Sum rate maximization for multicell downlink OFDMA with subcarrier pairbased opportunistic DF relaying
 Zhiwen Jin^{1}Email author,
 Tao Wang^{2},
 Jibo Wei^{3} and
 Luc Vandendorpe^{1}
https://doi.org/10.1186/16871499201424
© Jin et al.; licensee Springer. 2014
Received: 17 September 2013
Accepted: 25 January 2014
Published: 8 February 2014
Abstract
This paper considers multicell decodeandforward (DF) relayaided orthogonal frequency division multiaccess (OFDMA) downlink systems, in which all sources and relays are coordinated by a central controller for resource allocation (RA). The improved subcarrier pairbased opportunistic DF relaying protocol proposed and studied in the IEEE International Conference on Communications, Beijing, 3795–3800, 2008 and IEEE Trans. Signal Process. 61:2512–2524, 2013 is applied. This protocol has a high spectrum efficiency (HSE) in the sense that all unpaired subcarriers are utilized for data transmission during the second time slot (TS). In particular, the sum (over all cells and all destinations) rate maximized problem with a total power constraint in each cell+ is formulated. To solve this problem, an iterative RA algorithm is proposed to optimize mode selection (decision whether the relay should help or not), subcarrier assignment and pairing (MSSAP) and power allocation (PA) in an alternate way. As for the MSSAP stage of each iteration, the formulated problem is decoupled into subproblems with the tentative PA results. Each subproblem can be easily solved by using the optimal results of a linear assignment problem (LAP), which is then solved by the Hungarian Algorithm in polynomial time. As for the PA stage of each iteration, an algorithm based on singlecondensation and geometric programming (SCGP) is proposed to optimize PA in polynomial time with the tentative MSSAP results. The proposed algorithm is coordinate ascent (CA)based and therefore can reach a local optimum in polynomial time. Finally, the convergence and effectiveness of the proposed algorithm, the impact of relay position and total power on the system performance, and the benefits of using subcarrier pairing (SP) and the HSE protocol are illustrated through numerical experiments.
Keywords
1 Introduction
High data rate and ubiquitous coverage are strongly required in the nextgeneration wireless communication networks. To achieve this goal, orthogonal frequency division multiple access (OFDMA) technology is receiving a lot of interest due to its inherent ability to combat frequencyselective multipath fading and its flexibility in applying dynamic radio resource allocation (RA) for performance improvement. On the other hand, the emerging relaying technology is highly favored by both academia and industry due to its attractive feature of coverage extension and data rate improvement [1]. Therefore, it is promising to incorporate OFDMA with relaying technologies in nextgeneration cellular systems.
Concerning relay aided OFDM(A) transmissions, two lowcomplexity yet efficient types of relaying have been proposed in [2] and [3], namely amplifyandforward (AF) and decodeandforward (DF). Recently, the DF relaying is receiving a lot of interest due to its simple processing at the relay. To better apply it, [4] and [5] have further proposed the ML decoding to address the error propagation issue. With DF relaying, symbols are transmitted in two time slots (TSs). During the first TS, the source broadcasts symbols on all subcarriers with the relay keeping quiet. During the second TS, except for the relay, the source might also broadcast symbols on subcarriers not used by the relay, as will be elaborated later. Each subcarrier used by the relay during the second TS is paired with a subcarrier during the first TS. In particular, these two subcarriers are termed as paired subcarriers, referred to as the relayaided transmission mode hereafter. Adopting subcarrierbysubcarrier pairingbased DF relaying, RA problems for downlink OFDMA have been investigated intensively [6–25].
In particular, the works of [8–11] have considered RA for the DF relayaided OFDMA systems when the destination cannot or hardly hear from the source, meaning, the sourcetodestination (SD) link is unavailable. Every subcarrier during the first TS is paired with a subcarrier during the second TS using the relayaided transmission mode. Under a total power constraint, it has been proven that the optimum sum rate can be reached by ordered subcarrier pairing, i.e., the best sourcetorelay subcarrier should be paired with the strongest relaytodestination subcarrier, and so on.
