## 5 Simulation Results and Discussions  
The performance of the joint LSE schemes is assessed in the MIMO-OFDM ATSC system. The main system parameters of the ATSC 3.0 are summarized in Table 2. In this work, we have considered 8 k and 16 k modes. The signal bandwidth of 6 MHz and the sampling interval of \(T=0.14468\)\(\upmu\)s are considered [10]. In the simulations, 6-path Hilly Terrain channel model whose maximum delay is 17.2 \(\upmu\)s is adopted [40], and the multipath channel is implemented according to Equation (5). In this channel model, the multipath intensity profile of the channel is characterized by two clusters, where each cluster is exponentially decreasing in delay. Table 3 summarizes the channel profile of 6-path Hilly Terrain channel model. To demonstrate the exactness of the MSE analysis, Doppler effects due to mobility are excluded in the adopted channel model. To find the optimal subset simultaneously meeting the constraint of Equations (20) and (22), we perform an exhaustive search over all combinations of possible subsets, and it is thereby obtained that \(N_{p}=45\) for 8 k mode and \(N_{p}=92\) for 16 k mode. In the sequel, the selected mode-specific values of \(N_{p}\) are used for the proposed estimation scheme. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Complex Multiplication** & **Complex Addition** & **Real Multiplication** & **Real Addition** \\ \hline Conventional & \(3N_{R}N_{f}\) & \((N_{R}-1)N_{f}\) & \(N_{f}+1\) & \(2N_{f}-2\) \\ Proposed & \(3N_{R}N_{p}\) & \((N_{R}-1)N_{p}\) & \(N_{p}+1\) & \(2N_{p}-2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Arithmetic operations of the frequency-offset estimation schemes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **8 k Mode** & **16 k Mode** & **32 k Mode** \\ \hline FFT size \(N\) & 8192 & 16,384 & 32,768 \\ sampling time \(T\) (\(\upmu\)s) & 0.14468 & 0.14468 & 0.14468 \\ Subacrrier spacing \(1/NT\) (Hz) & 843.75 & 421.875 & 210.9375 \\ Number of used subcarriers \(N_{u}\) & 6913 & 13,825 & 27,649 \\ Number of GI samples \(N_{g}\) & 512 & 1024 & 2048 \\ Number of CPs \(N_{f}\) & 48 & 96 & 192 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Path No.** & **Propagation Delay** & **Path Power (dB)** & **Doppler PSD** \\ \hline
0 & 0.0 \(\upmu\)s & 0 & [PERSON] \\
1 & 0.1 \(\upmu\)s & \(-1.5\) & [PERSON] \\
2 & 0.3 \(\upmu\)s & \(-4.5\) & [PERSON] \\
3 & 0.5 \(\upmu\)s & \(-7.5\) & [PERSON] \\
4 & 15.0 \(\upmu\)s & \(-8.0\) & Gaus II \\
5 & 17.2 \(\upmu\)s & \(-17.7\) & Gaus II \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Channel profiles. Figure 2 depicts a comparison between the numerical and simulated results of the frequency-offset estimation schemes when \(\varepsilon=0.05\), \(\xi=50\) ppm, and 8 k mode is used. The probability of Rayleigh distributed random variable \(|H_{l}^{m,r}(k)|\) larger than a minimum level \(p_{min}\) is \(e^{-p_{min}/\sigma_{li}^{2}}\) so that 99.9% level of Rayleigh fading is realized in the case of \(p_{min}/\sigma_{H}^{2}=-30\) dB, which is used to obtain the MSE in the case of \(N_{R}=2\). One can find that there is an excellent match between the numerical results and those given by the simulation in the multipath fading channel except at low SNR values. It is evident that the use of optimal subsets makes the proposed LSE achieve a better performance over the conventional LSE. In the case of SFO, it is observed that the ICI becomes a dominant term in comparison with that of an unwanted bias from the use of CPs. Especially, the performance of the proposed scheme becomes slightly degraded due to the reduced number of pilot symbol when \(\text{SNR}<15\) dB, whereas its performance is improved thanks to the optimal subset selection at 15 dB or higher, compared to that of the conventional scheme. Figure 3 shows the MSE of the frequency-offset estimation schemes using 16k mode.