In this work, we investigate linear precoding for secure spatial modulation. With secure spatial modulation, the achievable secrecy rate does not have an easy-to-compute mathematical expression, and hence, has to be evaluated numerically, which leads to high complexity in the optimal precoder design. To address this issue, an accurate and analytical approximation of the secrecy rate is derived in this work. Using this approximation as the objective function, two low-complexity linear precoding methods based on gradient descent (GD) and successive convex approximation (SCA) are proposed. The GD-based method has much lower complexity but usually converges to a local optimum. On the other hand, the SCA-based method uses semi-definite relaxation to deal with the non-convexity in the precoder optimization problem and achieves near-optimal solution. Compared with the existing GD-based precoder design in the literature that directly uses the exact and numerically evaluated secrecy capacity as the objective function, the two proposed designs have significantly lower complexity. Our SCA-based design even achieves a higher secrecy rate than the existing GD-based design.

在这项工作中，我们研究了用于安全空间调制的线性预编码。利用安全的空间调制，可获得的保密率不具有易于计算的数学表达式，因此必须进行数值评估，这导致最佳预编码器设计的复杂性很高。为了解决这个问题，在这项工作中得出了保密率的准确和分析近似值。以这种近似为目标函数，提出了两种基于梯度下降（GD）和逐次凸近似（SCA）的低复杂度线性预编码方法。基于GD的方法具有较低的复杂度，但通常会收敛到局部最优值。另一方面，基于SCA的方法使用半定松弛来处理预编码器优化问题中的非凸性，并获得接近最优的解决方案。与文献中现有的基于GD的预编码器设计直接使用精确的和经过数字评估的保密能力作为目标函数相比，这两个提议的设计的复杂度大大降低。与基于GD的现有设计相比，基于SCA的设计甚至可以实现更高的保密率。