Considering the case when the SD link is available, [6, 7, 12–25] have studied RA in systems with opportunistic relaying (sometimes termed as selection relaying). To start with, a low spectrum efficiency (LSE) protocol was studied in [6, 12–16], when only the relay broadcasts symbols during the second TS. Specifically, every symbol can be transmitted in either the relayaided mode or the direct mode. In the relayaided mode, a subcarrier during the first TS is paired with a subcarrier during the second TS for the transmission. While in the direct mode, the symbol is transmitted on a subcarrier during the first TS directly to the targeted destination without the help of the relay. Note that using this LSE protocol, unpaired subcarriers during the second TS are actually unused, which causes a waste of the limited spectrum resource. To address this issue, [17–25] have proposed and studied improved high spectrum efficiency (HSE) protocols, which allow new symbols to be transmitted on the unpaired subcarriers during the second TS. Note that the HSE protocols do not really modify the implementation of DF relaying during the second TS but simply let the source use unpaired subcarriers during the second TS for directmode transmission. To further improve the system performance, a novel subcarrier pairbased opportunistic DF relaying protocol is proposed in [7]. Note that this protocol truly improves the implementation of DF relaying by using transmit beamforming at a subcarrier during the second TS. Specifically, this novel protocol is the same as the HSE protocol, except that the source and relay implement transmit beamforming to transmit the symbol at a paired subcarrier during the second TS.
Note that all these papers consider RA in singlecell situations with the cochannel interference (CCI) modeled as additive background noise, which is reasonable only when the frequency reuse factor is low. Here, the frequency reuse factor is 1/W, where W is the number of cell clusters which cannot use the same frequencies for transmission. However, in nextgeneration cellular systems, aggressive frequency reuse is recommended due to its ability to achieve higher system capacity [26]. When the frequency reuse factor is high or even up to 1, the CCI becomes a key factor affecting the system performance and thus cannot be ignored [27, 28].
Considering the CCI, the models for multicell DF relayaided OFDMA systems become much more interesting and challenging than those for singlecell DF relayaided OFDMA systems. For RA problems in such multicell systems, some algorithms have been proposed in [29] and [30] when powers are uniformly allocated to all stations. Several RA algorithms have been proposed in [31–34] for multicell OFDMA systems without DF relaying. However, these methods cannot be extended directly to solve RA problems jointly optimizing transmission mode selection, subcarrier assignment and pairing (MSSAP) as well as power allocation (PA) in multicell OFDMA systems with subcarrier pairbased opportunistic DF relaying. Paper [35] has recently considered and formulated the joint RA and scheduling problem in multicell DF relayed OFDMA systems, taking CCI as well as user data rate requirements into account. However, both opportunistic relaying and subcarrier pairing (SP) were not considered there. The CCI from the interfering sources to the considered destination and relay is ignored for practical and simplification consideration. Moreover, to solve this problem, an interference temperature constraint has been imposed to transform the original problem into a standard convex one.
Compared with the above existing works, the contributions of this paper are listed below:

We formulate the sum rate maximized joint RA problem in multicell OFDMA downlink systems aided by a DF relay station (RS) in each cell. In particular, the subcarrier pairbased opportunistic relaying is adopted. To start with, the HSE protocol with SP is considered here without signal combining and transmit beamforming. It will be shown that the system performance can be improved by using SP and the HSE protocol. Note that when modeling the intercell CCI of a subcarrier in a selected cell, instead of using an additional integer variable to indicate whether a node in an interfering cell transmits data on this subcarrier or not, we use the corresponding power value to do it. This choice is motivated to simplify the system sum rate expression and facilitate the algorithm design. Specifically, the rate expression would be a nonlinear function of the integer variables when using an integer variable to indicate whether a subcarrier k in a cell n is used or not, while it is a linear function of the integer variables when using the corresponding power value to do it.

We propose an iterative RA algorithm to solve the RA problem, which optimizes MSSAP and the PA in an alternate way with the sum rate that keeps on increasing until convergence. As for the MSSAP stage of each iteration, the formulated problem is decoupled into subproblems with the tentative PA results. Each subproblem can be easily solved by using the optimal results of a linear assignment problem (LAP), which is then solved by the Hungarian Algorithm in polynomial time. As for the PA stage of each iteration, an algorithm based on singlecondensation and geometric programming (SCGP) is proposed to optimize PA in polynomial time with the tentative MSSAP results. The proposed algorithm is coordinate ascent (CA)based and therefore can reach a local optimum in polynomial time.
The rest of this paper is organized as follows. First, the system model and the problem formulation for the considered system are presented in the next section. Then, the proposed algorithms are described in Section 3. Furthermore, the effectiveness and convergence of the proposed RA algorithm as well as the benefits of using SP and the HSE protocol are illustrated by numerical experiments in Section 4. Finally, some conclusions are drawn in Section 5.
