 Research
 Open Access
Decoupled beamforming techniques for distributed MIMO twoway multirelay networks with imperfect CSI
 Wei Duan^{1, 2},
 Xiaojun Zhu^{1, 4},
 Li Jin^{1},
 Wei Wang^{1, 4},
 Guoan Zhang^{1}Email author and
 Jeaho Choi^{3}
https://doi.org/10.1186/s1363801811261
© The Author(s) 2018
 Received: 3 August 2017
 Accepted: 24 April 2018
 Published: 18 May 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.
Keywords
 Twoway relay network
 MIMO
 Amplifyandforward
 Beamforming design
 LMI
 Imperfect CSI
1 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.
2 System model and optimal relay beamforming design
2.1 System model
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.
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 \(\sigma ^{2}_{R_{i}}\mathbf {I}_{N}\).
where n_{ t } denotes the noise vector at the source node S_{ t } with zero mean and variance \(\sigma _{S_{t}}^{2}\mathbf {I}_{N}\).
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.
2.2 Optimal relay beamforming design
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
where \(\mathbf {C}_{i}\in {\mathbb {C}^{N\times {N}}}\) is a matrix to be determined.
2.3 Objective problem
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).
3 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.

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 \(\prod _{t=1}^{2}\widehat {\gamma }^{(n)}_{t}\prod _{t=1}^{2}\widehat {\gamma }_{t}^{(n1)}\leq \xi \).
4 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.
5 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.
Declarations
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.
Authors’ contributions
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.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 S Han, C Yang, M Bengtsson, User scheduling for cooperative base station transmission exploiting channel asymmetry. IEEE Trans. Commun. 61(4), 1426–1435 (2013).View ArticleGoogle Scholar
 H Lin, F Gao, S Jin, G Ye Li, A new view of multiuser hybrid massive MIMO: nonorthogonal angle division multiple access. IEEE J. Sel. Areas Commun. 35(10), 2268–2280 (2017).View ArticleGoogle Scholar
 H Xie, F Gao, S Jin. An overview of lowrank channel estimation for massive MIMO systems.IEEE Access.4(99), 7313–7321 (2016).Google Scholar
 H Xie, F Gao, S Zhang, S Jin, A unified transmission strategy for TDD/FDD massive MIMO systems with spatial basis expansion model. IEEE Trans. Veh. Technol. 66(4), 3170–3184 (2017).View ArticleGoogle Scholar
 J Joung, J Choi, Linear precoder design for an AF twoway MIMO relay node with no source node precoding. IEEE Trans. Veh. Technol. 66(11), 10526–10531 (2017). to appear.View ArticleGoogle Scholar
 R Budhiraja, B Ramamurthi, Transceiver design for nonconcurrent twoway MIMO AF relaying with QoS guarantees. IEEE Trans. Veh. Technol. 65(12), 9651–9661 (2016).View ArticleGoogle Scholar
 Y Dai, X Dong, Power allocation for multipair massive MIMO twoway AF relaying with linear processing. IEEE Trans. Wireless Commun. 15(9), 5932–5946 (2016).View ArticleGoogle Scholar
 W Duan, M Wen, X Jiang, Y Yan, M Lee, Sumrate maximization and robust beamforming design for MIMO twoway relay networks with reciprocal and imperfect CSI. EURASIP J. Commun. and Network. (2016). Atical ID: 157.Google Scholar
 Q Cui, T Yuan, X Tao, AA Dowhuszko, R Jantti, Energy efficiency analysis of twoway DF relay system with nonideal power amplifiers. IEEE Commun. Lett. 18(7), 1254–1257 (2014).View ArticleGoogle Scholar
 J Gao, SA Vorobyov, H Jiang, J Zhang, M Haardt, Sumrate maximization with minimum power consumption for MIMO DF twoway relaying—part II: network optimization. IEEE Trans. Signal Process. 61(14), 3578–3591 (2013).MathSciNetView ArticleGoogle Scholar
 Z Zhao, M Peng, Z Ding, W Wang, H Chen, Denoiseandforward network coding for twoway relay MIMO systems. IEEE Trans. Veh. Technol. 63(2), 775–788 (2014).View ArticleGoogle Scholar
 M Masjedi, A Hoseini, S Gazor, Noncoherent detection and denoiseandforward twoway relay networks. IEEE Trans. Commun. 64(11), 4497–4505 (2016).View ArticleGoogle Scholar
 C Wang, H Chen, Q Yin, A Feng, AF Molisch, Multiuser twoway relay networks with distributed beamforming. IEEE Trans. Wireless Commun. 10(10), 3460–3471 (2011).View ArticleGoogle Scholar
