## 2 Preliminary Knowledge
### The Proposed Hybrid PAPR Reduction Scheme
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{b}_{1}\) & \(\chi_{1}\) & \(\{l_{1}^{c},l_{3}^{c}\}\) \\ \hline \(\mathbf{b}_{2}\) & \(\chi_{2}\) & \(\{l_{2}^{c},l_{4}^{c}\}\) \\ \hline \(\mathbf{b}_{3}\) & \(\chi_{3}\) & \(\{l_{1}^{c},l_{4}^{c}\}\) \\ \hline \(\mathbf{b}_{4}\) & \(\chi_{4}\) & \(\{l_{2}^{c},l_{3}^{c}\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A look-up table example between \(\mathbf{b}_{u}\) and \(\chi_{u}\) for \(V=4\), \(U=4\), and \(\beta=0.5\). Figure 7: Transmitter structure of the proposed hybrid scheme. Here, we take an example where \(V=3\). \(U=2\), \(\beta=0.25\), \(N=4\) to illustrate how candidates with different PAPR are constructed. It is assumed that \(\mathbf{X}=[X(0),0,0,0]\). If \(\mathbf{b}_{1}=[1,1,1,1]\) and \(\mathbf{b}_{2}=[-1,-1,-1,-1]\), \(\chi_{1}=\{I_{1}^{c},I_{3}^{c}\}=\{2,4\}\), \(\chi_{2}=\{I_{2}^{c},I_{3}^{c}\}=\{3,4\}\), the construction of candidates \(\mathbf{X}^{1}\) and \(\mathbf{X}^{2}\) after corresponding dither signals injection can be illustrated in Table 2.  
Therefore, the PAPR optimization problem for this hybrid method can be written as:  
\[ \begin{split}\underset{\zeta}{\text{min}}||\mathbf{F}_{\chi_{u }}^{H}\zeta+\mathbf{x}||_{\infty}^{2}\\ s.t.||\zeta||_{\infty}\leq R\end{split}  \tag{19}\]  
At the receiving side, the selected index \(\tilde{u}\) needs to be detected in order to recover the original transmitted signals. On the one hand, side information that represents which phase rotation factor \(\mathbf{b}\) is used can be transmitted as extra bits. However, this method will transmit at least \(\log_{2}U\) extra bits, which compromises the spectral efficiency. Moreover, the incorrect detection of side information bits may cause catastrophic error performance. Here, we consider a low-complex energy-based detection scheme. First, the indices of idle sub-carriers are required to be detected as the conventional OFDM-IM scheme. Then, the transmitted \(\mathbf{b}^{a}\) are detected based on the energy distribution of the received sequence. Here, we denote the received sequence and the estimated idle indicex set as \(\mathbf{Y}\) and \(I^{c}\), respectively. The detection process is summarized below. Step 1: Obtain all possible idle sub-carrier sets \(\hat{\chi}_{u}\) by utilizing \(\hat{I}^{c}\) and look-up tables. Step 2: Obtain the received signal blocks \(\mathbf{Y}_{\hat{\chi}_{u}}\) according to the received sequence \(\mathbf{Y}\) and \(\hat{\chi}_{u}\). Step 3: Compute the signal energy \(|\mathbf{Y}_{\hat{\chi}_{u}}|^{2}\), for \(u=1,2,\ldots,U\) and arrange them in descending order,  
\[|\mathbf{Y}_{\hat{\chi}_{1}}|^{2}\geq|\mathbf{Y}_{\hat{\chi}_{2}}|^{2}\geq \ldots\geq|\mathbf{Y}_{\hat{\chi}_{U}}|^{2} \tag{20}\]  
where \(u\) is the index of the \(u\)th largest energy signal block. Step 4: Find the set corresponding to \(\hat{\chi}_{1}\) as the idle sub-carriers set and determine the corresponding phase rotation factor \(\mathbf{b}\) utilizing the look-up table. The complexity of the energy-based detection method can be separated into two parts. First, the sorting complexity of Step 3 is \(O(U)\) if \(U\) candidates and appropriate sorting algorithms are considered. Then, the received signal corresponding to \(\hat{\chi}_{1}\) is detected. If the LLR detection method is utilized, the corresponding complexity can be denoted as \(O(\left\lfloor{n\choose k}M^{k}\right\rfloor_{2})\)[28]. Figure 8 illustrates the index error probability (IEP) of the proposed energy-based detection method with various \(U\), \(V\), and \(\beta\). It can be seen from the result that \(U\) and \(V\) are dominate factors that affect the IEP. It is reasonable that increasing \(U\) will make it more difficult for the detection method to identify the correct \(\tilde{u}\). However, it will be shown in the next section that with the same \(\beta\), increasing \(U\) will benefit the PAPR reduction performance.