# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 3 AN Maximizing Relay's SER Performance
### Selecting \(\theta_{z}\) for a Fixed \(|z|\)  
The focus of this subsection is to determine the optimal phase distribution of the AN symbol \(z\). In this case, the power \(|z|^{2}\) of the AN is assumed to be fixed. Then, for simplicity, we can omit \(|z|\) in the argument of \(\mathrm{SER}(\theta,|z|)\) in Equation (8) and denote the SER expression as \(\mathrm{SER}(\theta)\) in the remainder of this subsection. Due to the \(\frac{\pi}{4}\)-periodicity of the function \(\mathrm{SER}\left(\theta\right)\) and due to its symmetry, we only consider the interval \(\theta\in\left[0,\frac{\pi}{4}\right]\) to determine the maximum of \(\mathrm{SER}\left(\theta\right)\). We first numerically verify the following result. _Result 1:_\(\mathrm{SER}\left(\theta\right)\) is a quasi-convex function of \(\theta\) for all \(\theta\in\left[0,\frac{\pi}{4}\right]\), and either \(\theta=0\) or \(\theta=\frac{\pi}{4}\) maximizes the value of \(\mathrm{SER}\left(\theta\right)\). To illustrate this result, consider the constellation point \"\(5\)\" in Figure 2 as an example. Given the power of the AN symbol \(z\), in order to maximize the expected SER with square \(16\)-QAM modulation at the relay according to _Result 1_, the destination can either allocate all the power \(|z|^{2}\) to the direction of \"\(12\)\" or \"\(13\)\", or equally distribute this power between the directions of \"\(12\)\" and \"\(13\)\". Due to the similarity of this AN phase design to the constellation diagram, we refer to the case of \(\theta=0\) as the _rotated QAM_ phase selection and the latter case of \(\theta=\frac{\pi}{4}\) as the _QAM_ phase selection. Accordingly, the optimal phase selection of the AN \(z\) is computed as \(\theta_{z}=\theta_{h}-\theta_{\delta}\) or \(\theta_{z}=\frac{\pi}{4}+\theta_{h}-\theta_{g}\). Next we address the question as to in which case using the rotated QAM constellation for the AN maximizes the SER at the relay and in which case the \(M\)-QAM constellation is optimal. Towards this aim, we obtain Figure 3, which shows \(|z|/\sigma\) as a function of \(a/\sigma\). Note that \(|z|/\sigma\) and \(a/\sigma\) correspond to the levels of AN-to-natural noise ratio (ANR) and signal-to-noise ratio (SNR), respectively. Each curve in this figure is plotted by numerically solving the equation \(\mathrm{SER}\left(\pi/4\right)=\mathrm{SER}\left(0\right)\) for \(|z|/\sigma\) for given values of \(a/\sigma\). The QAM region and the rotated QAM region denote the regions of \(\mathrm{SER}\left(\pi/4\right)>\mathrm{SER}\left(0\right)\) and \(\mathrm{SER}\left(\pi/4\right)<\mathrm{SER}\left(0\right)\), respectively. Therefore, Figure 3 displays phase selection thresholds for different system parameters. In this figure, \(a/\sigma\) and \(|z|/\sigma\) indicate the energy of the source signal and the energy of the AN z, respectively. We focus on the relation between the signal energy and the AN energy, and thus ignore the effect of channel gains by assuming \(h=1\) and \(g=1\). The curves in Figure 3 show that for a given signal power, there exists a threshold for the AN power, under which the optimal phase selection is \(\theta=0\), and above which the optimal phase selection is \(\theta=\pi/4\). To explain the intuition behind this, consider the constellation point \"5\" in Figure 2. With the purpose of maximizing the SER performance at the relay, if the power level of the AN \(|z|/\sigma\) is low (as compared to a given signal power \(E_{m}\)), the AN transmitted from the destination may deviate the equalized information signals received at the relay towards the adjacent constellation points \"1\", \"4\", \"13\", and \"7\", whilst if the power level of the AN \(|z|/\sigma\) is sufficiently high, the AN may shift the receive symbol towards the farther constellation points \"0\", \"12\", \"3\", and \"15\". From Figure 3, one can also observe that for small values of \(a/\sigma\), the values of the threshold are very close to zero. This fact indicates that QAM phase selection is the preferable scheme if the received SNR at the relay is low.