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abstract: 'In this paper, Sphere Decoding (SD) algorithms for Spatial Modulation (SM) are developed to reduce the computational complexity of Maximum–Likelihood (ML) detectors. Two SDs specifically designed for SM are proposed and analysed in terms of Bit Error Ratio (BER) and computational complexity. Using Monte Carlo simulations and mathematical analysis, it is shown that by carefully choosing the initial radius the proposed sphere decoder algorithms offer the same BER as ML detection, with a significant reduction in the computational complexity. A tight closed form expression for the BER performance of SM–SD is derived in the paper, along with an algorithm for choosing the initial radius which provides near to optimum performance. Also, it is shown that none of the proposed SDs are always superior to the others, but the best SD to use depends on the target spectral efficiency. The computational complexity trade–off offered by the proposed solutions is studied via analysis and simulation, and is shown to validate our findings. Finally, the performance of SM–SDs are compared to Spatial Multiplexing (SMX) applying ML decoder and applying SD. It is shown that for the same spectral efficiency, SM–SD offers up to $84\%$ reduction in complexity compared to SMX–SD, with 