## 4 Simulation Results
### Performance Evaluation  
In Figure 2, the probability of detection failure defined as \(\text{Prob}\{(\hat{0},\hat{w})\
eq(\upsilon,w)\}\) is depicted for both the conventional SDICID and proposed detection algorithms versus SNR in the flat fading channel. To focus on verifying our theoretical analysis, we consider scenarios in which \(D_{f}=0\) Hz and RSTO are completely estimated. Despite not being shown in Figure 2, the error probabilities of both the JDICID and the proposed methods show a significant similarity because of the use of the identical cost function. It is noteworthy that the proposed NR-PSS detector shows a significant performance enhancement compared to the SDICID method with respect to the probability of detection failure. Moreover, the theoretical results exhibit a good correspondence with the results obtained through simulation, even when confronted with frequency offset values. These observations offer ample evidence to support the effectiveness of the proposed NR-PSS detection method and the exactness of the numerical analysis presented in Section 3.2. As expected, using average estimation improves the detection ability of the NR-PSS detection methods, and the trade-off for achieving this enhanced performance is the additional processing time and arithmetic load. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & \(f_{c}=6\) GHz \\ DFT size & \(M=2048\) \\ CP size & \(M_{g}=144\) \\ NR-PSS length & \(M_{s}=128\) \\ Channel bandwidth & \(B_{w}=40\) MHz \\ Subcarrier spacing & \(\Delta_{f}=30\) kHz \\ Sampling frequency & \(f_{s}=61.44\) MHz \\ Doppler frequency & \(D_{f}=300\) Hz \\ Channel model & Flat and Rayleigh fading \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 3 presents \(P_{f}\) of the various NR-PSS synchronization schemes versus SNR in the multipath fading channel model. By employing the ML strategy, the AMMSE method shows significant improvement over the proposed NR-PSS detection method. However, despite the adoption of approximated detection to decrease complexity, the arithmetic burden of the AMMSE method remains considerably high, thereby affecting its overall efficiency. As we can see in Figure 3, it is evident that the proposed NR-SSS detection method provides comparable or even superior detection performance compared to other baseline methods like JDICID and SDICID. Furthermore, the performance difference between the SDICID and proposed methods widens with an increase in the frequency selectivity of the channel, as observed in the Vld model. This enhancement can be attributed to the presence of multipath diversity, while the widening gap is a result of the degradation of the autocorrelation property of NR-PSS as the maximum channel delay spread increases. Based on these observations, it is clear that the proposed NR-PSS synchronization scheme constitutes an efficient method to strike a balance between estimation accuracy and arithmetic load. Regardless of the detection method, an irreducible error floor is observed because of the presence of the ICI term introduced by uncompensated RCFO, which informs us that RCFO-induced ICI is ultimately the limiting cause for performance in both multipath channel environments. Therefore, it is essential to periodically estimate the RCFO during the subsequent cell search procedure [16]. Figure 4 shows \(P_{f1}\) and \(P_{f2}\) of the various NR-PSS synchronization schemes versus SNR in the Vsd channel model. The simulation environment is the same as that used in Figure 3. It is evident from Figure 4 that the shape of the probability curves exhibits qualitative similarity to that illustrated in Figure 3. Examining the results in Figures 3 and 4, one can observe that the ICFO detector plays a predominant role in determining the overall capability of the NR-PSS synchronization receiver than the CIDS detector, with the latter exhibiting better detection performance.