2 System model and problem formulation
2.1 System model
We consider a cellular OFDMA system with N cells coordinated by a central controller for RA. In each cell, the downlink transmission is carried out from a source to U destinations with the help of a relay, which is assumed to be of the DF type. For each link, the channel is assumed to be frequency selective and transformed into K parallel subchannels by using OFDM with sufficiently long cyclic prefix. The data transmission is carried out in two TSs. During the first TS, a symbol is first broadcasted by the source at a subcarrier k, which is in either relayaided mode or direct mode. Both the relay and the targeted destination receive this symbol. If the relayaided mode is used, the relay decodes the received symbol and forwards it to the targeted destination over a subcarrier l with the source keeping quiet on this subcarrier during the second TS. Here, the subcarrier l is paired with the subcarrier k for transmitting the same symbol. The destination only decodes the symbol received during the second TS. If the direct mode is used, the targeted destination decodes the symbol received during the first TS. Also, another symbol is broadcast by the source at an unpaired subcarrier l during the second TS, which is received and only decoded by the destination selected for the second TS.
With OFDMA, each subcarrier is allocated to only one destination in each cell. For the relayaided transmission, subcarrier k during the first TS can be paired with only one subcarrier l during the second TS and vice versa. Throughout this paper, perfect decoding, timing, and carrier synchronization are assumed. The RA is carried out by a central controller which has perfect knowledge of the system channel state information. Moreover, the coherence time of each link is assumed to be sufficiently long for the RA to be implemented. Lastly, we assume that the optimized RA can be correctly distributed to all nodes. Note that an upper bound on the system performances is obtained by assuming the above idealities.
The data transmission procedure in every cell is identical. Therefore, we only analyze the downlink data transmission inside one selected cell n, which is impaired by cochannel interference from the other cells. Specifically in cell n, the symbols are transmitted in either relayaided mode or direct mode, as will be elaborated in the following.
where ${v}_{{\mathrm{r}}_{n}}^{k}$ denotes the additive white Gaussian noise (AWGN) at subcarrier k and r_{ n } during the first TS. ${h}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{r}}_{n}}^{k}$ denotes the channel frequency response (CFR) for subcarrier k from ${\mathrm{s}}_{{n}^{\prime}}$ to r_{ n }.
where ${v}_{{\mathrm{d}}_{\mathit{\text{un}}},{t}_{2}}^{l}$ denotes the AWGN corrupting d_{ un } at subcarrier l during the second TS, ${h}_{{\mathrm{r}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}$ denotes the CFR for subcarrier l from ${\mathrm{r}}_{{n}^{\prime}}$ to d_{ un }. ${h}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}$ denotes the CFR of subcarrier l from ${\mathrm{s}}_{{n}^{\prime}}$ to d_{ un }.
where ${f}_{{\mathrm{r}}_{n}}^{k}={\sigma}^{2}+\sum _{{n}^{\prime},{n}^{\prime}\ne n}{P}_{{\mathrm{s}}_{{n}^{\prime}},{t}_{1}}^{k}{G}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{r}}_{n}}^{k}$ denotes the sum power of the AWGN and the interference received by r_{ n } at subcarrier k during the first TS. ${G}_{{\mathrm{s}}_{n},{\mathrm{r}}_{n}}^{k}={h}_{{\mathrm{s}}_{n},{\mathrm{r}}_{n}}^{k}{}^{2}$ denotes the channel gain of subcarrier k from s_{ n } to r_{ n }.
where ${f}_{{\mathrm{d}}_{\mathit{\text{un}}},{t}_{2}}^{l}={\sigma}^{2}+\sum _{{n}^{\prime}=1,{n}^{\prime}\ne n}^{N}{P}_{{\mathrm{s}}_{{n}^{\prime}},{t}_{2}}^{l}{G}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}+\sum _{{n}^{\prime}=1,{n}^{\prime}\ne n}^{N}{P}_{{\mathrm{r}}_{{n}^{\prime}}}^{l}{G}_{{\mathrm{r}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}$ denotes the sum power of the AWGN and the interference received by d_{ un } at subcarrier l during the second TS. ${G}_{{\mathrm{r}}_{n},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}={h}_{{\mathrm{r}}_{n},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}{}^{2}$ denotes the channel gain of subcarrier l from r_{ n } to d_{ un }. ${G}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}={h}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{un}}}}^{l}{}^{2}$ denotes the channel gain of subcarrier l from ${\mathrm{s}}_{{n}^{\prime}}$ to d_{ un }.
in nats/two TSs.
where ${v}_{{\mathrm{d}}_{\mathit{\text{un}}},{t}_{1}}^{k}$ denotes the AWGN corrupting d_{ un } at subcarrier k during the first TS.
