Energyefficient resource allocation for OFDMAbased twoway relay channel with physicallayer network coding
 Min Zhou^{1}Email author,
 Qimei Cui^{1},
 Mikko Valkama^{2} and
 Xiaofeng Tao^{1}
https://doi.org/10.1186/16871499201266
© Zhou et al; licensee Springer. 2012
Received: 19 July 2011
Accepted: 27 February 2012
Published: 27 February 2012
Abstract
Physicallayer network coding (PNC) is a novel cooperative technique for twoway relay channel (TRC), where two users exchange information via intermediate relay node(s). On the other hand, the issue of green communications to reduce energy consumption has recently started to arouse much attention. This article studies the energyefficient resource allocation problem for orthogonal frequency division multiple access (OFDMA)based TRC with PNC. In particular, resource allocation for multiuser, multirelay OFDMA systems is vital for the optimization of power allocation, relay selection and subcarrier assignment. By applying convex optimization techniques, an optimal resource allocation scheme is proposed to minimize total transmit power with required rates. Numerical simulations show that the proposed scheme provides diversity gain compared to the single relay network, and PNC gain relative to the TRC without PNC.
1 Introduction
Wireless relay was introduced in 3GPPLTEAdvanced [1] for throughput enhancement and coverage extension without requiring large transmit powers. However, practical relay systems typically consider halfduplex mode [2], where relay nodes cannot transmit and receive simultaneously. For the purpose of overcoming the spectral loss of halfduplex relay, physicallayer network coding (PNC) [3–7] was proposed in twoway relay channel (TRC), where two users wanted to exchange information with each other via relay node(s).
Compared with the traditional oneway relay schemes that need four time slots to finish information exchange in TRC, network coding (NC) scheme [8] needs three time slots. PNC makes use of the additive nature of simultaneously arriving electromagnetic waves for equivalent network coding operation at physical layer. The number of required time slots of PNC is reduced to two: in the first time slot, two users transmit their signals simultaneously to relay node(s); then in the second time slot, relay node(s) broadcast the processed version of the received superimposed signal to the two users. Distinguished by the process function of relay node(s), PNC has several subprotocols, such as denoiseandforward (PNCDNF) [3, 4], decodeandforward (PNCDF) [5] and amplifyandforward (PNCAF) [5–7]. To the best of authors' knowledge, PNCAF is the only protocol of PNC that has been demonstrated in a practical system with a bit rate of 500 kbps [6]. Since PNCAF is most likely to be realized in practical systems also more widely, this article will focus on PNCAF.
OFDM/OFDMA is one of the most important transmission techniques for future wireless communication systems. In addition, TRC with PNC that employs OFDM/OFDMA has recently attracted considerable attention. Maximizing the system rate by varying the power allocation and tone permutation in TRC with single relay over OFDM was considered in [9]. Power allocation to maximize the sumrate under the total power constraint was developed for a TRC exchanging OFDM signals via a single relay in [10, 11]. The resource allocation problem for the OFDMAbased multiuser TRC system was investigated by [12], where an iterative algorithm was proposed to maximize the sumrate. In [13], adaptive subcarrier allocation was proposed to maximize the achievable sumrate for a multiuser multirelay OFDMAbased TRC. In summary, the past literatures mainly focus on maximizing sumrate to achieve full system load, which rarely happens in practical systems, even at peak traffic hours [14]. Hence, the systems optimized for full system load should be redesigned for arbitrary system load to achieve better energy efficiency.
In this article, an energyefficient resource allocation to minimize transmit power consumption with required link rates is considered for OFDMAbased TRC with PNC. Additionally, OFDMAbased TRC indicates that the subcarriers of an OFDM symbol can be shared by multiple relay nodes. The main contributions of our study are:

An optimal energyefficient resource allocation is proposed with joint power allocation, relay selection and subcarrier assignment. To the best of authors' knowledge, this problem is considered for the first time in OFDMAbased TRC with PNC and multiple relay nodes.

The best energyefficient subcarrier assignment for OFDMAbased TRC with PNC is proved to be an opportunistic subcarrier assignment, where each subcarrier is assigned to a unique relay node.

The closedform expressions of optimal power allocation, relay selection, subcarrier assignment and minimum transmit power consumption are derived through convex optimization techniques.
