# An Adaptive INS-Aided PLL Tracking Method for GNSS Receivers in Harsh Environments
## 4 Adaptive Adjustments of Integration Time and Bandwidth of INS-Aided PLL  
From Equation (34), we can find the relation between the phase tracking error of discriminator output and aiding information update time, integration time, noise bandwidth, \(C/N_{0}\), vehicle dynamics. Usually, a shorter INS prediction update time is selected as the aiding information update timeto improve the dynamic performance of INS-aided PLL [19]. Thus, for a certain \(C/N_{0}\) and vehicle dynamics condition, only a two-dimensional search of noise bandwidth and integration time is needed to find the minimum tracking error of discriminator output in theory, meanwhile the corresponding noise bandwidth and integration time are the optimal ones, respectively.  
According to Equations (18) and (30), \(\Delta a_{kINS}\) and \(\mathrm{var}[n_{\emptyset}^{osc}]\) can be obtained. And the method is proposed for adaptive adjustments of \(B_{n}\) and \(T\) as below in Algorithm 1.  
```
0: The variance of oscillator phase error, \(\mathrm{var}[n_{\emptyset}^{osc}]\); The LOS acceleration error, \(\Delta a_{kINS}\); Carrier to noise density ratio, \(C/N_{0}\); The output of the phase discriminator, \(e(k)\); The bandwidth and integration time of the last epoch, \(B_{n}\)' and \(T\)'; The choices of integration time: \(1\)ms, \(5\)ms, \(10\)ms and \(20\) ms; A range of bandwidth from \(1\) Hz to \(20\) Hz with an interval of \(2\) Hz;
0: The optimal bandwidth and integration time under certain vehicle dynamic and carrier to noise ratio, \([B_{n}\),\(T]\);
1: Considering the real-time performance and accuracy, a one-second sliding window is used to evaluate the mean of the PLL discriminators output \(E[\mathbf{\Psi}_{ss}]\) using Equation (32). Then the residual LOS acceleration error can be estimated;
2: According to \(E[\mathbf{\Psi}_{ss}]\), \(\Delta a_{kINS}\), \(\mathrm{var}[n_{\emptyset}^{osc}]\), \(B_{n}\)' and \(T\)', \(p\) can be calculated by Equation (17);
3: Based on the calculated \(C/N_{0}\), \(\Delta a_{kINS}\), \(\mathrm{var}[n_{\emptyset}^{osc}]\) and \(p\), we take different integration time and bandwidth as referred above, and calculate the phase tracking error with different integration time and bandwidth according to Equation (34);
4: When the tracking error is the minimum, select the optimal bandwidth and integration time, \([B_{n}\),\(T]\);
5: return\([B_{n}\), \(T]\);
```  
**Algorithm 1**The adaptive method for choosing the optimal noise bandwidth and integration time.  
According to Equation (34), the minimum phase tracking error of the discriminator output for different LOS jerk and \(C/N_{0}\) is interpreted in Figure 4, while the corresponding optimal choices of \(B_{n}\) and \(T\) are illustrated in Figure 5, where the carrier frequency is GPS L1 with a value of \(1575.42\) MHz, the aiding information update times \(T_{f}=10\) ms (the output frequency of the inertial sensor used here is \(100\) Hz), the damping factor of second-order is \(\zeta=0.707\), the parameters of a temperature controlled crystal oscillator (TCXO) are \(h_{0}=1.0\times 10^{-21}\), \(h_{-2}=1.0\times 10^{-20}\), and the biases of gyro and accelerometer are \(\delta w_{ib}^{b}=10^{\circ}/\mathrm{h}\) and \(\delta f_{ib}^{b}=1\) mg, respectively.  
As can be seen in Figures 4 and 5 the minimum phase tracking error increases with lower \(C/N_{0}\) and higher vehicle LOS jerk. The optimal choices of bandwidth and integration time vary with \(C/N_{0}\) and vehicle dynamical. For a certain \(C/N_{0}\), increasing the vehicle dynamics will increase the optimal bandwidth, but decrease the optimal integration time, and under certain dynamics, decreasing \(C/N_{0}\) will increase the optimal integration time, but decrease the optimal bandwidth. Hence, in order to obtain the minimum tracking error, we should choose appropriate integration time and bandwidth for certain vehicle LOS jerk and \(C/N_{0}\).