where ${v}_{{\mathrm{d}}_{\mathit{\text{vn}}},{t}_{2}}^{l}$ denotes the AWGN corrupting d_{ vn } at subcarrier l during the second TS.
denotes the SINR associated with the decoding of ${x}_{{\mathrm{s}}_{n},{t}_{2}}^{l}$ from ${y}_{{\mathrm{d}}_{\mathit{\text{vn}}},{t}_{2}}^{l}$ at d_{ vn } during the second TS. ${f}_{{\mathrm{d}}_{\mathit{\text{vn}}},{t}_{2}}^{l}={\sigma}^{2}+\sum _{{n}^{\prime}=1,{n}^{\prime}\ne n}^{N}{P}_{{\mathrm{s}}_{{n}^{\prime}},{t}_{2}}^{l}{G}_{{\mathrm{s}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{vn}}}}^{l}+\sum _{{n}^{\prime}=1,{n}^{\prime}\ne n}^{N}{P}_{{\mathrm{r}}_{{n}^{\prime}}}^{l}{G}_{{\mathrm{r}}_{{n}^{\prime}},{\mathrm{d}}_{\mathit{\text{vn}}}}^{l}$ denotes the sum power of the AWGN and the interference received by d_{ vn } at subcarrier l during the second TS.
2.2 Problem formulation
In order to formulate the RA problem, we now introduce binary variables ${a}_{\mathit{\text{uvn}}}^{\mathit{\text{kl}}}$ and ${b}_{\mathit{\text{un}}}^{\mathit{\text{kl}}}$ to describe the mode selection, subcarrier assignment, and pairing in both TSs. To be more specific, ${a}_{\mathit{\text{uvn}}}^{\mathit{\text{kl}}}=1$ indicates that subcarrier k is allocated for data transmission to d_{ un } in direct mode during the first TS and so is subcarrier l to d_{ vn } during the second TS. Note in direct mode, although used for transmitting two different symbols, subcarrier k and subcarrier l are virtually seen as a subcarrier pair for data transmission during two TSs. ${b}_{\mathit{\text{un}}}^{\mathit{\text{kl}}}=1$ indicates that subcarrier k used during the first TS is paired with subcarrier l used during the second TS, and they are both allocated for data transmission to d_{ un } aided by r_{ n }.
Here, C 1, C 2, and C 3 ensure that each subcarrier k during the first TS can be paired or virtually paired with only one subcarrier l during the second TS in each cell n. Also, subcarrier l can be paired or virtually paired with only one subcarrier k. For each subcarrier pair (k,l), only one mode (direct/relayaided) can be selected for data transmission. If the direct mode is used, unique destinations (d_{ un } and d_{ vn }) should be targeted for subcarrier k and subcarrier l, respectively. If the relayaided mode is selected, the subcarrier pair (k,l) can be allocated to only one destination. Moreover, C 4 and C 5 ensure that the consumed sum power for each cell is less than its available sum power. This type of power constraints gives an upper bound of the system performance. In practice, each node (source, relay) in each cell will have an individual power constraint. Finally, C 6, C 7, and C 8 guarantee that no data is transmitted on an unused subcarrier and subcarrier l is used by only one node (either the source or the relay) in each cell during the second TS.
3 Algorithm development
3.1 Algorithm overview
Note that problem (11), which contains both integer and continuous variables, is a mixed combinatorial and nonconvex optimization. Here, the nonconvexity is due to the existence of the intercell cochannel interference terms in (12). In general, an exhaustive search is needed to find the global optimum. Let us consider a system with N cells, U users, and K subcarriers in each cell. There are K^{ N K } possible subcarrier pairs, each of which can be assigned to one of the U possible users with two possible transmission modes. Thus, there are ${\left(2U\right)}^{{K}^{\mathit{\text{NK}}}}$ possible MSSAP, which limits the scalability.
In order to make the problem tractable, we opt for a CA approach made of two stages: the MSSAP stage and the PA stage. We propose an iterative algorithm which optimizes the two stages in an alternate way, as depicted by Algorithm 1. Integer m indicates the iteration number, and a variable with the superscript m denotes the value obtained at the end of iteration m.
Algorithm 1 Overall RA Optimization Algorithm
Specifically in the proposed RA algorithm, we first set m=0 and initialize the power P^{0} by uniform power allocation (UPA). Each iteration is made of the MSSAP stage followed by the PA stage.
