## 2 System Model and Power Amplifier Operations
### Energy Efficiency Model and Power Amplifier Operation  
The energy efficiency is defined to measure the EE gain and is usually expressed as follows:  
\[EE=R/P_{tot}, \tag{6}\]  
where \(R\) is the achievable rate in bits per second, and \(P_{tot}\) is the reference total power for the derivation of EE. For simplicity, we divide the \(P_{tot}\) into two parts as  
\[P_{tot}=P_{PA}+P_{R}, \tag{7}\]  
where \(P_{PA}\) is the power consumption of the PA, and \(P_{R}\) is the power consumption of the remaining relevant components. It is appropriate to define \(P_{tot}\) as (7) because the \(P_{PA}\) consumes most of the power in the current BS.  
When a Class B amplifier is used, the relationship between the TX power, \(P_{tx}\), and \(P_{PA}\) can be expressed as follows [20; 21]:  
\[P_{tx}=\eta P_{PA}, \tag{8}\]  
where \(\eta\) is the PA efficiency in percentage (%) and is defined as  
\[\eta=\frac{\pi}{4p}, \tag{9}\]  
where \(p\) is the square-root of the input back-off (IBO), \(\sqrt{IBO}\). The \(\eta\) depends heavily on the PAPR of the OFDM signal, as shown in Figure 1, and the PAPR of the OFDM signal also depends on the system bandwidth and/or number of subcarriers [20; 21]. According to the 3 GPP LTE system parameters [22], we can choose the \(20 MHz\) bandwidth with 1200 subcarriers/2048 FFT size. With these parameters, the PAPR of the OFDM signal is around \(12 dB\) from the complementary cumulative distribution function (CCDF = \(10^{-3}\) ) [23].  
Now let us describe how (8) can be changed for the ITXP and the RPC schemes. First, we define the reference TX power for the case in which PAPR reduction technique is not applied, and it is expressed as follows:  
\[P_{tx}^{ref}=\eta^{ref}P_{PA}^{ref}. \tag{10}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Precoding** & **MF** & **ZF** \\ \hline
**W** & \(N_{t}^{-1}\mathbf{H}^{H}\) & \(\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The precoding matrices of MF and ZF.  
When the PAPR reduction technique and the ITXP scheme are applied, (10) can be rewritten as follows:  
\[P_{tx}^{ITXP}=\eta^{PAPR}P_{PA}^{ref}, \tag{11}\]  
where \(P_{tx}^{ITXP}\) is the TX power with the ITXP scheme, and \(\eta^{PAPR}\) is the PA efficiency after PAPR reduction. (11) means that \(\eta^{ref}\) increased to \(\eta^{PAPR}\) as a result of the PAPR reduction technique, while \(P_{PA}^{ref}\) is kept as is. That is, \(P_{tx}^{ITXP}>P_{tx}^{ref}\), due to the increase in PA efficiency.  
When the PAPR reduction technique and the RPC scheme are applied, (10) can be rewritten as follows:  
\[P_{tx}^{RPC} = \eta^{PAPR}P_{PA}^{RPC}, \tag{12}\]  
where \(P_{tx}^{RPC}\) is the TX power and \(P_{PA}^{RPC}\) is the PA power consumption with the RPC scheme. (12) means that \(\eta^{ref}\) increases to \(\eta^{PAPR}\) due to the PAPR reduction technique, while reducing PA power consumption from \(P_{PA}^{ref}\) to \(P_{PA}^{RPC}\), and maintaining TX power. To make \(P_{tx}^{RPC}=P_{tx}^{ref}\), we should choose \(P_{PA}^{RPC}=\frac{P_{PA}^{ref}}{\eta^{PAPR}}\), and then, from (12), \(P_{tx}^{RPC}\) can be represented as follows:  
\[P_{tx}^{RPC} = \eta^{PAPR}\frac{P_{PA}^{ref}}{\eta^{PAPR}}\] \[= P_{tx}^{ref}. \tag{13}\]  
That is, by employing the RPC scheme, \(P_{tx}^{RPC}\) becomes equal to \(P_{tx}^{ref}\), because of the increase in the PA efficiency \(\eta\) and reduction in the PA power consumption \(P_{PA}\).  
For easy understanding, we present Table 2 for the transmission powers of each scheme.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reference** & **ITXP** & **RPC** \\ \hline \(P_{tx}^{ref}=\eta^{ref}P_{PA}^{ref}\) & \(P_{tx}^{ITXP}=\eta^{PAPR}P_{tx}^{ref}\) & \(P_{tx}^{RPC}=\eta^{PAPR}P_{PA}^{RPC}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Transmission Power for Each Scheme.  
Figure 1: PA efficiency, \(\eta(\%)\) versus PAPR(dB).