Research  Open  Published:
Decoupled beamforming techniques for distributed MIMO twoway multirelay networks with imperfect CSI
EURASIP Journal on Wireless Communications and Networkingvolume 2018, Article number: 126 (2018)
Abstract
This paper investigates decoupled beamforming techniques for the distributed multiinput multioutput (MIMO) twoway relay networks (TWRN) with imperfect channel state informations (CSIs). The objective of this paper is to maximize the weighted sum rate (SR) with semiinfinite relay power constraints. Since the objective problem is difficult to be solved directly, considering the high signaltoresidualinterferenceplusnoise ratio (SRINR) and employing the CauchySchwarz inequality and Slemma, the problem can be approximately converted into a source beamforming decoupled one. In addition, with the optimal relay beamforming design and maximum ratio combining (MRC) at the receiver, the MIMO channels are decoupled into parallel singleinput singleoutput (SISO) channels. By this way, the suboptimal relay beamforming matrix can be efficiently obtained with minimal relay power constraint. Specifically, the semiinfinite constraints can be reformulated into a linear matrix inequality (LMI), which can be efficiently solved by using an alternating optimization algorithm. Numerical results demonstrate that our proposed decoupled beamforming scheme outperforms the existing works in terms of the SR and the computational complexity with satisfactory convergence.
Introduction
Recently, cooperative and multipleinput multipleoutput (MIMO) systems [1–4] have been widely considered as a candidate for the fifth generation (5G) wireless communication due to their transmission reliability, where multiple users (MU) are simultaneously served at the same frequency band by the base station (BS) equipped with number of antennas. Cooperative twoway relay network (TWRN) technology has attracted significant interests due to the superior spectral efficiency. Several relaying schemes haven proposed, i.e., amplifyandforward (AF) [5–8], decodeandforward (DF) [9, 10], and denoiseandforward (DNF) [11, 12]. Specifically, the cooperative relaying beamforming design has been studied in [13–16], where the authors showed the optimal solutions of the source and/or relay beamforming matrices. Moreover, the spatially correlated fading channels are considered in [17, 18], which is a more practical assumption but makes the training problem more challenging. The authors of [17] presented training designs for estimation of spatially correlated MIMO AF twoway multirelay channels, where an optimal training structure is initially derived to minimize total meansquareerror (MSE) of the channel estimation. Based on [17], an optimal training scheme is efficiently designed to minimize the total MSE of the channel estimation under the transmit power constraints at the source nodes and at the relay in [18]. In addition, the authors considered a TWRN with an AF protocol over either two, three, or four time slots [19].
Considering inaccurate channel estimation and feedback delay, the perfect channel state information (CSI), which is proposed in the above works [5–18], is usually hard to obtain in practice. By taking account into the channel uncertainties, the imperfect CSI scenario has been studied in [20–25]. In [20], joint relay and jammer selection and power control for physical layer security issues in twoway relay networks are studied to maximize the secrecy capacity of the network. In [21], the authors addressed the robust multipleantenna relay design problem in TWRN and provided the robust multipleantenna relay design based on the channel estimates. A low complexity processing matrices are presented for a multipair twoway massive MIMO AF fullduplex relay system [22]. Specifically, in [23], the problem of optimal beamforming and power allocation for an AFbased TWRN is studied in the presence of interference and CSI uncertainty, where two different approaches, namely the total power minimization method and the signaltointerferenceplusnoiseratio (SINR) balancing technique, are proposed. Particularly, the robust twoway relay precoder design for a cognitive radio network is investigated in [24], where two different types of CSI errors with corresponding robust designs are proposed. It is worth noting, in [25], the joint optimal robust beamforming designs in multipair twoway nonregenerative relaying systems, where an insightful closed form solution is obtained that is not only robust to the imperfect CSI but also adjustable to various CSI circumstances at the users and the relay.