During the MSSAP stage of iteration m, problem (11) is solved with P=P^{m−1} and the output delivered is denoted as {A^{ m },B^{ m }}. As will be shown in Section 4.2, when P is fixed, problem (11) is decoupled into n subproblems which can be easily solved using the optimal results of n LAP. Note as each LAP can be solved by the Hungarian Algorithm in polynomial time [36], {A^{ m },B^{ m }} will be obtained in polynomial time. Finally, we have R(P^{m−1},A^{m−1},B^{m−1})≤R(P^{m−1},A^{ m },B^{ m }).
During the PA stage of iteration m, problem (11) is solved with A=A^{m−1} and B=B^{m−1}. The output delivered is denoted as P^{ m }. However, when {A,B} is fixed, problem (11) is still nonconvex and hard to solve. Fortunately, by using the algorithm based on SCGP proposed in Section 4.3, this nonconvex problem can be approximated into a series of standard geometric programs (GP), which can be solved by common GP solvers. As will be shown in Section 4.3, the solutions of the approximated GP problems converge to a local optimum, satisfying the KarushKuhnTucker (KKT) conditions of the nonconvex problem. Note as each standard GP can be solved in polynomial time [37], P^{ m } will be obtained in polynomial time after the convergence of the SCGP algorithm. Finally, we have R(P^{m−1},A^{ m },B^{ m })≤R(P^{ m },A^{ m },B^{ m }).
where E 1 and E 2 are due to the effects of the MSSAP stage and the PA stage, respectively. This means that Algorithm 1 yields nondecreasing sum rates with iterations. Moreover, the optimum sum rate is upper bounded due to the total power constraint in each cell. Therefore, the sum rate values will not increase indefinitely with iterations, meaning, that the iterations will eventually converge. Algorithm 1 will stop when the sum rate increase is below a prescribed value ε_{1} or when m reaches a prescribed value M.
Note that, as both the MSSAP stage and the SCGP stage can be solved in polynomial time, problem (11) can be solved in polynomial time. At the end of iteration m, a local optimum of (11) is obtained at the SCGP stage, which is then improved at the MSSAP stage of the next iteration. After that, a better local optimum of (11) can be calculated at the SCGP stage of iteration m+1. Finally, a good local optimum can be obtained after convergence.
3.2 MSSAP optimization
where ${R}_{{n}_{0},\mathit{\text{UB}}}=\sum _{k=1}^{K}\sum _{l=1}^{K}{c}_{{n}_{0}}^{\mathit{\text{kl}}}{Q}_{{n}_{0}}^{\mathit{\text{kl}}}$.
Let us now consider Theorem 1.
Theorem 1.
Theorem 1. An optimal solution of (13) (${\mathbf{A}}_{{n}_{0}}^{\ast}$ and ${\mathbf{B}}_{{n}_{0}}^{\ast}$) can be easily obtained using the optimal solution of (17) (${\mathbf{c}}_{{n}_{0},\mathit{\text{UB}}}^{\ast ,\mathit{\text{kl}}}$). Specifically,${\mathbf{A}}_{{n}_{0}}^{\ast}$ and ${\mathbf{B}}_{{n}_{0}}^{\ast}$ are obtained by assigning for every combination of k and l , all entries in $\left\{{a}_{{\mathit{\text{uvn}}}_{0}}^{\mathit{\text{kl}}}\right\forall u,v\}$ and $\left\{{b}_{{\mathit{\text{un}}}_{0}}^{\mathit{\text{kl}}}\right\forall u\}$ to zero, except for the one with the metric equal to ${Q}_{{n}_{0}}^{\mathit{\text{kl}}}$ to ${\mathbf{c}}_{{n}_{0},\mathit{\text{UB}}}^{\ast ,\mathit{\text{kl}}}$.
Proof 1
Proof 1. Let us view ${R}_{{\mathit{\text{uvn}}}_{0},1}^{\mathit{\text{kl}}}$ and ${R}_{{\mathit{\text{un}}}_{0},2}^{\mathit{\text{kl}}}$ as metric for ${a}_{{\mathit{\text{uvn}}}_{0}}^{\mathit{\text{kl}}}$ and ${b}_{{\mathit{\text{un}}}_{0}}^{\mathit{\text{kl}}}$, respectively. Note that $\forall {c}_{{n}_{0}}^{\mathit{\text{kl}}}=0$, inequality (15) is tightened when all entries of $\left\{{a}_{{\mathit{\text{uvn}}}_{0}}^{\mathit{\text{kl}}}\right\forall u,v\}$ and $\left\{{b}_{{\mathit{\text{un}}}_{0}}^{\mathit{\text{kl}}}\right\forall u\}$ are set to zero. While $\forall {c}_{{n}_{0}}^{\mathit{\text{kl}}}>0$, inequality (15) is tightened when all entries of $\left\{{a}_{{\mathit{\text{uvn}}}_{0}}^{\mathit{\text{kl}}}\right\forall u,v\}$ and $\left\{{b}_{{\mathit{\text{un}}}_{0}}^{\mathit{\text{kl}}}\right\forall u\}$ are set to zero, except that the one with the metric equal to ${Q}_{{n}_{0}}^{\mathit{\text{kl}}}$ is assigned to ${c}_{{n}_{0}}^{\mathit{\text{kl}}}$.
