Sum-rate maximization and robust beamforming design for MIMO two-way relay networks with reciprocal and imperfect CSI
- Wei Duan^{1},
- Miaowen Wen^{2},
- Xueqin Jiang^{3}Email author,
- Yier Yan^{4} and
- Moon Ho Lee^{1}
https://doi.org/10.1186/s13638-016-0655-8
© The Author(s) 2016
Received: 14 January 2016
Accepted: 10 June 2016
Published: 2 July 2016
Abstract
In this paper, we investigate the robust relay beamforming design for a multi-input multi-output (MIMO) two-way relay networks (TWRN) by considering imperfect and reciprocal channel state informations (CSIs). In order to maximize the sum-rate (SR) subject to the individual relay power constraint, we first equivalently convert the objective problem into a sum of the inverse of the signal-to-residual-interference-plus-noise ratio (SI-SRINR) problem. The SI-SRINR problem can be reformulated as a biconvex semi-definite programming (SDP) which employs bounded channel uncertainties as the worst-case model. Then, we convert residual-interference-plus-noise (RIN) and relay power constraints into linear matrix inequalities (LMIs). By this way, the objective problem can be tackled by the proposed efficient iterative algorithm. The analysis demonstrates the procedures of the proposed SI-SRINR robust design.
Keywords
1 Introduction
Recently, cooperative multi-input multi-output (MIMO) relaying system approach is popularized to increase the system capacity and improve the transmission reliability by leveraging spatial diversity. The MIMO relay network with perfect channel state information (CSI) has been studied in [1–3]. In [1], the authors developed a unified framework for optimizing two-way linear non-regenerative MIMO relay systems. In [2], the authors studied transceiver designs for a cognitive two-way relay network aiming at maximizing the achievable transmission rate of the secondary user. Based on iterative minimization of weighted mean-square error (MSE), a linear transceiver design algorithm for weighted sum-rate maximization has been investigated in the cellular network [3].
All the above works consider perfect CSI, which, however, is usually hard to obtain in practice, due to inaccurate channel estimation, feedback delay, and so on. To evaluate this imperfectness, by taking account into the channel uncertainties, the authors proposed a robust multi-branch Tomlinson-Harashima precoding (MB-THP) transceiver design in MIMO relay networks with imperfect CSI [4], where the Kronecker model is adopted for the covariance of the CSI mismatch. Moreover, the deterministic CSI uncertainty model has been widely used in the worst-case system [5–8]. In [5], the design of robust relay beamforming for two-way relay networks with channel feedback errors was studied, where each node is equipped with a single antenna. In [6], the authors investigated a robust beamforming scheme for the multi-antenna non-regenerative cognitive relay network with the bounded channel uncertainties which are modeled by the worst-case model. In [7], the authors investigated the multi-antenna non-regenerative relay network and addressed the joint source-relay-destination beamforming design problem under deterministic imperfect CSI model. In [8], based on the linear beamformer at the relay and QR successive interference cancelation (SIC) at the destination, the authors proposed a robust minimum MSE-regularized zero-forcing (MMSE-RZF) beamformer optimized in terms of rate. In this work, the authors assumed that the source can estimate the first-hop (form the source to the relay) CSI perfectly via the reciprocal channel when the high signal-to-noise (SNR) is training. In particular, in [9, 10], the authors considered the problem of robust minimum sum mean-square error (SMSE) relay precoder design for the two-way relay networking (TWRN).
For the one-way relay networks [4, 6–8, 10], the self-interference is not considered and the objective problem is easy to be converted into the convex version. For the single-antenna scenarios [5, 11], the equivalent channel can be expressed by employing the Hadamard product. For above references, the objective problem is actually only for the manipulation of a single node. By using ([12] Lemma 2), the robust minimum sum-MSE optimization problem [9, 10] can be easily converted into a convex problem. Therefore, the sum-rate (SR) maximization problem with multi-relay nodes for TWRN with the residual relay power constraint is more challenging and more general. In this paper, we propose a joint source and relay robust beamforming scheme for the MIMO TWRN where both the first- and second-hop CSIs are considered to be reciprocal and imperfectly known at each node. To get the accurate performance, the residual interference has been reserved. Since the considered sum-rate (SR) maximization problem is not only non-convex but also subject to the semi-infinite relay power constraints, we convert the objective problem into a sum of the inverse of the signal-to-residual-interference-plus-noise ratio (SI-SRINR) problem which is subject to the linear matrix inequality (LMIs) version of the constraints. In order to efficiently tackle the SI-SRINR problem, we first transform the problem into a biconvex semi-definite program (SDP) using the sign-definiteness lemma and then propose an alternating iterative algorithm with satisfactory convergence.
