# Timing and Frequency Synchronization Using CAZAC Sequences for OFDM Systems
## 5 Results and Discussion
### Simulation of the CCPD Algorithm  
The CCPD algorithm has two important parameters: the number \(m\) and length \(N\) of the ZC sequences included in the preamble. In the following, the impact of different \(m\) and \(N\) on the performance of the CCPD algorithm is simulated. A fixed CFO of 4 kHz is added to the simulation system, and the correct timing probability of the CCPD algorithm with different \(m\) and \(N\) is shown in Figure 10.  
From Figure 10, we can see that, on the whole, the synchronization accuracy of the CCPD algorithm increases as the SNR increases. In the case of a low SNR, the synchronization accuracy decreases as \(N\) decreases. This is because as \(N\) decreases, the amplitude of the correlation peaks also decreases. At a low SNR, the correlation peaks are submerged in the noise and, thus, are more difficult to examine, which can be seen in Figure 9. However, when the SNR increases above \(-2\) dB, there is no difference in the synchronization accuracy with different \(m\) and \(N\), and all synchronization accuracy values reach 100%. This is because as the SNR increases, the correlation peaks become more prominent. Therefore, in the case of a high SNR, the synchronization accuracy is not affected by the variation in \(m\) and \(N\). From the above discussion, we seem to be able to draw the following conclusion: the larger \(N\) is (and, correspondingly, the smaller \(m\) is), the better the performance of the CCPD algorithm will be. However, this conclusion is not comprehensive, because the performance of the frequency offset estimation has not been considered. In Section 4.3, we concluded that the range of the frequency offset estimation of the quadratic interpolation algorithm was \(\left[-\frac{f_{s}}{2N},\frac{f_{s}}{2N}\right)\), which was inversely proportional to the length \(N\) of the ZC sequences. Since the sampling frequency is 100 MHz, we can calculate that the frequency offset estimation range is \(\pm\)24.414 KHz when \(N\) = 2048, \(\pm\)48.828 KHz when \(N\) = 1024, \(\pm\)97.656 KHz when \(N\) = 512, \(\pm\)195.312 KHz when \(N\) = 256, and \(\pm\)390.625 KHz when \(N\) = 128. In the following, the normalized mean square error (MSE) of the frequency offset estimation is simulated for different \(m\) and \(N\). First, a small CFO of 4 KHz is added to the simulation system, which is far smaller than the upper bound of the minimum estimation range (mentioned above). The simulation results are shown in Figure 11.  
Figure 10: The correct timing probability of the CCPD algorithm with different \(m\) and \(N\) under a fixed CFO of 4 kHz. It can be seen from Figure 11 that when the CFO is much smaller than the estimation range, the accuracy values of the frequency offset estimations under different \(m\) and \(N\) are almost the same. Next, a CFO of 20 KHz is added to the simulation system, which is close to the upper bound of the estimation range for \(m=2\) and \(N=2048\). The simulation results are shown in Figure 12.  
Figure 11: The normalized mean square error (MSE) with different \(m\) and \(N\) under a fixed CFO of 4 kHz. Figure 12: The normalized MSE with different \(m\) and \(N\) under a fixed CFO of 20 kHz. It can be seen from Figure 12 that the accuracy of the frequency offset estimation when \(m=2\) and \(N=2048\) is worse than that in other cases. This is because the CFO is close to the upper bound of the estimation range, which causes the estimation performance to approach the limit, thus causing performance degradation. By combining Figures 10-12, it can be seen that the timing synchronization accuracy and the frequency offset estimation range are contradictory. Therefore, if we want to obtain the best performance, we must make a compromise. In a practical communication system, the optimal parameters need to be determined according to the possible frequency offset range. Under the simulation conditions presented in Table 1, the frequency offset range is \(\pm 97.656\) kHz when \(m=8\) and \(N=512\), which can meet the requirements of most communication scenarios.