Capacity Region Bounds and Resource Allocation for Two-Way OFDM Relay Channels
Fei He, Yin Sun, Limin Xiao, Xiang Chen, Chong-Yung Chi, Shidong Zhou
aa r X i v : . [ c s . I T ] M a r Capacity Region Bounds and ResourceAllocation for Two-Way OFDM RelayChannels
Fei He ⋆ , Yin Sun † , Limin Xiao ⋆ , Xiang Chen ⋆ , Chong-Yung Chi ‡ , andShidong Zhou ⋆ Abstract
In this paper, we consider two-way orthogonal frequency division multiplexing (OFDM) relaychannels, where the direct link between the two terminal nodes is too weak to be used for datatransmission. The widely known per-subcarrier decode-and-forward (DF) relay strategy, treats eachsubcarrier as a separate channel, and performs independent channel coding over each subcarrier. Weshow that this per-subcarrier DF relay strategy is only a suboptimal DF relay strategy, and present amulti-subcarrier DF relay strategy which utilizes cross-subcarrier channel coding to achieve a largerrate region. We then propose an optimal resource allocation algorithm to characterize the achievablerate region of the multi-subcarrier DF relay strategy. The computational complexity of this algorithm ismuch smaller than that of standard Lagrangian duality optimization algorithms. We further analyze theasymptotic performance of two-way relay strategies including the above two DF relay strategies and anamplify-and-forward (AF) relay strategy. The analysis shows that the multi-subcarrier DF relay strategytends to achieve the capacity region of the two-way OFDM relay channels in the low signal-to-noiseratio (SNR) regime, while the AF relay strategy tends to achieve the multiplexing gain region of thetwo-way OFDM relay channels in the high SNR regime. Numerical results are provided to justify allthe analytical results and the efficacy of the proposed optimal resource allocation algorithm.
Index Terms
Two-way relay, orthogonal frequency division multiplexing, capacity region, decode-and-forward,amplify-and-forward, resource allocation.
This work is supported by National Basic Research Program of China (2012CB316002), National S&T Major Project(2010ZX03005-003), National NSF of China (60832008), China’s 863 Project (2009AA011501), Tsinghua Research Funding(2010THZ02-3), International Science Technology Cooperation Program (2010DFB10410), National Science Council, Taiwan,under Grant NSC-99-2221-E-007-052-MY3, Ericsson Company and the MediaTek Fellowship. The material in this paper waspresented in part in IEEE ICC 2012, Ottawa, Canada [1]. ⋆ Fei He, Limin Xiao, Xiang Chen, and Shidong Zhou are with Department of Electronic Engineering, Research Institute ofInformation Technology, Tsinghua National Laboratory for Information Science and Technology (TNList), Tsinghua University,Beijing 100084, China. Xiang Chen, the corresponding author, who is also with Aerospace Center, Tsinghua University. E-mail:[email protected], { xiaolm,chenxiang,zhousd } @tsinghua.edu.cn. † Yin Sun is with the Department of Electrical and Computer Engineering, the Ohio State University, Columbus, Ohio 43210,USA. E-mail: [email protected]. ‡ Chong-Yung Chi is with the Institute of Communications Engineering and the Department of Electrical Engineering, NationalTsinghua University, Hsinchu, Taiwan 30013. E-mail: [email protected].
I. I
NTRODUCTION
Orthogonal frequency division multiplexing (OFDM) relaying is a cost-efficient techniqueto enhance the coverage and throughput of future wireless networks, and it has been widelyadvocated in many 4G standards, such as IEEE 802.16m and 3GPP advanced long term evolution(LTE-Advanced) [2], [3]. In practice, a relay node operates in a half-duplex mode to avoid strongself-interference. However, since the half-duplex relay node can not transmit all the time (or overthe entire frequency band), the benefits provided by the relay node are not fully exploited [4].Recently, two-way relay technique has drawn extensive attention, because of its potential toimprove the spectrum efficiency of one-way relay strategies [4]–[13]. If one utilizes traditionalone-way relay strategies to realize two-way communications, four phases are needed. To improvethe four-phase strategy, the two relay-to-destination phases can be combined into one broadcastphase [5], [6], and the yielded three-phase strategy can support the same data rates with lesschannel resource by exploiting the side information at the terminal nodes. One can furthercombine the two source-to-relay phases into one multiple-access phase to yield a two-phasestrategy (see Fig. 1) [7]. Hybrid strategies with more phases have been considered in [8]–[10] tofurther enlarge the achievable rate region. The diversity-multiplexing tradeoff for two-way relaychannels was studied in [11]–[13].Two-way relay strategies also have been in conjunction with OFDM techniques [14]–[20].With amplify-and-forward (AF) relay strategy, power allocation and subcarrier permutation havebeen studied in [14], [15], and its corresponding channel estimation problem has been thoroughlydiscussed in [16]. Resource allocation for two-way communications in an OFDM cellular networkwith both AF and decode-and-forward (DF) relay strategies was studies in [17]. A graph-basedapproach was proposed to solve the combinatorial resource allocation problem in [18]. Forpractical quality of service (QoS) requirements, the proportional fairness and transmission delayhave been considered for two-way DF OFDM relay networks in [19] and [20], respectively.All these studies of two-way OFDM relay channels with a DF strategy were almost centeredon a per-subcarrier DF relay strategy, which treats each subcarrier as a separate two-way relaychannel, and performs independent channel coding over each subcarrier. Such a per-subcarrierDF relay strategy is probably motivated by the fact that per-subcarrier channel coding can achievethe capacity of point-to-point OFDM channels. However, the story is different in OFDM relay Y R Y RN X X N X X N … … … … … Terminal T Relay T R Terminal T h h N h N h (a) The multiple-access phase. X R X RN Y Y N Y Y N … … … … … h ɶ h ɶ N h ɶ N h ɶ Terminal T Relay T R Terminal T (b) The broadcast phase.Fig. 1. System model of a two-way OFDM relay channel, consisting of (a) a multiple-access phase and (b) a broadcast phase. channels: per-subcarrier channel coding can no longer attain the optimal achievable rate regionof DF relaying for two-way OFDM relay channels. In other words, per-subcarrier DF relayingis merely a suboptimal DF relay strategy. More details are provided in Section III, where anexample is provided to show that a novel DF relay strategy achieves a larger rate region.This paper focuses on the two-way OFDM relay channel with a negligible direct link due tolarge path attenuation or heavy blockage. This is motivated by the fact that the relay node playsa more important role when the direct link is weak than when it is strong [13]. The optimaltwo-way relay strategy in this case consists of two phases, which are illustrated in Fig. 1. Weintend to answer the following questions in this paper: What is the optimal DF relay strategywhen the direct link is negligible? Under what conditions is the optimal DF relay strategy better(or worse) than the AF relay strategy, and vice versa? Is the optimal DF relay strategy able toachieve the capacity region of two-way OFDM relay channels in some scenarios? To addressthese questions, we also investigate the capacity region bounds of the two-way OFDM relaychannels and compare different relay strategies under the optimal resource allocation. The maincontributions of this paper are summarized as follows: • We present a multi-subcarrier DF relay strategy, which has a larger achievable rate regionthan the widely studied per-subcarrier DF relay strategy. Though this multi-subcarrier DFrelay strategy is merely a simple extension of the existing result [7], it is the optimal DFrelay strategy for two-way OFDM relay channels. To the best of our knowledge, this A strategy is the optimal DF relay strategy, meaning that its achievable rate region contains the rate region of any other DFrelay strategy. It is worth mentioning that relay strategies other than DF relay strategies may have a larger or smaller achievablerate region compared to this multi-subcarrier DF relay strategy in certain scenarios.
TABLE IC
OMPARISON OF PER - SUBCARRIER AND MULTI - SUBCARRIER DF TWO - WAY RELAY STRATEGIES .Strategy Achievable rate region Resource allocation complexityper-subcarrier DF small low [17]multi-subcarrier DF large very low multi-subcarrier DF relay strategy has not been reported in the open literature. We developan optimal resource allocation algorithm to characterize the achievable rate region of themulti-subcarrier DF relay strategy. We show that the optimal resource allocation solution hasa low-dimension structure. By exploiting this structure, the complexity of both primal anddual optimizations can be significantly reduced. The relative benefits of our multi-subcarrierDF relay strategy and its resource allocation algorithm are summarized in Table I. • We analyze the asymptotic performance of different relay strategies in the low and highsignal-to-noise ratio (SNR) regimes under optimal resource allocation. First, we show thatthe multi-subcarrier DF relay strategy tends to achieve the capacity region of two-wayOFDM relay channels in the low SNR regime. Then, we characterize the multiplexinggain regions of the two DF relay strategies, the AF relay strategy, and the cut-set outerbound under optimal resource allocation. We show that the AF relay strategy can achievethe multiplexing gain region of two-way OFDM relay channels in the high SNR regime.Numerical results are provided to justify our analytical results. The asymptotic performancecomparison of AF and DF strategies is summarized in Table II.The rest of this paper is organized as follows. Section II presents the system model. Section IIIpresents the multi-subcarrier two-way DF relay strategy and its achievable rate region. Theresource allocation algorithm of the multi-subcarrier DF relay strategy is developed in Section IV.The asymptotic performance analysis of different relay strategies is provided in Section V. Somenumerical results are presented in Section VI. Finally, Section VII draws some conclusions.