However, most existing works for the robust twoway relaying networks were unavoidable to focus on the joint design of source and relay beamforming matrices that may lead to a higher computational complexity. In addition, as shown in [26], for the multiple relays scheme, the performance of the capacity outperforms that of the single relay one. On the other hand, multiple relays scheme is more practical and challenging for the wireless communication scenarios. Moreover, the weighed scheme can be regarded as one kind of the resource allocations, which is general. Therefore, in order to reduce the complexity, our study aims to design a decoupled beamforming scheme for the distributed MIMO twoway relaying scheme with imperfect and reciprocal CSI. By converting the weighted sumrate (SR) maximization problem into a source beamforming decoupled one whose target is to maximize the signaltoresidualinterferenceplusnoise ratio (SRINR) by means of approximations, the Slemma, and the CauchySchwarz inequality, the objective problem can be efficiently solved by an alternating algorithm. With the decoupled source beamforming design and the maximum ratio combining (MRC) at the receiver, the optimal relay beamforming can be efficiently obtained which leads to that the MIMO channels can be decoupled into parallel singleinput singleoutput (SISO) channels. Numerical results are presented to corroborate that the performance of the decoupled scheme is improved and to compare it to the existing works with satisfactory convergence.
The rest of this paper is organized as follows. Section 2 describes the system model of the TWRN and objective problems. In Section 3, the proposed decoupled beamforming and optimal relay beamforming designs are investigated. Numerical results are presented to show the excellent performance of our proposed scheme in Section 4. Section 5 concludes this paper.
Notations: For an M×N matrix A, E(A), ∥A∥, A^{T}, tr(A), vec(A), and A^{†} denote the statistical expectation, Frobenius norm, transpose, trace, vectorization, and Hermitian transpose of A, respectively. I_{ N } represents an N×N identity matrix.
System model and optimal relay beamforming design
System model
Consider a simple twoway relay channels (TWRC) consisting of two user equipments, i.e., UE_{1} and UE_{2}, and L relay nodes {R_{1},R_{2},…,R_{ L }} as shown in Fig. 1. Each source and relay nodes are equipped with M and N antennas, respectively. Assume that all nodes operate in halfduplex mode and the direct link between two UEs does not exist. By taking the estimation error into account, the CSI is assumed to be reciprocal and partially known at each node. Denote the estimated channels from the UE_{ t } to the relay R_{ i }, from the relay R_{ i } to the UE_{ t } by $\widetilde {\mathbf {F}}_{i,t}\in {\mathbb {C}^{N\times {M}}}$ and $\widetilde {\mathbf {G}}_{t,i}\in {\mathbb {C}^{M\times {N}}}$, for t∈{1,2}, respectively, as follows:
and
with $\left \{\Delta _{\mathbf {F}_{i,t}},~\Delta _{\mathbf {G}_{t,i}}\right \}$ as the channel uncertainties. For a better readability and distinction, in this paper, we simply consider two channel matrices (F_{ i,t }, G_{ t,i }) with $\mathbf {F}_{i,t}=\mathbf {G}^{T}_{t,i}$ and $\mathbf {F}_{i,\overline {t}}=\mathbf {G}^{T}_{\overline {t},i}$. Since that, the channel uncertainty is provided by the inaccurate channel estimation, feedback delay, and so on. Therefore, the norm bounded error (NBE) of the channel uncertainty is normally considered as a small slack value [27], which is a reasonable assumption in a practical system. Therefore, in this paper, the channel uncertainties are considered as a NBE model for its simplicity:
where 0≤{α_{ i },β_{ i }}≪1. By this way, a worstcase design methodology can be adopted, resulting in the proposed system design which maximizes performance for the worst possible CSI realization as defined by the NBE.
It is assumed that two UEs exchange the message using two consecutive time slots. At the first time slot, the information data x_{ t }=[x_{1,t},x_{2,t},…,x_{ m,t }]^{T}, with $\mathrm {E}\left (\mathbf {x}_{t}\mathbf {x}_{t}^{\dag }\right)=\mathbf {I}_{M}$, for i∈{1,…,M}, is linearly processed by a precoding matrix $\mathbf {B}_{t}\in {\mathbb {C}^{M\times {M}}}$ with the power constraint ∥B_{ t }∥^{2}≤P_{ t } and then transmitted to the relay nodes. Thus, the received signal at R_{ i } can be expressed as
where $\mathbf {n}_{R_{i}}\sim {\mathcal {CN}\left (0,\sigma ^{2}_{R_{i}}\mathbf {I}_{N}\right)}$ denotes the additive white Gaussian noise (AWGN) vector with zero mean and variance $\sigma ^{2}_{R_{i}}\mathbf {I}_{N}$.
