# Robust Sampling Frequency Offset Estimation for OFDM over Frequency Selective Fading Channels
## 5 Simulation Results and Discussions
Interestingly, the different estimation performance can be observed according to the robustness mode-specific parameter as shown in Table 1. It is interesting to note that the proposed scheme (25) has no restriction to pilots similarly to (4). Figure 1: Mean square error (MSE) of the conventional frequency estimators in channel model (CM)1: (1) lines-analysis; (2) symbols-simulation. Figure 2: MSE of the proposed frequency estimator in CM1: (**a**) \(\Delta_{f}=10^{-2}\); (**b**) \(\Delta_{f}=3\times 10^{-2}\). Figure 3 presents the comparison of MSE between the conventional scheme (14) and the proposed scheme when DRM mode A is used and \(\Delta_{f}=10^{-2}\). For the sake of fairness, \(N_{s}=D_{t}+1\) is chosen for the conventional scheme (18), which uses the pilot symbols inserted into every OFDM symbol. In order to compute (19) and (20), a perfect knowledge of the CSI is assumed in the conventional scheme D. From the figure, it is observed that a slight performance degradation of the proposed scheme relative to the conventional scheme C in low SNR regions is observed, which is due to the SNR loss induced by the squaring of the complex conjugate operation in (22). As reported in [22; 23], the conventional scheme C exhibits a non-negligible bias with an increase in SNR, which stems from the fact that its performance strongly is affected by the frequency selectivity of the channel and the value of SFO. Conversely, the performance of the proposed estimation scheme and the conventional scheme D is almost independent of the increased frequency selectivity and SFO. For a relatively small values of \(\Delta_{s}\) and \(\Delta_{f}\), therefore, we conclude that the proposed SFO estimation scheme and the conventional scheme D are shown to be unbiased in the presence of the frequency selectivity. Under the assumption of perfect CSI, the MSE of the conventional scheme D is better than that of the proposed method at the cost of computational burdens. It can be seen that the number of flops used in the proposed estimation scheme is decreased by approximately 96% when compared with that used in the conventional scheme D.  
In Figures 4 and 5, we compare MSE performance of the proposed scheme with the conventional schemes (14) and (18) under the same conditions as in Figure 3 when \(\text{SNR}=25\) dB and \(\text{SNR}=40\) dB, respectively. It is observed that the MSE of the conventional method C heavily depends on the amount of SFO, whereas the proposed scheme and the conventional scheme D exhibit a remarkable robustness against the SFO. Notably, we find that the amount of dependence on SFO in the conventional scheme C is different according to the DRM modes, which is inversely proportional to the ratio \(N_{g}/N\). This primarily comes from the fact that the conventional estimation scheme C experiences substantially less frequency selectivity as the ratio \(N_{g}/N\) grows. Since SFO estimation is the focus here, the mobile speed is not taken into account. The DRM standard provides four different modes to cope with various transmission channel like delay spread and Doppler spread. In the case of analyzing the robustness of DRM modes against Doppler spread, the effect of mobile speed should be considered. As discussed in Figure 3, the MSE of the proposed scheme is slightly worse than that of the conventional scheme D, because the proposed scheme uses only two pilot symbols based on \(D_{t}\)-symbol delay estimation. If the estimated CSI is used in the conventional scheme D, the performance degradation at low SNR regions can be observed due to channel estimation error, while the performance gap between the ideal  
Figure 3: MSE of the conventional and proposed frequency estimators in DRM mode A: (**a**) \(\Delta_{s}=50\) ppm; (**b**) \(\Delta_{s}=100\) ppm. and estimated cases is expected to be negligible with the increase in SNR. As for the conventional estimation scheme C and the proposed estimation scheme, they do not require any CSI.