This article is organized as follows. In Section 2, the system model is given. In Section 3, energyefficient resource allocation problem is formulated as an optimization problem, which has been decomposed to N persubcarrier problems. In Section 4, the persubcarrier problem is solved and the joint power allocation, relay selection and subcarrier assignment scheme is proposed to achieve the minimum transmit power consumption. Section 5 gives numerical simulations and performance analysis under the energy efficiency evaluation framework (E^{3}F) provided in EARTH project [15]. Conclusions are given in the last section.
Notation: [●] ^{ T } denotes the transpose of vector. ● denotes the scalar norm. For vectors, ≽ and = are used to indicate the componentwise inequality and equality, respectively.
2 System model
where N_{0} is power spectral density of AWGN and W denotes the subcarrier bandwidth.
By taking ${\rho}_{i}^{\left(n\right)}$ and τ_{ i } into account, we use $\sqrt{{\tau}_{i}{\rho}_{i}^{\left(n\right)}{\alpha}_{i}^{\left(n\right)}}$as a general scale factor for R_{ i } on n th subcarrier. It is noted that the n th subcarrier of relay node R_{ i } is marked as free subcarrier (as shown in Figure 1), while ${\rho}_{i}^{\left(n\right)}$ is equal to zero.
3 Problem formulation and simplification
In this section, the optimal energyefficient resource allocation is formulated as a joint optimization of power allocation, relay selection and subcarrier assignment to achieve the minimum transmit power consumption with required link rates. With the mathematical analysis, the optimization problem has been decomposed into N subproblems. Then, the subproblems are simplified by using the KarushKuhnTucker (KKT) condition.
3.1 Problem formulation
The factor 1/2 comes from the two time slots required by messages exchange. However, Problem 3.1 has high computational complexity, while optimum values of K(2N+1)+2N variables need to be found. Some simplifications will be implemented in the following section.
3.2 Problem simplification
The overall transmit power consumption will be the sum of individual transmit power consumptions on each subcarrier.
where sgn$\left(x\right)=\left\{\begin{array}{c}\hfill 1\phantom{\rule{2.77695pt}{0ex}}\forall x>0\hfill \\ \hfill 0\forall x\le 0\hfill \end{array}\right..$
Hence, it is shown that ${\rho}_{i}^{\left(n\right)}={\tau}_{i}{\rho}_{i}^{\left(n\right)}$for the persubcarrier optimization on the n th subcarrier.
where $m=\left({2}^{\frac{2{\stackrel{\u0304}{r}}_{1}}{WN}}+{2}^{\frac{2{\stackrel{\u0304}{r}}_{2}}{WN}}2\right){N}_{0}W.$
Finally, the original Problem 3.1 with K(2N+1)+2N variables is reduced to a persubcarrier optimization Problem 3.3 with only 2K variables. Furthermore, Problem 3.3 has very simple constraints, where the only requirements are that the factors ${\alpha}_{i}^{\left(n\right)}$ should be nonnegative and ${\rho}_{i}^{\left(n\right)}\in \left\{0,1\right\}.$
Remark 1: Problem 3.3 is to find the minimum power consumption on the n th subcarrier. If Problem 3.3 is solved, the minimum power consumption on the other subcarriers can also be obtained with the similar steps. Accordingly, the overall minimum power consumption will be the sum of transmit power consumptions on each subcarrier. In the following section, more detailed mathematical analysis will be carried out around the persubcarrier optimization Problem 3.3.
4 Persubcarrier optimization
Problem 3.3 is a mixed binary integer programming problem, which considers the minimum power consumption on the n th subcarrier. However, it is prohibitive to find the global optimum in terms of computational complexity. To obtain the global optimum, an exhaustive search is needed throughout the subcarrier assignment vectors ρ^{(n)}and amplification factor vectors α^{(n)}to find the overall minimum transmit power. Before solving the Problem 3.3 with optimal subcarrier assignment vector ρ^{(n)}, we first propose an energyefficient resource allocation scheme with opportunistic subcarrier assignment in 4.1. Then, the proposed heuristic scheme is proved to be the optimal solution of Problem 3.3 in Section 4.2.
4.1 Energyefficient resource allocation with opportunistic subcarrier assignment
This section considers the Problem 3.3 with an opportunistic subcarrier assignment, where each subcarrier is assigned to an unique relay node. Hence, Problem 3.3 is divided into a power allocation and a opportunistic subcarrier assignment problem.
