The multiple-input multiple-output (MIMO) detection problem, a fundamental problem in modern digital communications, is to detect a vector of transmitted symbols from the noisy outputs of a fading MIMO channel. The maximum likelihood detector can be formulated as a complex least-squares problem with discrete variables, which is NP-hard in general. Various semidefinite relaxation (SDR) methods have been proposed in the literature to solve the problem due to their polynomial-time worst-case complexity and good detection error rate performance. In this paper, we consider two popular classes of SDR-based detectors and study the conditions under which the SDRs are tight and the relationship between different SDR models. For the enhanced complex and real SDRs proposed recently by Lu et al., we refine their analysis and derive the necessary and sufficient condition for the complex SDR to be tight, as well as a necessary condition for the real SDR to be tight. In contrast, we also show that another SDR proposed by Mobasher et al. is not tight with high probability under mild conditions. Moreover, we establish a general theorem that shows the equivalence between two subsets of positive semidefinite matrices in different dimensions by exploiting a special "separable" structure in the constraints. Our theorem recovers two existing equivalence results of SDRs defined in different settings and has the potential to find other applications due to its generality.

翻译：多重投入多重输出(MSIM)检测问题是现代数字通信中一个根本问题,即如何检测一个来自淡化的MSIM频道的杂音输出的传输符号矢量。最有可能的检测器可以被设计成一个与离散变数(一般来说是NP硬的)的复杂最不平方的问题。文献中提出了多种半无限制放松(SDR)方法,以解决由于其多元-最坏情况复杂和探测率差而出现的问题。本文中,我们考虑了两类基于特别提款权的检测器,并研究了特别提款权紧张的条件和不同特别提款权模式之间的关系。对于最近由Lu等人提出的强化的复杂和真实的特别提款权,我们改进了分析,并提出了复杂特别提款权紧缩的必要和充分条件,以及真正特别提款权紧缩的必要条件。相比之下,我们还表明,Mobasher等人提出的另一类特别提款权在温和条件下并不十分接近。此外,我们建立了一种一般的比重质,即表明特别提款权在两种不同程度的等值结构中,“通过正态的等值结构中,我们确定不同的等值结构在正反等值结构中可以找到其他等值结构。”