Lossless Dimension Reduction for Integer Least Squares With Application to Sphere Decoding

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.