At the second time slot, the relay node R_{ i } generates the transmit signal $\phantom {\dot {i}\!}\mathbf {W}_{i}\mathbf {y}_{R_{i}}$, for ∥W_{ i }∥^{2}≤ω_{ i }, with the N×N beamforming matrix W_{ i } and forwards to the UE_{1} and UE_{2} which results in $\phantom {\dot {i}\!}\mathbf {x}_{R_{i}}=\mathbf {W}_{i}\mathbf {y}_{R_{i}}$ with the average transmit power as
where we have assumed that the elements of the data stream vector x_{ t } transmitted by each user are independent, i.e., $\mathbf {E}\left \{\mathbf {x}_{t}\mathbf {x}_{t}^{\dag }\right \}=\mathbf {I}_{M}$. Thus, the received signal at source node S_{ t }, for t∈{1,2}, becomes
where n_{ t } denotes the noise vector at the source node S_{ t } with zero mean and variance $\sigma _{S_{t}}^{2}\mathbf {I}_{N}$.
In practical systems, however, achieving perfect SIC is a hard problem due to channel estimation errors. Thus, the selfinterference cannot be completely eliminated which leads to the residual selfinterference cannot be ignorable. In order to subside the adverse effect of the interference, with the estimated channel coefficients, the final observation of the residual selfinterference at the $\text {UE}_{\overline {t}}$ can be obtained as
It is clear that the covariance of the term $\mathcal {K}$ can be obtained as $\left \\Delta _{\mathbf {G}_{\overline {t},i}}\mathbf {W}_{i}\Delta _{\mathbf {F}_{i,\overline {t}}}\right \^{2}$. Since that, the norm bounded errors of $\left \{\Delta _{\mathbf {G}_{\overline {t},i}}, \Delta _{\mathbf {F}_{i,\overline {t}}}\right \}$ are assumed to be small slack values, which is a reasonable assumption in a practical system, i.e., 0≤{α_{ i },β_{ i }}≪1. Using ∥AB∥≤∥A∥∥B∥, we have $\left \\Delta _{\mathbf {G}_{\overline {t},i}}\mathbf {W}_{i}\Delta _{\mathbf {F}_{i,\overline {t}}}\right \^{2}\leq \left \{\alpha ^{4}_{i}\left \\mathbf {W}_{i}\right \^{2},\beta ^{4}_{i}\left \\mathbf {W}_{i}\right \^{2}\right \}$, which is very close to 0. Therefore, in this paper, the negligible term involving only CSI uncertainty is omitted, which is practical and relatively easy to achieve.
Since the sum of the individual rate in a MIMO system is difficult to be obtained, therefore, we turn to design the relay beamforming to recast the optimization problem into the one of achievable data rate for each data stream.
Optimal relay beamforming design
Suppose the singular value decomposition (SVD) of $\widetilde {\mathbf {G}}_{\overline {t},i}$ and $\widetilde {\mathbf {F}}_{i,t}$ as follows
and
where $\mathbf {V}_{\overline {t},i}\in {\mathbb {C}^{N\times {N}}}$, $\mathbf {U}_{i,t}\in {\mathbb {C}^{N\times {N}}}$, $\mathbf {\Pi }_{\overline {t},i}\in {\mathbb {C}^{M\times {M}}}$, and $\mathbf {\Omega }_{i,t}\in {\mathbb {C}^{M\times {M}}}$ are unitary matrices as well as $\mathbf {\Sigma }_{\overline {t},i}={\left [\begin {array}{cc} \left [\mathbf {G}^{\diamondsuit }_{\overline {t},i}\right ]_{M\times M} &\mathbf {0}_{M\times (NM)}\end {array}\right ]} $ and $\mathbf {\Gamma }_{i,t}={\left [\begin {array}{c} \left [\mathbf {F}^{\diamondsuit }_{i,t}\right ]_{M\times M}\\ \mathbf {0}_{(NM)\times M}\end {array}\right ]}. $
Theorem 1
Using the SVDs in (7) and (8), the optimal relay beamforming matrix as the solution to the objective problem T_{3} can be obtained as
where $\mathbf {C}_{i}\in {\mathbb {C}^{N\times {N}}}$ is a matrix to be determined.
