The Sign-Definiteness Lemma and Its Applications to Robust Transceiver Optimization for Multiuser MIMO Systems

Abstract

We formally generalize the sign-definiteness lemma to the case of complex-valued matrices and multiple norm-bounded uncertainties. This lemma has found many applications in the study of the stability of control systems, and in the design and optimization of robust transceivers in communications. We then present three different novel applications of this lemma in the area of multi-user multiple-input multiple-output (MIMO) robust transceiver optimization. Specifically, the scenarios of interest are: (i) robust linear beamforming in an interfering adhoc network, (ii) robust design of a general relay network, including the two-way relay channel as a special case, and (iii) a half-duplex one-way relay system with multiple relays. For these networks, we formulate the design problems of minimizing the (sum) MSE of the symbol detection subject to different average power budget constraints. We show that these design problems are non-convex (with bilinear or trilinear constraints) and semiinfinite in multiple independent uncertainty matrix-valued variables. We propose a two-stage solution where in the first step the semi-infinite constraints are converted to linear matrix inequalities using the generalized signdefiniteness lemma, and in the second step, we use an iterative algorithm based on alternating convex search (ACS). Via simulations we evaluate the performance of the proposed scheme.

Secure Communications for Dual-Polarized MIMO Systems

“…In the simulation study, we also apply these standard algorithms, specifically, the algorithms of [26], [28], [31], to the dual-polarized MIMO system and use their achievable security performance as the benchmarks. These standard designs are based on the full instantaneous CSI and, therefore, they outperform the dualstructured precoding.…”

“…where G k is the nominal channel and ∆ k is the norm-bounded channel uncertainty [31], [35]. By substituting (58) into the model (3), we further obtain the uncertainty model of H k as…”

“…We reformulate each MSE term into the form that is affine in ∆ ′ k . With the aid of linear matrix inequality (LMI) and the sign-definiteness lemma [31], an equivalent SDP transformation to the problem (63) is conducted. This transformed problem is still biconvex in P k and M k , and we use an iterative algorithm to alternately perform the convex search for P k and M k , respectively.…”

“…Although (72) is semi-infinite due to the channel uncertainty term ∆ ′ k , the sign-definiteness lemma [31], which is introduced in the following lemma, can be applied to transform it into a finite biconvex form.…”

“…Thus, in addition to providing more compact antenna array which is beneficial for practical implementation as well as imposing lower computational complexity in designing the dualstructured precoding, the dual-polarized MIMO offers the performance advantage, compared to the single-polarized MIMO. Additionally, we apply the standard MIMO precoding schemes [26], [28], [31] to the dual-polarized MIMO system and use their achievable security performance as the benchmarks for the three respective CSI scenarios. Although outperforming the proposed dual-structured precoding scheme, these standard MIMO designs require the full instantaneous CSI which is difficult to acquire in practice and they impose huge channel feedback overhead.…”