Minimum achievable complexity (MAC) for a maximum likelihood (ML) performance-achieving detection algorithm is derived. Using the derived MAC, we prove that the conventional sphere decoding (SD) algorithms suffer from an inherent weakness at low SNRs. To find a solution for the low SNR deficiency, we analyze the effect of zero-forcing (ZF) and minimum mean square error (MMSE) linearly detected symbols on the MAC and demonstrate that although they both improve the SD algorithm in terms of the computational complexity, the MMSE linearly detected point has a vital difference at low SNRs. By exploiting the information provided by the MMSE of linear method, we prove the existence of a lossless dimension reduction which can be interpreted as the feasibility of a detection method which is capable of detecting the ML symbol without visiting any nodes at low and high SNRs. We also propose a lossless dimension reduction-aided detection method which achieves the promised complexity bounds marginally and reduces the overall computational complexity significantly, while obtaining the ML performance. The theoretical analysis is corroborated with numerical simulations.

中文翻译：

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用于球体解码的整数最小二乘无损降维

导出了最大似然（ML）性能实现检测算法的最小可实现复杂度（MAC）。使用派生的 MAC，我们证明了传统的球形解码 (SD) 算法在低 SNR 下存在固有的弱点。为了找到低 SNR 缺陷的解决方案，我们分析了迫零 (ZF) 和最小均方误差 (MMSE) 线性检测符号对 MAC 的影响，并证明尽管它们都在计算方面改进了 SD 算法复杂性，MMSE 线性检测点在低 SNR 下具有重要差异。通过利用线性方法的 MMSE 提供的信息，我们证明了无损降维的存在，这可以解释为一种检测方法的可行性，该方法能够检测 ML 符号，而无需在低和高 SNR 下访问任何节点。我们还提出了一种无损降维辅助检测方法，该方法略微实现了承诺的复杂度界限，并显着降低了整体计算复杂度，同时获得了 ML 性能。理论分析得到了数值模拟的证实。