniques h as b een advocated by many industry stand ardization gro ups of n ext generation wir eless networks, such as IEEE 80 2.16m and 3GPP’ s LET -Advanced. In one-way OFDM relay networks, mu lti-subcarrier joint decode- and-for ward (DF) relaying was stud ied in [3]–[5], which can achieve h igher data rate than per-subcarrier DF relaying. For two-way OFDM relay n etworks, th e a mplify- and-fo rward (AF) relay strategies were comm only ad opted [6]–[8]. Howe ver, their perfor mance is quite poor in the low signal-to-n oise-ratio (SNR) r egion due to the amplified noises. The per-subcarrier DF relay strategies were considered in [9]– [11], which are essentially simple accumu lations of nar row- band two-way DF r elaying over the individual subcarriers. Unfortu nately , these strategies suffer from rate losses due to channel m ismatching. ∗ Fei He and Y in Sun cont ribute equally to th is work . This work is supported by National Basic Research Program of China (2012CB316002), Nationa l S&T Major Proje ct (2010ZX03005-0 03), National NSF of China (60832008), China’ s 863 Projec t (2009AA011501), T singhua Resea rch Fund- ing (2010THZ02-3), PCSIR T , Internationa l Science T echnol ogy Cooperation Program (2010DFB1041 0) and Tsinghua-Qua lcomm Joint Research Program. In this paper, we propose a novel mu lti-subcarrier DF relay strategy for two-way OFDM relay networks. By per- forming chan nel coding acr oss subcarrier s, this strategy can exploit frequen cy selective fading, and achieve higher data rate than the per-subcarrier DF relay strategy in [9]. W e further formulate a power alloca tion p roblem to maximize the exc hange r ate , which is defined as the maxim al data rate can be simultan eously ach ie ved in both direction s. An efficient dual decomp osition algorithm is pro posed to resolve this prob lem, which has a linear com plexity with respect to the num ber of subcarriers. Simulation r esults show that the propo sed multi-subcar rier DF re lay strategy o utperfor ms not only the conventional per-subcarrier DF relay strategy , but also the AF r elay strategy in a wide SNR region . I I . S Y S T E M D E S C R I P T I O N W e c onsider th e two-way OFDM relay network shown in Fig. 1: two terminal no des T 1 and T 2 exchange inform ation via an in termediate relay n ode T R . Assume that each n ode has a single an tenna and op erates in a half -duplex mod e, i.e., transmitting and recei v ing in o rthogon al time slots [1], [2]. All the nod es employ OFDM air interface with the same N subcarriers. Y R 1 Y RN X 11 X 1 N X 21 X 2 N … … … … … T 1 T R T 2 11 h 21 h 1 N h 2 N h (a) The MA phase. X R 1 X RN Y 11 Y 1 N Y 21 Y 2 N … … … … … T 1 T R T 2 11 h ɶ 21 h ɶ 1 N h ɶ 2 N h ɶ (b) The BC phase. Fig. 1. System model of two-way OFDM relay netw ork. The DF relay p rocedur e co mprises of a mu ltiple-access (MA) ph ase and a broad cast (BC) phase with out direct tr ans- missions, as shown in Fig. 1. W e set X i = ( X i 1 , . . . , X iN ) and Y i = ( Y i 1 , . . . , Y iN ) for i = 1 , 2 , R , wher e X in and Y in denote the norm alized tran smitted and recei ved sig nal in the n th subcar rier at T i . In the MA phase, T 1 and T 2 transmit X 1 and X 2 simultaneou sly to the relay n ode, and the relay T R perfor ms mu lti-user detectio n to f ully deco de X 1 and X 2 from the recei ved Y R ; in the BC