## 2 System Description
### Precoding  
Since \(N_{t}\) is much larger than \(K\), we should map \(K\) message signal to each antenna. In general, we call it precoding, and it is applied to the transmitter part to reduce inter-user interference (IUI). Since \(N_{t}\) is very large, the linear precoding should be used for real systems [28; 29; 30]. For regular MIMO systems, both nonlinear and linear precoding scheme could be considered. Nonlinear precoding techniques, such as dirty paper coding (DPC) [31], vector perturbation (VP) [32], and lattice-aided methods [33] are important techniques when \(N_{t}\) is not much larger than \(K\). However, with an increase in the number of antennas at the BS, linear precoders are shown to be near-optimal [28; 29]. When \(N_{t}\) is, say, two times \(K\), this gap is only 3 dB [29]. It is shown that with linear precoding, a sum rate as high as 98% of that of the DPC scheme can be achieved for two single antenna users served by 20 BS antennas [30]. Thus, it is more practical to use low-complexity linear precoding techniques in massive MIMO systems. Therefore, we mainly focus on linear precoding techniques. It is well-known in the literature that zero-forcing (ZF) and regularized zero-forcing (RZF) are effective linear precoding techniques [34]. Also, in [4], [PERSON] suggested even a simpler precoding technique, matched filtering (MF) precoding. The TX signal vector \(\mathbf{s}\) and the message signal vector \(\mathbf{x}\) are related as follows:  
\[\mathbf{s}=\zeta\mathbf{F}\mathbf{x}, \tag{6}\]  
where \(\zeta\) is the normalization factor of TX power and \(\mathbf{F}\) is a precoding matrix. Then, (1) can be represented as follows:  
\[\mathbf{y} = \sqrt{\rho_{t}}\mathbf{G}\zeta\mathbf{F}\mathbf{x}+\mathbf{n}, \tag{7}\]  
where \(\mathbf{F}\) is the \(N_{t}\times K\) precoding matrix. The three representative precoding matrices are denoted in Table 1.  
Figure 2: Normalized Interference Power versus Number of Trials: As \(N_{t}\) increases, the interference level converges to a certain point that can be easily estimated. MF precoding matrix is simply the conjugate transpose of the channel matrix. This is conceptually simplest approach to reduce the IUI. In MF precoding, it is seen that the signal-to-interference and noise ratio (SINR) can be made as high as desired by increasing the number of antennas. However, the MF precoding exhibits an error floor in the practical number of TX antennas. The main advantage of MF precoding is its low computational complexity. Another advantage is that it is effective for distributed antenna systems, because the Massive MIMO signal processing can be performed locally at each antenna [5]. Another approach for precoding is to invert the channel by means of the pseudo-inverse. This is referred to as ZF precoding. When \(N_{t}>K\), ZF precoding completely removes the IUI. Moreover, when SNR is high and/or \(N_{t}\gg K\), ZF precoding can achieve nearly optimum performance [29]. However, a disadvantage of ZF is that processing cannot be done distributedly at each antenna separately. With ZF precoding, all data must instead be collected at a central node that handles the processing [29]. MF precoding can outperform ZF precoding when SNR is very low, while ZF precoding outperform MF precoding when SNR is relatively high. When SNR is not so high, ZF precoding cannot give any meaningful performance. RZF precoding can combat this problem. When SNR is high, both ZF precoding and RZF precoding shows very similar performance, while when SNR is low, RZF precoding outperforms ZF precoding. When SNR is very low, RZF precoding show similar performance with MF precoding. A brief summary of the advantages and disadvantages of the MF, ZF, and RZF is shown in Table II of [30]. The normalization factor should be determined such that the total transmit power becomes \(\rho_{t}\). It can be expressed as follows:  
\[\|\zeta\mathbf{F}\mathbf{x}\|_{F}^{2}=1, \tag{8}\]  
where \(\|\|_{F}\) stands for Frobenius norm.