Remark 2: To summarize shortly, the energyefficient resource allocation algorithm on the n th subcarrier with opportunistic subcarrier assignment approach proposed above has two steps: First, the n th subcarrier is assigned to the best relay ${R}_{\stackrel{\u0303}{d}}$ by opportunistic subcarrier assignment criterion. Then, optimal transmit power determined by power allocation criterion will be allocated to S_{1}, S_{2}, and ${R}_{\stackrel{\u0303}{d}}$ on the n th subcarrier. The minimum transmit power consumption ${P}_{o}^{\left(n\right)}$ on the n th subcarrier of this scheme is given by ${P}_{o}^{\left(n\right)}={\stackrel{\u0304}{P}}^{\left(n\right)}\left(\stackrel{\u0303}{d}\right).$
4.2 Energyefficient resource allocation with optimal subcarrier assignment
In this section, it is proved that the above opportunistic twostep resource allocation algorithm is also the optimum method in terms of energyefficiency. This is formulated with the following Theorem:
Theorem 1 In multirelay OFDMAbased TRC with PNC, the opportunistic subcarrier assignment is the optimal energyefficient subcarrier assignment scheme, in which the nth subcarrier is assigned to the best relay node from the K candidate relay nodes.
Proof: Two steps are needed to complete the proof. We first consider the necessary and sufficient conditions of Theorem 1, and then solve its corresponding mathematical problem.
Hence, the necessary and sufficient condition of Theorem 1 is that the minimum value of f(Ξ_{1 × K}) is nonnegative in the feasible region.
where $\psi \left(i\right)=2\left{g}_{i}^{\left(n\right)}{}^{2}\right{h}_{i}^{\left(n\right)}{}^{2}\left(\mathsf{\text{m}}+{N}_{0}W\right)$ and $\vartheta \left(i,\phantom{\rule{2.77695pt}{0ex}}j\right)=\vartheta \left(j,\phantom{\rule{2.77695pt}{0ex}}i\right)=\left{g}_{i}^{\left(n\right)}{}^{2}\right{h}_{i}^{\left(n\right)}{}^{2}{N}_{0}W+\left{g}_{j}{}^{2}\right{h}_{j}{}^{2}{N}_{0}W+\left{g}_{i}^{\left(n\right)}{}^{2}\right{h}_{j}{}^{2}\mathsf{\text{m}}+\left{g}_{j}{}^{2}\right{h}_{i}^{\left(n\right)}{}^{2}\mathsf{\text{m}}.$
 (1)∇^{2} f(Ξ_{1 × K}) is PSM: It means all the secondorder principal minors of ∇^{2} f(Ξ_{1 × K}) must be zero. So, it is derived that ${g}_{1}^{\left(n\right)}={g}_{2}^{\left(n\right)}=\cdots ={g}_{k}^{\left(n\right)}={h}_{1}^{\left(n\right)}={h}_{2}^{\left(n\right)}=\cdots ={h}_{k}^{\left(n\right)}.$ In this condition, it is obvious that${f}_{min}\left({\mathrm{\Xi}}_{1\times K}\right)=0,\mathsf{\text{while}}\phantom{\rule{0.3em}{0ex}}{\nabla}^{2}f\left({\mathrm{\Xi}}_{1\times K}\right)\phantom{\rule{0.3em}{0ex}}\mathsf{\text{is PSM}}$(40)
 (2)∇^{2} f(Ξ_{1 × K}) is IM: According to [[20], Section III], function (25) has no extreme points, in other words the minimum value must be obtained on the domain boundary (one of the ζ_{ i } equals to 0 or γ). The values on boundary are expressed as b_{ γ } (w) = f _{min}(ζ _{1}, . . ., ζ _{w1}, ζ _{w+1}, . . . , ζ_{ K } ) and b _{0}(w) = f _{min}(ζ _{1}, . . . , ζ _{w1}, 0, ζ _{w+1}, . . . , ζ_{ K } ). Hence, the minimum value of f(Ξ_{1 × K}) is f _{min}(Ξ_{1 × K}) = min(b_{ γ } (1), b_{ γ } (2), . . . , b_{ γ } (K), b _{0}(1), b _{0}(2), . . . , b _{0}(K)). If we can prove that b_{ γ } (w) ≥ 0 and b _{0}(w) ≥ 0, ∀w = 1, 2, . . . , K, the minimum value of function f(Ξ_{1 × K}) will be positive. We can easily prove that$\begin{array}{cc}\hfill {b}_{\gamma}\left(w\right)& =f\left(0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}\gamma ,\phantom{\rule{2.77695pt}{0ex}}0,\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}0\right)\hfill \\ =m\left(1+\gamma {\left{g}_{w}^{\left(n\right)}\right}^{2}\right)\left(1+\gamma {\left{h}_{w}^{\left(n\right)}\right}^{2}\right)>0\hfill \end{array}$(41)
If we can now prove that b_{0}(K) = f(Ξ_{1 × (K1)}) ≥ 0, it can be extended to prove b_{0}(w) ≥ 0, ∀w = {1, 2, . . . , K} straightforwardly. Then, the problem degenerates to prove f(Ξ_{1 × (K1)}) ≥ 0.
