## 2 Preliminary Knowledge
### The Proposed Hybrid PAPR Reduction Scheme  
Adding dither signals on corresponding partial idle sub-carriers is helpful to prevent error performance degradation. However, choosing partial indices for a dither signal injection could affect PAPR reduction performance. Therefore, in order to compensate for the PAPR reduction performance degradation due to the insufficient use of partial sub-carriers, this section introduces a hybrid method that implements phase rotation factors for modulated symbols to generate different candidates with different PAPR. The system can select one with the lowest PAPR to transmit. Figure 7 shows the transmitting structure of the proposed hybrid PAPR reduction scheme. Symbol sequence **X** is generated from information bits after index modulation. The proposed scheme first defines an ensemble of phase rotation factors, as:  
\[\textbf{b}_{u}=[b_{u}(1),b_{u}(2),\ldots,b_{u}(N)]^{T} \tag{14}\]  
where \(u=1,2,\ldots,U\), that are known to both transmitter and receiver. The entries of \(\textbf{b}_{u}\) are formed as follows:  
\[b_{v}(\alpha)=e^{i\phi_{u}(\alpha)} \tag{15}\]  
where \(\phi_{u}(\alpha)\in[0,2\pi)\), \(\alpha=1,2,\ldots,N\). In general, binary elements are randomly selected for building \(b_{v}(\alpha)\), that is, \(b_{v}(\alpha)\in\{+1,-1\}\). In the proposed hybrid method, the component-wise multiplication of the input symbol vector **X** with \(U\) different phase rotation factors are firstly implemented. Then, \(U\) dither signals \(\textbf{d}_{u}\) whose idle sub-carriers for injection are different from each other are injected into \(\textbf{X}\textbf{b}_{u}\). These \(U\) dither signals can be obtained from convex programming. Then, after signal injection, \(U\) parallel IDFT (or IFFT) operations are implemented to obtain \(U\) time domain candidates \(\textbf{x}^{u}\). After obtaining \(U\) candidate sequences, the proposed PAPR reduction scheme chooses one candidate with the lowest PAPR for transmission. We note the corresponding signal as \(\textbf{x}^{\bar{u}}\) and \(\textbf{x}^{\bar{u}}\) can be selected based on the following equation:  
\[\textbf{x}^{\bar{d}}=\underbrace{\text{argmin}}_{u=1,2,\ldots,U}PAP\{\textbf{x }^{u}\} \tag{16}\]  
Figure 6: A CCDF comparison among the original OFDM-IM, S.1 scheme, and the _Dither_ scheme [17]. In order to improve the PAPR reduction performance and facilitate the detection procedure, the proposed scheme utilizes the \"interleaved\" method for idle sub-carriers' partitioning. For simplicity, in this part, we consider an available idle indices set \(l^{c}\) for all \(N\) sub-carriers, which is written as:  
\[l^{c}=\{i_{1},i_{2},\ldots,i_{N-K}\} \tag{17}\]  
where \(K=Gk\), and \(i_{\gamma}\in\{1,2,\ldots,N\}\), \(1\leq\gamma\leq N-K\). The idle index set needs to be divided into \(V\) disjoint subsets, such that:  
\[l^{c}=\sum_{v=1}^{V}l_{v}^{c} \tag{18}\]  
where:  
\[I_{1}^{c} =\{\underbrace{i_{1},i_{V+1}\ \ldots,\ i_{N-K-V+1}}_{(N-K)/V}\},\] \[I_{2}^{c} =\{\underbrace{i_{2},i_{V+2}\ \ldots,\ i_{N-K-V+2}}_{(N-K)/V}\},\] \[\vdots I_{V}^{c} =\{\underbrace{i_{V},i_{2V}\ \ldots,\ i_{N-K}}_{(N-K)/V}\}\]  
The idle sub-carriers set \(\chi_{u}\) for \(U\) candidates can be jointly determined through the combination of these \(V\) sub-blocks and the idle sub-carrier active ratio \(\beta\). To facilitate the detection procedure, in the proposed method, \(\chi_{u}\) and \(\mathbf{b}_{u}\) have a one-to-one correspondence. Once \(V\), \(\beta\), and \(U\) are determined, the idle sub-carriers set \(\chi_{u}\) can be determined by utilizing a look-up table. A mapping example between \(\mathbf{b}_{u}\) and \(\chi_{u}\) with \(V=4\), \(U=4\), and \(\beta=0.5\) is given in Table 1.