# Uplink NOMA-MIMO Systems with Rayleigh Power Distribution
## 1 Introduction  
Non-Orthogonal Multiple Access (NOMA) is a key technology for future wireless systems including 6G mobile networks [1; 2; 3; 4; 5; 6]. Compared with orthogonal multiple access technologies, NOMA have a new parameter of communication system-load factor.  
The load factor is a ratio between the number of subscribers (or the total number of data streams distributed using resources) with the number of orthogonal resources used. For systems with orthogonal access, it does not exceed 1 (<100%). For NOMA systems in full load mode, it must be greater than 1 (>100%). Therefore, for NOMA systems, this indicator is more often referred to as the overload coefficient of the communication system.  
MIMO (Multiple-Input Multiple-Output) is another known and efficient technology widely used in modern mobile communication systems [4; 5; 6; 7; 8; 9; 10; 11]. MIMO technology improves energy and spectral efficiency and allows an increase in the capacity and bandwidth of wireless systems and the data rate of each user.  
The load factor also can be used for the analysis single-user mode MIMO (Single User MIMO-SU-MIMO) and for multiuser mode MIMO (Multiuser MIMO-MU-MIMO) systems as well as for NOMA systems [12]. It is shown in [8] that MIMO channel can be represented as a combination of orthogonal channels, i.e., orthogonal resources, with different characteristics.  
The number of such orthogonal resources is determined by the minimum number \(\min(N_{tx,\Sigma},\,N_{rx,\Sigma})\), where \(N_{tx,\Sigma}\), \(N_{rx,\Sigma}\) is the total number of transmitting and receiving antennas in the MIMO system, respectively. In the single-user mode, overloading means that the number of streams distributed over the transmitting antennas is greater than the number of receiving antennas. In the multiuser mode, overloading occurs when the total number of distributed streams across all transmitting antennas is greater than the number of receiving antennas \(N_{tx,\Sigma}\) at the base station.  
Theoretical analysis of the MIMO channel has shown that the best results are obtained when the number of receiving antennas at the base station \(N_{rx}\) is greater or equal to the total number of multiplexed data streams of all users, which is usually equal to the total number of transmitting antennas of all users \(N_{tx,\Sigma}\)[6; 11; 12; 13; 14; 15; 16]. This conclusion has been originallyobtained for a single user system [8], where it is shown that when the condition \(N_{rx}\geq N_{tx}\) is met, the system bandwidth increases linearly from the number of transmitting antennas. Therefore, we have a conclusion that MIMO systems, where \(N_{tx,\Sigma}>N_{rx}\), are ineffective.  
Let us consider a case when \(N_{tx,\Sigma}=N_{rx}\) corresponds to 100% of the MIMO system load. Then the case \(N_{tx,\Sigma}>N_{rx}\) can be considered as the operation of the system in overload mode. In this case, the MU-MIMO system becomes a NOMA system. Moreover, this case can be considered as a NOMA system with power division (Power Domain NOMA-PD-NOMA) [6]. In this article it will be shown that MU-MIMO systems with independent Rayleigh fading in an uplink channel can work quite efficiently in overload mode because in this case NOMA technology is implemented with a natural separation of signals in the power domain.