We now denote by $\left\{{\mathbf{C}}_{{n}_{0}}^{\ast}\right\}$ the optimal variables of (17). Thus, $\exists {\mathbf{A}}_{{n}_{0}}^{\ast},{\mathbf{B}}_{{n}_{0}}^{\ast}$, so that ${R}_{{n}_{0}}\left({\mathbf{A}}_{{n}_{0}}^{\ast},{\mathbf{B}}_{{n}_{0}}^{\ast}\right)={R}_{{n}_{0}}^{\mathit{\text{UB}}}\left({\mathbf{C}}_{{n}_{0}}^{\ast}\right)$. Here, ${\mathbf{A}}_{{n}_{0}}^{\ast}$ and ${\mathbf{B}}_{{n}_{0}}^{\ast}$ are obtained by assigning for every combination of k and l, all entries in $\left\{{a}_{{\mathit{\text{uvn}}}_{0}}^{\mathit{\text{kl}}}\right\forall u,v\}$ and $\left\{{b}_{{\mathit{\text{un}}}_{0}}^{\mathit{\text{kl}}}\right\forall u\}$ to zero, except for the one with the metric equal to ${Q}_{{n}_{0}}^{\mathit{\text{kl}}}$ to ${\mathbf{c}}_{{n}_{0},\mathit{\text{UB}}}^{\ast ,\mathit{\text{kl}}}$. It is important to know that $\{{\mathbf{A}}_{{n}_{0}}^{\ast},{\mathbf{B}}_{{n}_{0}}^{\ast},{\mathbf{C}}_{{n}_{0}}^{\ast}\}$ belongs to the feasible set of (13).
We further note that ∀ feasible $\{{\mathbf{A}}_{{n}_{0}},{\mathbf{B}}_{{n}_{0}},{\mathbf{C}}_{{n}_{0}}\},{R}_{{n}_{0}}({\mathbf{A}}_{{n}_{0}},{\mathbf{B}}_{{n}_{0}})\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{R}_{{n}_{0}}^{\mathit{\text{UB}}}\left({\mathbf{C}}_{{n}_{0}}\right)\phantom{\rule{0.3em}{0ex}}\le \phantom{\rule{0.3em}{0ex}}{R}_{{n}_{0}}^{\mathit{\text{UB}}}\left({\mathbf{C}}_{{n}_{0}}^{\ast}\right)$. Recalling ${R}_{{n}_{0}}\phantom{\rule{0.3em}{0ex}}\left({\mathbf{A}}_{{n}_{0}}^{\ast},{\mathbf{B}}_{{n}_{0}}^{\ast}\right)={R}_{{n}_{0}}^{\mathit{\text{UB}}}\left({\mathbf{C}}_{{n}_{0}}^{\ast}\right)$, we obtain the optimal variables of problem (13) as $\{{\mathbf{A}}_{{n}_{0}}^{\ast},{\mathbf{B}}_{{n}_{0}}^{\ast},{\mathbf{C}}_{{n}_{0}}^{\ast}\}$. This concludes the proof of Theorem 1.□
Note that (18) is the same as (17) except that the feasible set of (18) F 1 is a subset of that of (17) F. (18) is actually a LAP. Hence, it can be solved by means of typical LAP algorithms, like the Hungarian Algorithm.
3.3 PA optimization
where variables l, u, and v are all constants. Note that R is a nonconvex function due to the presence of interfering power terms in the denominators of ${\Gamma}_{{\mathrm{d}}_{\mathit{\text{un}}},{t}_{1}}^{k}$, ${\Gamma}_{{\mathrm{d}}_{\mathit{\text{vn}}},{t}_{2}}^{l}$, ${\Gamma}_{{\mathrm{r}}_{n}}^{k}$, and ${\Gamma}_{{\mathrm{d}}_{\mathit{\text{un}},{t}_{2}}}^{l}$. To solve problem (11), we first replace it with an equivalent complementary geometric program (CGP) that is then addressed by means of the SCGP algorithm.