Notations: A ^{ T } and A ^{ H } denote the transpose and the Hermitian transpose of a matrix A, respectively. I _{ N } represents an N×N identity matrix. E(·), ⊗, and ∥·∥ stand for the statistical expectation, the Kronecker product, and the Frobenius norm.
2 System model and objective problem
where \(\mathbf {F}_{i,t}\in {\mathbb {C}^{N\times {M}}}, \forall {i}\in \{1,...,L\}\) represents the channel coefficient from the source node S _{ t } to the relay node R _{ i } and \(\mathbf {n}_{R_{i}}\sim {CN(0,\sigma ^{2}_{R_{i}}\mathbf {I}_{N})}\) denotes the additive white Gaussian noise (AWGN) vector with zero mean and variance \(\sigma ^{2}_{R_{i}}\mathbf {I}_{N}\).
where the slack values satisfy 0≤{α _{ i },β _{ i }}≪1, which is a reasonable assumption in a practical system.
where \(P_{R_{i}}\) is the maximum allocated power to the relay and the factor \(\frac {1}{2}\) is due to the half-duplex relay. Obviously, the objective problem T _{1} is non-convex since the variables W _{ i } and B _{ t } are in the numerator and the denominator of the SRINR_{ t }. Moreover, since the semi-infinite expressions of the optimal W _{ i }, B _{ t } are intractable, it is difficult to obtain the globally optimal solution. In the next section, we will propose a subpotimal solution to solve T _{1}.
3 Joint optimal beamformer design and proposed algorithm
3.1 Joint optimal beamformer design
Since the CSIs are imperfectly known, Q _{1} cannot be solved by using zero-gradient (ZG) algorithm [16]. In particular, due to a _{1} and a _{2} are both not only non-convex but also non-concave, the objective function a _{1}+a _{2}+a _{1} a _{2} is difficult to convert into the convex version. To efficiently solve T _{2}, we propose a biconvex SDP to obtain the suboptimal solution of the worst-case SR.
Proposition 1
where \(q_{t}=\frac {\|\widetilde {\chi }_{t}\mathbf {B}_{t}\|^{2}+N_{t}}{\left \|\mathbf {H}_{\overline {t}}\mathbf {B}_{\overline {t}}\right \|^{2}}\) and γ _{ t } is an auxiliary optimization variable which serves as upper bound of q _{ t } for t=1,2.
Proof.
Since the optimal solution of {max (1+A)(1+B)} is equivalent to the problem \(\left \{\text {min}~\left (\frac {1}{A}+\frac {1}{B}\right)\right \}\) [17] and the fact that log(·) is a monotonic function, letting \(q_{t}=\frac {\|\widetilde {\chi }_{t}\mathbf {B}_{t}\|^{2}+N_{t}}{\left \|\mathbf {H}_{\overline {t}}\mathbf {B}_{\overline {t}}\right \|^{2}}\), the SR maximization problem can be equivalently converted into {min (q _{1}+q _{2})}. Similar to [9], by introducing the auxiliary optimization variable γ _{ t }, the problem {min (q _{1}+q _{2})} can be recast in the epigraph form [18] as {min (γ _{1}+γ _{2})},s.t. q _{ t }≤γ _{ t } ^{2}, and we have the objective problem T _{3}.
Proposition 2
Proof.
See Appendix 1.
Proposition 3
where \(\Theta _{t}=\sum _{t=1}^{2}\text {vec}\footnotesize {\left (\mathbf {\widetilde {F}}_{i,t}\right)}\mathbf {\widehat {B}}_{t}\text {vec}\left (\mathbf {\widetilde {F}}_{i,t}\right)^{H}-\lambda _{1}{\omega _{i}^{2}}{\alpha _{i}^{2}}-\lambda _{2}{\omega _{i}^{2}}{\beta _{i}^{2}}-\widehat {P}_{R_{i}}\) with ω _{ i }=∥W _{ i }∥.
Proof.
See Appendix 2.
It is clear that the problem T _{5} is a biconvex SDP with linear objective function, which can be efficiently solved by an iterative algorithm. Furthermore, with fixed B _{ t }/ W _{ i }, T _{5} is convex with regard to W _{ i }/ B _{ t } which can be solved by CVX [18].
3.2 Proposed algorithm and computational complexity analysis
Now, we summarize the proposed beamforming method in Algorithm 1.
The proposed Algorithm 1 will converge to a sub-optimal solution as \(\sum _{t=1}^{2}\gamma _{t}^{(n)}-\sum _{t=1}^{2}\gamma _{t}^{(n-1)}\leq \xi \). Therefore, ξ is initialized to be a small value, and N _{max} is set to limit the number of iterations.
i.e., the objective function increases monotonically with the number of iterations. This observation, coupled with the fact that J({W _{ i }},{B _{ t }}) is upper-bounded, implies that the proposed algorithm converges to a limit as number n→∞.