From the definition of ${P}_{o}^{\left(n\right)},$ we have ${\epsilon}_{2}\le {P}_{o}^{\left(n\right)}\le {\epsilon}_{1},$ which implies $\phi \left({P}_{o}^{\left(n\right)}\right)\ge 0$ and then f_{min}(ζ_{ i } ) ≥ 0. Finally, it is known that b_{0}(K) = f(Ξ_{1 × (K1)}) ≥ 0, which can be extended to prove b_{0}(w) ≥ 0, ∀w = 1, 2, . . . , K straightforwardly.
From (40) and (46), it is then clear that the minimum value of f(Ξ_{1 × K}) is nonnegative. The necessary and sufficient conditions of Theorem 1 discussed in step 1 are thus satisfied. Theorem 1 is proved. ■
From Theorem 1, it is known that the proposed opportunistic subcarrier assignment criterion is the optimal energyefficient criterion in multiuser multirelay OFDMAbased TRC with PNCAF.
5 Performance analysis
where P_{total} is the overall transmit power required to deliver required rates ${\stackrel{\u0304}{r}}_{1}$ and ${\stackrel{\u0304}{r}}_{2}$, and the factor 1/2 comes from the two time slots required by messages exchange. We assume that the spectral density of noise, denoted by N_{0}, is equal to 174 dBm/Hz and all the subcarrier complex gains are realized independently through complex normal distribution of the form $\mathcal{C}\mathcal{N}\left(0,\frac{1}{{\left(1+{d}_{i,j}\right)}^{\mathrm{\Delta}/2}}\right),$ where d_{ i,j }is the distance between node i and node j, and the path loss exponent Δ is 3. We assume that L is the distance between two users and all the relay nodes are randomly distributed between them.
The ECI of each scheme is averaged over 1 × 10^{8} independent realizations of relay' location by MonteCarlo simulation.
6 Conclusion and discussion
In this article, we studied multiuser, multirelay OFDMAbased twoway relay network with PNCAF protocol. An optimal energyefficient resource allocation with joint power allocation, relay selection and subcarrier assignment to achieve the minimum transmit power consumption with required link rate pair were derived. From the analysis, we proved that the optimal subcarrier assignment criterion in energy efficiency sense is an opportunistic subcarrier assignment, in which a subcarrier is assigned to a unique relay node. Based on the proof, the closedform expressions of power allocation, relay selection, subcarrier assignment and minimum transmit power consumption were derived. The simulations confirmed the proposed scheme is far superior to the other existing schemes in terms of energy efficiency. It was also observed that energy efficiency of TRC is generally better with symmetric symmetric rate pair than asymmetric rate pair.
Beside the scenario considered in this article, energy efficiency can be further improved with more complex schemes, e.g., adaptive bit allocation on subcarriers and phase alignment operation at relay nodes. Furthermore, the analysis of energy efficiency is based on Shannon capacity expressions in this article, which do not take into account the impact of practical channel coding, modulation and retransmission methods. Hence, the consideration of energy efficiency in more practical scenarios will be another future research item.
Declarations
Acknowledgements
This article is supported by International Scientific and Technological Cooperation Program (2010DFA11060), ChinaEU International Scientific and Technological Cooperation Program (0902), National Nature Science Foundation of China (Grant No. 61001119, 61027003) and Fund for Creative Research Groups of China (Grant No. 61121001). This article is partially sponsored by the Finnish Funding Agency for Technology and Innovation (Tekes), under the project Energy and Cost Efficiency for Wireless Access.
