## 2 System Models  
Let \(\mathbf{X}~{}=~{}[X(0),X(1),\cdots,X(N-1)]^{\mathrm{T}}\) denote an input OFDM symbol block where \(X(k)\) for \(0\leq k\leq N-1\) is the complex data transmitted at the \(k\)th subcarrier and \(N\) is the total number of subcarriers of OFDM systems. In the OFDM block, there are \(N_{p}\) equally spaced pilot subcarriers for the purpose of channel estimation. By performing the \(N\)-point IFFT operation of \(\mathbf{X}\), the time domain signal vector \(\mathbf{x}~{}=~{}[x(0),x(1),\cdots,x(N-1)]^{\mathrm{T}}\) is obtained, which can be represented as  
\[\mathbf{x}~{}=~{}\mathbf{F}\mathbf{X} \tag{1}\]  
where \(\mathbf{F}\) is the \(N\times N\) IFFT matrix whose element is \(e^{j2\pi nk/N}/\sqrt{N}\) for \(0\leq n\leq N-1\) and \(0\leq k\leq N-1\). The \(n\)th element of \(\mathbf{x}\) can be given as  
\[x(n)~{}=~{}\frac{1}{\sqrt{N}}\sum_{k=~{}0}^{N-1}X(k)e^{j2\pi kn/N},\quad 0\leq n \leq N-1 \tag{2}\]  
The PAPR of the transmitted OFDM signal vector \(\mathbf{x}\) in Equation (1) can be defined as  
\[\mathrm{PAPR}(\mathbf{x})~{}=~{}\frac{\max_{0\leq n\leq N-1}\left|x(n)\right| ^{2}}{E\left[\left|x(n)\right|^{2}\right]} \tag{3}\]  
where \(E[\cdot]\) denotes the expectation operation. In the conventional SLM PAPR scheme, the phase rotation vectors can be described as \(\mathbf{P}^{(u)}~{}=~{}[P^{(u)}(0),P^{(u)}(1),\cdots,P^{(u)}(N-1)]^{\mathrm{T}}\) for \(0\leq u\leq U-1\), where the \(k\)th element of \(\mathbf{P}^{(u)}\) is \(P^{(u)}(k)~{}=~{}e^{j\theta^{(u)}(k)}\) with \(\theta^{(u)}(k)\in(-\pi,~{}\pi]\). Then, the \(u\)th candidate signal in the frequency domain is generated by performing carrier-wise multiplication of the data vector \(\mathbf{X}\) and the \(u\)th phase rotation vector \(\mathbf{P}^{(u)}\), and which can be given as  
\[\mathbf{X}^{(u)}~{}=~{}\left[P^{(u)}(0)X(0),\cdots,P^{(u)}(N-1)X(N-1)\right]^ {\mathrm{T}}~{}=~{}\mathbf{Q}^{(u)}\mathbf{X} \tag{4}\]  
where \(\mathbf{Q}^{(u)}~{}=~{}diag\{\mathbf{P}^{(u)}\}\) is a diagonal matrix whose diagonal element is \(P^{(u)}(k)\) for \(k~{}=~{}0,\cdots,N-1\). For better approximating the true PAPR in the discrete time case, usually \(G\) times oversampling is considered. It was shown in [15] that \(G\) = 4 is sufficient to capture the peak information. To obtain the oversampling vector \(\mathbf{x}\) in the time domain, \(N(G-1)\) zeros should be inserted in the middle of the input data vector \(\mathbf{X}\) to form a \(GN\)-dimensional data vector given by \(\left[X(0),\cdots,X(N/2-1),0,\cdots 0,X(N/2),\cdots,X(N-1)\right]^{\mathrm{T}}\). Obviously, if \(G\) times oversampling is considered, \(\mathbf{F}\) is of \(GN\times GN\) dimensions and \(\mathbf{P}^{(u)}\) is of \(GN\)-dimensional phase rotation vector with \(\mathbf{P}^{(u)}~{}=~{}[P^{(u)}(0),\cdots,P^{(u)}(N/2-1),C^{(u)}(0),\cdots C^{ (u)}((G-1)N-1),P^{(u)}(N/2),\cdots,P^{(u)}(N-1)]^{\mathrm{T}}\) where \(C^{(u)}(k)\) can be an arbitrary value for \(k~{}=~{}0,~{}\cdots,(G-1)N-1\). Finally, the \(u\)th candidate signal vectors \(\mathbf{x}^{(u)}~{}=~{}\left[x^{(u)}(0),x^{(u)}(1),\cdots,x^{(u)}(GN-1)\right] ^{\mathrm{T}}\) in the time-domain can be obtained and which can be expressed as  
\[\mathbf{x}^{(u)}~{}=~{}\mathbf{F}\mathbf{X}^{(u)}~{}=~{}\mathbf{F}(\mathbf{Q}^ {(u)}\mathbf{X}) \tag{5}\]  
Among these candidate signals \(\mathbf{x}^{(u)}\) for \(0\leq u\leq U-1\), the one with the lowest PAPR is selected to be transmitted, which can be given as follows  
\[\mathbf{x}^{(\overline{u})}~{}=~{}\underset{\mathbf{x}^{(u)},0\leq u\leq U-1} {\mathrm{argmin}}~{}\mathrm{PAPR}(\mathbf{x}^{(u)}) \tag{6}\]  
To avoid the inter-signal interference caused by the wireless channel, a cyclic prefix (CP) is inserted at the beginning of each transmitted signal whose length is always assumed to be larger than thatof the wireless channel.