## 4 Proposed Frequency-Offset Estimation Method
### Algorithm  
Assuming that the CFR remains constant during several successive OFDM symbols, the frequency-offset estimation is performed on a block-by-block basis. Let \(N_{a}\) be the size of the first block that includes \(N_{a}\) successive observation symbols. Since the proposed estimation scheme is based on the observation of \(N_{b}\) OFDM symbols, the size of the second block becomes equal to \(N_{b}-N_{a}\). With this configuration in mind, the proposed correlation is defined by:  
\[ \begin{split}\tilde{Y}_{l}(k)&=\left\{\sum_{m=l}^{N _{a}+l-1}R_{m}(k)\right\}^{*}\left\{\sum_{q=N_{a}+l}^{N_{b}+l-1}R_{q}(k)\right\} \\ &=\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l-1}R_{m}^{*}(k)R _{q}(k).\end{split}  \tag{20}\]  
Substituting (5) into (20) leads to:  
\[ \begin{split}\tilde{Y}_{l}(k)&=|\tilde{H}_{l}(k)|^{2 }E_{X}\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l-1}e^{l2\pi\beta\varphi(k )(q-m)}+\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l-1}\tilde{C}_{m,q}(k)\\ &+\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l-1}^{N_{b}+l-1}\tilde{Z}_{ m,q}(k),\;k\in\mathcal{S}_{f}\end{split}  \tag{21}\]  
where:  
\[ \begin{split}\tilde{C}_{m,q}(k)&=\tilde{H}_{m}^{*}(k )X_{m}^{*}(k)C_{q}(k)e^{-j2\pi\varphi(k)(qN_{a}+N_{g})/N}+\tilde{H}_{q}(k)X_{q }(k)C_{m}^{*}(k)e^{j2\pi\varphi(k)(mN_{a}+N_{g})/N}\\ &+C_{q}(k)Z_{m}^{*}(k)+C_{m}^{*}(k)Z_{q}(k)+C_{m}^{*}(k)C_{q}(k) \end{split}  \tag{22}\]  
and:  
\[ \begin{split}\tilde{Z}_{m,q}(k)&=\tilde{H}_{m}^{*}(k )X_{m}^{*}(k)Z_{q}(k)e^{-j2\pi\varphi(k)(mN_{a}+N_{g})/N}+\tilde{H}_{q}(k)X_{q }(k)Z_{m}^{*}(k)e^{j2\pi\varphi(k)(qN_{a}+N_{g})/N}\\ &+Z_{m}^{*}(k)Z_{q}(k).\end{split}  \tag{23}\]  
The double summation in the first term of the right-hand side (RHS) of (21) is formulated by:  
\[ \begin{split}\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l-1}e ^{j2\pi\beta\varphi(k)(q-m)}&=\underbrace{\frac{\sin(\pi\varphi(k )(N_{b}-N_{a})\beta)}{\sin(\pi\varphi(k)\beta)}\frac{\sin(-\pi\varphi(k)N_{a} \beta)}{\sin(-\pi\varphi(k)\beta)}}_{S(\varphi(k))}e^{j\pi\beta\varphi(k)N_{b} }.\end{split}  \tag{24}\]  
Plugging (24) into (21) produces:  
\[ \begin{split}\tilde{Y}_{l}(k)&=|\tilde{H}_{l}(k)|^{2 }E_{X}S(\varphi(k))e^{j\pi\beta\varphi(k)N_{b}}+\sum_{m=l}^{N_{a}+l-1}\sum_{q= N_{a}+l}^{N_{b}+l-1}\tilde{C}_{m,q}(k)+\sum_{m=l}^{N_{a}+l-1}\sum_{q=N_{a}+l}^{N_{b}+l- 1}\tilde{Z}_{m,q}(k),\;k\in\mathcal{S}_{f}.\end{split}  \tag{25}\]From (25), the argument of \(\tilde{Y}_{l}(k)\) can be expressed as:  
\[ \begin{split}\angle\tilde{Y}_{l}(k)&=\pi\beta\varphi(k )N_{b}+\tilde{I}_{l}(k)\\ &=\pi\beta\varepsilon N_{b}+\pi\beta k\xi N_{b}+\tilde{I}_{l}(k), \ k\in\mathcal{S}_{f}\end{split}  \tag{26}\]  
where the first term of the RHS is the phase rotation due to \(\varepsilon\), the second term of the RHS is the phase rotation due to \(\xi\) and \(\tilde{I}_{l}(k)\) is the appropriate interference component after taking an argument. Then, the total frequency offsets \(\varphi(k)\) can be estimated on the non-zero CP subcarriers:  
\[ \begin{split}\hat{\varphi}(k)&=\frac{1}{\pi\beta N _{b}}\angle\tilde{Y}_{l}(k)\\ &=\varphi(k)+\frac{1}{\pi\beta N_{b}}\tilde{I}_{l}(k),\ k\in \mathcal{S}_{f}.\end{split}  \tag{27}\]  
Assuming that \(X_{l}(k)\) are statistically independent for different \(k\)'s and \(l\)'s, from the central limit theorem, \(C_{l}(k)\) is viewed as a zero-mean Gaussian random variable (RV) [29]. Thus, we have:  
\[E\left\{\hat{\varphi}(k)\right\}=\varepsilon+k\xi,\ k\in\mathcal{S}_{f} \tag{28}\]  
where \(E\{x\}\) is the expectation of RV \(x\). The joint estimation of the CFO and SFO can be obtained by collecting \(\hat{\varphi}(k)\) from the non-zero CP subcarriers and performing LS regression on the estimates of the total frequency offsets in (27). Thus, the LLSE of the CFO is derived as:  
\[ \begin{split}\hat{\varepsilon}&=\frac{\sum_{k\in \mathcal{S}_{p}}\hat{\varphi}(k)}{\sum_{k\in\mathcal{S}_{p}}1}\\ &=\frac{1}{N_{p}}\sum_{k\in\mathcal{S}_{p}}\hat{\varphi}(k)\end{split}