# BER Minimization by User Pairing in Downlink NOMA Using Laser Chaos Decision-Maker
## 2 System Model and Problem Formulation  
In this paper, we consider a downlink single-cell NOMA system with one base station (BS) and multiple users. Figure 1 shows an overview of the NOMA system model considered in this study. We assume that the BS and all of the users have one antenna each. BS sends data to the users, and the users respond with ACK or NACK (negative acknowledgment) information to BS when the downlink data transfer is successful and non-successful, respectively.  
The outline of how the system works is described as follows. First, BS determines certain user pairings and transmits data to the users based on the determined pairing. The users receive and decode the data, followed by checking bit errors and sending a response to BS with ACK or NACK according to the bit errors. Next, BS updates user pairing based on the received ACK and NACK information. BS transmits data to users again based on the revised pairing. By repeating above processes, BS aims to achieve an optimal pairing to minimize BER in the NOMA system.  
Let \(U=\{U_{1},U_{2},\cdots,U_{i},\cdots,U_{j},\cdots,U_{N}\}\) be the set of \(N\) users in a circular cell and \(K=\{1,2,\cdots,k,\cdots,N/2\}\) be the index of each pair. \(N\) is the total number of users, which we assume is an even number. The total bandwidth is assumed to be \(B\). We assume that the bandwidth for each pair is equal. Then, the bandwidth for the \(k\)th pair can be represented as \(B_{k}=B/(N/2)\). The NOMA symbol \(x_{k}\) to the \(k\)th pair can be expressed as follows [38]:  
\[x_{k}=\sqrt{a_{k}P_{k}}x_{k}^{i}+\sqrt{(1-a_{k})P_{k}}x_{k}^{j}, \tag{1}\]  
where \(x_{k}^{i}\) and \(x_{k}^{j}\) are the signals for the \(i\)th and \(j\)th users that from the pair \(k\). \(x_{k}^{i}\)and \(x_{k}^{j}\) are \(\theta_{i}\) and \(\theta_{j}\) modulated by a certain modulation method and turned into OFDM symbols by IFFT, where \(\theta_{i}\) and \(\theta_{j}\) are data bit of the \(i\)th and \(j\)th users. \(a_{k}\) is the power allocation factor for the \(k\)th pair. We assume that the \(i\)th user is located closer to the BS than the \(j\)th user. \(P_{k}\) is the transmit power allocated to each pair for the \(k\)th pairs. The received signal \(y_{n}\) at the \(n\)th user in the \(k\)th pair can be expressed as follows [38]:  
\[y_{n}=d_{n}^{-\lambda}\hbar_{n}x_{k}+w_{n}, \tag{2}\]  
where \(d_{n}^{-\lambda}\) is the path loss between the BS and the \(n\)th user and \(\lambda\) is the path-loss exponent, \(\hbar_{n}\) is the Rayleigh fading of the \(n\)th, and \(w_{n}\) is the additive white Gaussian noise (AWGN) with zero mean and variance of \(\sigma_{n}^{2}\). \(\sigma_{n}^{2}\) can be expressed as \(\sigma_{n}^{2}=N_{0}B_{k}\), where \(N_{0}\) is the noise power spectral density. Herein, we assume that the farthest user in the \(x\)th user pairing is  
Figure 1: System model.  
the \(f\)th user. Then, the signal-to-noise ratio (SNR) of the farthest user \(U_{f}\) can be expressed as follows:  
\[\text{SNR}_{f}=\frac{|\hbar_{f}|d_{f}^{-\lambda}(1-a_{x})P_{x}}{|w_{f}|}. \tag{3}\]  
In this study, a laser chaos decision-maker is used in the BS to pair users. The object of our proposed scheme is to minimize the BER of the NOMA system by optimizing user pairing, which can be expressed as follows:  
\[\min_{v}\frac{\sum_{t=1}^{N}bit\text{-}error_{n}}{\sum_{t=1}^{N}\theta_{n}}, \tag{4}\]  
where \(bit\text{-}error_{n}\) denotes the number of bit errors at the \(n\)th user. \(v\) denotes the user pairing combinations, which will be explained in Section 4.