## 4 Proposed Frequency-Offset Estimation Method
### MSE Analysis  
This section derives the MSE of (32) and (33) in the AWGN channel. For the purpose of notational convenience, (25) is rearranged into:  
\[\bar{Y}_{l}(k)=E_{X}S(\varphi(k))e^{j\pi\beta\varphi(k)N_{b}}\left[1+\frac{ \left\{\mathcal{C}_{l}(k)+\mathcal{Z}_{l}(k)\right\}e^{-j\pi\beta\varphi(k)N _{b}}}{E_{X}S(\varphi(k))}\right] \tag{41}\]  
where:  
\[\mathcal{C}_{l}(k)=\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l-1}^{N_{b}+l-1}\bar{ \mathcal{C}}_{m_{d}}(k) \tag{42}\]  
and:  
\[\mathcal{Z}_{l}(k)=\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l-1}\bar{ \mathcal{Z}}_{m_{d}}(k). \tag{43}\]  
Under the high SNR assumption [32] and CP boosting factor of \(7/3\), we make the useful approximation that:  
\[\hat{\varphi}(k)\approx\varphi(k)+\frac{1}{\pi\beta N_{b}}\frac{\mathcal{C}_{ l}^{Q}(k)+\bar{Z}_{l}^{Q}(k)}{E_{X}S(\varphi(k))} \tag{44}\]  
where \(X^{Q}\) denotes the imaginary component of a complex number \(X\), \(\bar{\mathcal{C}}_{l}(k)=\mathcal{C}_{l}(k)e^{-j\pi\beta\varphi(k)N_{b}}\), and \(\bar{\mathcal{Z}}_{l}(k)=\mathcal{Z}_{l}(k)e^{-j\pi\beta\varphi(k)N_{b}}\). Notice that the statistics of \(\mathcal{C}_{l}(k)\) and \(\mathcal{Z}_{l}(k)\) are untouched after multiplying with \(e^{-j\pi\beta\varphi(k)N_{b}}\). Since \(L_{p}=0\) from the use of the optimal subset, substituting (44) into (29) and (30) produces the estimation error of the CFO and SFO estimator:  
\[\hat{\varepsilon}-\varepsilon=\frac{1}{\pi\beta N_{b}N_{p}}\sum_{k\in \mathcal{S}_{p}}\frac{\bar{\mathcal{C}}_{l}^{Q}(k)+\bar{\mathcal{Z}}_{l}^{Q} (k)}{E_{X}S(\varphi(k))} \tag{45}\]  
and:  
\[\hat{\xi}-\xi=\frac{1}{\pi\beta N_{b}M_{p}}\sum_{k\in\mathcal{S}_{p}}\frac{k \bar{\mathcal{C}}_{l}^{Q}(k)+\bar{\mathcal{Z}}_{l}^{Q}(k)}{E_{X}S(\varphi(k))}. \tag{46}\]  
After some straightforward manipulations, it turns out to be:  
\[E\left\{\left|\hat{\varepsilon}-\varepsilon\right|^{2}\right\}=\left(\frac{1} {\pi\beta N_{b}N_{p}E_{X}}\right)^{2}\sum_{k\in\mathcal{S}_{p}}\frac{E\left\{ \left|\hat{\mathcal{C}}_{l}^{Q}(k)\right|^{2}\right\}+E\left\{\left|\bar{ \mathcal{Z}}_{l}^{Q}(k)\right|^{2}\right\}}{S(\varphi(k))^{2}} \tag{47}\]  
and:  
\[E\left\{\left|\hat{\xi}-\xi\right|^{2}\right\}=\left(\frac{1}{\pi\beta N_{b}M _{p}E_{X}}\right)^{2}\sum_{k\in\mathcal{S}_{p}}\frac{k^{2}E\left\{\left|\hat{ \mathcal{C}}_{l}^{Q}(k)\right|^{2}\right\}+k^{2}E\left\{\left|\hat{\mathcal{Z }}_{l}^{Q}(k)\right|^{2}\right\}}{S(\varphi(k))^{2}}. \tag{48}\]From (22) and (23), we obtain their variances \(\sigma_{\zeta}^{2}=E_{X}\sigma_{C}^{2}+\sigma_{C}^{2}\sigma_{Z}^{2}+\sigma_{C}^{4 }/2\) and \(\sigma_{X}^{2}=E_{X}\sigma_{Z}^{2}+\sigma_{Z}^{4}/2\), respectively. Bearing in mind that \(E\{\tilde{C}_{l}^{Q}(k)\}=E\{\tilde{Z}_{l}^{Q}(k)\}=0\), the variances of \(\tilde{C}_{l}^{Q}(k)\) and \(\tilde{Z}_{l}^{Q}(k)\) are computed by:  
\[E\left\{\left|\tilde{C}_{l}^{Q}(k)\right|^{2}\right\}=N_{b}(N_{b}-N_{a})N_{a}( E_{X}\sigma_{C}^{2}+\sigma_{C}^{2}\sigma_{Z}^{2})/2+(N_{b}-N_{a})N_{a}\sigma_{C}^{4 }/2 \tag{49}\]  
and:  
\[E\left\{\left|\tilde{Z}_{l}^{Q}(k)\right|^{2}\right\}=N_{b}(N_{b}-N_{a})N_{a} E_{X}\sigma_{Z}^{2}/2+(N_{b}-N_{a})N_{a}\sigma_{Z}^{4}/2 \tag{50}\]  
which is put into (47) and (48). More importantly, \(S(\varphi(k))\approx(N_{b}-N_{a})N_{a}\) for small enough values of \(\varepsilon\) and \(\xi\). With those implications in mind, one gets:  
\[\mathcal{C}_{l}(k)=\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l-1}^{N_{b}+l-1}\bar{ \mathcal{C}}_{m_{d}}(k) \tag{42}\]0  
and:  
\[\mathcal{C}_{l}(k)=\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l-1}^{N_{b}+l-1}\bar{ \mathcal{C}}_{m_{d}}(k) \tag{42}\]1  
where \(\rho_{z}=E_{X}/\sigma_{Z}^{2}\) is the SNR and \(\rho_{c}=E_{X}/\sigma_{C}^{2}\) is the signal-to-ICI ratio.