## 2 System Model  
The system model of downlink MU system with combination of precoding and NOMA is presented in Figure 1, the BS is equipped with \(M\geq 2\) antennas and zero-forcing based precoding (ZFBP) method to serve the \(M\) user clusters, while every user has only a single antenna because of its limited size. All users are simply clustered by user location trackingalgorithms or spatial direction method such as Global Positioning System (GPS) technique. The number of users in every cluster, \(N\), is assumed to be the same. The precoding matrix \(\mathbf{w}_{m}\) according to ZFBP method is designed for the \(m\)th cluster in order to remove the inter-cluster interference as follows. The \(n\)th user in the \(m\)th cluster is called the \((m,n)\)th user, and the channel matrix between the \((m,n)\)th user and the BS is denoted by \(\mathbf{h}_{m,n}=[h_{m,n,1},h_{m,n,2},\cdots,h_{m,n,M}]^{T}\in\mathbb{C}^{M \times 1}\) with \(m\in\{1,\cdots,M\}\) and \(n\in\{1,\cdots,N\}\). \(\mathbf{h}_{m,n,i}\sim\mathcal{CN}(0,\Omega_{m,n})\) denotes the channel coefficient, and \(\mathbb{E}\{|\mathbf{h}_{m,n,i}|^{2}\}=\Omega_{m,n}\) is the variance of the channel gain. All channels are assumed to be quasi-static independent and identically distributed (i.i.d.) and block fading. For the purpose of maximizing the signal to noise ratio (SNR), the precoding matrix \(\mathbf{w}_{m}\) can be expressed as the projection of \(\mathbf{h}_{m}=[\mathbf{h}_{m,1},\mathbf{h}_{m,2},\cdots,\mathbf{h}_{m,N}]\in \mathbb{C}^{M\times N}\) in the direction of interference user in the \(m\)th cluster. Mathematically, the precoding matrix \(\mathbf{w}_{m}\) is represented in Equation (1). \[\mathbf{w}_{m}=\frac{\mathbf{B}_{m}\mathbf{h}_{m}}{\|\mathbf{B}_{m}\|_{\mathbf{ h}_{m}}\|}, \tag{1}\]  
where \(\mathbf{B}_{m}=\mathbf{I}_{N}-\mathbf{H}_{m}(\mathbf{H}_{m}^{H}\mathbf{H})_{m}^ {-1}\mathbf{H}_{m}^{H}\), \(\mathbf{I}_{N}\) denotes a \(N\times N\) identity matrix and the structure of channel matrix \(\mathbf{H}_{m}\) is given in detail as follows in Equation (2):  
\[\mathbf{H}_{m}=[\mathbf{h}_{1},\mathbf{h}_{2},\cdots,\mathbf{h}_{m-1}, \mathbf{h}_{m+1},\cdots,\mathbf{h}_{M}]^{T}. \tag{2}\]  
We assume that the CSI and the precoding at the BS are perfect. Hence, we have \(\mathbf{h}_{m}^{H}\mathbf{w}_{j}=0,\forall m\
eq j\). It means that, the ZFBP lets the complexity of the system be high. However, all users in the \(m\)th cluster don't receive signals of user in the other clusters, they receive signals of all users in the same cluster. Consequently, the NOMA method should be applied to cancel the inter-user interference (IUI). Based on the principle of NOMA method, all users are assumed to determine the order of decoding signal according to the channel gains. Without loss of generality, the distances between the BS and users are assumed to be sorted as \(d_{m,1}>d_{m,2},\cdots,>d_{m,N}\), therefore, the channel gains of these users satisfies the following condition in Equation (3). \[|\mathbf{w}_{m}\mathbf{h}_{m,1}|^{2}\leq|\mathbf{w}_{m}\mathbf{h}_{m,2}|^{2} \leq,...,\leq|\mathbf{w}_{m}\mathbf{h}_{m,N}|^{2}. \tag{3}\]  
Figure 1: System Model. Hence, according to the principle of NOMA, the power allocation coefficient of every user is sorted by Equation (4):  
\[a_{m,1}\geq a_{m,2}\geq...\geq a_{m,N}. \tag{4}\]  
The superposition code which consists of signals of all users in the \(m\)th cluster is defined by \(\mathbf{x}_{S,m}=[x_{m,1},\cdots,x_{m,N}]^{T}\), where \(x_{m,n}\) denotes the signal of the \((m,n)\)th user, and is normalized with zero mean unit variance. The superposition code are multiplied by the precoding matrix, \(\mathbf{w}_{m}\), at the output antenna following the principle of ZF method. The transmit power of all antennas of BS is fixed, \(\mathbb{E}\{|\mathbf{x}_{S,m}|^{2}\}\ =\ P_{\text{S}}\), the power allocation coefficient for the \((m,n)\)th user is denoted by \(a_{m,n}\) and satisfies condition \(\sum_{n=1}^{N}a_{m,n}=1\).