p hase, the re lay T R broadc asts X R = f ( ˆ X 1 , ˆ X 2 ) to T 1 and T 2 . Specifically , in the n th subcarrier, h 1 n and h 2 n denote th e channel co efficients from T 1 and T 2 to T R , respecti vely , ˜ h 1 n and ˜ h 2 n denote the cha nnel coe ffi cients f rom T R to T 1 and T 2 , respectively . Thus, th e rec ei ved signals Y in ’ s in the n th subcarrier at T i ’ s are given by Y Rn = p P 1 n h 1 n X 1 n + p P 2 n h 2 n X 2 n + Z Rn , (1) Y 1 n = p P Rn ˜ h 1 n X Rn + Z 1 n , (2) Y 2 n = p P Rn ˜ h 2 n X Rn + Z 2 n , (3) where P in denotes the transmit po wer in the n th subcarrier at T i , and Z in denotes indep endent comp lex additive white Gaussian no ises with zero mean an d unit variance, i.e., Z in ∼ C N (0 , 1) , for i = 1 , 2 , R . Ther efore, P in essentially den otes the corresp onding transmit SNR. W e assume that µ ∈ (0 , 1) d enotes the fixed propo rtion of time slot allocated to the MA phase, an d all the terminals are subject to separate power co nstraints P N n =1 P in ≤ P i max ( i = 1 , 2 , R ), where P i max denotes the max imum a vailable power for T i . I I I . A N OV E L D F R E L AY S T R A T E G Y F O R T W O - W A Y O F D M R E L A Y N E T W O R K S For two-way OFDM relay networks, the co n vention al DF relay strategies simply applied the n arrow-band DF techniqu e over each su bcarrier in dependen tly , and the overall throu ghput was the sum rate of all the subc arriers [9], [11]. Ho wever , these strategies suffer fr om rate losses due to channel m ismatching. In this section, we propose a novel multi-subcarrier DF relay strategy , which p erforms channel co ding across sub- carriers to le verage f requency selective fadin g, an d achie ve higher data rate. An achievable rate region for this strategy is provided in the following theo rem. Theorem 1. Any rate pair ( R 12 , R 21 ) satisfying the follow- ing inequa lities is a chievable for the two-wa y OFDM r elay network given b y (1) - (3) : R 12 ≤ µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n  , (4a) R 12 ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 2 n | 2 P Rn  , (4b) R 21 ≤ µ N X n =1 log 2  1 + | h 2 n | 2 P 2 n  , (4c) R 21 ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 1 n | 2 P Rn  , (4d) R 12 + R 21 ≤ µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n  , (4e) N X n =1 P in ≤ P i max , i = 1 , 2 , R, (4f) P in ≥ 0 , ∀ n ∈ N , i = 1 , 2 , R. (4g) wher e N , { 1 , . . . , N } , R 12 and R 21 denote th e a chievable data rates fr om T 1 to T 2 and fr o m T 2 to T 1 , r espective ly . Pr oof: An achiev able r ate region of a DF relay strategy for the discrete-memo ryless two-way relay n etwork has be en giv en by the set of ( R 12 , R 21 ) r ate pairs satisfy ing [ 12] R 12 ≤ min { µI ( X 1 ; Y R | X 2 ) , (1 − µ ) I ( X R ; Y 2 ) } , (5a ) R 21 ≤ min { µI ( X 2 ; Y R | X 1 ) , (1 − µ ) I ( X R ; Y 1 ) } , (5b ) R 12 + R 21 ≤ µI ( X 1 , X 2 ; Y R ) . (5c) Each mu tual inf ormation item in (5a)-(5c) cor responds to the achiev ab le rate of a par allel point-to-poin t chan nel. Similar to th e idea in the proo f of [3, Theorem 1], we choose the input signa ls for each subcarrier to be ind ependen t Gaussian distributed with unit v ar iance, i.e., X in ∼ C N (0 , 1) . Thu s, each mutual info rmation item in (5a)-(5c) is rep laced by the sum of N lo garithmic rate items decide d by (1)-(3) with separate power constraints. The proo f is c omplete. Remark 1: T he key idea of th is multi-sub carrier two-way DF relay strategy is in troducin g channel cod ing across sub - carriers to fully exploit fr equency selective fading. Imp licitly , the inform ation tr ansmitted over one subcarrier in the MA phase may be for warded over some other sub carriers in the BC pha se. By this, the pr oblem fro m mismatching o f wireless channels over sub carriers is resolved. The achie vable rate region of Theorem 1 is no smaller tha n that achieved by the per-subcarrier two-way DF re laying, wh ich is th e set of ra te pairs satisfyin g [9] R 12 ≤ N X n =1 min  µ log 2  1 + | h 1 n | 2 P 1 n  , (1 − µ ) log 2  1 + | ˜ h 2 n | 2 P Rn  , R 21 ≤ N X n =1 min  µ log 2  1 + | h 2 n | 2 P 2 n  , (1 − µ ) log 2  1 + | ˜ h 1 n | 2 P Rn  , R 12 + R 21 ≤ µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n  . Therefo re, multi-subcarrier two-way relay cha nnel is not a simple linear co mbination of multiple nar row-band sing le- subcarrier two-way relay subc hannels. Similar obser vations have b een fou nd for on e-way parallel relay networks [3]. I V . O P T I M A L P OW E R A L L O C AT I O N In this section, we investigate th e largest ac hiev able sym - metric exchange d ata rate o f o ur propo sed DF relay stratey , which can be ap proache d b y an optimal power allocatio n. A. Pr ob lem F ormulation By optimizing th e power allo cation strategy ( P 1 , P 2 , P R ) , our ob jecti ve is to maximize the e x change rate R X = min { R 12 , R 21 } , which is defined as th e data rate ca n be achieved simu ltaneously in b oth d irections, wh ere P i = [ P i 1 , P i 2 , . . . , P iN ] T denotes the power allocation vector at T i , for i = 1 , 2 , R . This c an be expressed a s the following co n vex optimization prob lem: max P 1 , P 2 , P R ,R X R X (7a) s . t . R X ≤ µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n  , (7b) R X ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 2 n | 2 P Rn  , (7c) R X ≤ µ N X n =1 log 2  1 + | h 2 n | 2 P 2 n  , (7d) R X ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 1 n | 2 P Rn  , (7e) R X ≤ µ 2 N X n =1 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n  , (7f) N X n =1 P in ≤ P i max , i = 1 , 2 , R, (7g) P in ≥ 0 , ∀ n ∈ N , i = 1 , 2 , R. (7h) It is readily o bserved that in the p roblem (7), P 1 and P 2 are only related to the constrain ts ( 7b) (7 d) ( 7f), while P R is only related to the constraints ( 7c) (7 e). This observation h elps to decomp ose our o riginal p ower allocation problem (7) into the following two su bproble ms: max P 1 , P 2 ,R MA R MA (8) s . t . R MA ≤ µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n  , R MA ≤ µ N X n =1 log 2  1 + | h 2 n | 2 P 2 n  , R MA ≤ µ 2 N X n =1 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n  , N X n =1 P 1 n ≤ P 1max , N X n =1 P 2 n ≤ P 2max , P 1 n ≥ 0 , P 2 n ≥ 0 , ∀ n ∈ N . max P R ,R BC R BC (9) s . t . R BC ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 1 n | 2 P Rn  , R BC ≤ (1 − µ ) N X n =1 log 2  1 + | ˜ h 2 n | 2 P Rn  , N X n =1 P Rn ≤ P R max , P Rn ≥ 0 , ∀ n ∈ N . W e can denote R ⋆ MA and R ⋆ BC as the optimal values for the MA subproblem (8) and the BC subpro blem (9), respec- ti vely . Eventually , the ma ximal p ractical exchange r ate for our propo sed DF