## 4 Simulation and Analysis
### Simulations
#### 4.1.4 Performance Comparison Results of the Proposed Algorithm and Other Algorithms  
To verify the signal parameter estimation accuracy corresponding to different SNR, we compare the proposed algorithm with the conventional second-order FLL assisted third-order PLL (FLL&PLL) carrier loop and EKF algorithm. The FLL&PLL displays both the high accuracy of PLL and the large dynamics of FLL. The specific parameters and corresponding discriminator algorithm of the second-order FLL assisted third-order PLL are given in Table 2. The EKF algorithm used for comparison in this paper is a pure EKF algorithm that is independent of MLE. The result of the coherent integration is used as the input observation signal parameter of the EKF. Since the carrier loop can be considered as a nonlinear discrete-time system. The purpose of using the EKF-based carrier loop is to linearize the state equation and the observation equation of carrier tracking loop to obtain a sub-optimal estimation method for the carrier loop. The algorithm proposed in this paper is to use MLE to estimate the interested parameters of the carrier loop, and then use KF to filter estimated parameters. Since the carrier loop can be considered as a nonlinear discrete-time system. The KF is a linear discrete system filter which cannot be directly used to the carrier loop. We use MLE to estimate the carrier loop parameters and do a non-linear approximation, and then use KF for filtering, which is different from the pure EKF algorithm. Two hundred Monte Carlo simulations are performed for each SNR with a fixed predetermined frequency error. The RMSE frequency error comparison results are shown in Figure 10. The results in Figure 10 show that the frequency estimation errors of the three algorithms all increase with SNR decreasing. Among them, the FLL&PLL has the largest error under low SNR, followed by the EKF. The proposed algorithm (MLE&KF) can still effectively estimate the frequency, which shows high frequency estimation accuracy and tracking performance for weak TC-OFDM signals. The simulation verifies the superiority of the proposed algorithm for various SNR. Figure 9: Frequency error comparison results by MLE and MLE&KF. Meanwhile, in order to further verify the tracking performance of the proposed algorithm, the comparison results of the tracking probability with other two algorithms are shown in Figure 11. Tracking probability is defined as the probability of successfully tracking the total number of Monte Carlo simulations. Tracking sensitivity is defined as the SNR at which tracking probability exceeds 50%. At present, the definition of tracking sensitivity can be divided into two types: the detection probability is 50% [31] and 90% [35]. In order to better illustrate the problem, this paper refers to [31] for the definition of tracking sensitivity. According to the results of the detection probability of the proposed algorithm in Figure 11, the SNR is adopted to be taken as the tracking sensitivity when the tracking probability is 50%, which can reflect the overall level of the performance of the algorithms. It can be seen that the tracking sensitivity of FLL&PLL is \(-31.1\) dB, followed next by EKF which is \(-33\) dB. The MLE&KF algorithm has better performance than the others, and the tracking sensitivity is \(-35.3\) dB, which is 2.3 dB lower than EKF and 4.2 dB lower than FLL&PLL. The Monte Carlo simulation results prove that proposed algorithm can significantly improve the tracking sensitivity for weak TC-OFDM signals.