Proof
The proof is similar to [8, 28]. □
Without loss of generality, C_{ i } can be further partitioned as follows:
On the one hand, since the channels from different relays, the estimated channel, and its corresponding channel uncertainties are independent, denoting $\widetilde {\chi }_{\overline {t}}=\sum _{i=1}^{L} \left (\widetilde {\mathbf {G}}_{\overline {t},i}\mathbf {W}_{i}\Delta _{\mathbf {F}_{i,\overline {t}}} + \Delta _{\mathbf {G}_{\overline {t},i}}\mathbf {W}_{i}\widetilde {\mathbf {F}}_{i,\overline {t}}\right)$, we have
where ϱ_{ t,i }=α_{ i } for t=1 as long as ϱ_{ t,i }=β_{ i } for t=2. On the other hand, since the channel uncertainty $\phantom {\dot {i}\!}\Delta _{\mathbf {F}_{i,t}}$ can be further partitioned following
upon substituting (9), (10), and (12) into the relay power constraint (4), we have
where $\mathbf {F}^{\sharp }_{i,t}\,=\,\mathbf {F}^{\diamondsuit }_{i,t}\,+\,\Delta ^{\diamondsuit }_{\mathbf {F}_{i,t}} \text {and}\,\mathbf {\Phi }_{i}=\sum _{t=1}^{2}\Delta _{\mathbf {F}_{i,t}}^{\flat }\mathbf {B}_{t}\left (\Delta _{\mathbf {F}_{i,t}}^{\flat }\mathbf {B}_{t}\right)^{\dag } +\sigma ^{2}_{R_{i}}\mathbf {I}_{NM}$. From (11) and (13), it is clear that, for any feasible C_{ i } with {X,Y,Z}≠0, one can always find
which can achieve the smaller relay and interference power constraints. By this way, the optimal expression of W_{ i } with minimal relay and interference power constraints can be represented as
Objective problem
It is easy to see that the actual equivalent channel from UE_{ t } to $\text {UE}_{\overline {t}}$ is with dimension M×M. Therefore, by employing the MRC at $\text {UE}_{\overline {t}}$, the equivalent received signals from each channel can be linearly added together. With the condition $\mathrm {E}\left (\mathbf {x}_{t}\mathbf {x}_{t}^{\dag }\right)=\mathbf {I}_{M}$, the objective weighted sumrate (SR) problem for the total system can be recast into the problem achievable data rate for each data stream, i.e., the MIMO channels are decoupled into M parallel SISO channels. With these observations, from (4), (6), and (15) similar to [29–31], the received SRINR over the mth channel (1≤m≤M) for $\text {UE}_{\overline {t}}$ can be simply written as
where $\overline {t}=2$ for t=1 while $\overline {t}=1$ for t=2. Based on which, we have the weighted SR maximization problem as
where $\mathbf {H}_{t}=\sum _{i=1}^{L}\mathbf {G}_{\overline {t},i}\mathbf {W}_{i}\mathbf {F}_{i,t}$, $N_{t}=\sigma ^{2}_{R_{i}}\left \\sum _{i=1}^{L}\mathbf {G}_{\overline {t},i}\mathbf {W}_{i}\right \^{2}+\sigma _{S_{t}}^{2}$, and $P_{R_{i}}$ is the maximum allocated power to the relay. In addition, the weight term λ_{ t } is determined depending on the required quality of service (QoS).
Decoupled source beamforming designs
Since the problem T_{1} is neither convex nor concave and the optimal B_{ t } is intractable, it is difficult to obtain the globally optimal solution. In this section, we propose a decoupled way to obtain the suboptimal solution of the worstcase weighted SR.
