# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 3 AN Maximizing Relay's SER Performance
### Assigning \(|z|\) for a Given \(E_{z}\)
This theorem means that only two types of AN symbols are generated from the destination, one with probability \((1-p)\) and one with probability \(p\), and the corresponding powers are \(x_{1}\) and \(x_{2}\), respectively. It is important to note that following Appendix, Theorem 1 is valid for _any_\(\widetilde{\mathrm{SER}}\left(\sqrt{x}\right)\) as long as \(\widetilde{\mathrm{SER}}\left(\sqrt{x}\right)\) is a monotonic increasing function. Inserting Equation (21) into Equation (20), the expected SER in Equation (20a) can be computed as,  
\[E_{z}=T_{m}\mathcal{E}\left\{\left|z\right|^{2}\right\}\leq\overline{E}. \tag{18}\]1 \[E_{z}=T_{m}\mathcal{E}\left\{\left|z\right|^{2}\right\}\leq\overline{E}. \tag{18}\]2 \[E_{z}=T_{m}\mathcal{E}\left\{\left|z\right|^{2}\right\}\leq\overline{E}. \tag{18}\]3  
From Equation (24), at high average AN power, i.e., \(\overline{P}\to\infty\), we observe that the expected SER in Equation (24) approaches \(\frac{M-1}{M}\), which corresponds to the non-informative error performance. The proof can be sketched as follows. From Equations (16) and (17), we know that \(\lim_{|z|\to\infty}\widetilde{\mathrm{SER}}\left(|z|\right)=\frac{M-1}{M}\). Given \(\overline{P}\to\infty\), by applying the _Optimal Power Allocation Theorem_, we have \(x_{2}\to\infty\) due to \(x_{2}\geq\overline{P}\). In this case, if \(x_{1}\to\infty\), we have \(\lim_{x_{1}\to\infty}\widetilde{\mathrm{SER}}\left(\sqrt{x_{1}}\right)=\frac{ M-1}{M}\) and \(\lim_{x_{2}\to\infty}\widetilde{\mathrm{SER}}\left(\sqrt{x_{2}}\right)=\frac{ M}{M}\), thereby \(\widetilde{\mathrm{SER}}(x_{1},x_{2})\) in Equation (24) approaches \(\frac{M-1}{M}\). If \(x_{1}\) is a finite value, \(x_{2}\approx\overline{P}\), then the term \(\frac{x_{2}-\overline{P}}{x_{2}-x_{1}}\widetilde{\mathrm{SER}}\left(\sqrt{x_ {1}}\right)\) in Equation (21) approaches zero, and the term \(\frac{\overline{P}-x_{1}}{x_{2}-x_{1}}\widetilde{\mathrm{SER}}\left(\sqrt{x_ {2}}\right)\) in Equation (21) converges to \(\frac{M-1}{M}\), which explains the above observation. Another interesting observation is that the maximum \(\widetilde{\mathrm{SER}}(x_{1},x_{2})\) is a monotonically decreasing function in \(|h|\). This is because \(\widetilde{\mathrm{SER}}(x_{1},x_{2})\) in Equation (24) is a linear combination of \(\widetilde{\mathrm{SER}}\left(\sqrt{x_{1}}\right)\) and \(\widetilde{\mathrm{SER}}\left(\sqrt{x_{2}}\right)\), each of which is a monotonically decreasing function of \(|h|\) as previously shown. Furthermore, maximizing \(\widetilde{\mathrm{SER}}(x_{1},x_{2})\) with respect to \(x_{1}\) and \(x_{2}\) preserves the monotonicity. Similarly, it can be shown that \(\widetilde{\mathrm{SER}}(x_{1},x_{2})\) in Equation (24) is a monotonically increasing function of \(|g|\). Therefore, by applying the optimal AN design, the SER at the relay increases with the relay-to-destination link quality and decreases with the source-to-relay link quality. The optimal values of \(x_{1}\) and \(x_{2}\) maximizing \(\overline{\text{SER}}(x_{1},x_{2})\) in Equation (24), denoted by \(x_{1}^{*}\) and \(x_{2}^{*}\), can be computed numerically based on \(h\), \(g\), \(\overline{P}\), \(M\), \(a\), and \(\sigma\). Substituting \(x_{1}^{*}\) and \(x_{2}^{*}\) back into Equation (24) yields the maximum expected SER, which is denoted as \(\overline{\text{SER}}_{\text{max}}\). For example, in the case of \(h=1\), \(g=1\), \(\overline{P}=3.9811\), \(M=16\), \(a=\sqrt{10}\), and \(\sigma=1/\sqrt{2}\), the optimal values can be computed as \(x_{1}=0\) and \(x_{2}=13.7098\), yielding \(\overline{\text{SER}}\left(\sqrt{x_{1}}\right)=0\) and \(\overline{\text{SER}}\left(\sqrt{x_{2}}\right)=0.5832\). The corresponding maximum expected SER is \(\overline{\text{SER}}_{\text{max}}=0.1694\). Interestingly, if we use a deterministic power usage \(\overline{P}\), i.e., the AN PDF written as \(f(x)=\delta(x-\overline{P})\), the corresponding expected SER is 0.0371. Therefore, applying Theorem 1 at the destination yields a SER increment of 357% at the relay compared with the deterministic power usage.