Notation:
Throughout this paper, we use bold lowercase letters to denote column vectors, andalso denote an n × column vector by ( x , . . . , x n ) . R + and R n + denote the set of nonnegative realnumbers and the set of n × column vectors with nonnegative real components, respectively. p (cid:23) means that each component of column vector p is nonnegative. I ( X ; Y ) denotes themutual information between random variables X and Y , and I ( X ; Y | Z ) denotes the conditionalmutual information of X and Y given Z . E [ · ] denotes the statistical expectation of the argument. TABLE IIA
SYMPTOTIC PERFORMANCE COMPARISON OF AF AND DF TWO - WAY RELAY STRATEGIES .Strategy Low SNR High SNRmulti-subcarrier DF achieving capacity region smaller multiplexing gainAF lower rate achieving multiplexing gain region
II. S
YSTEM M ODEL
We consider a two-way OFDM relay channel with N subcarriers, where two terminal nodes T and T exchange messages by virtue of an intermediate relay node T R . The wireless transmissionsin the two-way DF relay channel is composed of two phases: a multiple-access phase and abroadcast phase, as illustrated in Fig. 1. In the multiple-access phase, the terminal nodes T and T simultaneously transmit their messages to the relay node T R . In the broadcast phase, the relaynode T R decodes its received messages, re-encodes them into a new codeword, and broadcastsit to the terminal nodes T and T . The time proportion of the multiple-access phase is denotedas t for < t < , and thereby the time proportion of the broadcast phase is − t .In the multiple-access phase, the received signal Y Rn of the relay node T R over subcarrier n can be expressed as Y Rn = h n r p n t X n + h n r p n t X n + Z Rn , (1)where X in ( i ∈ { , } ) is the unit-power transmitted symbol of the terminal nodes T i oversubcarrier n , h in is the channel coefficient from T i to T R over subcarrier n , p in is the averagetransmission power, and Z Rn is the independent complex Gaussian noise with zero mean andvariance σ Rn .In the broadcast phase, the received signals of the terminal nodes T and T over subcarrier n are given by Y n = ˜ h n r p Rn − t X Rn + Z n , (2) Y n = ˜ h n r p Rn − t X Rn + Z n , (3)where X Rn and p Rn denote the unit-power transmitted symbol and the average transmissionpower of the relay node T R over subcarrier n , respectively, ˜ h in denotes the associated channelcoefficient from T R to T i over subcarrier n , and Z in is the independent complex Gaussian noisewith zero mean and variance σ in ( i ∈ { , } ).Each node is subject to an individual average power constraint, which is given by N X n =1 p in ≤ P i , i = 1 , , R, (4)where P i denotes the maximum average transmission power of node T i . Let us use P , ( P , P , P R ) to represent the maximum average powers of the three nodes, and use G , { g n , g n , ˜ g n , ˜ g n } Nn =1 to represent the channel state information (CSI), where g in , | h in | /σ Rn and ˜ g in , | ˜ h in | /σ in ( i ∈ { , } ) represent the normalized channel power gains. We assume that the perfect CSI G isavailable at the network controller to perform resource allocation throughout the paper.III. O PTIMAL T WO -W AY OFDM DF R
ELAY S TRATEGY
This section presents a multi-subcarrier DF relay strategy, which can realize the optimalachievable rate region of the DF relay strategy for two-way OFDM relay channels. We also showthat the per-subcarrier DF relay strategy considered in [17]–[20] can only achieve a suboptimalrate region.Let R and R denote the end-to-end data rates from T to T and from T to T , respectively.When the direct link between T and T is negligible, the optimal DF relay strategy of discretememoryless two-way relay channels was given by Theorem 2 in [7]. By applying this theorem totwo-way parallel Gaussian relay channel and considering the optimal channel input distribution,we can obtain the optimal achievable rate region as stated in the following lemma:
Lemma 1
Given the maximum transmission powers P of the three nodes and the CSI G , theoptimal achievable rate region of the two-way parallel Gaussian relay channel (1) - (3) with aDF strategy is given by: R DF ( P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ min (cid:26) N X n =1 t log (cid:16) g n p n t (cid:17) , N X n =1 (1 − t ) log (cid:16)
1+ ˜ g n p Rn − t (cid:17)(cid:27) ,R ≤ min (cid:26) N X n =1 t log (cid:16) g n p n t (cid:17) , N X n =1 (1 − t ) log (cid:16)
1+ ˜ g n p Rn − t (cid:17)(cid:27) ,R + R ≤ N X n =1 t log (cid:16) g n p n + g n p n t (cid:17) , < t < , N X n =1 p in ≤ P i , p in ≥ , i = 1 , , R, n = 1 , . . . , N ) . (5) Proof:
See Appendix A.In fact, the optimal rate region of (5) is the intersection of the capacity regions of a parallelmulti-access channel and a parallel broadcast channel with receiver side information [21]. Thisrate region can be achieved by the following multi-subcarrier DF relay strategy: In the multiple-access phase, the relay node decodes the messages from the two terminal nodes by eithersuccessive cancellation decoding with time sharing/rate-splitting, or joint decoding [22]–[24].In the broadcast phase, the relay node can utilize nested lattice codes, nested and algebraicsuperposition codes to transmit the messages to the intended destinations [9], [21]. Some relatedinformation theoretical random coding techniques were discussed in [6], [21], [25]. In either ofthe phases, channel encoding/decoding is performed jointly across all the subcarriers.On the other hand, the per-subcarrier DF relay strategy independently implements the DF relayscheme of [7] over each subcarrier [17]–[20]. The achievable rate region of the per-subcarriertwo-way DF relay strategy is given by R p,DF ( P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ N X n =1 min (cid:26) t log (cid:16) g n p n t (cid:17) , (1 − t ) log (cid:16)
1+ ˜ g n p Rn − t (cid:17)(cid:27) ,R ≤ N X n =1 min (cid:26) t log (cid:16) g n p n t (cid:17) , (1 − t ) log (cid:16)
1+ ˜ g n p Rn − t (cid:17)(cid:27) ,R + R ≤ N X n =1 t log (cid:16) g n p n + g n p n t (cid:17) , < t < , N X n =1 p in ≤ P i , p in ≥ , i = 1 , , R, n = 1 , . . . , N ) . (6)The only difference between R DF ( P , G ) and R p,DF ( P , G ) lies in the order of the function min {·} and the summation in (5) and (6), implying R p,DF ( P , G ) ⊆ R DF ( P , G ) . Therefore, theper-subcarrier DF relay strategy is only a suboptimal DF relay strategy. Similar results have beenreported in [26], [27] for one-way parallel relay channels.We now provide a toy example to compare these two DF relay strategies. Consider a two-wayOFDM relay channel with N = 2 subcarriers. The wireless channel power gains are given by ( g , g , g , g ) = (1 , , , and ˜ g in = g in for n, i ∈ { , } . The power and channel resources Here, the receiver side information means each user’s own transmitted message. R (bits/s/Hz) R ( b it s / s / H z ) (2 : ; : Multi-subcarrier DFPer-subcarrier DF (1 : ; : Fig. 2. Comparison of the achievable rate regions of different two-way OFDM relay strategies for a toy example with N = 2 , ( g , g , g , g ) = (1 , , , , ˜ g in = g in , p in = 0 . , t = 0 . . are fixed to be p in = 0 . and t = 0 . . According to Lemma 1, the achievable rate region of themulti-subcarrier DF relay strategy with fixed resource allocation is given by the set of rate pairs ( R , R ) satisfying R ≤ min { . , . } = 2 . bits/s/Hz , (7a) R ≤ min { . , . } = 2 . bits/s/Hz , (7b) R + R ≤ . (9) + log (19)] = 3 . bits/s/Hz . (7c)Similarly, by (6), the achievable rate region of the per-subcarrier DF relay strategy with fixedresource allocation is given by the set of rate pairs ( R , R ) satisfying R ≤ min { . , . } + min { , } = 1 . bits/s/Hz , (8a) R ≤ min { . , . } + min { , } = 1 . bits/s/Hz , (8b) R + R ≤ . (9) + log (19)] = 3 . bits/s/Hz , (8c)where the sum-rate constraint is actually inactive. The achievable rate regions in (7) and (8) areshown in Fig. 2, from which one can easily observe that the considered multi-subcarrier DF relaystrategy has a larger achievable rate region.An effective and computationally efficient approachfor the optimal resource allocation of the proposed two-way DF strategy will be presented inthe next section. IV. R
