## 4 The Proposed LSE Scheme
### Pilot Selection  
Let \(\mathcal{S}^{+}_{\hat{b}}\) and \(\mathcal{S}^{-}_{\hat{b}}\) be the searched optimal subsets for positive and negative subcarrier frequencies around DC, respectively. The argument over the pilot-compensated signal \(\tilde{Y}_{l}(k)\) is then decomposed as  
\[\begin{split}\Lambda_{1}&=\sum_{k\in\mathcal{S}^{+ }_{\hat{b}}}\angle\tilde{Y}_{l}(k)+\sum_{k\in\mathcal{S}^{-}_{\hat{d}}} \angle\tilde{Y}_{l}(k)\\ &=2\pi\rho eN_{p}+\sum_{k\in\mathcal{S}^{+}_{\hat{b}}\cup \mathcal{S}^{-}_{\hat{d}}}I_{l}(k)+2\pi\rho\xi A_{p}\end{split} \tag{16}\]  
with  
\[A_{p}=\sum_{k\in\mathcal{S}^{+}_{\hat{b}}}k+\sum_{k\in\mathcal{S}^{-}_{\hat{d }}}k \tag{17}\]  
where \(\mathcal{S}^{+}_{\hat{b}}\in\mathcal{S}^{+}\) contains the \(N^{+}_{\hat{b}}\) elements, \(\mathcal{S}^{-}_{\hat{d}}\in\mathcal{S}^{-}\) contains the \(N^{-}_{\hat{d}}\) elements, and \(N_{p}=N^{+}_{\hat{b}}+N^{-}_{\hat{d}}\) is the number of CPs utilized in the proposed estimation scheme. Similarly, we have  
\[\begin{split}\Lambda_{2}&=\sum_{k\in\mathcal{S}^{+ }_{\hat{b}}}k\angle\tilde{Y}_{l}(k)+\sum_{k\in\mathcal{S}^{-}_{\hat{d}}}k \angle\tilde{Y}_{l}(k)\\ &=2\pi\rho\xi M_{p}+\sum_{k\in\mathcal{S}^{+}_{\hat{b}}\cup \mathcal{S}^{-}_{\hat{d}}}kl_{l}(k)+2\pi\rho eA_{p}\end{split} \tag{18}\]  
where  
\[M_{p}=\sum_{k\in\mathcal{S}^{+}_{\hat{b}}}k^{2}+\sum_{k\in\mathcal{S}^{-}_{ \hat{d}}}k^{2}. \tag{19}\]In order to independently estimate the CFO and SFO using Equations (16) and (18), \(A_{p}\) should be zero because of non-zero valued SFO and CFO in the third term of the RHS of Equations (16) and (18), respectively. In this paper, the goal of the pilot subset selection scheme is to make \(A_{p}\) zero as well as maximize \(N_{p}\) and \(M_{p}\) at the same time. For this purpose, two subsets \(\mathcal{S}_{\hat{b}}^{+}\) and \(\mathcal{S}_{\hat{d}}^{-}\) are selected to meet the following criteria:  
\[\left(\hat{b},\hat{d}\right)=\underset{1\leq b\leq C_{+},1\leq d\leq C_{-}}{ \arg\ zero}\left\{\sum_{k\in\mathcal{S}_{b}^{+}}k+\sum_{k\in\mathcal{S}_{\hat{ d}}^{-}}k\right\} \tag{20}\]  
and  
\[\left(\hat{b},\hat{d}\right)=\underset{1\leq b\leq C_{+},1\leq d\leq C_{-}}{ \arg\max}\left\{\sum_{k\in\mathcal{S}_{b}^{+}}k^{2}+\sum_{k\in\mathcal{S}_{\hat {d}}^{-}}k^{2}\right\} \tag{21}\]  
where \(\arg\ zero\{\cdot\}\) returns the argument which makes the enclosed objective function zero, \(\mathcal{S}_{b}^{+}\subset\mathcal{S}^{+}\) and \(\mathcal{S}_{\hat{d}}^{-}\subset\mathcal{S}^{-}\) stand for the possible combinatorial subset whose elements are \(N_{b}^{+}\) and \(N_{d}^{-}\), respectively, and \(C_{+}\) and \(C_{-}\) are the sum of the number of combinations choosing \(g\) subcarriers from \(C_{+}\) and \(C_{-}\) subcarriers respectively given by  
\[C_{+}=\sum_{g=1}^{N^{+}}\binom{N^{+}}{g}\ \text{ and }\ C_{-}=\sum_{g=1}^{N^{-}}\binom{N^{-}}{g}. \tag{22}\]  
With the use of the searched subsets from Equations (20) and (22), the proposed LSE of CFO and SFO can be shown to be of the following forms:  
\[\hat{\varepsilon}=\frac{1}{2\pi\rho N_{p}}\left(\sum_{k\in\mathcal{S}_{\hat{b }}^{+}}\angle\tilde{Y}_{l}(k)+\sum_{k\in\mathcal{S}_{\hat{d}}^{-}}\angle \tilde{Y}_{l}(k)\right) \tag{23}\]  
and  
\[\hat{\xi}=\frac{1}{2\pi\rho M_{p}}\left(\sum_{k\in\mathcal{S}_{\hat{b}}^{+}}k \angle\tilde{Y}_{l}(k)+\sum_{k\in\mathcal{S}_{\hat{d}}^{-}}k\angle\tilde{Y}_{ l}(k)\right), \tag{24}\]  
which indicates that the MSE performance of the CFO estimation method is primarily enhanced in proportion to \(N_{p}\), whereas both the number and locations of CPs determine the performance of the SFO estimation method.