Sphere Decoding for Spatial Modulation

In this paper, Sphere Decoding (SD) algorithms for Spatial Modulation (SM) are developed to reduce the computational complexity of Maximum--Likelihood (ML--) optimum detectors, which foresee an exhaustive search of the whole search space and have a complexity that linearly increases with the product of number of transmit--antenna, receive--antenna, and size of the modulation scheme. Three SDs specifically designed for SM are proposed and analyzed in terms of Bit Error Probability (BEP) and computational complexity. By judiciously choosing some key parameters, e.g., the radius of the sphere centered around the received signal, it is shown that the proposed algorithms offer the same BEP as ML--optimum detection, with a significant reduction of the computational complexity. Also, it is shown that none of the proposed SDs is always superior to the others, but the best SD to use depends on the system setup, i.e., the number of transmit--antenna, receive--antenna, and the size of the modulation scheme. The computational complexity trade--off offered by the proposed solutions is studied via analysis and simulation, and numerical results are shown to validate our findings.