## 3 Enhancing Security Through Joint Precoding and Artificial Noise Design
### Secrecy Rate Optimization Through Joint Precoding and Artificial Noise Design  
In this subsection, we propose a low-complexity approach to solve the optimization problem in (9). We consider an alternating optimization method. When \(\zeta\) is fixed, we derive a lower bound for the secrecy rate and optimize this bound to enhance the secrecy rate. When \(\mathbf{P}\) is fixed, we use a method based on maximizing the product of the SNR at Bob and the interference-to-signal-plus-noise ratio (ISNR) at Eve to obtain a suboptimal solution for \(\zeta\). This alternating optimization approach is commonly used in many classical algorithms, such as variational Bayesian methods. Since each alternating optimization step guarantees an increase in the secrecy rate, the convergence of the algorithm is naturally ensured. In the following, we are committed to solving the optimization problems for the two parameters separately. (1) Optimizing \(\mathbf{P}\) with fixed \(\zeta\): For fixed \(\zeta\), combining [25] with Jensen's inequality, and through a tedious yet straightforward derivation, we obtain the lower bound of [PERSON]'s rate \(R_{b}\) and [PERSON]'s rate \(R_{e}\) as  
\[ \begin{split} R_{b}\geq&\ k-\frac{1}{2^{k}}\sum_{ \mathbf{s}_{l}^{m}\in\mathcal{S}}\log_{2}\left(\sum_{\mathbf{s}_{l}^{m}\in \mathcal{S}}\mathbb{E}_{\mathbf{m}_{\
u}}[\exp(\mathbf{\Omega}_{b})]\right)\\ =&\ k-N_{b}(\log_{2}e-1)-\frac{1}{2^{k}}\sum_{ \mathbf{s}_{l}^{m}\in\mathcal{S}}\log_{2}\left(\sum_{\mathbf{s}_{l}^{m}\in \mathcal{S}}\exp\left(\mathbf{\Omega}_{b}^{\prime}\right)\right),\end{split}  \tag{10}\]  
\[ \begin{split} R_{e}\geq&\ k-\frac{1}{2^{k}}\sum_{ \mathbf{s}_{l}^{m}\in\mathcal{S}}\log_{2}\left(\sum_{\mathbf{s}_{l}^{m}\in \mathcal{S}}\mathbb{E}_{\mathbf{m}}[\exp(\mathbf{\Omega}_{e})]\right)\\ =&\ k-N_{b}(\log_{2}e-1)-\frac{1}{2^{k}}\sum_{ \mathbf{s}_{l}^{m}\in\mathcal{S}}\log_{2}\left(\sum_{\mathbf{s}_{l}^{m}\in \mathcal{S}}\exp\left(\mathbf{\Omega}_{e}^{\prime}\right)\right)\end{split}  \tag{11}\]  
with \(\mathbf{\Omega}_{b}^{\prime}=\frac{-\left\|\zeta\mathbf{H}(\Theta_{b}) \mathbf{P}\left(\mathbf{s}_{l}^{m}-\mathbf{s}_{l}^{m}\right)\right\|^{2}}{2 \sigma_{e}^{2}}\) and \(\mathbf{\Omega}_{e}^{\prime}=-\frac{1}{2}\left\|\zeta\mathbf{\Sigma}^{-1/2} \mathbf{H}(\Theta_{e})\mathbf{P}\left(\mathbf{s}_{l}^{m}-\mathbf{s}_{l}^{m} \right)\right\|^{2}\). According to [25], the lower bound of those two secrecy rate expressions, when increased by a constant \(N_{b}(\log_{2}e-1)\) and \(N_{e}(\log_{2}e-1)\), respectively, provide a tighter lower bound that can be approximated as the secrecy rate. Therefore, we have the following:  
\[R_{b,L}=k-\frac{1}{2^{k}}\sum_{\mathbf{s}_{l}^{m}\in\mathcal{S}}\log_{2}\left( \sum_{\mathbf{s}_{l}^{m}\in\mathcal{S}}\exp\left(\mathbf{\Omega}_{b}^{\prime} \right)\right), \tag{12}\]\[R_{e,L}=k-\frac{1}{2^{k}}\sum_{\mathbf{s}_{i}^{m}\in\mathcal{S}}\log_{2}\left( \sum_{\mathbf{s}_{i}^{m}\in\mathcal{S}}\exp\left(\mathbf{\Omega}_{e}^{\prime} \right)\right). \tag{13}\]  
Thus, we can obtain the lower bound \(R_{L}(\mathbf{P})\) for the secrecy rate \(R(\mathbf{P})\) as  
\[R_{L}(\mathbf{P})=\left[R_{b,L}-R_{e,L}\right]^{+}=\left[\frac{1}{2^{k}}\sum_{ \mathbf{s}_{i}^{m}\in\mathcal{S}}\log_{2}\left(\frac{\sum_{\mathbf{s}_{i}^{m} \in\mathcal{S}}\exp\left(\mathbf{\Omega}_{e}^{\prime}\right)}{\sum_{\mathbf{s} _{i}^{m}\in\mathcal{S}}\exp\left(\mathbf{\Omega}_{b}^{\prime}\right)}\right) \right]^{+} \tag{14}\]  
The subproblem centers around finding the optimal \(\mathbf{P}\) that maximizes the lower bound of the secrecy rate \(R_{L}(\mathbf{P})\), which is formulated as  
\[ \begin{split}&\max_{\mathbf{P}}\quad R_{L}(\mathbf{P})\\ &\text{s.t}\quad\text{tr}\left(\mathbf{P}\mathbf{P}^{H}\right) \leq P_{t}.\end{split}  \tag{15}\]  
We resort to a gradient descent to solve the above problem, and we further need to compute the gradient of the objective function with respect to \(\mathbf{P}\), which is given by  
\[