Signal Shaping for Generalized Spatial Modulation and Generalized Quadrature Spatial Modulation

This paper investigates the generic signal shaping methods for the multiple-data-stream generalized spatial modulation (GenSM) and the generalized quadrature spatial modulation (GenQSM). Three cases with different channel state information at the transmitter (CSIT) are considered, including no CSIT, statistical CSIT, and perfect CSIT. A unified optimization problem is formulated to find the optimal transmit vector set under size, power, and sparsity constraints. We propose an optimization-based signal shaping (OBSS) approach by solving the formulated problem directly and a codebook-based signal shaping (CBSS) approach by finding the sub-optimal solutions in discrete space. In the OBSS approach, we reformulate the original problem to optimize the signal constellations used for each transmit antenna combination (TAC). Both the size and the entry of all signal constellations are optimized. Specifically, we suggest the use of a recursive design for the size optimization. The entry optimization is formulated as a non-convex large-scale quadratically constrained quadratic programming (QCQP) problem and can be solved by the existing optimization techniques with rather high complexity. To reduce the complexity, we propose the CBSS approach using a codebook generated by the quadrature amplitude modulation (QAM) symbols and a low-complexity selection algorithm to choose the optimal transmit vector set. The simulation results show that the OBSS approach exhibits the optimal performance in comparison with existing benchmarks. However, the OBSS approach is impractical for large-size signal shaping and adaptive signal shaping with instantaneous CSIT due to the demand of high computational complexity. As a low-complexity approach, the CBSS shows comparable performance and can be easily implemented in large-size systems.