Transmit Precoded Spatial Modulation: Maximizing the Minimum Euclidean Distance Versus Minimizing the Bit Error Ratio

In this paper, we investigate a pair of transmit precoding (TPC) algorithms conceived for spatial modulation (SM) systems communicating over flat-fading multiple-input multiple-output (MIMO) channels. In order to retain all the benefits of conventional SM, we design the TPC matrix to be diagonal and introduce two design criteria for optimizing the elements of the TPC matrix. Specifically, we first investigate a TPC design based on maximizing the minimum Euclidean distance d(min) (max-d(min)) between the SM signal points at the receiver side. A closed-form solution of the optimal max-d(min)-based TPC matrix is derived. Then, another TPC design algorithm is proposed for directly minimizing the bit error ratio (BER) upper bound of SM, which is capable of jointly optimizing the overall Euclidean distance between all received signal points. In the minimum BER (min-BER)-based TPC algorithm, the theoretical gradient of the BER with respect to the diagonal TPC matrix is derived and a simplified iterative conjugate gradient (SCG) algorithm is invoked for TPC optimization. Our simulation results demonstrate that the proposed max-d(min)-based TPC algorithm is optimal in terms of the minimum distance. However, increasing d(min) does not achieve a further BER improvement. We also confirm that the min-BER-based TPC outperforms the max-d(min)-based TPC schemes in terms of the achievable BER performance.