# A Carrier Estimation Method Based on MLE and KF for Weak GNSS Signals
## 3 MLE Discriminator
### Dynamic Characteristics  
The LM method is easy to converge to local optimal values if the initial point is not correct. This means the initial point must be in a pull-in range, i.e., the main peak of the cost function, in each update. The dynamic tolerance of the LM algorithm means the frequency change does not exceed half the width of the main peak during the update interval. Ignoring the noise term in (9), it can be simplified as:  
\[ \begin{array}{ll}L_{s}&=\sum\limits_{n=0}^{N-1}\left(\begin{array}{c}\cos (2\pi fnT+\varphi_{0})\cdot\cos(2\pi(f+\Delta f)nT+\varphi_{0}+\Delta\varphi) \\ +\sin(2\pi fnT+\varphi_{0})\cdot\sin(2\pi(f+\Delta f)nT+\varphi_{0}+\Delta \varphi)\end{array}\right)\\ &=\sum\limits_{n=0}^{N-1}\cos(2\pi\Delta fnT+\Delta\varphi)\end{array}  \tag{22}\]  
where \(\Delta f\) and \(\Delta\varphi\) are the frequency and phase error, respectively. The cost function in (22) is the accumulation of the cosine function whose phase is from \(\Delta\varphi\) to \(2\pi\Delta f(N-1)T+\Delta\varphi\). It is obvious that \(Ls\) reaches the minimum when \(2\pi\Delta f(N-1)T+\Delta\varphi\) is equal to \(3\pi/2\) or \(-3\pi/2\). Therefore, the main peak of L is in the range of \(\left[\frac{-3\pi/2-\Delta\varphi}{2\pi NT},\frac{3\pi/2-\Delta\varphi}{2\pi NT }\right]\) which is inversely proportional to the discriminator update time and related to the phase error. Figure 5 shows the one-dimensional cost function in (22) with respect to the frequency error with various phase errors. The parameters are the same as those in Figure 3. It can be seen that the position of the extreme point (the red spot) changes with the phase error changing. When the phase error is more than \(\pi/2\) or less than \(-\pi/2\), the main peak becomes a negative peak. To avoid converging to local optimal values, the initial point of LM algorithm must be in the range of the main peak. Assuming the phase is tracked precisely (\(\Delta\varphi\approx 0\)), the pull-in range of LM algorithm is \([-3/4 NT,3/4 NT]\). This means the tolerable Doppler rate is \(3/4(NT)^{2}\) Hz/s which is inversely proportional to the square of the discriminator update time. Therefore, increasing the discriminator  
Figure 4: Cramér–Rao bound (CRB) of \(f\) and \(\varphi\) with respect to carrier-to-noise spectral density ratio (\(C/N_{0}\)) of the IF signal. (**a**) CRB of \(f\). (**b**) CRB of \(\varphi\). update time will lead to reduction in dynamic tolerance. However, to improve the sensitivity of the receiver, increasing the update time is necessary. It is a compromise between dynamic tolerance and sensitivity. When the discriminator update time is more than 20 ms, the integration results must be squared to remove the data bits. This will lead to two effects on the performance of the loop. Firstly, the Doppler frequency and phase residuals are doubled, resulting in halved dynamic tolerance. Secondly, the noise term is squared too, resulting in SNR loss (square loss) [15]. The SNR gain and dynamic tolerance should be considered simultaneously to choose the optimal discriminator update time. It is not difficult to find that (9) is equivalent to the coherent integration (\(NT\leq 20\) ms) or non-coherent integration (\(NT>20\) ms) when the Doppler frequency and phase residual are zero. The SNR gain at the maximum point of \(L\) is equal to coherent or non-coherent integration gain, which have been derived in [16]. Table 1 shows the dynamic tolerance and SNR gain of the proposed algorithm with respect to the different discriminator update time. The \(C/N_{0}\) of the received signal is 30 dB-Hz. It can be seen that the dynamic tolerance decreases rapidly with \(NT\) increasing. The SNR gain reaches to 76.1 dB when the discriminator update time is 20 ms. However, it drops to 72.9 dB when the discriminator update time increases from 20 ms to 40 ms because of the square loss. Until the discriminator update time reaches 100 ms, the SNR gain goes back to 76.9 dB with a low dynamic tolerance of 7.1 m/s\({}^{2}\).