## 3 MLE Parameter Estimation Model
### KF Model  
MLE can obtain the minimum error variance of parameters without relying on the prior distribution of signal estimation parameters. For MLE, the choice of the number of observations \(N\) is significant. The reason is that if a small value of \(N\) is adopted, the observed sample value of MLE is small, thus the estimated result cannot reflect the true data very well. Conversely, if the  
Figure 4: LM algorithm flow chart. observed value of \(N\) is large, excessive observation noise will be introduced, which is not beneficial to obtaining accurate results. Therefore, if we use the estimated frequency directly in weak signal environment, the non-negligible estimated error in the carrier loop will occur, and finally restrict the improvement of tracking sensitivity and accuracy under weak signals. Since KF is the optimal filter in linear discrete systems [33]. KF is adopted in this paper to smooth the estimation of the parameters after a non-linear estimation by MLE. Therefore, for the carrier frequency tracking problem of TC-OFDM system, the combination algorithm of MLE and KF is the optimal solution. The proposed carrier loop structure is shown in Figure 5.  
The main idea is to use MLE to get the estimated value of \(\Delta f\) and \(\Delta\varphi\), and KF to smooth the MLE parameter estimation error to get more accurate \(\Delta f\) and \(\Delta\varphi\). The output value of the KF is used to adjust the carrier NCO. The MLE uses the LM iteration to get the next value of \(\Delta f\) and \(\Delta\varphi\). In practical application, the form of KF is closely related to the state equation and observation equation. The state equation describes the behaviour of the state vector. In order to adjust the carrier loop accurately, the state vector in the selected carrier loop is expressed as:  
\[X=[ \begin{array}{cc}\varphi&\omega_{d}&\omega_{d}^{\prime}\end{array} ] \tag{22}\]  
where \(\varphi\) is the carrier phase estimation value, \(\omega_{d}\) = 2\(\pi\Delta f\) denotes the angular frequency estimation value, \(\omega_{d}^{\prime}\) represents the angular frequency rate. The state equation can be expressed as:  
\[X_{k}=\Phi X_{k-1}+W_{k-1} \tag{23}\]  
where \(W_{k-1}\) is the input Gaussian white noise with mean zero and variance \(Q\), \(\Phi\) is the state transition matrix and the form can be written as:  
\[\Phi=\left[ \begin{array}{ccc}1&T&T^{2}/2\\ 0&1&T\\ 0&0&1\end{array} \right] \tag{24}\]  
where \(T\) = \(NT_{coh}\) is the loop update period. The observation equation describes the relationship between observations and state vectors. The observation equation can be expressed as:  
\[Y_{k}=HX_{k}+V_{k} \tag{25}\]  
Figure 5: The proposed carrier loop structure based on MLE and KF. where \(H\) is the observation matrix, the form can be written as:  
\[H=\left[ \begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array} \right] \tag{26}\]  
The value of the rate of change of Doppler in the observation matrix \(H\) is a predetermined experienced value. The preset value is reasonable which satisfies the current applications in the TC-OFDM system. And this kind of pre-setting for the rate of change of Doppler is beneficial to us to further study the TC-OFDM receiver algorithm in high dynamic scenario. \(V_{k}\) is the observed noise, the mean is zero and the variance matrix is \(R\). The phase difference \(\Delta\theta\) between the input signal and the local signal can be expanded by the Taylor series:  
\[\left\{ \begin{array}{l}\Delta\theta(k+1)=\Delta\theta(k)+T\Delta\omega_{0}(k )+\frac{T^{2}}{2}\Delta\omega_{1}(k)+\xi_{1}(k)\\ \Delta\omega_{0}(k+1)=\Delta\omega_{0}(k)+T\Delta\omega_{1}(k)+\xi_{2}(k)\\ \Delta\omega_{1}(k+1)=\Delta\omega_{1}(k)+\xi_{3}(k)\end{array} \right. \tag{27}\]  
where \(\Delta\theta\) is the carrier phase, \(\Delta\omega_{0}\) is the residual carrier and \(\Delta\omega_{1}\) is the rate of change of Doppler.