Semidefinite Programming Relaxation Based Virtually Antipodal Detection for MIMO Systems Using Gray-Coded High-Order QAM

An efficient generalized semidefinite programming relaxation (SDPR) based virtually antipodal (VA) detection approach is proposed for Gray-coded high-order rectangular quadrature-amplitude modulation (QAM) signaling over multiple-input–multiple-output (MIMO) channels. The decomposition of symbol-based detection to a bit-based detection is desirable owing to its reduced complexity and increased flexibility. However, Gray-mapping is nonlinear, hence the direct bit-based detection of Gray-coded-QAM MIMO systems constitutes a challenging problem. In this paper, we find a way to exploit the structural regularity of Gray-coded high-order rectangular QAM and to transform the classic symbol-based MIMO detection model to a low-complexity bit-based detection model. As an appealing benefit, the conventional three-step “signal-to-symbols-to-bits” decision process can be substituted by a simpler “signal-to-bits” decision process for the classic Gray-mapping-aided high-order rectangular QAM; hence, any bit-based detection method becomes potentially applicable. As an application example, we propose a direct-bit-based VA-SDPR (DVA-SDPR) MIMO detector, which is capable of directly making binary decisions concerning the individual information bits of the ubiquitous Gray-mapping-aided high-order rectangular QAM while dispensing with symbol-based detection. Furthermore, the proposed model transformation method facilitates the exploitation of the unequal error protection (UEP) property of high-order QAM with the aid of the low-complexity bit-flipping-based “hill climbing” method. As a result, the proposed DVA-SDPR detector achieves the best bit error ratio (BER) performance among the known SDPR-based MIMO detectors in the context considered, while still maintaining the lowest possible worst-case complexity order of $O[(N_{T}\log_{2}M + \hbox{1})^{3.5}]$.