## 2 System Model
### Precoding  
Since we consider that in system \(N_{t}\) is much larger than \(K\), we need to map \(K\) message signals to each antenna; coincidently, the \(K\) message signals should less interfere each other at RXs. For this, the BS should transmit signals with some kind of precoding. Since \(N_{t}\) is very large, linear precoding is tractable for real systems. It is well-known that zero-forcing (ZF) and regularized zero-forcing (RZF) are effective linear precoding techniques [18; 19]. Also, in [6], [PERSON] suggested even simpler precoding technique--matched filtering (MF) or conjugate transpose precoding. Note that MF precoding is only effective for LSASs, while ZF/RZF precoding is also useful for traditional multi-antenna systems. The relationship between the TX signal vector \(\mathbf{s}\) and the message signal vector \(\mathbf{x}\) can be expressed as follows:  
\[\mathbf{s}=\zeta\mathbf{F}\mathbf{x}, \tag{2}\]  
where \(\zeta\) is the TX power normalization factor, and \(\mathbf{F}\) is the \(N_{t}\times K\) precoding matrix.  
Then, (1) can be rewritten as:  
\[\mathbf{y} = \sqrt{P_{tx}}\mathbf{H}\zeta\mathbf{F}\mathbf{x}+\mathbf{n}. \tag{3}\]  
All of the three precoding matrices are provided in Table 1, where the superscript \"\(H\)\" denotes conjugate transpose, \((\cdot)^{-1}\) is the inverse operator, and \(\mathbf{I}_{K}\) is the an \(K\times K\) identity matrix. Note that RZF precoding can be the same as ZF and/or minimum mean square error precoding, if we choose \(\
u=0\).  
As \(N_{t}\) grows without bounds, the autocorrelation of channel vectors can be simplified as follows [6]:  
\[\mathbf{H}\mathbf{H}^{H}\to N_{t}\mathbf{I}_{K}. \tag{4}\]  
Since the channel has a characteristic of mean zero and unit variance in a rich scattering environment, the autocorrelation of channel vectors become \(N_{t}\) and the cross correlation of channel vectors becomes 0. Thus, simply we call (4) as the LSAS channel characteristic.  
By applying (4), minimum mean square error and ZF can be simplified as  
\[\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H}+\frac{KN_{0}}{N_{t}P_{tx}}\mathbf{I} _{K})^{-1}\approx\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\rightarrow \mathbf{H}^{H}(N_{t}\mathbf{I}_{K})^{-1}=N_{t}^{-1}\mathbf{H}^{H}, \tag{5}\]  
and it means that all representative linear precoding techniques show similar performance with an excessive number of TX antennas and limited number of RX antennas. It is well-known that ZF is near  
\begin{table}
\begin{tabular}{c c c c} \hline  & **MF** & **ZF** & **RZF** \\ \hline F & \(N_{t}^{-1}\mathbf{H}^{H}\) & \(\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\) & \(\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H}+\
u\mathbf{I}_{K})^{-1}\) \\ \hline \end{tabular}
\end{table}
Table 1: Precoding matrices of MF, ZF, and RZF.  
optimal in very high SNR regions. If we consider the complexity of nonlinear precoding techniques and high effective signal-to-interference plus noise ratio (SINR) of the LSAS, linear precoding would show enough performance with low complexity.  
The normalization factor should be determined such that the total transmitted power becomes \(P_{tx}\), and it is expressed as follows:  
\[\|\zeta\mathbf{F}\mathbf{x}\|_{F}^{2}=1, \tag{6}\]  
where \(\|\|_{F}\) stands for Frobenius norm.  
Since \(\mathbf{F}\) and \(\mathbf{x}\) are independent, and we already chose the average power of the signal constellation in unity, \(\zeta\) can be represented as  
\[\zeta\approx\sqrt{\frac{N_{t}}{K}}. \tag{7}\]  
Both ZF and the minimum mean square error precodings show very similar performance, even in the relatively practical number of large scale TX antennas (i.e., \(N_{t}\approx 10K\)). Due to this reason, we only consider MF and ZF as precoding techniques in this paper.