## 2 System Model and Power Amplifier Operations  
### Is-Mimo-Ofdm and PAPR  
A single isolated cell with a BS using the LS-MIMO-OFDM system and K terminals is considered. The BS has \(N_{t}\) TX antennas, and each terminal has only one RX antenna. Let the frequency domain of an OFDM signal of the \(t\)-th antenna be \(\mathbf{X}_{t}(=[X_{t}[0],X_{t}[1],\ldots,X_{t}[N-1]]^{T})\). Then, the time domain signal \(\mathbf{x}_{t}\) can be represented as follows:  
\[\mathbf{x}_{t}=\mathbf{F}_{N}^{H}\mathbf{X}_{t},t=1,\ldots,N_{t}, \tag{1}\]  
where \(\mathbf{F}_{N}^{H}\) is \(N\)-point IFFT matrix, which can be represented as \(F_{n,k}=\frac{1}{\sqrt{N}}e^{-j2\pi nk/N},0\leq n,k\leq N-1\), and \(x_{t}(n)=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}X_{t}[k]e^{j2\pi nk/N}\). To measure the PAPR of the OFDM signal, oversampling is usually considered to catch the PAPR of the analog domain. Note that most of the nonlinear distortions due to a high PAPR are caused in the analog domain, but most of the signal processes for the PAPR reduction are performed in the digital domain. The PAPR of the digital domain is not necessarily the same as the PAPR of the analog domain. However, 4-times of oversampling is well-known to be sufficient in satisfactorily approximating the PAPR of the analog domain [19]. If we consider the oversampled OFDM signal, (1) can be represented as follows:  
\[\mathbf{x}_{t}^{o}=(\mathbf{F}^{o})_{N}^{H}\mathbf{x}_{t}^{o},t=1,\ldots,N_{t}, \tag{2}\]  
where \(\mathbf{x}_{t}^{o}\) is the oversampled signal of \(\mathbf{x}_{t}\), \(\mathbf{x}_{t}^{o}\) is the zero-padded signal of \(\mathbf{X}_{t}\) to generate the oversampled time domain signal, \((\mathbf{F}^{o})_{N}^{H}\) is \(LN\)-point IFFT matrix, which can be represented as \(F_{n,k}^{o}=\frac{1}{\sqrt{LN}}e^{-j2\pi nk/LN},0\leq n,k\leq LN-1\), and \(x_{t}^{o}(n)=\frac{1}{\sqrt{LN}}\sum\limits_{k=0}^{LN-1}X_{t}[k]e^{j2\pi nk/ LN}\), and \(L\) is the oversampling factor. Then, the PAPR of the OFDM signal for the \(t\)-th antenna can be expressed as follows:  
\[PAPR=\frac{LN\cdot\|\mathbf{x}_{t}^{o}\|_{\infty}^{2}}{E[\|\mathbf{x}_{t}^{o} \|_{2}^{2}]}, \tag{3}\]  
where \(E[\cdot]\), \(\|\mathbf{x}_{t}^{o}\|_{\infty}\), and \(\|\mathbf{x}_{t}^{o}\|_{2}\) denote the expectation of \([\cdot]\), the \(l_{\infty}\)-norm, and \(l_{2}\)-norm of vector \(\mathbf{x}_{t}^{o}\), respectively. From now on, we omit the oversampling superscription for simplicity. Cyclic prefix (CP) is necessary to remove the inter-symbol interference (ISI). If the CP is longer than the multipath delay spread, the individual frequency carrier experiences a flat fading channel, and the frequency domain representation of the received \(k\)th signal vector, \(\mathbf{Y}[k](=[Y_{1}[k],Y_{2}[k],\ldots,Y_{K}[k]]^{T})\), can be represented as follows:  
\[\mathbf{Y}[k]=\sqrt{P_{tx}}\mathbf{H}[k]\mathbf{X}[k]+\mathbf{N}[k], \tag{4}\]  
where \(\mathbf{Y}[k]\) is the \(K\times 1\) received vector for each terminal, \(P_{tx}\) is the total TX power for a forward link, \(\mathbf{H}[k]\) is the \(K\times N_{t}\) small scale i.i.d. Rayleigh fading channel matrix between \(N_{t}\) BS antennas and \(K\) terminals, \(\mathbf{X}[k]\) is the \(N_{t}\times 1\) signal vector for each antenna, and \(\mathbf{N}[k]\) is the \(K\times 1\) noise (AWGN) vector. The precoding process is necessary to reduce the IUI, and there are two representative precoding schemes for the LS-MIMO: matched filtering (MF) and zero forcing (ZF) schemes [1]. Both schemes are considered in this paper, and the precoding matrices for MF and ZF are summarized in Table 1.  
Generally the performance of the ZF precoding scheme is known to be superior to that of the MF precoding scheme. However, the MF scheme is also known to be simple and effective especially when the number of TX antennas is very large. Moreover, the MF scheme is suitable for a distributed antenna system because each distributed antenna system does not need to know the channel information of the other antenna systems [1].