Problem (21) is made of an objective function and bounding constraints which all are ratios of two posynomials, making the problem belong to the class of CGP [38]. More details about CGP can be found in Appendix Appendix. Problem (21) can not be made convex and is NP hard [37]. In order to solve it, the SCGP algorithm is now proposed. The proposed SCGP algorithm approximates the nonconvex problem (21) into a series of standard GPs. Therefore, it belongs to the class of successive convex approximation methods [39]. The SCGP is described in Algorithm 2. Integer t indicates the current iteration number, and a variable with superscript m^{′} denotes the value obtained at the end of iteration m^{′}.
Algorithm 2 SCGP Algorithm
Specifically in the proposed SCGP algorithm, we first set m^{′}=0 and initialize the PA vector P^{0} with P_{ini}. Each iteration contains an SC stage followed by a standard geometric programming (SGP) stage.
 1.Bounding condition: ∀Θ,$\begin{array}{l}{h}_{i}^{n,k}\left(\mathbf{\Theta}\right)\le {\stackrel{~}{h}}_{i}^{n,k}\left(\mathbf{\Theta}\right),\phantom{\rule{2.77626pt}{0ex}}\forall i=1,2,3.\end{array}$
 2.Tightness condition: At the beginning of iteration m ^{′},$\begin{array}{l}++{h}_{i}^{n,k}\left({\mathbf{\Theta}}^{m,{m}^{\prime}1}\right)={\stackrel{~}{h}}_{i}^{n,k}\left({\mathbf{\Theta}}^{m,{m}^{\prime}1}\right),\phantom{\rule{0.3em}{0ex}}\forall i=1,2,3.\end{array}$
 3.Differential condition: At the beginning of iteration m ^{′}, ∀θ∈Θ,$\begin{array}{l}\phantom{\rule{20.0pt}{0ex}}\frac{\partial {h}_{i}^{n,k}\left({\mathbf{\Theta}}^{m,{m}^{\prime}1}\right)}{\mathrm{\partial \theta}}=\phantom{\rule{1em}{0ex}}\mathrm{frac\partial}{\stackrel{~}{h}}_{i}^{n,k}\left({\mathbf{\Theta}}^{m,{m}^{\prime}1}\right)\mathrm{\partial \theta},\phantom{\rule{2.77626pt}{0ex}}\forall i=1,2,3.\phantom{\rule{2em}{0ex}}\end{array}$
Algorithm 3 Condensation Rule
After the SC stage, problem (21) is formulated by a standard GP. The SGP stage amounts to solving the GP by means of a standard GP solver, e.g., the software provided at [40]. The output provided by this stage corresponds to ${\mathbf{P}}^{{m}^{\prime}}$.
Thanks to the three conditions fulfilled during the SC stage, our proposed SCGP algorithm is a general inner approximation algorithm [39], which will converge to a local optimum satisfying the KKT conditions of problem (21) according to the corollary 1 of [39]. In practice, the iterations of the SCGP algorithm will be stopped when $\left\right{\mathbf{P}}^{{m}^{\prime}}{\mathbf{P}}^{{m}^{\prime}1}\left\right\le {\epsilon}_{2}$ or when m^{′} exceeds a prescribed value M^{′}.
4 Numerical experiments
In order to illustrate the benefits of using SP and the highefficiency protocol, respectively, we now compare the proposed protocol, named P1, with two other protocols, named P2 and P3, respectively. Specifically, P2 is the same as P1 except that subcarrier k in TS1 is always paired with subcarrier l in TS2. P3 is also the same as P1 except that all sources keep silent during TS2. For both P2 and P3, Algorithm 1 can easily be applied by adding additional constraints. Both UPA and best power allocation (BPA) are used to initialize the proposed algorithm. Here, BPA is the PA method used in [7], where CCI is set to 0.
The impact of relay deployment, on average sum rate, is first studied and proper relay positions are selected for further experiments. Then, RA and the sum rate results are presented for one particular channel realization. The convergence of the proposed RA algorithm is illustrated for all three protocols. Finally, the results are provided and discussed for the performance of those protocols averaged over many channel realizations.
