## 2 Preliminary Knowledge
### The PAPR Reduction Scheme Utilizing Partial Idle Sub-Carriers Based on CSI  
The previous works that utilize dither signals explore all idle sub-carriers. Such schemes are able to keep solid PAPR reduction performances. However, from the perspective of error performance, these methods may not be the best. Therefore, we propose and analyze the PAPR reduction method, which utilizes partial idle sub-carriers according to the CSI. This scheme picks the idle sub-carriers whose own low channel gains add dither signals for PAPR reduction. Figure 4 shows the transmitter structure of the proposed PAPR reduction scheme utilizing partial idle sub-carriers based on CSI (denoted as \"S.1\"). It is assumed that the frequency domain channel response \(\mathbf{H}=[H(1),H(2),\ldots,H(N)]^{T}\), and the percentage of partial idle sub-carriers that require dither signal injection is \(\beta\). The indices for dither signals depend on the channel gain. Hence, in the S.1 scheme, the channel response \(\mathbf{H}\) needs to be sorted first, which can be written as:  
\[\mathbf{\bar{H}}=[\bar{H}(1),\ldots,\bar{H}(\alpha),\ldots,\bar{H}(N)]^{T} \tag{12}\]  
where \(1\leq\alpha\leq N\), and \(|\bar{H}(1)|<\ldots|\bar{H}(\alpha)|<\ldots|\bar{H}(N)|\). The system selects the sub-carriers with the least \(\beta(N-K)\), where \(K=Gk\), channel gains for dither signal injection. If the indix set of positions which own the least \(\beta(N-K)\) channel gains is denoted as \(I_{h}^{c}\), the PAPR optimization problem for this method can be written as:  
\[ \begin{array}{l}\underset{\bar{\zeta}}{\min}||\mathbf{F}_{I_{h}^{c}}^{H} \zeta+\mathbf{x}||_{\infty}^{2}\\ \underset{s.t.}{s.t.}||\bar{\zeta}||_{\infty}\leq R\end{array}  \tag{13}\]  
Symbol sequence \(\mathbf{X}\) is generated from information bits after index modulation. The dither signal \(\mathbf{d}_{h}\) is obtained through convex programming. The positions of \(\mathbf{d}_{h}\) are according to the channel response \(\mathbf{H}\). Then, \(\mathbf{X}\) and \(\mathbf{d}_{h}\) are added to the frequency domain. Finally, the transmitted time domain signal vector \(\mathbf{x}_{h}\) is obtained by an IDFT operation. Figure 5 shows the BER performance among the proposed S.1 scheme with a different number of \(\beta\) (\(\beta\in\{0.25,0.5\}\)), the scheme utilizing dither signals in [17] (denoted as \"_Dither_\"), and the original OFDM-IM signals (the ideal case). It is illustrated from Figure 5 that the S.1 scheme utilizing partial idle sub-carriers according to CSI can achieve a very close performance to that of the original OFDM-IM signals. Furthermore, due to the injected noise, it is obvious that the more idle sub-carriers are utilized, the worse BER performance is achieved. Therefore, using partial idle sub-carriers for dither signals is an effective way for PAPR reduction without compromising the error performance of the system. Figure 4: Transmitter structure of the S.1 scheme. However, utilizing partial idle sub-carriers may compromise the PAPR reduction performance due to the insufficient use of idle sub-carriers. The PAPR reduction performance comparison among the proposed S.1 scheme and _Dither_[17] scheme is shown in Figure 6, using different number of \(\beta\) (\(\beta\in\{0.25,0.5\}\)) with \(N=128\) sub-carriers. This result highlights that although using partial sub-carriers for dither signals can significantly improve the error performance, the PAPR reduction capability degradation compared with the conventional _Dither_ scheme cannot be overlooked. Furthermore, obtaining an instantaneous channel response is not always available in practical cases thus, the S.1 scheme in this section can only be regarded as an ideal case. This ideal case can be regarded as the best error performance that our proposed hybrid method can achieve. In the next subsection, a hybrid PAPR reduction scheme that combines phase rotation factors is proposed. The proposed hybrid scheme can significantly improve the system's PAPR reduction capability as well as keeping a solid error performance.