Lagrangian Detection for Generalized Space-Shift Keying MIMO Systems

Generalized space-shift keying (GSSK) has recently established itself as a promising technology for massive multiple-input multiple-output (MIMO) systems. However, the computational complexity of maximum likelihood (ML) detection is too high, and it increases significantly as the number of transmit antennas and active antennas increases. In this correspondence, we propose a low-complexity suboptimal detection for massive GSSK-MIMO systems. The ML detection of GSSK can be posed as a 0-1 quadratic programming with an equality constraint. First, we employ the Lagrange multiplier to transform the 0-1 quadratic programming with a linear equality constraint into a standard 0-1 quadratic programming. Most of the conventional methods for determining the Lagrange multiplier are derived from Karush-Kuhn-Tucker (KKT) conditions, which are usually valid for continuous variable programming rather than the discrete one. However, in our problem, the optimization variables are binary. Therefore, we propose a theorem that can determine the Lagrange multiplier iteratively by an 1-D binary search rather than KKT conditions and, finally, detect the GSSK transmission symbols. Simulation results demonstrate that the proposed method can achieve an excellent signal detection performance for massive GSSK-MIMO systems with low computational complexity.