# Secure PD-NOMA with Multi-User Cooperation and User Clustering in Both Uplink and Downlink PD-NOMA
## 2 System Model for Multi-User Downlink and Uplink PD-NOMA
### Multi-User Downlink PD-NOMA  
The multi-user downlink PD-NOMA is shown in Figure 1. A gNodeB is used to superpose and transmit multiple distinct signals over the same frequency and time but with different powers non-orthogonally. For simplicity, we consider four users in a single cluster surrounding the gNodeB. All the four users \(R_{1}\), \(R_{2}\), \(R_{3}\), and \(R_{4}\) have distinct channel gains. At each receiver, the desired signal is decoded once all the high-powered signals are eliminated from the superposed signal using the SIC technique. In the downlink PD-NOMA, the signals of strong users are assigned low power, while the signals of weak users are assigned high power. At the gNodeB, all the users' signals are combined, and the composite signal \(X\) can be given as follows.  
\[X(t)=\sum_{i=1}^{4}\sqrt{P_{i}}x_{i}(t), \tag{1}\]  
where \(P_{i}\) is the power factor that has been normalized, and \(x_{i}(t)\) is the signal for the \(i\)th user. The gNodeB total transmit power is denoted by \(P_{b}\), and the sum of the power assigned to all users is less than or equal to the total transmit power, such as,  
\[P_{1}+P_{2}+P_{3}+P_{4}\leq P_{b}. \tag{2}\]  
Allocation of power is based on the distances and channel gains of each user. Users \(R_{1}\), \(R_{2}\), \(R_{3}\), and \(R_{4}\) have channel gains in the order of \(h_{1}>h_{2}>h_{3}>h_{4}\). Depending on the channel conditions, the power factors are in the range of \(P_{1}<P_{2}<P_{3}<P_{4}\).  
After passing through the channel, the received signal \(Y(t)\) for each user is given as follows.  
\[Y_{i}(t)=h_{i}X(t)+n_{i}(t), \tag{3}\]  
Figure 1: Block diagram of multi-user downlink PD-NOMA with the channel gains order \(h_{1}>h_{2}>h_{3}>h_{4}\).  
where \(h_{i}\) is the channel gains, which are the reciprocal of the path losses, and \(n_{i}\)(t) represents the additive white Gaussian noise (AWGN) for the \(i\)th user. The Path Loss (PL) is employed in the New York University simulator model (NYUSIM), which is expressed as follows.  
\[PL(f,d)[dB]=FSPL(f,1m)[dB]+10 nlog_{10}(d)+AT[dB]+X_{\sigma}, \tag{4}\]  
where \(f\) and \(d\) are the frequency in gigahertz and the 3D distance between gNodeB and the \(i\)th user, respectively. \(n\) is the path loss exponent, and \(AT\) is the attenuation factor by the atmosphere. \(X_{\sigma}\) is a Gaussian random variable having a mean of zero and a standard deviation of \(\sigma\) dB. The \(FSPL\) (\(f,1m\)) is the free space path loss in decibels at the carrier frequency \(f\) and a separation distance of 1 meter. This parameter is defined by,  
\[FSPL(f,1m)[dB] =20 log_{10}(\frac{4\pi\times f\times 10^{9}}{c}), \tag{5}\] \[=32.4[dB]+20 log_{10}(f),\]  
where, \(c\) denotes the speed of light. To decode user signals, the channel gains (\(h_{i}\)) order are used. The highest channel gain user will decode all the users' signals which are lower in channel gains. For the channel gain order, \(h_{1}>h_{2}>h_{3}>h_{4}\): \(R_{1}\) will perform three-level SIC to eliminate the effect of \(R_{2}\), \(R_{3}\), and \(R_{4}\) signals. Similarly, \(R_{2}\) will employ two-level SIC to eliminate \(R_{3}\) and \(R_{4}\) signals, \(R_{3}\) will employ single-level SIC to eliminate \(R_{4}\) signals, and \(R_{4}\) will decode its signal directly due to its low channel gain and high power. In the multi-user downlink scenario, the strong users eliminate the interference caused by the weak users through the SIC technique and hence experience the maximum throughput compared to the weak users. The throughput for user \(R_{i}\) in the multi-user downlink PD-NOMA scenario is given by [35].  
\[R_{i}=log_{2}\Big{(}1+\frac{P_{i}|h_{i}|^{2}}{\sum_{j=1}^{i-1}P_{j}|h_{i}|^{2 }+N_{i}}\Big{)},\qquad\forall\quad i=1,2,3,4. \tag{6}\]  
where \(N_{i}\) is the noise power spectral density for the \(i\)th user.