# Uplink NOMA-MIMO Systems with Rayleigh Power Distribution
## 2 System Model  
Let us consider a multiuser system with a MIMO channel, in which every \(u\)-th user has \(N_{tx,u}\) transmitting antennas. The base station has \(N_{rx}\) receiving antennas. Each user uses spatial multiplexing with the maximum use of the number of transmitting antennas, i.e., the number of multiplexed streams of each user is equal to the number of transmitting antennas. In this case, the system model is described by the following equation:  
\[Y=\sum_{u=1}^{U}\frac{\sqrt{P_{u}}}{\sqrt{N_{tx,u}}}H_{u}X_{u}(\Theta_{u})+\eta \tag{1}\]  
where \(U\)-number of users; \(P_{u}\), \(N_{tx,u}\)-power and number of transmitting antennas of \(u\)-th user; \(Y\)-complex vector of received signal dimension of \(N_{rx}\); \(H_{u}\)-matrix of \(u\)-th user MIMO channel dimension of \((N_{rx}\times N_{tx,u})\); \(X_{u}(\Theta_{u})=\left[ \begin{array}{cccc}x_{1,u}(\theta_{1,u})&x_{2,u}( \theta_{2,u})&\ldots&x_{N_{tx,u},\mu}(\theta_{N_{tx,u},\mu})\end{array} \right]^ {T}\)-vector of QAM symbols of \(u\)-th user dimension of \(N_{tx,u}\), where each element is a function \(x_{i,u}(\theta_{i,u})\) of discrete parameter \(\theta_{i,u}\in\Omega_{i,u}\), \(\Omega_{i,u}\)-set of values of \(i\)-th discrete parameter of \(u\)-th user, \(i=\overline{1,N_{tx,u}}\), \(u=\overline{1,U}\); \(\Theta_{u}=\left[ \begin{array}{cccc}\theta_{1,u}&\ldots&\theta_{N_{tx,u}, \mu}\end{array} \right]^{T}\)-vector of random discrete values dimension of \(N_{tx,u}\); \(\eta\sim\mathbb{CN}\Big{(}0,2\sigma_{\eta}^{2}I_{N_{rx}}\Big{)}\)-complex vector of additive white Gaussian noise in channel dimension of \(N_{rx}\). Functional dependencies \(x_{i,u}(\cdot)\) describing the constellations of used QAM symbols can be either the same for all \(i\) and \(u\) or different. The discrete parameter \(\theta_{i,u}\) is a function of binary symbols vector (information bits). The dimension of the set \(\Omega_{i,u}\) is determined by the number of bits \(k_{b,i,u}\) transmitted by the \(i\)-th symbol of \(u\)-th user and is equal to \(2^{k_{b,i,u}}\). For future simplification, we assume that the number of transmitting antennas for each user is the same \(N_{tx,u}=N_{tx}\), in addition, all users use the same power of the emitted signals and the same constellations, i.e., \(P_{u}=P\), \(x_{i,u}(\theta_{i,u})=x(\theta_{i,u})\), \(k_{b,i,u}=k_{b}\), \(\Omega_{i,u}=\Omega\). Using this simplification, Equation (1) can be rewritten as follows:  
\[Y=\frac{\sqrt{P}}{\sqrt{N_{tx}}}\mathbf{H}X(\Theta)+\eta \tag{2}\]  
where \(\mathbf{H}\triangleq\left[ \begin{array}{cccc}H_{1}&\ldots&H_{U}\end{array} \right]\) -extended matrix of uplink multiuser MIMO channel, \(X(\Theta)\triangleq[x(\theta_{1,1})\ldots x(\theta_{N_{tx,1}})x(\theta_{1,2} )\ldots x(\theta_{N_{tx,2}})\ldots x(\theta_{1,U})\ldots x(\theta_{N_{tx,U}},\mu)]\)T-extended vector of QAM symbols of all users dimension of \(N_{tx,\Sigma}=U\!N_{tx}\). The number of possible vector states \(X(\Theta)\in\Omega^{N_{tx,\Sigma}}\) is equal to the number of states of the total vector of discrete values, which is equal to \(M=2^{N_{tx,\Sigma}k_{b}}=2^{N_{tx,b}U}\). Capacity of MIMO channel obtained in [7; 8] is determined by the mutual information of the continuous channel:  
\[C=\underset{p(X)}{\max}I(X;Y) \tag{3}\]where \(p(X)\)-a priori distribution of continuous vector \(X\); \(I(X;Y)\)-mutual information between the input vector \(X\) and output vector of MIMO-channel \(Y\), which is defined as:  
\[I(X,Y)=\int\limits_{Y}\int\limits_{X}p(Y|X)p(X)\log\biggl{(}\frac{p(Y|X)}{p(Y)} \biggr{)}dYdX \tag{4}\]  
Maximization in Equation (3) is implemented over all possible a priori distributions. As is known, the maximum of this expression is observed with a Gaussian a priori distribution.