# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 5 Numerical and Simulation Results  
In this section, we present simulation results to validate the obtained analytical results derived in the previous sections. In all figures, the average SNR to the transmitted information symbol is chosen as \(E_{m}/N_{0}=10\) dB. In order to depict the SER performance at the relay using the optimal AN design in Section 3, we plot Figure 8, where the signal symbol \(m\) is randomly selected from the square 4-QAM constellation, and the channel gains \(g\) and \(h\) are simply assigned as 1. We observe that all analytical results coincide very well with the corresponding simulation results. For the sake of comparison, the expected SER values for Gaussian AN are also plotted in Figure 8. In the Gaussian case, we assume that the AN symbol \(z\) is generated according to a complex Gaussian distribution with the same average symbol energy \(E_{z}\). The line denoted as \"non-informative\" represents the non-informative SER performance, which has been defined after Equation (17). The curve \"w/o AN\" in the figure depicts the SERs at the relay without AN, which lie at a value of 0.0016 for the given parameters. Figure 8 shows that the SER at the relay can be significantly increased by applying our AN design. Even when the ANR \(E_{z}/N_{0}\) is small, e.g., \(E_{z}/N_{0}=2\) dB, the SER is increased from 0.0016 to 0.05, i.e., an increment of 303%. Moreover, Figure 8 clearly demonstrates that the Gaussian distribution is not optimal for the AN symbol generation, since our scheme described in Section 3 can induce much larger SER at the relay. For the case shown in Figure 8, if the ANR \(E_{z}/N_{0}\) is above 16 dB, the maximum SER, i.e., \(\overline{\text{SER}}_{\text{max}}\), achieves \(3/4\). This fact indicates that if the AN \(z\) is properly generated and \(E_{z}\) is above a certain threshold, applying ideal coherent demodulation at the relay does not lead to better performance than the non-informative case, and secure data transmission can therefore be ensured by the proposed AN design. Figure 9 shows the SER performance for square 16-QAM constellations. Comparing Figure 9 with Figure 8, similar observations can be made as for the case of 4-QAM. Now, we investigate the ASER performance at the relay using the optimal AN design in Section 4. Figure 10 depicts the ASER performance for square 4-QAM and 16-QAM signals. In this figure, for each QAM constellation, the curve \"\(\overline{\text{ASER}}_{\text{max}}\) Ana\". is computed by numerically maximizing \(\overline{\text{ASER}}(x_{1},x_{2})\) in Equation (26) with respect to \(x_{1}\) and \(x_{2}\). The maximization revealed that the optimal power allocation is \(x_{2}^{*}=\overline{P}\) with probability \(p=1\) for both of 4-QAM and 16-QAM signals. In other words, different from the AN design based on the instantaneous CSI in Section 3, constant power usage is optimal to design the AN based on the statistical CSI. Intuitively, due to the averaging effect over several channel realizations, the ASER performance is not sensitive to the different instantaneous power allocations. For comparison, the ASER curves (\(\overline{\text{ASER}}_{\text{max}}\) Ana./\(\overline{\text{ASER}}_{\text{max}}\) Sim.) for various power distributions are plotted, i.e., the uniform and the exponential power distribution. For each curve, the numerical results are obtained from computing \(\int_{0}^{\infty}\text{ASER}(\sqrt{x})f(x)\text{d}x\), where,  
\[f(x)=\frac{1}{2\overline{P}},\ \ \text{for}\ 0\leq x\leq 2\overline{P} \tag{27}\]  
and,  
\[f(x)=\frac{1}{\overline{P}}\exp\left(-\frac{x}{\overline{P}}\right),\ \ \text{for}\ x\geq 0 \tag{28}\]  
are the PDFs of the AN power with uniform and exponential distributions, respectively. Note that there exist numerous AN designs with exponentially distributed power, such as the Gaussian distributed AN. Figure 10 shows that our proposed scheme with the SER-maximizing power distribution yields the largest ASER at the relay.