# Securing Relay Networks with Artificial Noise: An Error Performance-Based Approach +
## 2 System Model and Problem Formulation  
Let us consider a relay communication channel as shown in Figure 1. A legitimate transmitter (the source) sends information symbols to a legitimate receiver (the destination) assisted by an untrusted relay. We assume that a direct link between the legitimate terminals is not available due to, for example, high path loss. All the terminals are configured with a single antenna (Note that this work can be generalized to the scenario where at least one of the source and destination has multiple antennas. Whilst beyond the scope of this work, the corresponding precoding design at each terminal remains an interesting topic for future extension). We assume quasi-static Rayleigh fading channels and the channel coefficients for the source-to-relay and relay-to-destination links are respectively denoted by \(h\) and \(g\), where \(h\sim\mathcal{CN}\left(0,\sigma_{h}^{2}\right)\) and \(g\sim\mathcal{CN}\left(0,\sigma_{g}^{2}\right)\). All channels are assumed to be reciprocal [42] and constant within the transmission duration from the source to the destination. The transmission from the source to destination can be partitioned into two time slots. During the first time slot, the source broadcasts the symbol \(m\). Meanwhile, to prevent the relay from deciphering the information, the destination sends the AN symbol \(z\) to increase the noise level at the relay. For this purpose, all the terminals are assumed to be perfectly synchronized. Similar to physical-layer network coding [43], the synchronization is an important issue, which requires further investigation. At the end of the first time slot, the received baseband signal at the relay can be mathematically expressed as,  
\[y=hm+gz+n \tag{1}\]  
where the additive white Gaussian noise (AWGN) symbol \(n\) is assumed to be zero mean and have the one-sided power spectral density \(N_{0}\) in Watts/Hz. During the second time slot, the source is muted, and the relay applies the amplify-and-forward protocol [37] to forward a scaled version of its received signal to the destination. Upon receiving this signal, the destination removes the AN \(z\) prior to decoding \(m\). For the afore-mentioned communication protocol, the destination is required to have the perfect CSI of \(h\) and \(g\) in order to maximize the system's transmission rate [44]. To be more specific, the destination performs the channel estimation of the relay-to-destination link and the compound source-to-relay-to-destination link, and the channel coefficient of the source-to-relay link is then obtained [40]. While exploiting the untrusted relay to help the data transmission, the system is designed to ensure that the untrusted relay cannot decipher the source's symbol \(m\). In addition, we assume that the relay can obtain knowledge about the modulation scheme of the source, e.g., by tracking the common control channel of the network. Moreover, it is important to note that the AN \(z\) in Equation (1) is designed and known solely by the destination so that the relay is not able to distinguish between the AN and the regular channel noise \(n\). Therefore, when decoding the symbols transmitted by the source, we assume that the relay treats the AN from the destination as noise. Furthermore, we assume that the source transmits demodulation reference signals (DM-RS) [36] so that the relay can perfectly estimate \(h\) to perform ideal coherent demodulation as,  
\[\tilde{y}=\frac{h^{*}}{|h|}y=|h|m+\frac{h^{*}g}{|h|}z+\frac{h^{*}}{|h|}n. \tag{2}\]  
Forwarded by the relay, the destination also has access to \(h\) and the DM-RS of the source in order to demodulate the signal \(m\). For the source's symbol \(m\), we consider square \(M\)-QAM modulation types with \(M=4^{k}\) and \(k=1,2,\dots\), which are frequently used in the current and upcoming communication standards [36]. We also make the common assumption that constellation points are uniformly distributed [45].