Robust Optimization for Amplify-and-Forward MIMO Relaying From a Worst-Case Perspective

Abstract

Abstract-In this paper, we consider robust optimization of amplify-and-forward (AF) multiple-input multiple-output (MIMO) relay precoders in presence of deterministic imperfect channel state information (CSI), when the CSI uncertainty lies in a norm bounded region. Two widely used performance metrics, mutual information (MI) and mean square error (MSE), are adopted as design objectives. According to the philosophy of worst-case robustness, the robust optimization problems with respect to maximizing the worst-case MI and minimizing the worstcase MSE are formulated as maximin and minimax problems, respectively. Due to the fact that these two problems do not have a concave-convex or convex-concave structure, we cannot rely on the conventional saddle point theory to find the robust solutions. Nevertheless, by exploiting majorization theory, we show that the formulated maximin and minimax problems both admit saddle points. We further analytically characterize the saddle points, and provide closed-form solutions to robust relay precoder designs. Interestingly, we find that, under both MI and MSE metrics, the robust relay optimization leads to a channeldiagonalizing structure, meaning that eigenmode transmission is optimal from the worst-case robustness perspective. The proposed robust designs can improve the spectral efficiency and reliability of AF MIMO relaying against CSI uncertainties at the similar cost of computational complexity as the existing non-robust schemes.Index Terms-Multiple-input multiple-output (MIMO) cooperative transmission, mutual information (MI), mean square error (MSE), worst-case robust design, relay precoding, imperfect channel state information (CSI).

Robust Transceiver Design for Downlink Multiuser MIMO AF Relay Systems

“…complex Gaussian random variables with zero mean and unit variance. Suppose that the uncertainty sizes are related to the quality of channels [22], [35], [36], e.g., the radius of uncertainty regions can be set to 1). Simulation results are averaged over 1000 independent channel realizations, and 5000 QPSK symbols are transmitted for each data stream per realization.…”

“…In [32]- [34], the norm-bounded CSI errors were considered in a multipointto-multipoint system and the MIMO relay was designed to minimize the consumed power under different quality of service (QoS) constraints. The authors in [35], [36] proved that the robust design for single user MIMO AF relay leads to the channel-diagonalizing structure, and the optimal solution can be obtained only through the scalar operations. The signdefiniteness lemma was generalized to the case of complexvalued variables and multiple semi-infinite constraints, and its applications to robust optimization were investigated in [37].…”

Optimization of a MIMO amplify-and- forward relay system with channel state information estimation error and feedback delay

“…Details of transformation from (11) to (13) are presented in Appendix B. The optimization problem (13) is easier to solve than (11). Once the optimal 1 is derived, Q opt and W opt are computed using (12) and (10), respectively.…”

Robust Energy Efficiency Maximization in Cognitive Radio Networks: the Worst-Case Optimization Approach

“…As pointed out in [33]- [35], the errors caused by quantization is bounded. Furthermore, there are several ways to determine ε j in practice [36], such as measuring the difference between the true and estimate of channel gain generated by a channel emulator or calculating ε j based on the specific distribution of channel errors to guarantee that the norms of possible errors are upper bounded with a certain probability [33]. Furthermore, we can choose M j based on the specific shape of G j , such as the identity matrix corresponding to a circle.…”