# Data-Aided Frequency Offset Estimation for CE-OFDM Broadband Satellite Systems
## 6 Simulation Results and Analysis  
The one-sided phase noise masks in [10] were used in the simulation. Table 12 shows these masks. We used the mean squared error (MSE) and the BER to evaluate the estimation performance. The MSE is defined as \(\Delta f_{M}=\frac{1}{F}\sum\limits_{j=1}^{F}\left(\Delta f-\Delta f\right)^{2}\), where \(F\) is the simulation frames per CNR. The simulation parameters are given in Table 13. Since we focused on the CFO estimation performance when we simulated the BER performance, the phase noise was assumed to be known at the receiver and was compensated after the CFO compensation. In order to compare with the existing CFO estimation scheme, the Minn algorithm in [10], the maximum likelihood (ML) algorithm in [11], and the decision feedback estimator (DFE) in [10] were chosen as the comparison schemes. The CRB of PSCE was compared with the Cramer-Rao lower bound (CRLB) in [10]. Figure 14 shows the MSE performance of the PSCE under different CFOs and pilot overheads when the modulation index of the CE-OFDM signal was 0.8. It can be seen that the threshold effect exists at a low CNR. As the maximum Doppler frequency offset was \(\pm\)400 kHz to \(\pm\)1600 kHz with the operating frequency changing from the Ka-band to W-band, the normalized CFO was set as 0.5 and 80.5. When \(\Delta f\) was 0.5, the threshold was about 10 dB. When \(\Delta f\) was 80.5, the threshold was about 12 dB. It \begin{table}
\begin{tabular}{c c} \hline \hline
**Frequency** & **[100 Hz, 1 kHz, 10 kHz, 100 kHz, 1 MHz,10 MHz]** \\ \hline Typical & [\(-\)25, \(-\)50, \(-\)73, \(-\)93, \(-\)103, \(-\)114] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Aggregate phase noise masks (dBc/Hz). Figure 15: The relationship between the value of \(\left|V_{i,1}\right|^{2}\) and the frequency position of the pilot symbol. \begin{table}
\begin{tabular}{c c} \hline \hline
**Modulation Technique** & **QPSK** \\ \hline Number of symbols \(N_{sym}\) & 31 \\ Number & 512 \\ Rangel of CNR & [0:2:30] \\ Number of frames per CNR & 10000 \\ Channel model & Rice channel (Rice factor 6) \\ Subcarrier Spacing & 20 kHz \\ Sampling Frequency & 10.24 MHz \\ Frame length & 50 us \\ Signal Power & 0 dB/ \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. Figure 16: The relationship between the value of \(\left|V_{i,1}\right|^{2}\) and the frequency position of the pilot symbol. can be seen that when the CNR was greater than the threshold and \(N_{p}\) was 31, the MSE performance of PSCE was better than that of the DFE algorithm. Therefore, adapting the frequency offset estimation algorithm to the characteristics of the CE-OFDM system can improve the estimation performance. Furthermore, when the CNR was greater than the threshold and \(N_{p}\) was 31, the MSE performance of PSCE was better than that of the Minn algorithm and was almost equal to that of the ML algorithm. At this time, the pilot overhead was the same as the Minn algorithm and the ML algorithm. When \(N_{p}\) was 2, the gap between the PSCE and the Minn algorithms as well as the ML algorithm was less than 1 dB, but the pilot overhead of PSCE was only two frequency domain symbols, which was far less than the other algorithms. When \(N_{p}\) was 0, the PSCE was transformed into a blind estimator, and its MSE performance was limited. Figure 7 shows the CRB of PSCE under different pilot overheads. When \(N_{p}\) was 31, the CRB of PSCE coincided with the CRLB, which proves that the CFO estimator is the MVU estimator. When the pilot overhead was reduced, the CRB of the proposed algorithm gradually increased. Figure 7 shows the BER performance of PSCE. Except for the blind estimation case (\(N_{p}\) is 0) when the CFO was equal to 0.5 or 80.5, the corresponding BER performance curve coincided with the ideal BER performance curve. Therefore, the PSCE can effectively reduce the pilot overhead while ensuring the BER performance. Figure 8: The Cramer-Rao bound (CRB) of PSCE. Figure 7: The mean squared error (MSE) performance of PSCE.