## 4 Proposed SFO Estimation Scheme
### Estimation Algorithm  
The presence of unknown \(X_{l}(k)\) and \(H_{l}(k)\) may make accurate SFO estimation difficult. To solve for this problem, the proposed frequency estimation scheme adopts two-step correlation, namely temporal correlation and frequential correlation, which are used to eliminate the use of CSI and dedicated pilots, respectively. First, to cancel out influence of \(X_{l}(k)\) without decoding it, we utilizes the fact that each non-masked subcarrier forms at least one pair of subcarriers satisfying \(X_{l}(k)=X_{l}(k+F_{m})\) in the HomePlug ROBO mode, leading to \(X_{l}^{*}(k)X_{l}(k+F_{m})=|X_{l}(k)|^{2}\). By using this property of the repetition code, the frequential correlation \(C_{l}(k)\) is constructed by  
\[C_{l}(k)=R_{l}^{*}(k)R_{l}(k+F_{m}),\;m=1,2,\cdots,N_{s},\;k\in S_{m} \tag{13}\]  
where \(N_{s}\) is the number of subsets containing distinctive \(F_{m}\)'s, \(\mathcal{S}_{m}\) is the \(m\)-th subset of the first data subcarriers among two same data spaced \(F_{m}\) subcarriers apart, and \(k\in\mathcal{S}_{m}\) means the data subcarrier in the \(m\)-th subset. Recognizing that \(X_{l}^{*}(k)X_{l}(k+F_{m})=|X_{l}(k)|^{2}\), \(\hat{C}_{l}(k)\) is further derived as  
\[C_{l}(k)=H_{l}^{*}(k)H_{l}(k+F_{m})|X_{l}(k)|^{2}e^{|2\pi F_{m}\Delta_{s}(IN_{u }+N_{g})/N}+\hat{l}_{l}(k)+\hat{Z}_{l}(k) \tag{14}\]  
where \(\hat{l}_{l}(k)\) stands for the combined ICI given by \[ \begin{split}\hat{I}_{l}(k)&=H_{l}^{*}(k)X_{l}^{*}(k)I_{ l}(k+F_{m})e^{j2\pi\phi(k+F_{m})(IN_{u}+N_{g})/N}\\ &+H_{l}(k+F_{m})X_{l}(k+F_{m})I_{l}^{*}(k)e^{-j2\pi\phi(k)(IN_{u}+ N_{g})/N}\\ &+I_{l}(k+F_{m})Z_{l}^{*}(k)+I_{l}^{*}(k)Z_{l}(k+F_{m})+I_{l}(k+F_{m})I_{l}^{*}(k), \end{split}  \tag{15}\]  
and \(\hat{Z}_{l}(k)\) represents the AWGN given by  
\[ \begin{split}\hat{Z}_{l}(k)&=H_{l}^{*}(k)X_{l}^{*}( k)Z_{l}(k+F_{m})e^{-j2\pi\phi(k+F_{m})(IN_{u}+N_{g})/N}\\ &+H_{l}(k+F_{m})X_{l}(k+F_{m})Z_{l}^{*}(k)e^{j2\pi\phi(k)(IN_{u}+ N_{g})/N}+Z_{l}(k+F_{m})Z_{l}^{*}(k).\end{split}  \tag{16}\]  
The second step is employed to remove the need for an estimated channel. For this purpose, we use the following temporal correlation  
\[\tilde{C}_{l}(k)=C_{l}^{*}(k)C_{l+D}(k) \tag{17}\]  
where we can make a reasonable assumption that \(H_{l}(k)=H_{l+D}(k)\) thanks to the characteristics of indoor PLC channels [5; 13]. Then, \(\tilde{C}_{l}(k)\) can be expressed by  
\[ \begin{split}\tilde{C}_{l}(k)&=|H_{l}(k)|^{2}|H_{l} (k+F_{m})|^{2}|X_{l}(k)|^{2}|X_{l+D}(k)|^{2}e^{j2\pi F_{m}D_{u}N_{u}/N}+\tilde {I}_{l}(k)+Z_{l}(k),\ m=1,2,\cdots,N_{s}\end{split}  \tag{18}\]  
where  
\[ \begin{split}\tilde{I}_{l}(k)&=H_{l}(k)H_{l}^{*}(k+ F_{m})\tilde{I}_{l+D}(k)|X_{l}(k)|^{2}e^{-j2\pi F_{m}\Delta_{s}(IN_{u}+N_{g})/N}\\ &+H_{l+D}^{*}(k)H_{l+D}(k+F_{m})\tilde{I}_{l}^{*}(k)|X_{l+D}(k)| ^{2}e^{j2\pi F_{m}\Delta_{s}((l+D)N_{u}+N_{g})/N}\\ &+\hat{I}_{l}^{*}(k)\hat{Z}_{l+D}(k)+\hat{Z}_{l}^{*}(k)\hat{I}_{l+ D}(k)+\hat{I}_{l}^{*}(k)\hat{I}_{l+D}(k)\end{split}  \tag{19}\]  
and  
\[ \begin{split}\hat{Z}_{l}(k)&=H_{l}(k)H_{l}^{*}(k+F_{ m})\hat{Z}_{l+D}(k)|X_{l}(k)|^{2}e^{-j2\pi F_{m}\Delta_{s}(IN_{u}+N_{g})/N}\\ &+H_{l+D}^{*}(k)H_{l+D}(k+F_{m})\hat{Z}_{l}^{*}(k)|X_{l+D}(k)|^{2 }e^{j2\pi F_{m}\Delta_{s}((l+D)N_{u}+N_{g})/N}+\hat{Z}_{l}^{*}(k)\hat{Z}_{l+D} (k).\end{split}  \tag{20}\]  
From former work [22], \(I_{l}(k)\) can be treated as zero-mean Gaussian noise with variance \(\sigma_{l}^{2}\). Since \(E\{\hat{I}_{l}(k)\}=E\{\hat{Z}_{l}(k)\}=0\), it immediately follows from Equations (19) and (20) that \(E\{\hat{I}_{l}(k)\}=E\{\hat{Z}_{l}(k)\}=0\). Thus, it turns out to be  
\[\arg\left\{E\left\{\tilde{C}_{l}(k)\right\}\right\}=2\pi F_{m}D\Delta_{s}N_{u }/N \tag{21}\]  
where \(E\{x\}\) is the expectation of \(x\). It is evident from Equation (21) that the estimation performance is proportional to the number of statistically significant correlation samples \(\tilde{C}_{l}(k)\) for \(m\)'s and \(k\)'s.