# Physical-Layer Security in Power-Domain NOMA Based on Different Chaotic Maps
## 2 System Model  
This section describes the SISO DL power-domain NOMA system model (Figure 1). One-cell NOMA combines M users and one BS, each with one antenna. The channel response is perfectly known at the base station. The BS transmits the superposed signal to all mobile users on the transmitter side of the DL-NOMA system; this is a superposition of several users' required signals with varying fixed power-coefficient allocations based on their distances from the BS [13]. The SIC procedure is supposed to be implemented sequentially on each user's receiver side until the user's signal is retrieved. The user with the greatest transmission power retrieves its signal with no execution of any SIC procedure and treats the other users' signals as noise. However, other users must perform the SIC procedure. In the SIC procedure, every user's receiver first discovers signals that are stronger than the user's required signal. These signals are then subtracted from the received signal, and the procedure is repeated until the user's required signal is identified. Finally, through consideration of users with lower power coefficients as noise, every user decodes its signal [5]. As in [13], the transmitted superposed signal at the BS is expressed as follows:  
\[x_{s}=\sum_{i=1}^{M}\sqrt{a_{i}P_{t}}\;x_{i} \tag{1}\]  
where \(x_{i}\) is the data of user \(i\) (\(U_{i}\)) with unit energy, \(P_{t}\) is the transmitted power at the BS, and \(a_{i}\) is the power-allocation coefficient for user \(i\); the allocation of power levels depends on the distance of every user so that \(a_{i}^{\prime}=\frac{d_{i}}{d_{max}}\) (\(i=1,\;2,\;\ldots M\)), where \(a_{i}^{\prime}\) is the absolute power factor and \(d_{i}\) is the distance between the BS and the \(ith\) user, with \(d_{max}=200\) m (a typical BS radius in 5G networks). The power factors are normalized  
Figure 1: SISO DL-NOMA system model. between 0 and 1 as follows: \(a_{i}=\frac{a_{i}^{\prime}}{\sum\limits_{n=1}^{M}a_{i}^{\prime}}\) (\(v=1,\ 2,\ \dots M\)) [14]. In other words, \(\sum\
olimits_{i=1}^{M}a_{i}=1\) and \(1>a_{1}\geq\ a_{2}\geq\dots\ a_{M}>0\) because without loss of generality, the channel gains are supposed to be arranged as \(\left|h_{1}\right|^{2}\ \leq\ \left|h_{2}\right|^{2}\ \leq\ \dots\ \left|h_{M}\right|^{2}\), where \(h_{i}\) is the Rayleigh-fading channel coefficient among the BS and the \(m^{\text{th}}\) user. As in [13], the received signal at the \(m^{\text{th}}\) user can be written as follows:  
\[y_{m}=h_{m}x_{s}+n_{m}=h_{m}\sum\limits_{i=1}^{M}\sqrt{a_{i}P_{i}}x_{i}+n_{m} \tag{2}\]  
where \(n_{m}\) is complex, zero-mean AWGN with a variance of \(\sigma^{2}\) and is denoted as \(n_{m}\ \sim\ CN\ \big{(}0,\sigma^{2}\big{)}\). Notably, users use SIC to obtain desired signals from the received signal, \(y_{m}\). In particular, \(U_{m}\) starts by decoding the stronger \(U_{1}\) signal through treatment of the remaining part of the signal as interference. Thereafter, to acquire a signal free of the \(U_{1}\) signal, \(U_{m}\) remodulates the decoded signal and removes it from the received signal, \(y_{m}\). Executing an identical process, \(U_{m}\) progressively cancels the whole signals corresponding to users \(U_{2}\), \(U_{3}\),..., \(U_{m-1}\) from the received signal, \(y_{m}\), lastly decoding its signal through treatment of the rest of the signal parts, corresponding to users \(U_{m+1}\), \(U_{m+2}\),..., \(U_{M}\), as interference [15]. From Equation (2), the \(m^{\text{th}}\) user's signal to interference and noise ratio (SINR) is used to discover the \(j^{th}\) user, \(j\ \leq\ m\), with \(j\ \
eq\ M\), expressed as follows [13]:  
\[SINR_{j\to m}=\frac{a_{j}\rho_{s}\left|h_{m}\right|^{2}}{\rho_{s}\left|h_{m} \right|^{2}\sum\
olimits_{i=j+1}^{M}a_{i}+1} \tag{3}\]  
where \(\rho_{s}=P_{t}/\sigma^{2}\) indicates the signal-to-noise ratio (SNR). To obtain the \(m^{\text{th}}\) user's required data, the SIC procedure is performed on the signal of user \(j\ \leq\ m\).