Considering the high signaltointerferenceplusnoise ratio (SINR), the objective weighted SR R_{sum} in (17) can be approximately reexpressed as
It is interesting to see that, since log(·) is a monotonic function and assuming the nominal CSI and the channel errors are independent, with fixed λ_{ t }, from (31), the objective problem T_{1} can be equivalently converted into the following problem as
where $q_{t}(\mathbf {B}_{t})=\frac {\left (\left \\mathbf {H}_{t}\mathbf {B}_{t}\right \^{2}\right)^{\lambda _{\overline {t}}}}{\left (\\widetilde {\chi }_{t}\mathbf {B}_{t}\^{2}+N_{t}\right)^{\lambda _{t}}}$ and γ_{ t } is a slack value, where, similar to [29, 32–34], by introducing a slack value a, the optimization problem $\left \{\text {max}~\mathcal {A}\right \}$ can be equivalently rewrite as $\left \{\text {max}~a\right \}~s.t. \mathcal {A}\geq a$. In order to relax N_{ t } in q_{ t }(B_{ t }), after introducing the slack value ς_{ t }, we have
which leads to N_{ t }≥ς_{ t }, where ϱ_{ t,i }=α_{ i } for t=1 as long as ϱ_{ t,i }=β_{ i } for t=2, and τ_{ t } is a slack value. With this observation and employing the CauchySchwarz inequality, we have
which decouples the source beamforming matrix B_{ t }. Denoting that $\frac {\left (\left \\mathbf {H}_{t}\right \^{2}\right)^{\lambda _{\overline {t}}}}{\left (\\widetilde {\chi }_{t}\^{2}+\frac {\varsigma _{t}}{P_{t}}\right)^{\lambda _{t}}} $ and $\widehat {\gamma }_{t}=\gamma _{t}P^{2(\lambda _{t}\lambda _{\overline {t}})}_{t}$, for the objective problem T_{2}, the SRINR constraint can be equivalently expressed as $\widehat {q}_{t}(\mathbf {B}_{t})\geq \widehat {\gamma }_{t}$. It is clear that, after decoupling the source beamdforming B_{ t } in the objective problem T_{2}, B_{ t } only exists at the relay power constraint (4). By considering $\mathbf {\widehat {B}}_{t}=\text {vec}(\widetilde {\mathbf {B}}_{t})\text {vec}(\widetilde {\mathbf {B}}_{t})^{\dag }$ with $\text {vec}(\widetilde {\mathbf {B}}_{t})=\left [\text {vec}(\mathbf {B}_{t})^{T}~~ \mathbf {0}_{1\times M(NM)}\right ]^{T}$, similar to Appendix 2 in [8], we have the following linear matrix inequality (LMI) of the individual relay power constraints:
where $\aleph _{t}=\sum _{t=1}^{2}\text {vec}{\left (\widetilde {\mathbf {F}}_{i,t}\right)}^{\dag }\mathbf {\widehat {B}}_{t}\text {vec}\left (\widetilde {\mathbf {F}}_{i,t}\right)\lambda _{1}\omega _{i}\alpha _{i}^{2}\lambda _{2}\omega _{i}\beta _{i}^{2}\widehat {P}_{R_{i}}$ and $\widetilde {\mathbf {Q}}_{t}=\mathbf {W}_{i}\widetilde {\mathbf {F}}_{i,t}$. Based on which, the optimization problem T_{2} can be approximately converted into the following version as
It is easy to see that the problem T_{3} is still hard to obtain the optimal solution straightforwardly, therefore, we try to convert it into LMI version. It is clear that the denominator of $\widehat {q}_{t}(\mathbf {B}_{t})$ can be approximately reexpressed as
where τ_{ t } is obtained from
Thus, letting $\varepsilon _{t}=\sqrt [\lambda _{\overline {t}}]{\widehat {\gamma }_{t}\left (\tau _{t}+\frac {\varsigma _{t}}{P_{t}}\right)^{\lambda _{t}}}$, the SRINR constraint becomes ∥H_{ t }∥^{2}≥ε_{ t } which has the LMI form as follows:
Since
with
and
substituting (25) back into (24) and using the SLemma [35], the LMI version of the SRINR constraint can be finally denoted as