4.1 System setup
The channel impulse response (CIR) of each link is drawn randomly from an eighttap delay line model, where each tap i has a circular complex gaussian distribution with zero mean and variance ${\sigma}_{i}^{2}$. We further assume $\frac{{\sigma}_{i}^{2}}{{\sigma}_{i+1}^{2}}={e}^{3}$, meaning that the tap power decreases exponentially with a coefficient of 3. Moreover, $\sum _{i}{\sigma}_{i}^{2}={d}^{2.5}$, meaning that the received power decreases exponentially with distance d and the propagation exponent equals to 2.5. Finally, the CFR of each link is computed from its CIR using the Kpoint FFT.
4.2 Impact of relay deployment on average sum rate
We have chosen a set of values for D sr1 and D sr2, ranging from 100 to 900 m. For each possible combination of D sr1 and D sr2, 100 sets of random realizations of channels are generated and the average optimized sum rate is computed by using the proposed algorithm with the UPA initialization method. We then compare the obtained average sum rates of all combinations to illustrate how the deployment of relays influences the system performance.
In the following simulations, we set D sr1=300 m and D sr2=300 m.
4.3 Results for a random realization of channels
4.3.1 Convergence of the proposed RA algorithm
In order to illustrate the convergence of the proposed RA algorithm, we plot the sum rates calculated during the iterations with the three protocols and two initialization methods when P_{ T }=10 dBm and P_{ T }=40 dBm.
Initialized by the UPA method, the optimized sum rate of P1 is much higher then that of P3, both when P_{ T }=10 dBm and when P_{ T }=40 dBm. Thus, the benefit of using the HSE protocol is very high. When P_{ T }=10 dBm, the optimized sum rate of P1 is slightly higher than that of P2. However, when P_{ T }=40 dBm, P1 and P2 leads to the same optimized sum rate. Thus, the benefit of using subcarrier pairing is limited. This can be understood as most of the subcarriers are in direct mode, for which pairing does not have any impact.
4.3.2 Benefits illustration of the HSE protocol
Finally, according to Figure 7c,d, no resources are allocated to the remote destinations (d_{11} in cell 1 and d_{21} in cell 2) for both protocols. Actually, this confirms the unfair nature of sum rate maximization. Therefore, fairness will be considered in our future work.
4.4 Results averaged over channel distribution
In order to illustrate the average performance of our proposed RA algorithm, 100 random realizations of channels are generated. Three benchmark algorithms (BA1, BA2, and BA3) are introduced. Here, BA1 applies the RA algorithm proposed in [7], where the CCI is always set to 0. BA2 uses UPA together with our algorithm proposed for MSSAP. BA3 also uses UPA but with the destinations randomly selected for each subcarrier. We denote the distance between two parallel cells as Dcell and increase it from 200 to 5,200 m.
Note that, in low SNR scenario, when P_{ T }=10 dbm, the sum rates are similar regardless of the distance between two cells. This is because the CCI is already low even when two cells are close to each other. However, in high SNR scenario, when P_{ T }=40 dbm, the sum rates of all the algorithms increase as Dcell increases. This is because the CCI is quite high when two cells are close to each other, and it decreases as they move away from each other. Note that when D c e l l=5200 m, meaning two cells are quite far from each other, BA 1 performs similarly to our proposed algorithms. This is because BA 1 is nearoptimal when the CCI is small.
5 Conclusion
We have considered a multicell OFDMA downlink system where the BS transmissions are aided by the cell specific RS. The adopted HSE protocol enables SP. Assuming a central controller, we have formulated the sum rate maximization problem under a percell total power constraint, for which an iterative RA algorithm has been proposed to find a local optimum in polynomial time. Through numerical experiments, the convergence and effectiveness of the proposed algorithm, the impact of relay position and total power on the system performance as well as the benefits of using SP and the HSE protocol has been illustrated. In the future, combining schemes and multirelay cases will be considered.
Appendix
Complementary GP
where the A, B, C, and D are all posynomials and some of them may be absent.
Actually, a CGP is an extension of a standard GP. As a matter of fact, when f_{ i }(x),i=0,1,…,I, are posynomials and h_{ j }(x),j=1,2,…,J, are monomials, problem (43) becomes a standard GP. For GPs, the global optimum can always be found in an efficient manner. For CGPs, only a local optimum can be obtained efficiently. But prohibitive computations are required to find the global optimum [41].
Declarations
Acknowledgements
The authors would like to thank Prof. S. Boyd for sharing the GGPLAB toolbox online, China Scholarship Council (CSC) for funding this work and BELSPO for the support of the IAP BESTCOM network. The work of Tao Wang is supported by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Innovation Program of Shanghai Municipal Education Commission (14ZZ096), Innovation Fund of Shanghai University and SRFDP (20133108120015).
Authors’ Affiliations
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