ESOURCE A LLOCATION A LGORITHM
We now develop a resource allocation algorithm to characterize the boundary of the achievablerate region R DF ( P , G ) in (5). We will show that the optimal resource allocation solution has alow-dimension structure, and thereby the number of dual variables to be optimized is reduced;see Propositions 1 and 2 below for more details. The complexity of our resource allocationalgorithm turns out to be much lower than that of the standard Lagrangian dual optimizationalgorithm and the existing resource allocation algorithm reported in [17]. A. Resource Allocation Problem Formulation
Let ρ ∈ (0 , ∞ ) denote the rate ratio of the two terminal nodes, i.e., ρ , R /R . (9)Then, a boundary point ( R , R ) = ( R , ρR ) of the achievable rate region R DF ( P , G ) isattained by maximizing R within R DF ( P , G ) for a given rate ratio ρ . Therefore, the boundarypoint of R DF ( P , G ) is characterized by the following resource allocation problem: max p , p , p R (cid:23) , R , t R (10a)s.t. R ≤ t N X n =1 log (cid:16) g n p n t (cid:17) , (10b) R ≤ tρ N X n =1 log (cid:16) g n p n t (cid:17) , (10c) R ≤ tρ + 1 N X n =1 log (cid:16) g n p n + g n p n t (cid:17) , (10d) R ≤ (1 − t ) N X n =1 log (cid:16)
1+ ˜ g n p Rn − t (cid:17) , (10e) R ≤ − tρ N X n =1 log (cid:16)
1+ ˜ g n p Rn − t (cid:17) , (10f) N X n =1 p in ≤ P i , i = 1 , , R, (10g) < t < , (10h)where p i , ( p i , p i , . . . , p iN ) ∈ R N + denotes the power allocation of node T i for i =1 , , R .Problem (10) is a convex optimization problem, which can be solved by standard interior-pointmethods or by using general purpose convex solvers such as CVX [28]. However, these methods quickly become computationally formidable as the number of subcarriers N increases, becausetheir complexity grows in the order of O ( N . ) [29], [30, p. 8 and Eq. (11.29)]. Since N canbe quite large in practical OFDM systems, we will develop a more efficient resource allocationalgorithm for large values of N in the sequel. B. Phase-Wise Decomposition of Problem (10)Let us first fix the value of t . Then, problem (10) can be decomposed into two power allocationsubproblems for the multi-access phase and the broadcast phase, respectively. Note that thetransmission powers of the terminal nodes p and p are only involved in the rate constraints(10b)-(10d) for the multiple-access phase, while the transmission power of the relay node p R is only involved in the rate constraints (10e) and (10f) for the broadcast phase. Let R MA and R BC denote the achievable rates for the multiple-access and broadcast phases, respectively. Forany fixed t , problem (10) can be decomposed into the following two subproblems, one for themultiple-access phase R ⋆ MA ( t ) , max p , p (cid:23) , R MA R MA (11a)s.t. R MA ≤ r k ( p , p ) , k = 1 , , , (11b) N X n =1 p in ≤ P i , i = 1 , , (11c)and the other for the broadcast phase R ⋆ BC ( t ) , max p R (cid:23) , R BC R BC (12a)s.t. R BC ≤ r k ( p R ) , k = 4 , , (12b) N X n =1 p Rn ≤ P R , (12c)where the rate functions r k ( p , p ) , k = 1 , , , and r k ( p R ) , k = 4 , , are defined by r ( p , p ) = t N X n =1 log (cid:16) g n p n t (cid:17) , (13a) r ( p , p ) = tρ N X n =1 log (cid:16) g n p n t (cid:17) , (13b) r ( p , p ) = tρ + 1 N X n =1 log (cid:16) g n p n + g n p n t (cid:17) , (13c) r ( p R ) = (1 − t ) N X n =1 log (cid:16)
1+ ˜ g n p Rn − t (cid:17) , (13d) r ( p R ) = 1 − tρ N X n =1 log (cid:16)
1+ ˜ g n p Rn − t (cid:17) . (13e)Then, the optimal objective value of problem (10) is given by R ⋆ = max 1) Structure of the Optimal Dual Solution λ ⋆ : Prior to the presentation of our power allocationalgorithm for solving the problems (15) and (17), we first present an important result that theoptimal dual solution λ ⋆ satisfies the following structural property: Proposition 1 There exists one optimal solution ( λ ⋆ , α ⋆ ) to the dual problem (17) , where λ ⋆ =(1 − λ ⋆ , , λ ⋆ ) or λ ⋆ = (0 , − λ ⋆ , λ ⋆ ) and ≤ λ ⋆ ≤ . Proof: See Appendix B. (cid:4) Proposition 1 is very useful for developing our power allocation algorithm, because the searchregion for λ ⋆ can be confined to a set Λ S Λ , where Λ and Λ are two one-dimensional dualsets defined by Λ , { λ ∈ R | λ = (1 − λ , , λ ) , ≤ λ ≤ } , (18a) Λ , { λ ∈ R | λ = (0 , − λ , λ ) , ≤ λ ≤ } . (18b)In the sequel, we will show that finding solutions to both problems (15) and (17) can besubstantially simplified by virtue of Proposition 1. 2) Primal Solution to Problem (15) : As the first important application of Proposition 1, weshow that the structure of λ ⋆ can be exploited to simplify the primal solution to problem (15).For any given dual variables ( λ , α ) , the optimal power allocation solution ( p ⋆ n , p ⋆ n ) to problem(15) is determined by the following Karush-Kuhn-Tucker (KKT) conditions: ∂L MA ∂p n = α − tg n λ / ( ρ + 1)( t + g n p ⋆ n + g n p ⋆ n ) ln 2 − tg n λ ( t + g n p ⋆ n ) ln 2 ≥ , if p ⋆ n = 0;= 0 , if p ⋆ n > , (19a) ∂L MA ∂p n = α − tg n λ / ( ρ + 1)( t + g n p ⋆ n + g n p ⋆ n ) ln 2 − tg n λ /ρ ( t + g n p ⋆ n ) ln 2 ≥ , if p ⋆ n = 0;= 0 , if p ⋆ n > . (19b)According to Proposition 1, at least one of λ ⋆ and λ ⋆ is 0, which can be utilized to simplifythe KKT conditions (19). The attained optimal ( p ⋆ n , p ⋆ n ) is provided in the following four cases: Case 1 : p ⋆ n > , p ⋆ n > . If λ = (1 − λ , , λ ) , then p ⋆ n = t (1 − λ )( α − g n g n α ) ln 2 − tg n , (20a) p ⋆ n = tλ ( ρ + 1) α ln 2 − t (1 − λ )( g n g n α − α ) ln 2 ; (20b)otherwise, if λ = (0 , − λ , λ ) , then p ⋆ n = tλ ( ρ + 1) α ln 2 − t (1 − λ ) ρ ( g n g n α − α ) ln 2 , (21a) p ⋆ n = t (1 − λ ) ρ ( α − g n g n α ) ln 2 − tg n . (21b)Case 1 happens if p ⋆ n p ⋆ n > is satisfied in (20) or (21). Case 2 : p ⋆ n > , p ⋆ n = 0 . Then the solutions to (19a) and (19b) are given by p ⋆ n = t [( ρ + 1) λ + λ ]( ρ + 1) α ln 2 − tg n , (22a) p ⋆ n = 0 . (22b) This case happens if p ⋆ n > is satisfied in (22a). Case 3 : p ⋆ n = 0 , p ⋆ n > . Then the solutions to (19a) and (19b) are given by p ⋆ n = 0 , (23a) p ⋆ n = t [( ρ + 1) λ + ρλ ] ρ ( ρ + 1) α ln 2 − tg n . (23b)This case happens if p ⋆ n > is satisfied in (23b). Case 4 : p ⋆ n = 0 , p ⋆ n = 0 . This is the default case when the above 3 cases do not happen. Remark 1 If the structural property of λ ⋆ in Proposition 1 is not available, one can still obtainan alternative closed-form solution to (19) [1]. However, this solution involves solving a moredifficult cubic equation, which is presented in Appendix C. Nevertheless, these two closed-formsolutions are much simpler than the iterative power allocation procedure proposed in [17] forthe per-subcarrier DF relay strategy. Remark 2 Since the Lagrangian (16) is not strictly convex with respect to the primal powervariables at some dual points, the power allocation solution in (20)-(23) may be non-unique atthose dual points. For example, if λ = 1 and α = g n g n α , the power allocation solution in eitherof (20) and (21) is indeterminate due to the presence of form. In fact, any nonnegative ( p ⋆ n , p ⋆ n ) satisfying g n p ⋆ n + g n p ⋆ n = tg n α ( ρ +1) ln 2 − is a solution to (19) in this case. Nevertheless, anyone of the optimal primal power solutions can be used to derive the subgradient for solvingthe dual problem (17) [32, Section 6.1]. After the optimal dual point ( λ ⋆ , α ⋆ ) is obtained, westill need to recover a feasible solution to the primal problem (11) [32]–[34]. According to [32,Proposition 5.1.1], this can be accomplished by incorporating the feasibility conditions (49h) andthe complementary slackness conditions (49i), which are contained in the KKT conditions ofproblem (11) given in Appendix B, into the power allocation solutions (20)-(23), which involvessolving a system of linear equations and inequalities. 