# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 3 AN Maximizing Relay's SER Performance
### SER Expression for a Given \(z\)  
The SER performance of the square \(M\)-QAM modulated signaling over AWGN [45] and over fading channels [46] has been widely studied. To provide a comprehensive study on the SER  
Figure 1: The considered system model in this work. performance at the relay, we first investigate the AN design using the instantaneous CSI in this section, and our SER analysis is based on the procedure presented in ([45], Chapter 5). As an example, the constellation diagram for the square 16-QAM modulation is illustrated in Figure 2. Denote the minimum distance between two constellation points as \(2a\) (\(a>0\)), then for a general \(M\)-QAM constellation, the average energy per signal symbol \(E_{m}\) can be expressed as ([45], Equation (5-2-76))  
\[E_{m}=\frac{2}{3}a^{2}T_{m}\left(M-1\right) \tag{3}\]  
where \(T_{m}\) denotes the symbol duration. Since the AN design in this section is based on \(h\) and \(g\), for simplicity of notation, we define \(s:=\frac{h^{*}g}{|h|}z\) and \(\bar{n}:=\frac{h^{*}}{|h|}n\) in Equation (2), and denote the real and imaginary parts of \(s\) and \(\bar{n}\) as \(s_{r}\), \(s_{i}\) and \(\bar{n}_{r}\), \(\bar{n}_{i}\), respectively. Thus, we have that \(s=s_{r}+js_{i}\) and \(\bar{n}=\bar{n}_{r}+j\bar{n}_{i}\) with \(j=\sqrt{-1}\). Note that the equivalent noise \(\bar{n}\) is identically distributed with \(n\), and thus \(\bar{n}\sim\mathcal{CN}(0,2\sigma^{2})\), where \(\sigma^{2}=\frac{N_{0}}{2T_{m}}\). For a vertex such as the point \"0\" in Figure 2, the received symbol \(\bar{y}\) in Equation (2) lies outside the decision region of the point \"0\" if \(s_{r}+\bar{n}_{r}\geq|h|a\) or \(s_{i}+\bar{n}_{i}\leq-|h|a\). Thus, given \(s\), the error probability of the point \"0\" is given by \(P_{e,0}=1-\left(1-\Pr\left(s_{r}+\bar{n}_{r}\geq|h|a\right)\right)\left(1-\Pr \left(s_{i}+\bar{n}_{i}\leq-|h|a\right)\right)\), which can be computed as,  
\[P_{e,0}=1-\left(1-Q\left(\frac{|h|a-s_{r}}{\sigma}\right)\right)\left(1-Q \left(\frac{|h|a+s_{i}}{\sigma}\right)\right) \tag{4}\]  
where \(Q\left(\cdot\right)\) is the Gaussian \(Q\)-function ([46], Equation (4.1)). Similarly, the error probability for the \(i\)th constellation point \(P_{e,l}\) (\(l=0,1,\ldots,M-1\)) is obtained by computing the probability that \(\tilde{y}\) in Equation (2) lies outside its decision region. Based on the assumption that the constellation points are uniformly distributed, for a given \(z\), or equivalently for a given \(s\), the SER of the square \(M\)-QAM signal at the relay can be derived by averaging the respective error probability expressions for all symbols under the assumption of a uniform symbol distribution, resulting in,  
\[\text{SER}\left(s\right)= c\bigg{[}Q\left(\frac{|h|a-s_{r}}{\sigma}\right)+Q\left(\frac{|h|a+s_{ r}}{\sigma}\right)+Q\left(\frac{|h|a-s_{i}}{\sigma}\right)+Q\left(\frac{|h|a+s_{ i}}{\sigma}\right)\bigg{]}\] \[-c^{2}\left[Q\left(\frac{|h|a-s_{r}}{\sigma}\right)+Q\left(\frac{ |h|a+s_{r}}{\sigma}\right)\right]\left[Q\left(\frac{|h|a-s_{i}}{\sigma}\right) +Q\left(\frac{|h|a+s_{i}}{\sigma}\right)\right] \tag{5}\]  
Figure 2: The constellation diagram of square 16-QAM with Gray encoding. QAM: quadrature amplitude modulation. where \(c=\frac{\sqrt{M}-1}{\sqrt{M}}\). Using Equation (3), we have that,  
\[\frac{a}{\sigma}=\sqrt{\frac{3E_{m}}{N_{0}\left(M-1\right)}}. \tag{6}\]  
Note that in the case of \(z=0\), the SER expression in Equation (5) coincides with that of the AWGN channel ([45], Equation (5-2-79)). Before proceeding, it is interesting to note that the SER expression in Equation (5) depends on the real and imaginary parts of \(s\), which motivates us to express \(s\) as,  
\[s=|g||z|\exp\left(j\theta\right) \tag{7}\]  
where we have expressed \(z\), \(g\) and \(h\) as \(z=|z|\exp(j\theta_{z})\), \(g=|g|\exp(j\theta_{g})\), \(h=|h|\exp(j\theta_{h})\), respectively, and \(\theta=\theta_{g}-\theta_{h}+\theta_{z}\).