## 2 Preliminary Knowledge
### The Conventional PAPR Reduction with Dither Signals  
In this sub-section, the classical PAPR reduction scheme with constant amplitude dither signals [17] is briefly introduced. Figure 3 is a block diagram that illustrates a PAPR reduction scheme utilizing constant amplitude dither signals. It is assumed that the set of indices of the inactive sub-carriers of \(\mathbf{x}\) is denoted as \(F\), where \(c\) means the complement. We note that the cardinality of is \(I^{c}\), \(N-K\), where \(K=kG\). In Figure 3, for a dither signal-based PAPR reduction scheme in frequency domain, \(N-K\) dither signals with amplitude constraint \(R\) are injected to the \(N-K\) inactive sub-carriers. Since dither signals are random, the injected signals can be seen as an extra noise to the original signals. Therefore, small constant \(R\) is regulated as the maximum amplitude of the dither signals in order to control the impact on demodulation errors. It is shown in (6) that minimizing the PAPR of \(\mathbf{x}\) is equivalent to minimizing the numerator of (6), which is equivalent to minimizing the infinity norm of the vector \(\mathbf{x}\). Therefore,  
according to (6), the optimal dither signals can be found through the convex programming. The PAPR optimization problem for such a method can be written as:  
\[ \begin{array}{l}\min_{\zeta}||\mathbf{F}_{l}^{H}\zeta+\mathbf{x}||_{\infty}^{2} \\ s.t.||\zeta||_{\infty}\leq R\end{array}  \tag{8}\]  
where the input time-domain signal vector is denoted as \(\mathbf{x}=\mathbf{F}^{H}\mathbf{X}\). \(\mathbf{F}_{l}^{H}\in\mathbb{C}^{N\times(N-K)}\) represents the matrix \(\mathbf{F}^{H}\) by deleting the columns whose indices belong to \(I\). The dither signals in the frequency domain is written as \(\zeta\in\mathbb{C}^{(N-K)\times 1}\). The \(\zeta\) can be obtained through the CVX package [26], which is a specific algorithm package used for convex programs. The illustration of the super constellation after injection dither signals is given in Figure 1 of [17]. It is illustrated that despite the fixed signals for the active sub-carriers, the dither signals on idle sub-carriers with the signal amplitude constraint \(R\) are around the origin, which can be regarded as a kind of adding noise that compromises both error performance and efficiency. Then, the frequency domain symbol vector \(\mathbf{X}_{d}\), which is injected by the optimized dither signals, are implemented by IDFT operations. In practical scenarios, the operations of IDFT are usually replaced by inverse Fast Fourier transform (IFFT). At the receiving side, both maximum likelihood (ML) [4] and the log-likelihood ratio (LLR) detector can be applied for the active sub-carrier [27] detection. The detection procedure of the dither signal-based PAPR reduction scheme is similar to the conventional OFDM-IM, where the extra dither signals can be treated as a kind of adding noise. The signal mode (active or inactive) of each sub-carrier can be determined by looking at the logarithm of the ratio between the a posteriori probabilities of the sub-carriers being modulated by the information symbols and zero values, respectively, which is formulated as:  
\[\lambda(\alpha)=\ln\frac{\sum_{j=1}^{M}Pr(X(\alpha)=S(j)|Y(\alpha))}{Pr(X( \alpha)=0|Y(\alpha))} \tag{9}\]  
where the size of constellation \(S\) is denoted as \(M\). \(Y(\alpha)\) is the \(\alpha\)th \((1\leq\alpha\leq N)\) received signal after inverse operations (CP removal and DFT). It is obvious that if \(\lambda(\alpha)\) is larger than 0, the \(\alpha\)th sub-carrier is more likely to be modulated by information symbols instead of being inactive. We note that since the inactive sub-carriers are occupied by small amplitude signals instead of zeros, the error performance of the dither signal-based PAPR reduction schemes are usually degraded. The Hybrid PAPR Reduction Scheme with Phase Rotation Factors and Dither Signals on Partial Subcarriers  
According to existing works, it is known that applying dither signals in OFDM-IM is an effective way to reduce its high PAPR.