3) Dual Solution to Problem (17) : We now solve the dual problem (17) by a two-leveloptimization approach [17], which first fixes λ and searches for the optimal solution α ⋆ ( λ ) tothe maximization problem G MA ( λ ) , max α (cid:23) D MA ( λ , α ) , (24)and then optimizes λ by λ ⋆ , arg max λ (cid:23) G MA ( λ ) . (25) The inner-level optimization problem (24) is solved by an ellipsoid method [35] summarized inAlgorithm 1, where the subgradient of the dual problem D MA ( λ , α ) with respect to α is givenby [32, Proposition 6.1.1] η ( λ , α ) = N X n =1 p ⋆ n − P , N X n =1 p ⋆ n − P ! , (26)where ( p ⋆ n , p ⋆ n ) is the optimal power allocation solution obtained by (20)-(23). More detailsabout the initialization of α and the matrix A in Algorithm 1 are given in Appendix D.By Proposition 1, the outer-level optimization problem (25) can be solved by searching for λ ⋆ over the set Λ S Λ , i.e., λ ⋆ = arg max λ ∈ Λ S Λ G MA ( λ ) . (27)In order to solve the reduced outer-level optimization problem (27), we first need the sub-gradient of the objective function G MA ( λ ) . According to [32] and [36, Corollary 4.5.3], onesubgradient of G MA ( λ ) in (24) is given by ξ ( λ ) = ( R ⋆ MA − r , R ⋆ MA − r , R ⋆ MA − r ) , (28)where R ⋆ MA = min { r , r , r } , and r k ( k =1 , , are the rate functions (13a)-(13c) associatedwith the optimal primal power allocation solution (20)-(23) obtained at the dual point ( λ , α ⋆ ( λ )) ,respectively, and α ⋆ ( λ ) is the optimal solution to (24).With the subgradient ξ ( λ ) of G MA ( λ ) , we are ready to solve the outer-level optimizationproblem (27). Instead of searching both the sets Λ and Λ , we propose a simple testing methodto determine whether λ ⋆ ∈ Λ or λ ⋆ ∈ Λ . Noticing that Λ T Λ = { (0 , , } , let us consider atesting method at the dual point λ = (0 , , . By the concavity of the dual function D MA ( λ , α ) , G MA ( λ ) is also concave in λ , which implies [32, Eq. (B.21)] G MA ( λ ) ≤ G MA ( λ ) + ( λ − λ ) T ξ ( λ ) , ∀ λ ∈ Λ S Λ . (29)Suppose that λ ⋆ is an optimal solution to (27), i.e., G MA ( λ ⋆ ) ≥ G MA ( λ ) . Then, by (29), wemust have ( λ ⋆ − λ ) T ξ ( λ ) ≥ (30)for the optimal dual point λ ⋆ . In other words, if a dual point λ satisfies ( λ − λ ) T ξ ( λ ) < ,then λ cannot be an optimal solution to problem (27). Due to this and (28), we establish thefollowing proposition: Algorithm 1 The ellipsoid method for solving the inner-level problem (24) Input CSI { g n , g n } Nn =1 , average powers { P , P } , rate ratio ρ , time proportion t , and λ . Initialize α and a × positive definite matrix A that define the ellipsoid E ( α , A ) = { x ∈ R | ( x − α ) T A − ( x − α ) ≤ } . repeat Compute the optimal ( p ⋆ n , p ⋆ n ) by (20)-(23) for given ( λ , α ) and t . Compute the subgradient η ( λ , α ) with respect to α by (26). Update the ellipsoid: (a) e η := √ η T A η η ; (b) α := α − A e η ; (c) A := (cid:0) A − A e η e η T A (cid:1) . until α converges to α ⋆ ( λ ) . Output the optimal dual variable α ⋆ ( λ ) for given λ . Proposition 2 Let r k ( k =1 , , denote the values of the terms used in the subgradient ξ ( λ ) in (28) with λ = λ . If r ≥ r , then λ ⋆ ∈ Λ . If r ≥ r , then λ ⋆ ∈ Λ . If both r ≥ r and r ≥ r , then λ ⋆ = λ = (0 , , .Proof: See Appendix E. (cid:4) The procedure for solving (27) is given as follows: First, we utilize the preceding testingmethod stated in Proposition 2 to determine whether λ ⋆ ∈ Λ or λ ⋆ ∈ Λ . Then, we use thebisection method to find the optimal dual variable λ ⋆ . If λ ⋆ =(1 − λ ⋆ , , λ ⋆ ) ∈ Λ , the directionalsubgradient ζ ( λ ) of G MA ( λ ) along the direction of Λ is determined by ζ ( λ ) = ξ ( λ ) T ∂ λ ∂λ = ξ ( λ ) T ( − , , 1) = r − r ; (31)otherwise, if λ ⋆ = (0 , − λ ⋆ , λ ⋆ ) ∈ Λ , the directional subgradient ζ ( λ ) along the direction of Λ is determined by ζ ( λ ) = ξ ( λ ) T (0 , − , 1) = r − r . (32)Since G MA ( λ ) is concave in λ , it is also concave along the direction of Λ (or Λ ). Thus, ζ ( λ ) is monotonically non-increasing with respect to λ . Therefore, we can use the bisectionmethod to search for the optimal solution λ ⋆ to (25), which satisfies ζ ( λ ⋆ ) = 0 , if < λ ⋆ < ; ζ ( λ ⋆ ) ≤ , if λ ⋆ = 0 ; or ζ ( λ ⋆ ) ≥ , if λ ⋆ = 1 . The obtained algorithm for solving subproblem(11) is summarized in Algorithm 2. Algorithm 2 Proposed duality-based algorithm for solving subproblem (11) Input CSI { g n , g n } Nn =1 , average powers { P , P } , rate ratio ρ , and time proportion t . Check whether λ ⋆ ∈ Λ or λ ⋆ ∈ Λ by Proposition 2. If λ ⋆ = λ = (0 , , , go to Step 10;otherwise, find λ ⋆ by the bisection method in Steps 3-9. Initialize λ , min = 0 , λ , max = 1 . repeat Update λ = ( λ , min + λ , max ) . Derive α ⋆ ( λ ) for the inner-level dual optimization problem (24) by Algorithm 1. Compute the subgradient ξ ( λ ) by (28) and the subgradient ζ ( λ ) by either (31) or (32). If the subgradient ζ ( λ ) < , then update λ , max = λ ; else update λ , min = λ . until λ converges. Obtain the optimal { p ⋆ , p ⋆ } by (20)-(23) and Remark 2. Output the optimal power allocation solution { p ⋆ , p ⋆ } and the optimal rate R ⋆ MA ( t ) . D. Lagrange Dual Optimization for Subproblem (12)The partial Lagrange dual function of subproblem (12) is defined as D BC ( λ , λ , α ) , min p R (cid:23) ,R BC L BC ( p R , R BC , λ , λ , α ) , (33)where λ , λ and α are the nonnegative dual variables associated with two rate inequalityconstraints in (12b) and one power inequality constraint (12c), respectively, and L BC ( p R , R BC , λ , λ , α ) = − R BC + X k =4 λ k (cid:2) R BC − r k ( p R ) (cid:3) + α (cid:16) N X n =1 p Rn − P R (cid:17) . (34)Then, the corresponding dual optimization problem is defined as max λ ≥ ,λ ≥ ,α ≥ D BC ( λ , λ , α ) . (35)The KKT conditions associated with (33) are given by ∂L BC ∂p Rn = α − (1 − t )˜ g n λ (1 − t + ˜ g n p ⋆Rn ) ln 2 − (1 − t )˜ g n λ ρ (1 − t + ˜ g n p ⋆Rn ) ln 2 ≥ , if p ⋆Rn = 0;= 0 , if p ⋆Rn > , (36a) ∂L BC ∂R BC = λ ⋆ + λ ⋆ − . (36b) Algorithm 3 Proposed duality-based algorithm for solving subproblem (12) Input CSI { ˜ g n , ˜ g n } Nn =1 , average power P R , rate ratio ρ , and time proportion t . Initialize λ , min = 0 , λ , max = 1 . repeat Update λ = ( λ , min + λ , max ) and initialize α , min , α , max with given λ . repeat Update α = ( α , min + α , max ) . Obtain the optimal p ⋆R by solving (37) at the dual point (1 − λ , λ , α ) . If P Nn =1 p ⋆Rn < P R , then update α , max = α ; else update α , min = α . until α converges to α ⋆ ( λ ) . Obtain the optimal p ⋆R by solving (37) at the dual point (1 − λ , λ , α ⋆ ( λ )) . If r ( p ⋆R ) < r ( p ⋆R ) , then update λ , max = λ ; else update λ , min = λ . until λ converges. Obtain the optimal p ⋆R by (37). Output the optimal power allocation solution p ⋆R and the optimal rate R ⋆ BC ( t ) .If p ⋆Rn > , then the equality in (36a) holds, and the optimal p ⋆Rn can be shown to be the positiveroot x of the following quadratic equation [17] (1 − t )˜ g n λ − t + ˜ g n x + (1 − t )˜ g n λ ρ (1 − t + ˜ g n x ) = α ln 2 . (37)If (37) has no positive root, then p ⋆Rn = 0 . By (36b), we have λ ⋆ = 1 − λ ⋆ . Thus, the optimal dualvariables ( α ⋆ , λ ⋆ ) can be derived by a two-level bisection optimization method, and the obtainedalgorithm for solving subproblem (12) is summarized in Algorithm 3. More details about theinitialization of α , min and α , max in Algorithm 3 are given in Appendix D.As previously mentioned, after solving the power allocation subproblems (11) and (12),problem (14) can be solved by utilizing the efficient one-dimensional search methods in [31,Chapter 8], thereby yielding Algorithm 4 for solving problem (10). E. Computational Complexity Analysis The computational complexity of Algorithm 2 is given by O ( L (2) KN C ) , where L ( n ) = O (2( n +1) ln(1 /ǫ )) is the number of iterations in the ellipsoid method for an n -variable non- Algorithm 4 Proposed resource allocation algorithm for solving problem (10) Input CSI { g n , g n , ˜ g n , ˜ g n } Nn =1 , average powers { P , P , P R } , rate ratio ρ . repeat Solve the power allocation subproblems (11) and (12) by Algorithm 2 and Algorithm 3,respectively, where t is a given parameter. Update t using the one-dimensional search method for problem (14). until t converges. Output the optimal resource allocation { p ⋆ , p ⋆ , p ⋆R , t ⋆ } and the optimal rate R ⋆ .smooth optimization problem [37, p. 155], ǫ is the accuracy of the obtained solution, K = O (ln(1 /ǫ )) is the complexity (abbreviated for the complexity order) of one-dimensional searchmethods such as the bisection method, C is the complexity for computing the closed-formpower allocation solution (20)-(23). The computational complexity of Algorithm 3 is given by O ( K N C ) , where C is the complexity for computing the closed-form power allocation solutionto (37). Therefore, the overall computational complexity of the proposed resource allocationalgorithm (Algorithm 4) is given by O ( L (2) K N C + K N C ) .The complexity of the resource allocation algorithm for the per-subcarrier DF relay strategyin [17] is given by O ( L (2) L (3) KN ( I + C )) , where I is the complexity of the iterative powerallocation algorithm in Eq. (26) and (27) in [17], C is the complexity of the closed-form powerallocation solution in Eq. (28) in [17]. The complexities of the algorithm in [17] and Algorithm 4both grow linearly with the number of subcarriers N , and therefore they are quite appropriate forpractically large values of N . In addition, the computational complexity of the iterative powerallocation algorithm I is much larger than that of the closed-form power allocation solution C . The computational complexity of the ellipsoid method L (3) is much larger than that ofone-dimensional search methods K . Therefore, the computational complexity of Algorithm 4 ismuch smaller than that of the resource allocation algorithm in [17].V. A SYMPTOTIC P ERFORMANCE A NALYSIS In this section, we analyze the asymptotic rate regions of different relay strategies for two-wayOFDM channels, including both the per-subcarrier and the proposed multi-subcarrier DF relay strategies, the AF relay strategy, and the cut-set outer bound, in order to compare their achievablerate regions in both low and high SNR regimes and their respective performance merits.The cut-set outer bound for the capacity region of the two-way OFDM relay channels (1)-(3)is obtained by removing the sum-rate constraints in (5), which is given by [7] R out ( P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ min (cid:26) N X n =1 t log (cid:16) g n p n t (cid:17) , N X n =1 (1 − t )log (cid:16) 1+ ˜ g n p Rn − t (cid:17)(cid:27) ,R ≤ min (cid:26) N X n =1 t log (cid:16) g n p n t (cid:17) , N X n =1 (1 − t )log (cid:16) 1+ ˜ g n p Rn − t (cid:17)(cid:27) , < t < , N X n =1 p in ≤ P i , p in ≥ , i = 1 , , R, n = 1 , . . . , N ) . (38)The achievable rate region for the AF relay strategy is given by [17] R AF ( P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ N X n =1 12 log (cid:16) 1+ 2 p n g n ˜ g n a n g n a n (cid:17) ,R ≤ N X n =1 12 log (cid:16) 1+ 2 p n g n ˜ g n a n g n a n (cid:17) , N X n =1 p in ≤ P i , p in ≥ , i = 1 , , R, n = 1 , . . . , N ) , (39)where a n = p Rn p n g n + p n g n +1 is the amplification factor of the relay node in subcarrier n and thetime proportion t is fixed to be . due to the incompressible nature of the AF relay strategy.Suppose that ¯ P , ( ¯ P , ¯ P , ¯ P R ) is a column vector constituted by nominal values of P , P and P R . Then the available transmission powers of the three nodes can be expressed as P = x ¯ P , (40)where x is a positive scalar variable. Note that the average SNRs of all the wireless links areproportional to x , and so we will analyze the achievable rate regions of the two-way relaystrategies under consideration for small x and large x instead. A. Low SNR Regime In the low SNR region (small x ), the function log (1 + ax ) with a > can be expressed as log (1 + ax ) = a ln 2 x + O ( x ) . (41)Using (41), we can show the following proposition: Proposition 3 For sufficiently small x > and any rate pair ( R , R ) ∈ R out ( x ¯ P , G ) , thereexists some ( ˆ R , ˆ R ) ∈ R DF ( x ¯ P , G ) such that R = ˆ R + O ( x b ) and R = ˆ R + O ( x b ) for b ≥ . The regions R DF ( x ¯ P , G ) and R out ( x ¯ P , G ) tend to be the same as x → . Therefore, the multi-subcarrier DF relay strategy tends to achieve the capacity region of two-wayOFDM relay channels (1)-(3) as x → . Proof: See Appendix F.On the other hand, it can be easily shown that the achievable rate region of the AF relaystrategy will deflate in a much faster speed than the other two-way strategies for small x , dueto the noise amplification and propagation effects. B. High SNR Regime In the high SNR region (large x ), the function log (1 + ax ) with a > satisfies log (1 + ax ) = log ( ax ) + O (1 /x ) = log ( x ) + O (1) . (42)Let us define the multiplexing gain region of the multi-subcarrier DF relay strategy [38]: r DF , lim x →∞ R DF ( x ¯ P , G )log ( x ) . (43)Using (42), we can prove the following proposition: Proposition 4 The multiplexing gain region of the multi-subcarrier DF relay strategy is givenby r DF = n ( r , r ) (cid:12)(cid:12)(cid:12) r + 2 r ≤ N, r + r ≤ N, r , r ≥ o . (44) Proof: To prove (44), it is sufficient to find two rate regions R ( x ¯ P , G ) and R ( x ¯ P , G ) ,such that R ( x ¯ P , G ) ⊂ R DF ( x ¯ P , G ) ⊂ R ( x ¯ P , G ) , and the corresponding multiplexing gainregions of R ( x ¯ P , G ) and R ( x ¯ P , G ) are both given by (44). The detailed proof is given inAppendix G.Actually, the multiplexing gain region r DF given by (44) depends on the time proportion allo-cation but not upon the power allocation, which can be observed from the proof of Proposition 