## 2 Preliminary Knowledge
### Interference Analysis  
Consider a case that a symbol vector \(\mathbf{X}_{g}\) in sub-block \(g\) is erroneously determined to be \(\hat{\mathbf{X}}_{g}\). We assume that the optimized dither signals are \(\xi_{g}^{\prime}=[\zeta_{1,g},\zeta_{2,g},\ldots,\zeta_{3,g}]\). Consider the fact that \(\mathbf{Y}_{g}=\mathrm{diag}\{\mathbf{X}_{g}+\xi_{g}\}\mathbf{H}_{g}+\mathbf{W }_{g}\), where \(\mathbf{H}\) is the frequency channel response, \(\mathbf{H}=[\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{g}]^{T}\), \(\mathbf{H}_{g}=[H_{g}(1),H_{g}(2),\ldots,H_{g}(n)]^{T}\), and \(\mathbf{W}\) is the added White Gaussian noise (AWGN). The conditional pair-wise error probability (PEP) can be written as:  
\[P(\mathbf{X}_{g}\rightarrow\hat{\mathbf{X}}_{g}|\mathbf{H}_{g}, \mathbf{Y}_{g},\mathbf{w}/\mathrm{dithersignal})\] \[=P(||\mathbf{Y}_{g}-\mathrm{diag}\{\hat{\mathbf{X}}_{g}\}\mathbf{ H}_{g}||^{2}\leq||\mathbf{Y}_{g}-\mathrm{diag}\{\mathbf{X}_{g}\}\mathbf{H}_{g}||^{2})\] \[=P(||\mathrm{diag}\{\mathbf{X}_{g}-\hat{\mathbf{X}}_{g}\}\mathbf{ H}_{g}+\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}+\mathbf{W}_{g}||^{2})\] \[\leq||\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}+\mathbf{W} _{g}||^{2} \tag{10}\]  
where the second equality is from the fact \(\mathbf{Y}_{g}=\mathrm{diag}\{\mathbf{X}_{g}+\xi_{g}^{\prime}\}+\mathbf{H}_{g}+ \mathbf{W}_{g}\). The error occurrence of \(\mathbf{X}_{g}\rightarrow\hat{\mathbf{X}}_{g}\) can be categorized into two cases. The first case is that the system successfully detects the active sub-carriers, while the constellation symbols are erroneously detected. The second case is that \(\mathbf{X}_{g}\) and \(\hat{\mathbf{X}}_{g}\) have different detected active sub-carriers. It is obvious that when the amplitude of dither signals \(R\) is large, the idle sub-carriers are more likely to be detected as active ones. Therefore, we discuss two cases here according to the amplitude of \(R\). _Case 1, \(R\) is small_: If \(R\) is small, it is more likely for the system to detect the correct active sub-carrier set. In such a scenario, \(\hat{\mathbf{X}}_{g}(i)\
eq 0\) and \(\hat{\mathbf{X}}_{g}(i)=0\) if \(X_{g}(i)=0\), for \(i=1,2,\ldots,n\). Moreover, if \(X_{g}(i)\
eq 0\), then \(\zeta_{i,g}=0\), according to the rules of dither signal injection. Therefore, (10) is calculated as:  
\[P(||\mathrm{diag}\{\mathbf{X}_{g}-\hat{\mathbf{X}}_{g}\}\mathbf{ H}_{g}+\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}+\mathbf{W}_{g}||^{2}\] \[\quad\leq||\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}+ \mathbf{W}_{g}||^{2})\] \[\quad=P(||\mathrm{diag}\{\mathbf{X}_{g}-\hat{\mathbf{X}}_{g}\} \mathbf{H}_{g}+\mathbf{W}_{g}||^{2}\leq||\mathbf{W}_{g}||^{2})\] \[\quad=P(\mathbf{X}_{g}\rightarrow\hat{\mathbf{X}}_{g}|\mathbf{H}_ {g},\mathbf{Y}_{g},\mathbf{w}/\mathrm{o~{}dithersignals}) \tag{11}\]  
which is equivalent to the conditional PEP without considering dither signals. Therefore, the conditional PEP is the same in both cases with and without the PAPR processing when \(R\) is small. _Case 2, \(R\) is large_: However, when the amplitude of dither signal \(R\) is large, the interference of \(\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}\) cannot be ignored. When the signal-to-noise ratio (SNR) is high, the main interference becomes \(\mathrm{diag}\{\xi_{g}^{\prime}\}\mathbf{H}_{g}\) instead of \(\mathbf{W}_{g}\). Therefore, the idle sub-carriers are more possible to be detected as an active one. Hence, the conditional PEP will be compromised with the increase of \(R\), and an error floor can be found in [17] in high SNR regions due to the interference brought by dither signals. Considering the fact that the dither signals can affect the error performance and energy efficiency of the system, a very straightforward idea for keeping such properties is to utilize partial instead of idle sub-carriers. Moreover, it can be expected that when utilizing the sub-carriers, which have low frequency channel responses, the error performance degradation can be further limited.