where $\mathbf {q}_{2}\,=\,\left [1,\mathbf {0}_{1\times M^{2}}\right ]$ and ϕ_{ j }≥0 are the slack variables for j=1,…,2l. $\mathbf {\Xi }_{i}\,=\,\left [\mathbf {0}_{M^{2}\times 1},\mathbf {M}_{\mathbf {G}_{\overline {t},i}}^{H}\right ]$, and $\xi _{i,t}=\left \\Delta _{\mathbf {G}_{\overline {t},i}}\right \$ for i=1,…,l, as well as $\mathbf {\Xi }_{i}\,=\,\left [\mathbf {0}_{M^{2}\times 1},\mathbf {M}_{\mathbf {F}_{i,t}}^{H}\right ]$, and $\xi _{i,t}=\left \\Delta _{\mathbf {F}_{i,\overline {t}}}\right \$ for i=l+1,…,2l, with
Thus, optimization problem T_{3} is recast as
Since the objective problem T_{3} is a biconvex semidefinite program (SDP) which can be efficiently solved by the following alternating algorithm by using CVX [34]:

1. Initialize: ξ, N_{ max }, $\mathbf {B}_{t}^{(n)}$ and $\mathbf {W}^{(n)}_{i}$, set n=0;

2. Repeat: For n=0 to N_{ max }

1: with fixed $\mathbf {B}_{t}^{(n1)}$ update $\mathbf {W}^{(n)}_{i}$ by solving T_{4};

2: for given $\mathbf {W}^{(n)}_{i}$ update $\mathbf {B}^{(n)}_{t}$ by solving T_{4};

3: If $\prod _{t=1}^{2}\widehat {\gamma }^{(n)}_{t}\prod _{t=1}^{2}\widehat {\gamma }_{t}^{(n1)}\leq \xi $, break;

By initializing the small ξ and setting the limitation of the number of iterations N_{ max }, the suboptimal solution can be obtained when $\prod _{t=1}^{2}\widehat {\gamma }^{(n)}_{t}\prod _{t=1}^{2}\widehat {\gamma }_{t}^{(n1)}\leq \xi $.
Numerical results
In this section, we examine the performance of the proposed decoupled scheme in terms of the average weighted SR and the convergent performance compared with the case with perfect CSI, the maximalratio transmission (MRT) and the SISRINR scheme in [8]. In all simulations, spatially uncorrelated Rayleigh fading channels with unit variance are assumed. The number of relay nodes is set to be L=3 with M=N=2, and the noise variance $\sigma _{R_{i}}^{2}$ and $\sigma _{S_{t}}^{2}$ are equally given as σ^{2}=1. We further set up that, in the proposed alternating algorithm, N_{max}=400, ξ=10^{−3}, and the relay beamforming matrices are initialed at random, respectively. In the following figures, we use “Optimal,” “Proposed perfect,” “Proposed α_{ i }/ β_{ i },” and “SNR” to denote the performance upper bound, proposed scheme with perfect SCI, proposed scheme with NBEs of α_{ i }/ β_{ i }, and transmit SNR, respectively. Moreover, the performance upper bound “Optimal” is obtained by using the exhaustive search as a benchmark.
Figure 2 depicts the weighted SR performance versus transmit SNR in two considered cases: (1)λ_{1}=6,λ_{2}=1; (2)λ_{1}=3,λ_{2}=1, which compares to the optimal, perfect CSI and MRT schemes, respectively, with α_{ i }=β_{ i }=0.01 and α_{ i }=β_{ i }=0.1. In addition, it is worth noting that the perfect one is obtained with the NBE as α_{ i }=β_{ i }=0 while the robust solutions are obtained by using the proposed alternating algorithm. It is clear that our proposed scheme has a relative advantage over the MRT maximization scheme and close to the optimal and perfect ones, especially for the smaller NBE case. In addition, the performance gain grows as the ratio between λ_{1} and λ_{2} increases. This is reasonable, since considering the high transmit SNR, (31) is very close to the original one which has the the negligible performance lose.