Discussion 1: For two-way relay networks, once the second transmission phase finishes, the signal transmitted by the transceiver nodes reappears as self-interference. Without eliminating the self-interference, with the condition of the imperfect CSI, the exactly optimal solution is difficult to obtain. In spite of this, the proposed suboptimal solution is very close to the exactly optimal solution when the CSI uncertainty is small enough and the number of the iterations n→∞ in the proposed algorithm.
Computational complexity of the proposed SI-SRINR algorithm
Step | Operations | Block dimensions (a _{ k }) | Number of variables (n) |
---|---|---|---|
1 | \(\widehat {P}_{R_{i}}\) | 2M M _{ b }+1 | 2M M _{ b }+2N ^{2}+6L+12 |
2 | ς _{ t } | (l+1)M N+1 | 2N ^{2}+3L+6 |
3.1 | τ _{ t } | l | (2N ^{2}+3L+6) |
×(2M M _{ b }+3L+6) | |||
3.2 | γ _{ t } | 2l M N+M M _{ b }+1 | (2N ^{2}+3L+6) |
×(2M M _{ b }+3L+6) |
where L(A) denotes the LMI version of A. Obviously, the constraints L(ε _{ i }≤a _{ i }) and \(L\left (\left \|\mathbf {x}_{R_{i}}\right \|^{2}\leq P_{R_{i}}\right)\) have the same computational complexity to the proposed SI-SRINR method as shown in Table 1. In addition, since the constraints (20) and (21) in Q _{3} not only request more FOLPs but also lead to lower convergence performance, our proposed SI-SRINR method outperforms non-SI-SRINR one.
4 Simulation results
In this section, we study the performance of the proposed SI-SRINR robust beamforming design for TWRN. The channel estimates \(\mathbf {\widetilde {G}}_{t,i},\mathbf {\widetilde {F}}_{i,t}\) are assumed to be reciprocal and identically distributed complex Gaussian random variables. The proposed scenario is considered with two source nodes and L=2 relay nodes. The source and relay nodes are equipped with M _{ b }=M=N=4 antennas. We further assume that the noise variances \(\sigma ^{2}_{R_{i}}, \sigma ^{2}_{S_{t}}\) for i=1,...,L and t=1,2, are equally given as σ ^{2}=1. All results are averaged over N _{max}=1000 channel realizations with ξ=10^{−4}.
5 Conclusions
In this paper, we considered MIMO TWRN with the robust relay beamforming design and proposed an efficient iterative algorithm to solve the SR maximization problem. The worst-case robust design problem was first converted into a SI-SRINR problem. After then, by utilizing the sign-definiteness lemma, the objective problems were represented as the tractable ones which are obtained through the SDP-based iterative optimization. Numerical results showed that the performance of the proposed SI-SRINR robust design is improved compared to the non-SI-SRINR one and non-robust one.
6 Appendix 1
where \(\alpha _{i}=\left \|\Delta _{\mathbf {F}_{i,1}}\right \|=\left \|\Delta _{\mathbf {G}_{1,i}}\right \|\), \(\beta _{i}=\left \|\Delta _{\mathbf {F}_{i,2}}\right \|=\left \|\Delta _{\mathbf {G}_{2,i}}\right \|\) are the norm-bounded errors (NBEs) of channel uncertainties, and 0 _{ MN } denotes M N×M N zero matrix. This completes the proof.
7 Appendix 2
where \(\Theta _{t}=\sum _{t=1}^{2}\text {vec}\footnotesize {\left (\mathbf {\widetilde {F}}_{i,t}\right)}\mathbf {\widehat {B}}_{t}\text {vec}\left (\mathbf {\widetilde {F}}_{i,t}\right)^{H}-\lambda _{1}{\omega _{i}^{2}}{\alpha _{i}^{2}}-\lambda _{2}{\omega _{i}^{2}}{\beta _{i}^{2}}-\widehat {P}_{R_{i}}\). This completes the LMI version of the individual relay power constraint.
Declarations
Acknowledgements
This work was supported by MEST 2015R1A2A1A05000977, NRF, South Korea, National Natural Science Foundation of China by Grants 61501190 and the Natural Science Foundation of Guangdong Province by Grant 2014A030310389, Shanghai Rising-Star Program (15QA1400100), Innovation Program of Shanghai Municipal Education Commission (15ZZ03), DHU Distinguished Young Professor Program (16D210402), and Key Lab of Information Processing & Transmission of Guangzhou 201605030014.
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
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