## 5 Simulation Results  
In this section, the performance of the frequency-offset estimation schemes is assessed in accordance with the DVB-C2 standard using MATLAB software. Tables 1 and 2 summarize the simulation parameters and channel profiles used in our experiments, respectively. The DVB-C2 OFDM system has \(N=4096\), \(N_{g}=64\) and \(N_{f}=30\) subcarriers. The bandwidth of 8 MHz, the sampling time of \(T_{s}=7/64\) us and 16 QAM are considered [10]. We simulate under AWGN and two echo channels specified in [11], which are based on the HFCchannel model. The CIR of echo channels can be described as:  
\[h(t)=1+\bar{\gamma}\sum_{i=1}^{N_{e}}\gamma_{i}e^{-|\theta_{i} }\delta(t-\tau_{i}) \tag{57}\]  
where \(\bar{\gamma}=(\sum_{i=1}^{N_{e}}\gamma_{i}^{2})^{-1/2}\), \(N_{e}\) is the number of echoes, the path gain \(\gamma_{i}\), the path delay \(\tau_{i}\) and the path phase \(\theta_{i}\), which are given in Table 2. Since \(N_{g}=64\) is used, an echo delay of up to 7 us can be tolerated. For CFO and SFO, \(\varepsilon=0.02\) and \(\bar{\varepsilon}=20\) ppm. The performance and complexity of the proposed WLSE method and the conventional Methods B and C are evaluated based on known CFR, while other schemes perform without relying on known CFR. To estimate the CFO and SFO in a decoupled manner using (32) and (33), \(L_{p}\) should be zero because of non-zero valued SFO and CFO in the first term of the RHS of (35) and (36), respectively. The goal of the pilot subset selection scheme is to make \(L_{p}\) zero, as well as to maximize \(N_{p}\) and \(M_{p}\) at the same time. For this purpose, an optimal subset \(\mathcal{S}_{\beta}\) is selected using the following procedure. For every possible number of combinations \(N_{c}\), we search for a subset that satisfies (37), where the number of subsets can be one or more if \(L_{p}=0\). In this case, the subset with the largest \(N_{p}\) is chosen. If there is only one subset satisfying (37), this subset is the optimal one. Even after this process, there may be several subsets for the same \(N_{p}\). Since the selected subsets all satisfy (38), if \(L_{p}=0\), it is obvious that the subset giving the minimum MSE of the SFO estimator is the subset with the largest \(M_{p}\). Through this exhaustive search, the optimal subset satisfying both (37) and (38) is found, and its value is \(N_{p}=27\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Bandwidth (Hz) & 8 M \\ FFT size \(N\) & 4096 \\ Sampling time \(T_{s}\) (μs) & 7/64 \\ Subcarrier spacing \(1/NT_{s}\) (Hz) & 2232 \\ Number of used subcarriers \(N_{z}\) & 3408 \\ Number of GI samples \(N_{g}\) & 64 \\ Number of CPs \(N_{f}\) & 30 \\ Subcarrier modulation & 16 QAM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. GI, guard interval; CP, continual pilot. Figure 1 presents the comparison between the theoretical and simulation results of the proposed LLSE for various configurations in the AWGN. Since the size of the first block \(N_{a}\) falls inside \(0<N_{a}<N_{b}\), the range of the x-axis is \(1\leq N_{a}\leq N_{b}-1\). It is evident that the analytical results are in perfect agreement with the simulated results in the AWGN channel. Hence, one can see that the optimal \(N_{a}\) obtained from (56) exactly accords with the value simulated in Figure 1. It is worthwhile to notice that the performance is symmetric with respect to \(N_{a}=N_{b}/2\). This is due to the fact that the representation of block-by-block correlation (20) in the symmetric position around \(N_{a}=N_{b}/2\) is the same except for the conjugate locations because the sum of the sizes of the two blocks is \(N_{b}\). From now on, the optimal value \(N_{a}\) will be used in the proposed scheme. Figure 2 depicts the comparison between the theoretical and simulation results of the proposed LLSE method versus SNR in the AWGN channel. The CRBs using (53) and (54) are also presented as a baseline.