strategy is given by R ⋆ X = min { R ⋆ MA , R ⋆ BC } . B. Pr op osed Dual Decomp osition Algo rithm The interior-point methods can be u sed to solve both of the conve x optim ization p roblems (8) and (9), h owe ver , they quickly become com putationally in tractable as N increases, because they h a ve a com plexity of O ( N 3 ) at least when solving the search direction in each iteration [1 3]. T herefore , we present a lo w-complexity dual decompo sition alg orithm for the subpr oblems ( 8) an d (9), to efficiently obtain the o ptimal solution to (7). Next, we will take the subp roblem (8) as an example to illustrate this algor ithm. Note th at problem (8) is strictly feasible. Then , accord ing to the Slater ’ s conditio n [ 13], it is equiv a lent with the following dual optimization prob lem: max λ , α  0  min P 1 , P 2  0 ,R MA L ( P 1 , P 2 , R MA , λ , α )  , (10) where L ( P 1 , P 2 , R MA , λ , α ) = − R MA + λ 1 " R MA − µ N X n =1 log 2  1 + | h 1 n | 2 P 1 n  # + λ 2 " R MA − µ N X n =1 log 2  1 + | h 2 n | 2 P 2 n  # + λ 3 " R MA − µ 2 N X n =1 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n  # + α 1 N X n =1 P 1 n − P 1max ! + α 2 N X n =1 P 2 n − P 2max ! = N X n =1  α 1 P 1 n + α 2 P 2 n − µλ 1 log 2  1 + | h 1 n | 2 P 1 n  − µλ 2 log 2  1 + | h 2 n | 2 P 2 n  − µλ 3 2 log 2  1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n   + ( λ 1 + λ 2 + λ 3 − 1 ) R MA − α 1 P 1max − α 2 P 2max (11) is the partial L agrangian of (8), and λ = [ λ 1 , λ 2 , λ 3 ] T , α = [ α 1 , α 2 ] T are nonn egati ve dual variables associated with th e three rate constra ints and two power constrain ts, respectively . According to ( 11), the inner min imization prob lem of (10) can be decomposed as N independent per-subcarr ier power allocation prob lems. Hence , the compu tational co mplexity for solving the inne r problem is only linear with respec t to N . In ad dition, the op timal ( P 1 n , P 2 n ) must satisfy the following Karush-Kuhn- T ucker (KKT) condition s for g iv en dual variables ( λ , α ) [13]: ∂ L ∂ P 1 n = α 1 − µλ 3 | h 1 n | 2 2 ln 2(1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n ) − µλ 1 | h 1 n | 2 ln 2(1 + | h 1 n | 2 P 1 n )  ≥ 0 if P 1 n = 0 = 0 if P 1 n > 0 , (12) ∂ L ∂ P 2 n = α 2 − µλ 3 | h 2 n | 2 2 ln 2(1 + | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n ) − µλ 2 | h 2 n | 2 ln 2(1 + | h 2 n | 2 P 2 n )  ≥ 0 if P 2 n = 0 = 0 if P 2 n > 0 . (13) Thus, it must belon g to o ne o f the f ollowing four ca ses: Case 1: P 1 n > 0 , P 2 n > 0 . Th en the for mulas (12) and (13) ho ld with equality . I t is hard to solve (12) and (13) dir ectly since they are both q uadratic equation s of two variables P 1 n and P 2 n . Ho wever , we can utilize an aux iliary variable defin ed as x = | h 1 n | 2 P 1 n + | h 2 n | 2 P 2 n to simplify them. M ore specifically , fr om (12) and (13), one can o btain that | h 1 n | 2 P 1 n = 2 µλ 1 | h 1 n | 2 2 ln 2 · α 1 − µλ 3 | h 1 n | 2 / (1 + x ) − 1 , (14) | h 2 n | 2 P 2 n = 2 µλ 2 | h 2 n | 2 2 ln 2 · α 2 − µλ 3 | h 2 n | 2 / (1 + x ) − 1 . (15) T aking the sum of the above two equ ations, we o btain a cubic eq uation o f x , w hich has c losed-form solution s given by Car d ano’s F ormula [14]. After deriving th e positive ro ot x of this cu bic equation, we can easily obtain the op timal P 1 n and P 2 n from (14) and ( 