# Design of an Energy Efficient Future Base Station with Large-Scale Antenna System
## 2 System Model
### Large-Scale Multiple-Input Multiple-Output Model
#### 2.2.1 Precoding Method  
Precoding is an MIMO TX signal processing method that reduces the interference among the users. Since \(N_{t}\) would be very large in a future BS equipped with LS-MIMO, linear precoding is tractable for real systems. ZF and regularized zero-forcing (RZF) are well known as effective linear precoding techniques [31; 32]. Furthermore, [PERSON] suggested an even simpler precoding technique, MF precoding [29]. MF precoding is only effective for LS-MIMO, while ZF/RZF precoding is also useful for traditional MIMO. The relation between the TX signal vector, \(\mathbf{s}\), and the massage signal vector, \(\mathbf{x}\), is expressed as follows:  
\[\mathbf{s}=\zeta\mathbf{F}\mathbf{x} \tag{10}\]  
where \(\zeta\) is the TX power normalization factor and \(\mathbf{F}\) is the \(N_{t}\times K\) precoding matrix.  
Then, (8) can be rewritten as:  
\[\mathbf{y} = \sqrt{P_{tx}}\mathbf{H}\zeta\mathbf{F}\mathbf{x}+\mathbf{n} \tag{11}\]  
All three precoding matrices are presented in Table 2.  
In the table, superscript \"\(H^{\prime\prime}\) denotes the conjugate transpose, \((\cdot)^{-1}\) is the inverse operator and \(\mathbf{I}_{K}\) is the \(K\times K\) identity matrix. RZF precoding can be the same as ZF and/or minimum mean square error (MMSE) precoding if we choose \(\
u=0\) and/or \(\
u=\frac{KN_{0}}{N_{t}P_{tx}}\).  
As \(N_{t}\) grows without bounds, the autocorrelation of the channel vectors can be simplified as follows [29]:  
\[\mathbf{H}\mathbf{H}^{H}\to N_{t}\mathbf{I}_{K} \tag{12}\]  
Simply, we call (12) an LS-MIMO channel characteristic. If we apply (12) to the MMSE and the ZF, the MMSE and the ZF can be simplified as [16]:  
\[\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H}+\frac{KN_{0}}{N_{t}P_{tx}}\mathbf{I} _{K})^{-1}\approx\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\to N_{t}^{-1} \mathbf{H}^{H} \tag{13}\]  
This means that with an excessive number of TX antennas with a limited number of RX antennas, all representative linear precoding techniques show similar performance. ZF is well known to be near optimal in a region with very high SNR. If we consider the complexity of the nonlinear precoding techniques and high effective signal-to-interference plus noise ratio (SINR) of the LS-MIMO systems, linear precoding would show sufficient performance with low complexity.  
\begin{table}
\begin{tabular}{c c c c} \hline  & **MF** & **ZF** & **RZF** \\ \hline \(\mathbf{F}\) & \(N_{t}^{-1}\mathbf{H}^{H}\) & \(\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\) & \(\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H}+\
u\mathbf{I}_{K})^{-1}\) \\ \hline \end{tabular}
\end{table}
Table 2: The precoding matrices of matched filtering (MF), zero-forcing (ZF) and regularized zero-forcing (RZF).  
Now, let us determine the normalization factor \(\zeta\). The normalization factor should be determined such that the total transmit power becomes \(P_{tx}\). This means:  
\[\|\zeta\mathbf{F}\mathbf{x}\|_{F}^{2}=1 \tag{14}\]  
where \(\|\|_{F}\) stands for the Frobenius norm.  
Since \(\mathbf{F}\) and \(\mathbf{x}\) are independent, and we already choose the average power of the signal constellation in unity, \(\zeta\) can be represented as [16]:  
\[\zeta\approx\sqrt{\frac{N_{t}}{K}} \tag{15}\]  
The ZF and MMSE precoding show a very similar performance, even with a relatively practical number of LS TX antennas (i.e., \(N_{t}\approx 10\) K). For this reason, we only use MF and ZF as precoding techniques in this paper.