Authors’ Affiliations
References
 Parkvall S, Dahlman E, Furuskar A, Jading Y, Olsson M, Wanstedt S, Zangi K: LTEAdvanced  Evolving LTE towards IMTAdvanced. In IEEE 68th Vehicular Technology Conference, 2008. VTC 2008Fall. Calgary, Canada; 2008:15.View ArticleGoogle Scholar
 Peters SW, Panah AY, Truong KT, Heath RW: Relay architectures for 3GPP LTEAdvanced. EURASIP J Wirel Commun Netw 2009., 2009:Google Scholar
 Zhang S, Liew SC, PLam P: Hot topic: physical layer network coding. In Proc International conference on Mobile computing and networking (MobiCom'06). Los Angeles, USA; 2006:358365.View ArticleGoogle Scholar
 Popovski P, Yomo H: Physical network coding in twoway wireless relay channels. In IEEE International Conference on Communications. Glasgow, Scotland; 2007:707712.Google Scholar
 Popovski P, Yomo H: The antipackets can increase the achievable throughput of a wireless multihop network. IEEE International Conference on Communications, 2006 2006, 9: 38853890.View ArticleGoogle Scholar
 Katti S, Gollakota S, Katabi D: Embracing wireless interference: analog network coding. In ACM SIGCOMM 2007. Volume 9. Kyoto, Japan; 2007:397408.Google Scholar
 Louie R, Li Y, Vucetic B: Practical physical layer network coding for twoway relay channels: performance analysis and comparison. IEEE Trans Wirel Commun 2010, 9(2):764777.View ArticleGoogle Scholar
 Wu Y, Chou PA, Kung SY: Information exchange in wireless networks with network coding and physicallayer broadcast. In Tech Rep MSRTR200478. Microsoft Research, Redmond, Wash, USA; 2004.Google Scholar
 Ho CK, Zhang R, Liang YC: Twoway relaying over OFDM: optimized tone permutation and power allocation. IEEE International Conference on Communications, 2008. ICC'08 2008, 39083912.View ArticleGoogle Scholar
 Dong M, Shahbazpanahi S: Optimal spectrum sharing and power allocation for OFDMbased twoway relaying. 2010 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP) 2010, 33103313.View ArticleGoogle Scholar
 Jang YU, Jeong ER, Lee Y: A twostep approach to power allocation for OFDM signals over twoway amplifyandforward relay. IEEE Trans Signal Process 2010, 58(4):24262430.MathSciNetView ArticleGoogle Scholar
 Jitvanichphaibool K, Zhang R, Liang YC: Optimal resource allocation for twoway relayassisted OFDMA. IEEE Trans Veh Technol 2009, 58(7):33113321.View ArticleGoogle Scholar
 Shin H, Lee JH: Subcarrier allocation for multiuser twoway OFDMA relay networks with fairness constraints. 2010 IEEE 71st Vehicular Technology Conference 2010, 15.View ArticleGoogle Scholar
 Correia LM, Zeller D, Oliver B, Ferling D, Jading Y, Godor I, Auer G, Van Der Perre L: Challenges and enabling technologies for energy aware mobile radio networks. IEEE Commun Mag 2010, 48: 6672.View ArticleGoogle Scholar
 D2.3  Energy efficiency analysis of the reference systems, areas of improvements and target breakdown EARTH project 2010, 3940. [https://bscw.ictearth.eu/pub/bscw.cgi/d31515/EARTH_WP2_D2.3.pdf]
 Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge, UK; 2004.View ArticleMATHGoogle Scholar
 Troutman JL: Variational Calculus and Optimal Control: Optimization with Elementary Convexity. Springer, US; 1995.MATHGoogle Scholar
 Luenberger DG, Ye Y: Linear and Nonlinear Programming. Springer, US; 2009.MATHGoogle Scholar
 Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, UK; 1985.View ArticleMATHGoogle Scholar
 Luo B, Cui Q, Wang H, Tao X: Optimal joint waterfilling for coordinated transmission over frequencyselective fading channels. IEEE Commun Lett 2011, 15(2):190192.View ArticleGoogle Scholar
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