4 in Appendix G. For instance, the simple equal power allocation scheme can achieve this mul-tiplexing gain region, and thereby the achievable rate region gap between this power allocationscheme and the optimal power allocation scheme asymptotically converges to a constant regiongap for sufficiently large x .Following similar ideas for the proof of Proposition 4, one can derive the multiplexing gainsfor the per-subcarrier DF relay strategy, the AF relay strategy, and the cut-set outer bound asstated in the following proposition (with the proof omitted): Proposition 5 Let r p,DF , r AF , and r out denote the multiplexing gain regions of the per-subcarrierDF relay strategy, the AF relay strategy and the cut-set outer bound, respectively. Then r p,DF = r DF (given by (44) ) and r AF = r out = n ( r , r ) (cid:12)(cid:12)(cid:12) r ≤ N , r ≤ N , r , r ≥ o . Proposition 5 implies that the AF relay strategy can achieve the multiplexing gain region ofthe two-way OFDM relay channels, while the performance of both DF relay strategies is worsethan that of the AF relay strategy in the high SNR regime. An illustrative example for theseanalytical results is given in Fig. 3. To the best of our knowledge, the multiplexing gain regionof the cut-set outer bound was derived in [11], while the multiplexing gain regions of the DFand AF relay strategies have not been reported in the open literature before. All the analyticalresults as presented in Propositions 3-5 will be confirmed by our numerical results in the nextsection. VI. N UMERICAL R ESULTS We now provide some numerical results to compare the performance of different two-wayOFDM relay strategies under optimal resource allocation. The wireless channels are generatedby using M = 4 independent Rayleigh distributed time-domain taps. The number of subcarriersin the OFDM channel is N = 16 . We assume that the wireless channels between T i ( i ∈ { , } ) and T R are reciprocal, i.e., g in = ˜ g in , for all i = 1 , , n = 1 , . . . , . The maximum averagetransmission powers for all the nodes are assumed to be the same, i.e., P = P = P R = P .Therefore, the average SNR of the wireless links between T i ( i ∈ { , } ) and T R is given bySNR i = E [ g in ] PN .We consider the following two-way relay strategies in our numerical comparisons: the multi-subcarrier DF relay strategy proposed in Lemma 1, the per-subcarrier DF relay strategy [17], r / N (bits/s/Hz) r / N ( b it s / s / H z ) ( = ; = r DF = r p,DF r AF = r out Fig. 3. Comparison of multiplexing gain regions of different two-way OFDM relay strategies in the high SNR regime. the AF relay strategy [17], and the cut-set outer bound [7]. The associated rate regions of thesestrategies are given by R DF ( P , G ) in (5), R p,DF ( P , G ) in (6), R AF ( P , G ) in (39), and R out ( P , G ) in (38), respectively. The optimal resource allocation of the multi-subcarrier DF relay strategy isobtained by Algorithm 4, the optimal resource allocation of the per-subcarrier DF relay strategyand the AF relay strategy are carried out based on the power allocation algorithms proposed in[17], and the optimal resource allocation of the cut-set outer bound is obtained by a simplerversion of Algorithm 4.Figures 4(a)-4(d) provide the rate regions of these relay strategies for four symmetric SNRscenarios with SNR = SNR = SNR = 0 , , , dB, respectively. Some observations fromthese figures are worth mentioning: First, the achievable rate region of the multi-subcarrierDF relay strategy is always larger than that of the per-subcarrier DF relay strategy. Second, asthe SNR decreases, the achievable rate region of the multi-subcarrier DF relay strategy tends toreach the cut-set outer bound. Third, the achievable rate region of the AF relay strategy growswith SNR, but it is still a subset of those of the DF relay strategies for SNR ≤ dB; this is nolonger true for SNR = 30 dB. Finally, in the high SNR region, the rate regions of these strategiestend to be dominated by the multiplexing gain region, thereby consistent with Propositions 4 and5. To be more specific, the shape of the outer bound tends to be a rectangle depending on theSNR. The achievable rate region of the AF strategy is closer to the outer bound for the higher R (bits/s/Hz) R ( b it s / s / H z ) (a) SNR = 0 dB. R (bits/s/Hz) R ( b it s / s / H z ) (b) SNR = 10 dB. R (bits/s/Hz) R ( b it s / s / H z ) (c) SNR = 20 dB. R (bits/s/Hz) R ( b it s / s / H z ) 56 Cut-set Outer Bound Multi-subcarrier DFAFPer-subcarrier DF2 424 (d) SNR = 30 dB.Fig. 4. Achievable rate regions of four two-way OFDM relay strategies for four symmetric SNR scenarios (i.e.,SNR = SNR = SNR), including (a) SNR = 0 dB, (b) SNR = 10 dB, (c) SNR = 20 dB, and (d) SNR = 30 dB. SNR, but that of the two DF strategies are not. However, for the low SNR, only the proposedmulti-subcarrier DF strategy can approach the outer bound.Figure 5 provides the optimal channel resource allocation result t ⋆ versus the rate ratio ρ = R /R of the multi-subcarrier DF strategy and the cut-set outer bound for SNR = SNR =20 dB. When < ρ < . or ρ > , the optimal t ⋆ of the multi-subcarrier DF strategy and the . ½ = R =R t ? Cut-set Outer Bound Multi-subcarrier DF Fig. 5. The optimal time proportion of the multiple-access phase t ⋆ versus the rate ratio ρ = R /R of the multi-subcarrierDF strategy and the cut-set outer bound for SNR = SNR = 20 dB. cut-set outer bound are the same; when . < ρ < , the optimal t ⋆ of the multi-subcarrier DFstrategy is larger than that of the cut-set outer bound due to the additional sum-rate constraint,and the maximal t ⋆ is achieved at ρ = 1 . Figure 6 shows the achievable rate region of the multi-subcarrier DF relay strategy obtained by solving problem (10) using CVX, and that obtained byusing Algorithm 4, justifying that they yield the same numerical results as expected.Figure 7(a) and 7(b) illustrate the rate regions of these relay strategies for two asymmetric SNRscenarios, including ( SNR , SNR ) = (10 dB , dB ) and ( SNR , SNR ) = (30 dB , dB ) . Similarobservations from Figure 4 can be seen in Figure 7 as well.Finally, Figure 8 shows some results (the achievable rate versus average SNR) of theserelay strategies for the symmetric SNR symmetric rate scenario, i.e., SNR = SNR = SNR and R = R . The numerical results in Fig. 8 were obtained by averaging over 500 fading channelrealizations. One can see from this figure that, in the low SNR regime, the multi-subcarrierDF relay strategy tends to have the same performance as the cut-set outer bound, and thatthe multi-subcarrier DF relay strategy performs better than the AF relay strategy in the low tomoderate SNR regime, i.e., SNR ≤ dB. Moreover, the multi-subcarrier DF relay strategy withthe optimal resource allocation performs better than with the equal power allocation and theoptimal t ⋆ used; it also outperforms the per-subcarrier DF strategy.By Proposition 5, in the high SNR regime, the multiplexing gains of the AF relay strategy and R (bits/s/Hz) R ( b it s / s / H z ) Algorithm 4CVX Fig. 6. Achievable rate region of the multi-subcarrier DF relay strategy obtained by using Algorithm 4 and that obtained bysolving problem (10) using the convex solver CVX for SNR = SNR = 20 dB. the cut-set outer bound are the same; the multiplexing gains of the two DF relay strategies arealso the same; the multiplexing gain of the AF relay strategy is larger than that of the DF relaystrategy (implying better performance for the former than the latter for sufficiently high SNR);both the equal power allocation and the optimal power allocation for the multi-subcarrier DFstrategy achieve the same multiplexing gain, and the rate gap between them tends to a constantvalue as SNR increases. All these analytical results have been substantiated by the numericalresults shown in Figures 4-8. VII. C ONCLUSION We have analytically shown that the widely studied per-subcarrier DF relay strategy is only asuboptimal DF relay strategy for two-way OFDM relay channels in terms of achievable rateregion as the direct link between the two terminal nodes is too weak to be used for datatransmission. We have presented a multi-subcarrier DF relay strategy that can achieve a largerrate region than the per-subcarrier DF relay strategy. Although the optimal resource allocationfor the proposed multi-subcarrier DF relay strategy can be formulated as a convex optimizationproblem which can be solved by off-the-self convex solvers, we have presented a computationallyefficient algorithm (Algorithm 4) for obtaining the optimal resource allocation together withits complexity analysis. Then we have presented an analysis of asymptotic performance forthe above two DF strategies, the AF strategy, and the cut-set outer bound, justifying that the R (bits/s/Hz) R ( b it s / s / H z ) (a) SNR = 10 dB , SNR = 5 dB. R (bits/s/Hz) R ( b it s / s / H z ) (b) SNR = 30 dB , SNR = 5 dB.Fig. 7. Achievable rate regions of four two-way OFDM relay strategies for two asymmetric SNR scenarios, including (a)SNR = 10 dB , SNR = 5 dB, and (b) SNR = 30 dB , SNR = 5 dB. proposed multi-subcarrier DF relay strategy is suitable for the low to moderate SNR regime,while the AF strategy is suitable for the high SNR regime (as stated in Propositions 3 to 5).Some numerical results have been presented to demonstrate all the analytical results and theefficacy of the proposed optimal resource allocation algorithm.A PPENDIX AP ROOF OF L EMMA ( R , R ) satisfying R ≤ min { tI ( X ; Y R | X ) , (1 − t ) I ( X R ; Y ) } , (45a) R ≤ min { tI ( X ; Y R | X ) , (1 − t ) I ( X R ; Y ) } , (45b) R + R ≤ tI ( X , X ; Y R ) , (45c)where X i and Y i ( i =1 , , R ) are the input and output symbols of the channel at the terminal andrelay nodes, respectively.In the two-way parallel Gaussian relay channel, the channel input and output symbols aregiven by the vectors X i = ( X i , . . . , X iN ) and Y i = ( Y i , . . . , Y iN ) , respectively. The mutualinformation terms in (45a)-(45c) can be maximized simultaneously with the following channelinput distributions [22, Section 9.4]: -10 -5 0 5 10 15 20 25 3000.51.01.52.02.53.03.54.04.5 Average SNR (dB) R = R ( b it s / s / H z ) Multi-subcarrier DF with equal PA andAF with optimal RAMulti-subcarrier DF with optimal RACut-set outer bound with optimal RAPer-subcarrier DF with optimal RA t ? Fig. 8. Achievable rate performance comparison of two-way OFDM relay strategies, where SNR = SNR = SNR, R = R ,“RA” stands for resource allocation, and “equal PA” stands for equal power allocation in the legend. 1) The elements of channel input X in should be statistically independent for different n ;2) The elements of channel input X in should be Gaussian random variables with zero meanand unit variance.By applying these channel input distributions, and by further considering the power and channelresource constraints, the achievable rate region (5) is attained.A PPENDIX BP ROOF OF P ROPOSITION ( p ⋆ , p ⋆ ) and the optimal dual variables ( λ ⋆ , α ⋆ ) to problem (11)must satisfy the KKT condition ∂L MA ∂R MA = λ ⋆ + λ ⋆ + λ ⋆ − , (46)and the complementary slackness conditions λ ⋆k (cid:2) R ⋆ MA − r k ( p ⋆ , p ⋆ ) (cid:3) = 0 , k = 1 , , . (47)By (46), the optimal dual variable λ ⋆ has at most two independent variables, i.e., λ ⋆ = 1 − λ ⋆ − λ ⋆ .For convenience, r k ( p ⋆ , p ⋆ ) is simply denoted as r ⋆k for k = 1 , , . If r ⋆ = r ⋆ , then by the complementary slackness conditions in (47), the optimal dual variable λ ⋆ must satisfy either λ ⋆ = (1 − λ ⋆ , , λ ⋆ ) with λ ⋆ = 0 or λ ⋆ = (0 , − λ ⋆ , λ ⋆ ) with λ ⋆ = 0 , and the asserted statementis thus proved. Therefore, we only need to consider the case of r ⋆ = r ⋆ .It can be easily seen from (13a)-(13c) that r ⋆ + ρr ⋆ = t N X n =1 log (cid:16) g n p ⋆ n + g n p ⋆ n t + g n g n p ⋆ n p ⋆ n t (cid:17) ≥ ( ρ + 1) r ⋆ , (48)and the equality holds in (48) if and only if p ⋆ n p ⋆ n = 0 for n = 1 , . . . , N . This leads to twocases to be discussed as follows: Case 1 : r ⋆ + ρr ⋆ > ( ρ + 1) r ⋆ .Since r ⋆ = r ⋆ , we have that r ⋆ < ρ +1 r ⋆ + ρρ +1 r ⋆ = r ⋆ = r ⋆ . If λ ⋆ > and λ ⋆ > , then thecomplementary slackness conditions in (47) imply R ⋆ MA = r ⋆ = r ⋆ > r ⋆ , which contradicts withthe rate constraint R ⋆ MA ≤ r ⋆ . Therefore, λ ⋆ and λ ⋆ can not be both positive, and Proposition 1is thus proved in Case 1. Case 2 : r ⋆ + ρr ⋆ = ( ρ + 1) r ⋆ .Since r ⋆ = r ⋆ , we have R ⋆ MA = r ⋆ = r ⋆ = r ⋆ . If problem (17) has an optimal dual solution ( λ ⋆ , α ⋆ ) with λ ⋆ λ ⋆ = 0 , the optimal dual variable λ ⋆ already satisfies either λ ⋆ = (1 − λ ⋆ , , λ ⋆ ) or λ ⋆ = (0 , − λ ⋆ , λ ⋆ ) . Suppose that there is an optimal dual point λ ⋆ satisfying λ ⋆ > and λ ⋆ > , we will construct another optimal dual solution with the desired structure stated inProposition 1.By (48), r ⋆ + ρr ⋆ = ( ρ +1) r ⋆ happens only if the optimal primal solution satisfies p ⋆ n p ⋆ n = 0 forall n . Let us define I ⊆ N , { , . . . , N } as the index set of subcarriers with p ⋆ n ≥ , p ⋆ n = 0 ,and I ⊆ N with p ⋆ n = 0 , p ⋆ n ≥ . The optimal primal variables ( p ⋆ , p ⋆ ) and the optimal dualvariables ( λ ⋆ , α ⋆ ) to problem (11) must satisfy the following KKT conditions: ∂L MA ∂R MA = λ ⋆ + λ ⋆ + λ ⋆ − , (49a) ∂L MA ∂p n = α ⋆ − tg n [ λ ⋆ + ( ρ + 1) λ ⋆ ]( ρ + 1)( t + g n p ⋆ n ) ln 2 ≥ , if p ⋆ n = 0= 0 , if p ⋆ n > , n ∈ I , (49b) ∂L MA ∂p n = α ⋆ − tg n [ ρλ ⋆ + ( ρ + 1) λ ⋆ ] ρ ( ρ + 1)( t + g n p ⋆ n ) ln 2 ≥ , if p ⋆ n = 0= 0 , if p ⋆ n > , n ∈ I , (49c) λ ⋆k ≥ , k = 1 , , , (49d) R ⋆ MA − r ⋆k ≤ , k = 1 , , , (49e) λ ⋆k (cid:0) R ⋆ MA − r ⋆k (cid:1) = 0 , k = 1 , , , (49f) α ⋆i ≥ , i = 1 , , (49g) P Nn =1 p ⋆in − P i ≤ , i = 1 , , (49h) α ⋆i (cid:0)P Nn =1 p ⋆in − P i (cid:1) = 0 , i = 1 , . (49i)If λ ⋆ ≥ ρ λ ⋆ > , we define a new dual point e λ = (cid:0) λ ⋆ − ρ λ ⋆ , , λ ⋆ + ρ +1 ρ λ ⋆ (cid:1) . Since e λ + e λ + e λ = λ ⋆ + λ ⋆ + λ ⋆ , e λ +( ρ +1) e λ = λ ⋆ +( ρ +1) λ ⋆ , ρ e λ +( ρ +1) e λ = ρλ ⋆ +( ρ +1) λ ⋆ , and R ⋆ MA = r ⋆ = r ⋆ = r ⋆ , the dual point ( e λ , α ⋆ ) and the primal point ( p ⋆ , p ⋆ ) also satisfy the KKT con-ditions (49a)-(49i). Therefore, e λ is an optimal dual solution of problem (17) that satisfies e λ = (1 − e λ , , e λ ) .If < λ ⋆ < ρ λ ⋆ , similarly we can define another dual point ˆ λ = (cid:0) , λ ⋆ − ρλ ⋆ , λ ⋆ +( ρ +1) λ ⋆ (cid:1) .Since ˆ λ +ˆ λ +ˆ λ = λ ⋆ + λ ⋆ + λ ⋆ , ˆ λ +( ρ +1)ˆ λ = λ ⋆ +( ρ +1) λ ⋆ , ρ ˆ λ +( ρ +1)ˆ λ = ρλ ⋆ +( ρ +1) λ ⋆ ,and R ⋆ MA = r ⋆ = r ⋆ = r ⋆ , the dual point ( ˆ λ , α ⋆ ) and the primal point ( p ⋆ , p ⋆ ) also satisfy theKKT conditions (49a)-(49i). Therefore, ˆ λ is an optimal dual solution of problem (17) that satisfies ˆ λ = (0 , − ˆ λ , ˆ λ ) . Hence, the statement of Proposition 1 has been proved for Case 2.A PPENDIX CC LOSED - FORM S OLUTION TO (19) WITHOUT U SING P ROPOSITION ISCUSSED IN R EMARK p ⋆ n > , p ⋆ n > . In this case, the KKT conditions(19a) and (19b) both hold with equality. Therefore, we need to solve a system of quadraticequations with two variables. To simplify this problem, we define an auxiliary variable x , g n p ⋆ n + g n p ⋆ n . (50)Then, by (19) and through some derivations, we obtain p ⋆ n = t ( ρ + 1) λ α ( ρ + 1) ln 2 − tg n λ / ( t + x ) − tg n , (51a) p ⋆ n = t ( ρ + 1) λ /ρα ( ρ + 1) ln 2 − tg n λ / ( t + x ) − tg n . (51b)By substituting (51a) and (51b) into (50), we end up with the following cubic equation of x : tg n ( ρ + 1) λ α ( ρ + 1) ln 2 − tg n λ / ( t + x ) + tg n ( ρ + 1) λ /ρα ( ρ + 1) ln 2 − tg n λ / ( t + x ) = x + 2 t. (52)It is widely known that the closed-form solutions of the cubic equation x + ax + bx + c = 0 are given by Cardano’s formula [39], i.e., x = e jθ q(cid:12)(cid:12) − q/ √ ∆ (cid:12)(cid:12) + e jθ q(cid:12)(cid:12) − q/ − √ ∆ (cid:12)(cid:12) − a/ , (53a) x = ωe jθ / q(cid:12)(cid:12) − q/ √ ∆ (cid:12)(cid:12) + ω e jθ / q(cid:12)(cid:12) − q/ − √ ∆ (cid:12)(cid:12) − a/ , (53b) x = ω e jθ / q(cid:12)(cid:12) − q/ √ ∆ (cid:12)(cid:12) + ωe jθ / q(cid:12)(cid:12) − q/ − √ ∆ (cid:12)(cid:12) − a/ , (53c)where p = − a / b, q = 2 a / − ab/ c, ω = − / j √ / , ∆ = p / 27 + q / , θ =angle ( − q/ √ ∆) , θ = angle ( − q/ − √ ∆) , and angle ( · ) denotes the phase angle of ancomplex number. If ∆ ≥ , the cubic equation has one real root and a pair of conjugate complexroots; if ∆ < , the cubic equation has three real roots.After obtaining the positive real root x of (52), we can easily obtain the optimal p ⋆ n and p ⋆ n by substituting x into (51), which is the closed-form power allocation solution.A PPENDIX DB OUNDS FOR P OWER D UAL V ARIABLES The optimal α ⋆ ( λ ) = ( α ⋆ , α ⋆ ) for problem (24) must satisfy the KKT conditions (19), andthere must exist n (1 ≤ n ≤ N ) such that p n > . Thus, by the feasible conditions p ⋆in ≥ i =1 , , n =1 , . . . , N ) and (19a) with n = n , we can obtain an upper bound for α ⋆ , α ⋆ = tg n λ / ( ρ + 1)( t + g n p ⋆ n + g n p ⋆ n ) ln 2 + tg n λ ( t + g n p ⋆ n ) ln 2 ≤ g n [( ρ + 1) λ + λ ]( ρ + 1) ln 2 ≤ ( ρ + 1) λ + λ ( ρ + 1) ln 2 max n { g n } , and a trivial lower bound α ⋆ ≥ .Similar discussions can be applied for α ⋆ and the optimal α ⋆ ( λ ) for problem (35), and thuswe can attain that α , min ≤ α ⋆ ≤ α , max = ( ρ + 1) λ + λ ( ρ + 1) ln 2 max n { g n } , (54a) α , min ≤ α ⋆ ≤ α , max = ( ρ + 1) λ + λ ( ρ + 1) ln 2 max n { g n } , (54b) α , min ≤ α ⋆ ≤ α , max = max n (cid:26) ρ ˜ g n (1 − λ )+˜ g n λ ρ ln 2 (cid:27) . (54c)By (54a) and (54b), we can initialize α and the matrix A in Algorithm 1 as α = ( α , max / , α , max / , A = α , max / α , max / , (55)and the initialization of α , min and α , max in Algorithm 3 is given by (54c). A PPENDIX EP ROOF OF P ROPOSITION r + ρr ≥ ( ρ + 1) r . This leads to two cases to be discussed: Case 1 : r + ρr > ( ρ + 1) r .In this case, if r ≥ r , we have ρ ( r − r ) > r − r ≥ . Hence, r > r . Assume λ ⋆ =(0 , − λ ⋆ , λ ⋆ ) ∈ Λ \{ λ } , which means ≤ λ ⋆ < , and then we have (1 − λ ⋆ )( r − r ) < . (56)On the other hand, since λ ⋆ = (0 , − λ ⋆ , λ ⋆ ) is an optimal dual point, by (28) and (30), it mustbe true that ( λ ⋆ − λ ) T ξ ( λ ) = (1 − λ ⋆ )( r − r ) ≥ , (57)which leads to a contradiction with (56). Thus, λ ⋆ / ∈ Λ \{ λ } . By Proposition 1, we must have λ ⋆ ∈ Λ .Similarly, if r ≥ r , we can show that λ ⋆ ∈ Λ . Case 2 : r + ρr = ( ρ + 1) r .If only one of the inequalities of r ≥ r and r ≥ r is satisfied, similar to Case 1, we canshow that λ ⋆ ∈ Λ if r ≥ r and λ ⋆ ∈ Λ if r ≥ r .If both r ≥ r and r ≥ r , we have r = r = r = R ⋆ MA by the condition of Case 2. Thus,according to (28), the subgradient ξ ( λ ) = . Substituting this into (29) yields G MA ( λ ) ≤ G MA ( λ ) , ∀ λ ∈ Λ S Λ , (58)which means that λ itself is an optimal solution to (27). i.e., λ ⋆ = λ .A PPENDIX FP ROOF OF P ROPOSITION x , since p in ≤ P i = x ¯ P i , p in is of the order O ( x b ) for b ≥ , and p in isof the order O ( x b ) . By this and (41), each rate pair ( R , R ) ∈ R DF ( x ¯ P , G ) satisfies R ≤ (cid:26) N X n =1 g n p n , N X n =1 ˜ g n p Rn (cid:27) + O ( x b ) , (59a) R ≤ (cid:26) N X n =1 g n p n , N X n =1 ˜ g n p Rn (cid:27) + O ( x b ) , (59b) R + R ≤ N X n =1 ( g n p n + g n p n ) + O ( x b ) . (59c) By taking the summation of (59a) and (59b), it is easy to see that the sum-rate constraint (59c)always holds if (59a) and (59b) are true. In other words, the achievable rate region R DF ( x ¯ P , G ) can be expressed by (59a) and (59b) for sufficiently small x . On the other hand, the cut-set outerbound region R out ( x ¯ P , G ) is also described by (59a) and (59b). Taking resource allocation intoaccount, both R DF ( x ¯ P , G ) and R out ( x ¯ P , G ) are in the form of R ( x ¯ P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ (cid:26) N X n =1 g n p n , N X n =1 ˜ g n p Rn (cid:27) + O ( x b ) ,R ≤ (cid:26) N X n =1 g n p n , N X n =1 ˜ g n p Rn (cid:27) + O ( x b ) , N X n =1 p in ≤ P i , p in ≥ , i = 1 , , R, n = 1 , . . . , N ) , (60)for sufficiently small x . Therefore, for any rate pair ( R , R ) ∈ R out ( x ¯ P , G ) , there exists some ( ˆ R , ˆ R ) ∈ R DF ( x ¯ P , G ) such that R = ˆ R + O ( x b ) and R = ˆ R + O ( x b ) for b ≥ . Nowlet x → , the rate regions R DF ( x ¯ P , G ) and R out ( x ¯ P , G ) tend to be the same.A PPENDIX GP ROOF OF P ROPOSITION p in = P i /N = x ¯ P i /N , which leads to arate region given by R ( x ¯ P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ N X n =1 t log (cid:16) g n x ¯ P tN (cid:17) ,R ≤ N X n =1 t log (cid:16) g n x ¯ P tN (cid:17) ,R + R ≤ N X n =1 t log (cid:16) g n x ¯ P + g n x ¯ P tN (cid:17) ,R ≤ N X n =1 (1 − t ) log (cid:16) 1+ ˜ g n x ¯ P R (1 − t ) N (cid:17) ,R ≤ N X n =1 (1 − t ) log (cid:16) 1+ ˜ g n x ¯ P R (1 − t ) N (cid:17) , < t < ) . (61)Since this equal power allocation scheme is feasible, the rate region (61) satisfies R ( x ¯ P , G ) ⊂ R DF ( x ¯ P , G ) . (62)All the rate functions in (61) are of the form log (1 + ax ) for a > . According to (42), it canbe expressed by log ( x ) + O (1) for sufficiently large x . By this, for sufficiently large x , we canobtain that R ( x ¯ P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ N X n =1 t log ( x ) + O (1) ,R ≤ N X n =1 t log ( x ) + O (1) ,R + R ≤ N X n =1 t log ( x ) + O (1) ,R ≤ N X n =1 (1 − t ) log ( x ) + O (1) ,R ≤ N X n =1 (1 − t ) log ( x ) + O (1) , < t < ) . (63)Hence, similar to the definition (43), the multiplexing gain region of R ( x ¯ P , G ) is given by n ( r , r ) (cid:12)(cid:12)(cid:12) r ≤ tN, r ≤ tN, r + r ≤ tN,r ≤ (1 − t ) N, r ≤ (1 − t ) N, < t < , r , r ≥ o . (64)After some simple manipulations, (64) can be simplified as (44).Then, we consider an infeasible power allocation scheme p in = P i = x ¯ P i , which results inthe following rate region: R ( x ¯ P , G ) = ( ( R , R ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) R ≤ N X n =1 t log (cid:16) g n x ¯ P t (cid:17) ,R ≤ N X n =1 t log (cid:16) g n x ¯ P t (cid:17) ,R + R ≤ N X n =1 t log (cid:16) g n x ¯ P + g n x ¯ P t (cid:17) ,R ≤ N X n =1 (1 − t ) log (cid:16) 1+ ˜ g n x ¯ P R − t (cid:17) ,R ≤ N X n =1 (1 − t ) log (cid:16) 1+ ˜ g n x ¯ P R − t (cid:17) , < t < ) . (65)It can be easily seen from (5) that the rate region (65) satisfies R DF ( x ¯ P , G ) ⊂ R ( x ¯ P , G ) , (66)and the multiplexing gain region of R ( x ¯ P , G ) is also given by (64), and thus (44). 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