## 5 Simulation Results
There is a small gap between the analysis and simulation results at low SNR because of the approximation used in (44). For a wide range of SNR values, however, the analysis and simulation results are very close to each other. We observe that the proposed scheme comes close to the CRB in low and medium SNR regions, whereas the CRB is not attainable even as the SNR increases because the ICI is no longer negligible. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Echo Channel 1**} & \multicolumn{3}{c}{**Echo Channel 2**} \\ \cline{2-7} \(i\) & **Power \(\gamma_{i}\) (dB)** & **Delay \(\tau_{i}\) (ns)** & **Phase \(\theta_{i}\) (rad)** & **Power \(\gamma_{i}\) (dB)** & **Delay \(\tau_{i}\) (ns)** & **Phase \(\theta_{i}\) (rad)** \\ \hline
1 & \(-11\) & \(38\) & \(0.95\) & \(-11\) & \(162\) & \(0.95\) \\
2 & \(-14\) & \(181\) & \(1.67\) & \(-14\) & \(419\) & \(1.67\) \\
3 & \(-17\) & \(427\) & \(0.26\) & \(-17\) & \(773\) & \(0.26\) \\
4 & \(-23\) & \(809\) & \(1.20\) & \(-23\) & \(1191\) & \(1.20\) \\
5 & \(-32\) & \(1633\) & \(1.12\) & \(-32\) & \(2067\) & \(1.12\) \\
6 & \(-40\) & \(3708\) & \(0.81\) & \(-40\) & \(13\),\(792\) & \(0.81\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Channel profiles. Figure 1: Performance of the proposed linear LSE (LLSE) method versus \(N_{a}\) in the AWGN channel: (a) carrier frequency offset (CFO) estimation; (b) sampling frequency offset (SFO) estimation. Figures 3 and 4 show the MSE of the frequency-offset estimation methods in Echo Channels 1 and 2, having a maximum delay of \(3.7~{}\upmu\) and \(13.8~{}\upmu\), respectively. In the example, the conventional Scheme D corresponds to (10) and (11) when \(\mathcal{S}_{\beta}\) is used instead of \(\mathcal{S}_{f}\), which is a simple extension of the work in [25] and needs \(N_{\beta}(8N_{b}-7)-1\) real flops. As reported in the literature, non-negligible ICI leads to an error floor in the frequency-offset estimation schemes when the SNR is high. The conventional Scheme A suffers from severe irreducible error floor owing to non-symmetrically-distributed CPs, whereas the conventional Scheme D benefits from the selected pilot subset, so that its performance approaches that of the proposed scheme for higher SNR values. In Echo Channel 1, the MSE of the proposed LLSE method is slightly better than that of the conventional Method B, particularly at low SNR values. This phenomenon becomes insignificant as \(N_{b}\) increases. The amounts of phase rotation of the conventional Schemes A, B and C caused by frequency offsets are all \(2\pi\beta\), whereas that of the proposed scheme is \(\pi\beta N_{b}\). If the number of observed OFDM symbols is \(N_{b}>2\), the amount of phase rotation of the proposed scheme is greater than that of the conventional schemes, which makes the proposed estimator less sensitive to the noise, especially for a low SNR regime. With the increase of the frequency selectivity in Echo Channel 2, the performance gain of the conventional Scheme B over the proposed LLSE scheme is observed for SNR \(>10\) dB. Such a gain of the conventional Scheme B is paid for by far higher computational processing. When \(N_{b}=8\), the total number of real flops used in the proposed LLSE using \(N_{\hat{p}}=27\) is saved by \(66.9\%\), \(94.9\%\) and \(63.2\%\) compared to that of the existing Methods A, B and D, respectively. Figure 2: Performance of the proposed LLSE method versus SNR in the AWGN channel: (a) \(N_{b}=8\); (b) \(N_{b}=16\). Figure 5 depicts the performance of the existing and proposed WLSE method in Echo Channel 1 and Echo Channel 2, using the same simulation parameters as in Figures 3 and 4. In this example, the conventional WLSE scheme (18) using \(\mathcal{S}_{p}\) instead of \(\mathcal{S}_{f}\) is included, whereas the proposed scheme corresponds to (39). As predicted, we observe that the proposed WLSE approach still outperforms the conventional WLSE scheme with the aid of block-by-block estimation.