## 4 Proposed SFO Estimation Scheme
### MSE Analysis  
This section derives the MSE expression of the proposed estimation scheme presented in Equation (24). By denoting \(v=e^{-j2\pi F_{m}DN_{u}N_{u}/N}\), it is found from Equation (24) that the argument of the phase-rotated partial correlation \(\Omega_{m}v\) can be written as  
\[\tan^{-1}\left\{\frac{\Im\left\{\Omega_{m}v\right\}}{\Re\left\{ \Omega_{m}v\right\}}\right\} =\tan^{-1}\left\{\frac{\Im\left\{\Omega_{m}\right\}}{\Re\left\{ \Omega_{m}\right\}}\right\}+\tan^{-1}\{e^{-j2\pi F_{m}DN_{u}N_{u}/N}\} \tag{25}\] \[=2\pi F_{m}DN_{u}/N(\hat{\Delta}_{s}-\Delta_{s})\]  
for \(m=1,2,\cdots,N_{s}\). Since \(\hat{\Delta}_{s}\rightarrow\Delta_{s}\) for high signal-to-noise ratio (SNR), it is obvious from Equation (25) that  
\[ \begin{split}\frac{\Im\left\{\Omega_{m}v\right\}}{\Re\left\{ \Omega_{m}v\right\}}&=\tan\left\{2\pi F_{m}DN_{u}/N(\hat{\Delta} _{s}-\Delta_{s})\right\}\\ &\approx 2\pi F_{m}DN_{u}/N(\hat{\Delta}_{s}-\Delta_{s}).\end{split}  \tag{26}\]  
Under the high SNR assumption [27], it can be safely assumed that  
\[\Re\left\{\Omega_{m}v\right\}\gg\Re\left\{\sum_{k\in\mathcal{S}_{m}}\left[ \tilde{I}_{l}(k)+\tilde{Z}_{l}(k)\right]v\right\}. \tag{27}\]  
With this assumption in mind, the denominator of the left-hand side (LHS) of Equation (26) is well approximated by  
\[\Re\left\{\Omega_{m}v\right\}\approx\sum_{k\in\mathcal{S}_{m}}|H_{l}(k)|^{2}| H_{l}(k+F_{m})|^{2}|X_{l}(k)|^{2}|X_{l+D}(k)|^{2}. \tag{28}\]  
For relatively large \(N_{m}\) and frequency selectivity, one can make the useful approximation  
\[\Re\left\{\Omega_{m}v\right\}\approx N_{m}E\{|H_{l}(k)|^{2}\}E\{|H_{l}(k+F_{m })|^{2}\}E_{X}^{2} \tag{29}\]  
where \(E_{X}=|X_{l}(k)|^{2}=|X_{l+D}(k)|^{2}\). Plugging Equation (29) into (26) produces  
\[\hat{\Delta}_{s}-\Delta_{s}\approx\frac{N}{2\pi F_{m}DN_{u}}\frac{\Im\left\{ \Omega_{m}v\right\}}{N_{m}\sigma_{H}^{4}E_{X}^{2}},\ m=1,2,\cdots,N_{s} \tag{30}\]  
where \(\sigma_{H}^{2}=E\{|H_{l}(k)|^{2}\}=E\{|H_{l}(k+F_{m})|^{2}\). Since \(E\{\tilde{Z}_{l}(k)\}=E\{\tilde{I}_{l}(k)\}=0\), the mean and variance of \(\Im\left\{\Omega_{m}v\right\}\) are respectively calculated by  
\[ \begin{split} E\left\{\Im\left\{\Omega_{m}v\right\}\right\}& =E\left\{\sum_{k\in\mathcal{S}_{m}}\Im\left\{\tilde{C}_{l}(k)v \right\}\right\}\\ &=\sum_{k\in\mathcal{S}_{m}}E\left\{\Im\left\{\tilde{I}_{l}(k)v +\tilde{Z}_{l}(k)v\right\}\right\}=0\end{split}  \tag{31}\]  
and\[ \begin{split}\sigma^{2}&=N_{m}E\left\{\left|\Im\left\{\bar{l}_ {l}(k)v+\bar{Z}_{l}(k)v\right\}\right|^{2}\right\}\\ &=N_{m}\left(\sigma_{H}^{4}E_{X}^{2}\sigma_{I}^{2}+\sigma_{I}^{2 }\sigma_{Z}^{2}+\frac{\sigma_{I}^{2}}{2}\right)+N_{m}\left(\sigma_{H}^{4}E_{X }^{2}\sigma_{Z}^{2}+\frac{\sigma_{Z}^{4}}{2}\right)\end{split}  \tag{32}\]  
where \(\sigma_{I}^{2}\) is the variance of Equation (15) calculated by \(\sigma_{I}^{2}=2\sigma_{H}^{2}E_{X}\sigma_{I}^{2}+2\sigma_{I}^{2}\sigma_{Z}^{2 }+\sigma_{I}^{4}\) and \(\sigma_{Z}^{2}\) is the variance of Equation (16) given by \(\sigma_{Z}^{2}=2\sigma_{H}^{2}E_{X}\sigma_{Z}^{2}+\sigma_{Z}^{4}\). From Equation (2), \(\sigma_{I}^{2}\) can be calculated by  
\[\sigma_{I}^{2}=\sigma_{H}^{2}E_{X}\sum_{n=0,n\
eq k}^{N-1}\frac{\sin^{2}(\pi( \phi(n)+n-k)}{N^{2}\sin^{2}(\pi(\phi(n)+n-k)/N)}. \tag{33}\]  
Using first order Taylor series of each term in summation in Equation (33) with respect to \(n-k\), we have  
\[ \begin{split}\sigma_{I}^{2}&\approx\sigma_{H}^{2} E_{X}\sum_{n=0,n\
eq k}^{N-1}\frac{\pi^{2}\phi^{2}(n)}{N^{2}\sin^{2}(\pi(n-k)/N)} \\ &=\sigma_{H}^{2}E_{X}\Delta_{s}^{2}\sum_{n=0,n\
eq k}^{N-1}\frac{ \pi^{2}n^{2}}{N^{2}\sin^{2}(\pi(n-k)/N)}\end{split}  \tag{34}\]  
which says that \(\sigma_{I}^{2}\) in the presence of a typical values of \(\Delta_{s}\) is very small in comparison to \(\sigma_{Z}^{2}\)[14]. Therefore, the first term of the LHS of Equation (32) can be omitted.