# Robust Sampling Frequency Offset Estimation for OFDM over Frequency Selective Fading Channels
## 5 Simulation Results and Discussions  
In order to verify the correctness of the MSE analysis as well as the effectiveness of the proposed estimation scheme, the simulations are conducted on DRM systems, taking into account different DRM robustness modes, whose main parameters are summarized in Table 1. To make a fair comparison, \(D=D_{t}\) is chosen for the conventional and proposed schemes. To show the performance, we consider the six channel models (CMs) provided in the DRM standard [2], which are listed in Table 2. In order to focus on the effect of SFO on the MSE performance of the DRM receiver in the presence of the frequency selectivity of the channel, the mobile speed is not included in the considered CMs. Firstly, the complexity comparison of the proposed scheme (25) to the conventional schemes in (14) and (18) is discussed in terms of the number of real floating point operations (flops) needed to complete one-shot estimation. A complex multiplication and a complex addition count six and two flops, respectively [25]. For fair comparison, we consider only the GRC-based estimation schemes. We assume that the CSI has already been estimated at the receiver so that the complexity analysis in the conventional scheme D is presented for the case of known CSI. Since \(b_{l}(k)\in\{1,-1\}\) does not require actual multiplication, its calculation to obtain \(\hat{R}_{l}(k)\) is not included in the complexity comparison. Table 3 lists the computational complexity between the conventional and proposed estimation schemes. For each one-shot estimate, the total number of flops of the conventional scheme C is \(8N_{p}-2\), while the conventional scheme D consumes \(N_{p}(12N_{s}^{2}-13N_{s}+98)/2+7\) flops. On the other hand, the total flops needed to obtain (25) in the proposed scheme is \(19N_{p}/2\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mode A** & **Mode B** & **Mode C** & **Mode D** \\ \hline \(N\) & 288 & 256 & 176 & 112 \\ \(N_{g}\) & 32 & 64 & 64 & 88 \\ \(N_{p}\) & 12 & 34 & 36 & 28 \\ \(D_{t}\) & 5 & 3 & 2 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters in Digital Radio Mondiale (DRM) modes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **CM1** & **CM2** & **CM3** & **CM4** & **CM5** & **CM6** \\ \hline Number of path & 1 & 2 & 4 & 2 & 2 & 4 \\ Maximum delay spread (ms) & 0 & 1 & 2.2 & 2 & 4 & 6 \\ Maximum Doppler spread (Hz) & 0 & 0.1 & 2 & 1 & 2 & 7.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main parameters in DRM channel models. The MSE of the conventional estimation schemes in DRM channel model 1 (CM1) is shown in Figure 1 according to DRM robustness modes. The CM1 corresponds to AWGN channel. In this example, DRM modes B and D are considered, \(\Delta_{f}=10^{-2}\), and \(\Delta_{s}=50\) ppm. First of all, one can see that the analytical curves in (6), (10), and (15) are shown to be close to the simulated curves. Note that the estimation scheme (14) can be generalized only to uniformly distributed pilot tones in the frequency domain, whereas (7) is valid for a specifically designed pilot symbol in the DRM. On the other hand, the estimation scheme (4) has no restriction to pilots, thereby can be generalized to arbitrary pilot configurations as similarly done in [21]. The MSE of the proposed estimator in CM1 is reported in Figure 2 for \(\Delta_{s}=50\) ppm. It is evident that the analytic curves in (32) are parallel to the simulated curves in CM1. It can be observed that the MSE of the proposed estimator is unbiased for relatively small \(\Delta_{f}\). With the increase in \(\Delta_{f}\), however, this does not hold at high SNR conditions, because the ICI is no longer negligible. For typical values of \(\Delta_{s}\) and \(\Delta_{f}\), therefore, the ICI is negligible in comparison with the Gaussian noise under medium SNR conditions. It is also found that the DRM system is shown to contain sufficient information to perform synchronization.