15). By th is p rocedur e, the qu adratic equations (12) an d (13) ar e solved an alytically by co n verting to an equiv alent c ubic equation. Fin ally , we n eed to c heck whether P 1 n and P 2 n satisfy the con ditions P 1 n > 0 , P 2 n > 0 . Case 2: P 1 n > 0 , P 2 n = 0 . Then the so lutions to ( 12) and (13) ca n be d erived as P 1 n = µ (2 λ 1 + λ 3 ) 2 ln 2 · α 1 − 1 | h 1 n | 2 , (16) P 2 n = 0 . (17) This case happ ens only if P 1 n > 0 an d the KKT conditio n (13), 2 ln 2 · α 2 ≥ 2 µλ 2 | h 2 n | 2 + µλ 3 | h 2 n | 2 1+ | h 1 n | 2 P 1 n , is satisfied. Case 3: P 1 n = 0 , P 2 n > 0 . Then the KKT con ditions can be r eformulated as P 1 n = 0 , (18) P 2 n = µ (2 λ 2 + λ 3 ) 2 ln 2 · α 2 − 1 | h 2 n | 2 . (19) This case happ ens only if P 2 n > 0 an d the KKT conditio n (12), 2 ln 2 · α 1 ≥ 2 µλ 1 | h 1 n | 2 + µλ 3 | h 1 n | 2 1+ | h 2 n | 2 P 2 n , is satisfied. Case 4: P 1 n = 0 , P 2 n = 0 . This is the d efault case whe n the above three ca ses do not happen . Then, w e optimize the dual variables ( λ , α ) fo r th e outer maximization problem of (10). W e red efine ν = [ λ 1 , λ 2 , λ 3 , α 1 , α 2 ] T . Furth er , con sidering the KKT condition for the o ptimal d ata rate R MA , we h a ve ∂ L ∂ R MA = λ 1 + λ 2 + λ 3 − 1 = 0 . (20) In view of that the o bjectiv e f unction is n ot differentiable with respect to ( λ , α ) , we consider to update ν using the subgrad ient metho d [15], [16]. S pecifically , i n the k th iteration, the sub gradient method updates ν k by ν k +1 =  ν k + s k η ( ν k )  P , (21) where [ ν ] P represents the orthogona l p r ojection of ν to the dual feasible set { ν | 1 T λ = 1 , λ , α  0 } based on a finite algorithm in [ 17], s k is th e step size of the k th iteratio n, and η ( ν k ) is the sub gradient of the outer problem of (1 0) at ν k , which can be ch osen a s η ( ν k ) =           − µ P N n =1 log 2  1 + | h 1 n | 2 P ⋆ 1 n  − µ P N n =1 log 2  1 + | h 2 n | 2 P ⋆ 2 n  − µ 2 P N n =1 log 2  1 + | h 1 n | 2 P ⋆ 1 n + | h 2 n | 2 P ⋆ 2 n  P N n =1 P ⋆ 1 n − P 1max P N n =1 P ⋆ 2 n − P 2max           , ( 22) where P ⋆ 1 n and P ⋆ 2 n are the optimal solution of the inn er minimization problem in th e k th iter ation. It has b een shown that the subgr adient upd ates in ( 21) can conver ge to the optimal dual po int ν ⋆ as k → ∞ , provided that the step size s k is ch osen a ccording to a d iminishing step size rule [16]. Let C MA ,i ( P 1 , P 2 )( i = 1 , 2 , 3) de note the the right-h and sides of three ra te constrain ts in (8), respectively , and thu s we obtain th e o ptimal R ⋆ MA = min { C MA ,i ( P ⋆ 1 , P ⋆ 2 ) , i = 1 , 2 , 3 } . The propo sed d ual deco mposition algo rithms fo r the MA subprob lem (8) ar e summarized in Algor ithm 1 . Similarly , the BC subprob lem (9) can be solved with the same technique s. Their complexity g row in the or der of O ( N ) , which ar e m uch lower than th e classic conv ex optimization software package based o n interio r-point methods. Th erefore, o ur p roposed algorithm is more fav or able for large value o f N , which is quite typical in OFDM sy stems. Algorithm 1 Propo sed dual deco mposition algorithm for (8) 1: Input the system parameter s { N , P 1max , P 2max } , the chan- nel coefficients { h 1 n , h 2 n } N n =1 , and a so lution accuracy ǫ . 2: Set k = 1 ; Initialize dual variables ν 1 = 1 . 