## 3 Enhancing Security Through Joint Precoding and Artificial Noise Design
### Problem Formulation  
Suppose that the joint maximum likelihood detector is employed for both [PERSON] and [PERSON]; the achievable rates between [PERSON] and both [PERSON] and [PERSON] can be computed separately. Define \(\mathcal{S}\) as the set of all transmitted symbols and \(\left\|\cdot\right\|\) as the Frobenius norm, with [PERSON]'s rate \(R_{b}\) being [22,24]  
\[R_{b}=k-\frac{1}{2^{k}}\sum_{\mathbf{s}_{1}^{m}\in\mathcal{S}}\mathbb{E}_{ \mathbf{n}_{r}}\left[\log_{2}\left(\sum_{\mathbf{s}_{1}^{m}\in\mathcal{S}} \exp(\mathbf{\Omega}_{b})\right)\right], \tag{6}\]  
where \(\mathbf{\Omega}_{b}=\frac{-\left\|\zeta\mathbf{H}(\mathbf{\Theta}_{b})\mathbf{ P}\left(\mathbf{s}_{1}^{m}-\mathbf{s}_{2}^{m}\right)+\mathbf{n}_{b}\right\|^{2}+ \left\|\mathbf{n}_{b}\right\|^{2}}{\sigma_{b}^{2}}\).  
The AN makes the rate calculation for [PERSON] slightly more complex. In order to derive [PERSON]'s rate, we multiple \(\mathbf{\Sigma}^{-1/2}\) on both sides of Equation (5), and then we can similarly obtain the expression of [PERSON]'s rate as  
\[R_{e}=k-\frac{1}{2^{k}}\sum_{\mathbf{s}_{1}^{m}\in\mathcal{S}}\mathbb{E}_{ \mathbf{n}}\left[\log_{2}\left(\sum_{\mathbf{s}_{1}^{m}\in\mathcal{S}}\exp( \mathbf{\Omega}_{e})\right)\right], \tag{7}\]  
where \(\mathbf{\Omega}_{e}=-\left\|\zeta\mathbf{\Sigma}^{-1/2}\mathbf{H}(\mathbf{ \Theta}_{e})\mathbf{P}\left(\mathbf{s}_{1}^{m}-\mathbf{s}_{1}^{m}\right)+ \mathbf{\Sigma}^{-1/2}\mathbf{n}\right\|^{2}+\left\|\mathbf{\Sigma}^{-1/2} \mathbf{n}\right\|^{2}\).  
The secrecy rate \(R\) is defined as the difference between the achievable rates of Bob and Eve, and it can be written as  
\[R=\left[\frac{1}{2^{k}}\sum_{\mathbf{s}_{1}^{m}\in\mathcal{S}}\mathbb{E}_{ \mathbf{n}_{b},\mathbf{n}}\left[\log_{2}\left(\frac{\sum_{\mathbf{s}_{1}^{m} \in\mathcal{S}}\exp(\mathbf{\Omega}_{e})}{\sum_{\mathbf{s}_{1}^{m}\in\mathcal{ S}}\exp(\mathbf{\Omega}_{b})}\right)\right]\right]^{+}, \tag{8}\]  
where \(\left[\cdot\right]^{+}=\max\{\cdot,0\}\).  
As a result, the secrecy rate optimization problem through joint precoding and power-control factor design is formulated as  
\[\begin{split}\underset{\mathbf{P},\zeta}{\text{max}}& R(\mathbf{P},\zeta)\\ \text{s.t}&\text{tr}\Big{(}\mathbf{P}\mathbf{P}^{H} \Big{)}\leq P_{t}\\ & 0\leq\zeta\leq 1,\end{split} \tag{9}\]  
Note that the optimization problem is non-convex and contains highly complex terms, making it intractable to derive a globally optimal closed-form solution. Furthermore, the expectation operator requires averaging over multiple noise realizations, which addssignificant computational complexity. For this optimization problem, a more efficient approach is necessary to avoid the difficulties associated with solving explicit expressions.  
In this scenario, both [PERSON] and [PERSON] are assumed to have identical reception capabilities, representing the \"worst-case\" scenario in practical systems. Under these conditions, [PERSON]'s achievable rate serves as an \"upper bound\", while the system's secrecy rate represents the \"lower bound\" of secure communication. The optimization design aims to maximize the \"lower bound\" of the secrecy rate while minimizing (approaching zero) the \"upper bound\" of [PERSON]'s achievable rate. By developing an optimization algorithm in this context, even if [PERSON] has weaker reception capabilities than Bob (e.g., if [PERSON] does not fully understand the GSDM symbol mapping rules), the actual achievable secrecy rate will exceed the optimized value, thus enhancing the system's security. This approach ensures that the system's security rate exceeds the optimized security rate derived in this paper under practical scenarios where [PERSON] possesses limited or incomplete information.