# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 4 AN Maximizing Relay's ASER Performance
### ASER Expression for a Given \(z\)  
In this section, the channel coefficients \(h\) and \(g\) are assumed to be random variables. Moreover, the envelope of the relay-to-destination channel coefficient \(g\) is assumed to be Rayleigh-distributed. For a given AN symbol \(z\), the received AN at the relay \(gz\) can be easily shown to be distributed as \(gz\sim\mathcal{CN}\left(0,\sigma_{g}^{2}|z|^{2}\right)\). This indicates that the received AN at the relay is an extra source of the AWGN \(n\). Hence, as can be observed from the signal model in Equation (1), the overall received noise at the relay is a superposition of the AN and the natural AWGN. The relay-to-destination channel coefficient \(g\) and the AWGN symbol \(n\) are independent, and it can be easily shown that the received noise \(gz+n\) is distributed as \(gz+n\sim\mathcal{CN}\left(0,\sigma_{g}^{2}|z|^{2}+2\sigma^{2}\right)\), where \(2\sigma^{2}\) is the variance of the AWGN \(n\). For Rayleigh-faded source-to-relay channel gain \(|h|\) and the overall noise symbol \(gz+n\), the performance of the \(M\)-QAM modulation at the relay can be obtained using ([46], Equation (8.106)) as,  
\[\text{ASER}(z)=2c\left(1-\sqrt{\frac{1.5\overline{\gamma}_{s}(z)} {M-1+1.5\overline{\gamma}_{s}(z)}}\right)-c^{2}\bigg{[}1- \sqrt{\frac{1.5\overline{\gamma}_{s}(z)}{M-1+1.5\overline{\gamma}_{s}(z)}}\] \[\times\left(\frac{4}{\pi}\tan^{-1}\sqrt{\frac{M-1+1.5\overline{ \gamma}_{s}(z)}{1.5\overline{\gamma}_{s}(z)}}\right)\bigg{]} \tag{25}\]  
where \(\overline{\gamma}_{s}(z)=\frac{\sigma_{s}^{2}E_{m}}{\sigma_{s}^{2}|z|^{2}+2 \sigma^{2}}\) represents the average received SNR per symbol.  
Similar to Propositions 1 and 2 made on Equation (8), the effects of channel statistics on the ASER given by Equation (25) can be summarized as follows.  
**Proposition 3** (ASER monotonicity): _The ASER in Equation (25) is a monotonically increasing function of \(\sigma_{S}^{2}\), as well as a monotonically decreasing function of \(\sigma_{h}^{2}\)._  
This proposition is intuitive and is similar to Propositions 1 and 2 in Section 3: Stronger relay-to-destination channels help to deteriorate the ASER performance at the relay, whereas stronger source-to-relay channels improve the average decoding performance at the relay.  
**Proposition 4** (high signal and AN power performance): _At high SNR and high ANR, i.e., \(E_{m}\gg\sigma^{2}\) and \(|z|^{2}\gg\sigma^{2}\), \(\overline{\gamma}_{s}(z)\approx\frac{\sigma_{h}^{2}}{\sigma_{S}^{2}}\frac{E_{m }}{|z|^{2}}\)._  
Proposition 4 reveals that provided an adequately high signal and AN power is available, the _relative_ strength of the source-to-relay and relay-to-destination links affects the ASER performance of the QAM signals at the relay.  
**Proposition 5** (high AN power performance): _In the high AN power regime, i.e., for \(|z|^{2}\to\infty\), \(\overline{\gamma}_{s}\to 0\), we have \(\lim_{|z|\to\infty}\mathrm{ASER}(z)=2c-c^{2}=\frac{M-1}{M}\)._  
This proposition shows that given sufficiently high AN power, the ASER at the relay is close to the non-informative error performance.  
Furthermore, we observe that the ASER expression in Equation (25) is independent of the phase of the AN, which is different from the AN design in Section 3 in the case of instantaneous CSI. Following a similar approach as in Section 3.3, our objective in the following subsection is to derive the SER-maximizing power distribution for the AN under an average power constraint.