3: repeat 4: Compute the op timal { P 1 n , P 2 n } a ccording to (12) and (13) for ∀ n ∈ N ; 5: Update the dual variables ν k accordin g to (21); 6: k := k + 1 ; 7: until k ν k − ν k − 1 k ≤ ǫ k ν k − 1 k . 8: Output the optimal primal solution { P ⋆ 1 , P ⋆ 2 } and R ⋆ MA = min { C MA ,i ( P ⋆ 1 , P ⋆ 2 ) , i = 1 , 2 , 3 } . V . S I M U L A T I O N R E S U LT S W e co nsider an OFDM system with N = 32 subcar riers. The fr equency-do main ch annels are g enerated using 8 inde- penden tly a nd identically distributed Rayleigh d istributed time-dom ain taps with unit v ar iance [6]. The separate power constraints are set as P 1max = P 2max = P R max , and µ = 0 . 5 . Our propo sed multi-subcar rier DF relay strategy is denoted as “T ype 1 DF” scheme. T wo ref erence schem es are considered in our simulatio ns: Th e first one is the per-subcarrier tw o- way DF OFDM relay strategy in [9], which is d enoted as “T ype 2 DF” scheme; the seco nd one is the two-way AF OFDM relayin g scheme with optimized tone p ermutation in [6]. W e d i vide the sum ra te ( approx imated by the lower bou nd 2 R X in T ype 1/2 DF scheme) by N and u se this per-subcarr ier sum rate to ev alu ate p erforma nce at different average SNRs, which are o nly related with the power constrain ts P i max ’ s. Fig. 2 presents the p erforman ce of different two-way OFDM relay stra tegies. Th e best performanc e is ach iev ed by T ype 1 DF scheme with optimal power allo cation (P A). At the spectral efficiency of 2 bits/s/Hz, T ype 1 DF scheme with op timal P A provides a cod ing gain of ab out 2.5 dB compared with T ype 2 DF sche me, by perfo rming channe l co ding across subcar riers. The P A gain between op timal P A and uniform P A of T ype 1 DF schem e is given by 1. 6 dB. It is in teresting th at T y pe 1 DF scheme with unifo rm P A ev e n ou tperform s T ype 2 DF scheme with optimal P A, when the average SNR is in the region [0 dB, 20 dB]. Although T ype 1 DF scheme has no ad vantage over th e AF scheme in the h igh SNR region due to its addition al fully decodin g r equiremen t at T R , it o utperfo rms th e AF sch eme in th e low an d med ian SNR region . The intersection of th e curves for T ype 1 DF schem e and the AF scheme is at abo ut 17.5 d B, which is 5 dB h igher than that f or T yp e 2 DF sche me and the AF scheme. V I . C O N C L U S I O N W e h av e prop osed a novel DF relay strategy f or two-way OFDM relay networks and der i ved its a chiev able rate region. The key ide a is making u se o f cro ss-subcarrier chan nel codin g to fully exploit freq uency selective fading. An efficient duality- based power allocatio n algorithm i s also proposed to maximize the symmetric exchange d ata rate in both directions. Our simulation results su ggest that the prop osed DF strategy has better p erforman ce than existing DF or AF two-way OFDM relay strategies in the modera tely low SNR region . W e believe this two-way DF strategy tends to be optimal, i.e., achieving the capacity region outer bou nd, in the mo derately low SNR region. The o ptimality of the proposed two-way DF strategy and the effect of chan nel uncertain ty are currently u nder ou r in vestigation . R E F E R E N C E S [1] B. Rankov and A. Witt neben, “ Achie vab le rate regions for the two-way relay channel, ” in Pr oc. 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