Figure 3 compares the convergence performance of the weighted SR for the proposed method with the SISRINR in [8], where we have fixed $\alpha ^{2}_{i}=\beta ^{2}_{i}=0.01$. Two cases are considered: (1) λ_{1}=6,λ_{2}=1 with SNR=30 dB and (2) λ_{1}=3,λ_{2}=1 with SNR=20 dB. Results reveal that our proposal has an obvious satisfactory convergence over the SISINR scheme. This is because, for our proposed method, the weighted SR problem is approximately converted into a source beamforming decoupled one which significantly reduces the computational complexity. Specifically, by employing the CauchySchwarz inequality and Slemma, the optimal relay beamforming W_{ i } is efficiently relaxed in q_{ t }(B_{ t }) which also simplifies the corresponding algorithm. In addition, it is observed that, for the higher transmit SNR, the convergence rate of our proposed scheme overwhelms the SISRINR one and both these two schemes provide the optimal solution with more iterations.
Figure 4 exhibits the average SR performance for the proposed method with the optimal scheme versus the transmission number L with fixed α_{ i }=β_{ i }=0.01 and λ_{1}=λ_{2}=1. Two cases are considered: (1) SNR=30 dB and (2) SNR=25 dB. It is clear that, from Fig. 3, it is easy to see that the performances of our proposed scheme have a negligible performance loss compared to the optimal scheme, especially for the smaller number of the relay, which supports the practical utility of our design. Remarkably, the performance gap between our proposed scheme and the optimal one is smaller for a lower SNR case.
Denoting d_{1,i} and d_{2,i} as the distance between {UE_{1}, UE_{2}} and the relay R_{ i }, respectively, the average transmit SNRs, which take into account the relay positions, can be written as
where $\overline {\zeta }_{{1,i}}=\frac {P_{1}}{\sigma ^{2}_{R_{i}}}$, $\overline {\zeta }_{{2,i}}=\frac {P_{2}}{\sigma ^{2}_{R_{i}}}$ and δ is the path loss exponent. Figure 5 depicts the maximum weighted SR of our proposed decoupled beamforming design for different values of d_{1,i} with fixed λ_{1}=λ_{2}=1, δ=3 in two considered system setups: (1) case 1: $\overline {\zeta }_{1}=20~\text {dB},~ \overline {\zeta }_{2}=15~\text {dB}$; (2) case 2: $\overline {\zeta }_{1}=20~\text {dB},~ \overline {\zeta }_{2}=25~\text {dB}$. For simplicity, we further assume that d_{1,1}=d_{1,2}=d_{1,3}=d_{1} and d_{2,1}=d_{2,2}=d_{2,3}=d_{2} with d_{1}+d_{2}=1. One can see that the achieved SR for case 2 shows a higher performance than case 1 due to a lager source power considered for UE_{2}. Moreover, the optimal relay positions occur when d_{1} is close to 0.1 for case 1 and d_{1}→0.95 for case 2. It is reasonable since that the power at UE_{1} is greater than the power at UE_{2} for case 1, while an opposite consideration is performed in case 2.
Conclusions
In this paper, we investigated the decoupled beamforming techniques in the distributed twoway multirelay networks with imperfect and reciprocal CSI. In order to maximize the weighted SR, the objective problem is first converted into a decoupled one by employing the CauchySchwarz inequality and Slemma, and then, the optimal relay beamforming design is also investigated with the LMI versions of the semiinfinite constraints. Based on which, the objective problem can be efficiently solved by the proposed alternating algorithm. By means of the numerical results, it has been shown that our proposed scheme not only has the advantage in the term of the SR but also is with the satisfactory convergence compared to the existing works.
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Acknowledgements
The authors would gratefully acknowledge the grants from the National Natural Science Foundation of China (61371113 and 61401241), Nantong UniversityNantong Joint Research Center for Intelligent Information Technology (KFKT2016B04 and KFKT2017B02), the Science and Technology Program of Nantong (GY22017013), and the Brain Korea BK21 plus.
Finally, we would like to thank the Editor for your constructive remarks and careful reading of our paper, which were essential in improving the overall presentation of the paper.
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WD, JC, and GZ conceived and designed the study. WD and XZ performed the simulations. WD and GZ wrote the paper. LJ, WW, JC, and GZ reviewed and edited the manuscript. All authors read and approved the final manuscript.
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Correspondence to Guoan Zhang.
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Keywords
 Twoway relay network
 MIMO
 Amplifyandforward
 Beamforming design
 LMI
 Imperfect CSI