## 3 The Conventional LSE Scheme  
Since many pilot symbols are dedicated to the purpose of time and frequency synchronization in the ATSC 3.0 system, we focus on the pilot-based LSE scheme [28; 29; 30; 31] as a reference to the proposed method. In this paper, we consider the common CPs as pilot symbols, which are continuously inserted at a fixed subcarrier position. The channel is assumed to remain stationary during two symbol periods so that \(H_{l}^{m,r}(k)=H_{l+1}^{m,r}(k)\). We omit the superscript \(m\) in \(X_{l}^{m}(k)\) assuming that the pilot symbols are the same among the transmit antennas. Based upon two consecutive CPs, the pilot-compensated signals in the MIMO configuration is represented by  
\[\begin{split}\tilde{Y}_{l}(k)&=\sum_{r=1}^{N_{R}}Y _{l}^{*}(k)X_{l}(k)Y_{l+1}^{*}(k)X_{l+1}^{*}(k)\\ &=\sum_{r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}E_{X}^{2}e^{\beta 2\pi\varphi(k) \rho}+\sum_{r=1}^{N_{R}}\tilde{C}_{l}^{r}(k)+\sum_{r=1}^{N_{R}}\tilde{Z}_{l}^ {r}(k),\;k\in\mathcal{S}_{f}\end{split} \tag{10}\]  
where \(\rho=N_{s}/N\), \(\mathcal{S}_{f}\) is the set of pilot subcarrier indices, \(\tilde{C}_{l}^{r}(k)\) is the ICI component written as  
\[\begin{split}\tilde{C}_{l}^{r}(k)&=E_{X}H_{l}^{r*}( k)X_{l+1}^{*}(k)C_{l+1}^{r}(k)e^{-\beta 2\pi\varphi(k)(IN_{s}+N_{g})/N}\\ &+E_{X}H_{l+1}^{r}(k)X_{l}(k)C_{l}^{r*}(k)e^{2\pi\varphi(k)((l+1) N_{s}+N_{g})/N}+C_{l+1}^{r}(k)Z_{l}^{r*}(k)X_{l}(k)X_{l+1}^{*}(k),\\ &+C_{l}^{r*}(k)Z_{l+1}^{r}(k)X_{l}(k)X_{l+1}^{*}(k)+C_{l+1}^{r}(k )C_{l}^{r*}(k)X_{l}(k)X_{l+1}^{*}(k)\end{split} \tag{11}\]  
and \(\tilde{Z}_{l}^{r}(k)\) is the AWGN component written as  
\[\begin{split}\tilde{Z}_{l}^{r}(k)&=E_{X}H_{l}^{r*}( k)X_{l+1}^{*}(k)Z_{l+1}^{r}(k)e^{-\beta 2\pi\varphi(k)(IN_{s}+N_{g})/N}\\ &+E_{X}H_{l+1}^{r}(k)X_{l}(k)Z_{l}^{r*}(k)e^{2\pi\varphi(k)((l+1 )N_{s}+N_{g})/N}+Z_{l}^{r*}(k)Z_{l+1}^{r}(k)X_{l}(k)X_{l+1}^{*}(k).\end{split} \tag{12}\]  
For notation simplicity, the argument of \(\tilde{Y}_{l}(k)\) takes the form  
\[\angle\tilde{Y}_{l}(k)=2\pi\rho\varphi(k)+I_{l}(k) \tag{13}\]  
where \(I_{l}(k)\) means an appropriate noise term after taking an argument. When the common CPs are entirely used for synchronization, the CFO and SFO are obtained by applying the LS regression to \(\varphi(k)=\varepsilon+k\xi\) as  
\[\begin{split}\hat{\varepsilon}&=\frac{1}{2\pi \rho N_{f}}\sum_{k\in\mathcal{S}^{+}\cup\mathcal{S}^{-}}\angle\tilde{Y}_{l}(k) \\ &=\varepsilon+\frac{1}{2\pi\rho N_{f}}\sum_{k\in\mathcal{S}^{+} \cup\mathcal{S}^{-}}I_{l}(k)+\frac{\xi A_{f}}{N_{f}}\end{split} \tag{14}\]  
and\[\begin{split}\hat{\xi}&=\frac{1}{2\pi\rho M_{f}}\sum_{k\in \mathcal{S}^{+}\cup\mathcal{S}^{-}}k\angle\tilde{Y}_{l}(k)\\ &=\xi+\frac{1}{2\pi\rho M_{f}}\sum_{k\in\mathcal{S}^{+}\cup \mathcal{S}^{-}}kl_{l}(k)+\frac{\varepsilon A_{f}}{M_{f}}\end{split} \tag{15}\]  
where \(\mathcal{S}^{+}\) denotes the set of positive CP subcarriers containing \(N^{+}\) members, \(\mathcal{S}^{-}\) denotes the set of negative CP subcarriers containing \(N^{-}\) members, \(N_{f}=N^{+}+N^{-}\) stands for the total number of pilot subcarriers present in \(\mathcal{S}_{f}=\mathcal{S}^{+}\cup\mathcal{S}^{-}\), \(A_{f}=\sum_{k\in\mathcal{S}^{+}\cup\mathcal{S}^{-}}k\), and \(M_{f}=\sum_{k\in\mathcal{S}^{+}\cup\mathcal{S}^{-}}k^{2}\). Note that the second term of the right-hand side (RHS) of Equations (14) and (15) is the intrinsic interference caused by AWGN and ICI, whereas the third term is the extra bias introduced by the existence of non-symmetrically located CPs. Among them, the interferences due to ICI and CPs do not vanish even as signal-to-noise ratio (SNR) grows.  
Since CPs are randomly distributed in the ATSC 3.0 system, it is not guaranteed that \(A_{f}=0\). In such a case, an unwanted bias can be occurred by non-symmetrically distributed CPs, which leads to an irreducible MSE floor. To account for this issue, an effective LSE scheme is suggested based on a pilot subset selection in the following section.