 V Nassab, S Shahbazpanahi, A Grami, Optimal distributed beamforming for twoway relay networks. IEEE Trans. Signal Process. 58(3), 1238–1250 (2010).MathSciNetView ArticleGoogle Scholar
 Y Rong, Joint source and relay optimization for twoway linear nonregenerative MIMO relay communications. IEEE Trans. Signal Process. 60(12), 6533–6546 (2012).MathSciNetView ArticleGoogle Scholar
 W Cheng, M Ghogho, Q Huang, D Ma, J Wei, Maximizing the sumrate of amplifyandforward twoway relaying networks. IEEE Commun. Lett. 18(11), 635–638 (2011).Google Scholar
 JM Kang, HM Kim, Training designs for estimation of spatially correlated fading channels in MIMO amplifyandforward twoway multirelay networks. IEEE Commun. Lett. 20(4), 772–775 (2016).View ArticleGoogle Scholar
 JM Kang, IM Kim, HM Kim, Optimal training design for MIMOOFDM twoway relay networks. IEEE Trans. Commun. 65(9), 3675–3690 (2017).View ArticleGoogle Scholar
 R Louie, Y Li, B Vucetic, Practical physical layer network coding for twoway relay channels: performance analysis and comparison. IEEE Trans. Wireless Commun. 9(2), 764–777 (2010).View ArticleGoogle Scholar
 F Jiang, C Zhu, J Peng, W Liu, Z Zhu, Y He, Joint relay and jammer selection and power control for physical layer security in twoway relay networks with imperfect CSI. Wireless Pers. Commun. 85:, 841–862 (2015).View ArticleGoogle Scholar
 C Wang, X Dong, Y Shi, Robust relay design for twoway multiantenna relay systems with imperfect CSI. J. Commun. and Networks.16(1), 45–55 (2014).View ArticleGoogle Scholar
 F Jiang, C Zhu, J Peng, W Liu, Z Zhu, Y He, Multipair twoway massive MIMO AF fullduplex relaying with imperfect CSI over Ricean fading channels. IEEE Access. 4:, 4933–4945 (2016).View ArticleGoogle Scholar
 S Salari, MZ Amirani, I Kim, D Kim, J Yang, Distributed beamforming in twoway relay networks with interference and imperfect CSI. IEEE Trans. Wireless Commun. 15(6), 4455–4469 (2016).View ArticleGoogle Scholar
 P Ubaidulla, S Aissa, Robust twoway cognitive relaying: precoder designs under interference constraints and imperfect CSI. IEEE Trans. Wireless Commun. 13(5), 2478–2489 (2014).View ArticleGoogle Scholar
 C Song, H Park, H Lee, I Lee, Robust beamforming designs for nonregenerative multipair twoway relaying systems. IEEE Trans. Wireless Commun. 65(9), 7802–7808 (2016).Google Scholar
 M Gastpar, M Vetterli, On the capacity of large Gaussian relay networks. IEEE Trans. Inf. Theory. 51(3), 765–779 (2005).MathSciNetView ArticleMATHGoogle Scholar
 Y Huang, D Palomar, S Zhang, Lorentzpositive maps and quadratic matrix inequalities with applications to robust MISO transmit beamforming. IEEE Trans. Signal Process. 61(5), 1121–1130 (2013).MathSciNetView ArticleGoogle Scholar
 Y Rong, Joint source and relay optimization for twoway linear nonregenerative MIMO relay communications. IEEE Trans. Signal Process. 60(12), 6533–6546 (2003).MathSciNetView ArticleGoogle Scholar
 Q Zhang, Q Li, J Qin, Beamforming design for OSTBCbased AFMIMO twoway relay networks with simultaneous wireless information and power transfer. IEEE Trans. Veh. Technol. 65(9), 7285–7296 (2016).View ArticleGoogle Scholar
 X Xie, H Yang, AV Vasilakos, Robust transceiver design based on interference alignment for multiuser multicell MIMO networks with channel uncertainty. IEEE Access. 5:, 5121–5134 (2017).View ArticleGoogle Scholar
 Y Tang, J Xiong, D Ma, X Zhang, Robust artificial noise aided transmit design for MISO wiretap channels with channel uncertainty. IEEE Commun. Lett. 17(1), 2096–2099 (2013).View ArticleGoogle Scholar
 P Ubaidulla, S Aissa, Robust twoway cognitive relaying: precoder designs under interference constraints and imperfect CSI. IEEE Trans. Wireless Commun. 13(5), 2478–2489 (2014).View ArticleGoogle Scholar
 Q Li, Q Zhang, J Qin, Robust beamforming for cognitive multiantenna relay networks with bounded channel uncertainties. IEEE Trans. Commun. 62(2), 478–487 (2014).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex optimization (Cambridge University Press, Cambridge, 2004).View ArticleMATHGoogle Scholar
 J Liu, F Gao, Z Qiu, Robust transceiver design for downlink multiuser MIMO AF relay systems. IEEE Trans. Wireless Commun. 14(4), 2218–2231 (2015).View ArticleGoogle Scholar