# An Adaptive INS-Aided PLL Tracking Method for GNSS Receivers in Harsh Environments
## 3 Tracking Error Analysis of Delta Doppler Aided PLL
### Dynamic Stress Error  
In [20], the traditional PLL tracking error in discrete-time domain has been analyzed, so we can obtain the dynamic stress error of traditional second-order PLL by analyzing the steady-state phase error of the loop as:  
\[E[\Psi_{ss}]=\frac{a}{\omega_{n}^{2}}(1+\zeta\omega_{n}T+\frac{1}{4}(\omega_{n }T)^{2}) \tag{3}\]  
Figure 1: The delta Doppler aided PLL architecture. where \(E[\cdot]\) is mathematical expectation function, \(\mathbf{\Psi}_{ss}\) is the steady-state of the phase error, \(a\) is the line-of-sight (LOS) vehicle acceleration, \(T\) is the integration time of the PLL, \(\zeta\) is the damping factor, and \(\omega_{n}\) is characteristic frequency of the loop. The noise bandwidth \(B_{n}\) is related with \(\omega_{n}\) and \(\zeta\) as follows:  
\[B_{n}=\frac{\omega_{n}}{2}(\zeta+\frac{1}{4\zeta}) \tag{4}\]  
From Equation (3) we know that dynamic stress is proportional to the LOS acceleration, and is inversely proportional to the square of the bandwidth. The structure of delta Doppler-aided PLL in the discrete-time domain is illustrated in Figure 2. The process of delta Doppler-aided tracking is actually the integral process of aiding information \(A_{dopp}(k)\). Due to the random nature of the input phase noise, the output of the discriminator should be a random process as well. From Figure 2, we have the following equations:  
\[e(k)=g\{\psi(k)\}+n_{\Theta}(k),e(k)\in[-\pi,\pi] \tag{5}\]  
\[\psi(k)=\theta(k)-\hat{\theta}(k),\Psi(k)=g\{\psi(k)\}\in[-\pi,\pi] \tag{6}\]  
\[\hat{\theta}\left(z\right)=D\left(z\right)f_{dopp}(z) \tag{7}\]  
\[f_{dopp}(z)=F(z)\Psi\left(z\right)+\frac{1}{1-z^{-1}}A_{dopp}\left(z\right) \tag{8}\]  
where \(e(k)\) is the output of the phase discriminator, \(\theta(k)\) and \(\hat{\theta}(k)\) are the phases of the incoming signal and the local NCO signal at the center of the interval, respectively, \(\psi(k)\) is the phase tracking error due to a noise-free incoming signal, \(\mathrm{n}_{\Theta}(k)\) is the phase disturbance due to the input noise, \(g[\cdot]\) is a characteristic function of the phase discriminator, \(\Psi(k)\) is the restricted phase error, the NCO transfer function is \(D(z)=T\cdot z^{-1}/(1-z^{-1})\), \(y(k)\) is the output of the loop filter at the end of the \(kth\) correlation interval, and \(F(z)\) is transfer function of second-order loop filter that can be written as:  
\[F(z)=\frac{\left(G_{1}+G_{2}\right)-G_{1}z^{-1}}{T\cdot(1-z^{-1})} \tag{9}\]  
where \(G_{1}\) and \(G_{2}\) denote the parameters of the loop filter in discrete-time domain which can be expressed as:  
\[\left\{ \begin{array}{l} G_{1}=\frac{1}{K}\frac{8\zeta\omega_{ n} T}{4+4\zeta\omega_{n}T+\left(\omega_{n}T\right)^{2}}\\ G_{2}=\frac{1}{K}\frac{4(\omega_{n}T)^{2}}{4+4\zeta\omega_{n}T+ \left(\omega_{n}T\right)^{2}}\end{array} \right. \tag{10}\]  
By substituting Equations (7)-(9) into Equation (6), we obtain:  
\[\Psi\left(z\right)=\frac{(1-z^{-1})\theta(z)}{(1-z^{-1})^{2}+\left(G_{1}+G_{2 }\right)z^{-1}-G_{1}z^{-2}}-\frac{A_{dopp}(z)Tz^{-1}}{(1-z^{-1})^{2}+\left(G_{ 1}+G_{2}\right)z^{-1}-G_{1}z^{-2}} \tag{11}\]  
Figure 2: Structure of delta Doppler-aided PLL in discrete-time domain. Using the analytical method of [20], we can obtain the differential equation of the phase tracking error of INS-aided PLL, which can be expressed as:  
\[ \begin{array}{c}\Psi\left(k\right)-2\Psi\left(k-1\right)+\Psi\left(k-2\right)= \theta\left(k\right)-2\theta\left(k-1\right)+\theta\left(k-2\right)\\ -\left(G_{1}+G_{2}\right)\Psi\left(k-1\right)+G_{1}\Psi\left(k-2\right)-A_{ dopp}\left(k-1\right)T\end{array}  \tag{12}\]  
Under ideal condition, the aiding Doppler rate \(df_{dopp}(k)\) is equal to second-order derivative of incoming signal phase \(\ddot{\theta}\left(k\right)\). However, there will be an aiding delay \(\Delta t\) caused by the asynchronous update rates of GNSS and INS, which is related to the aiding information update time \(T_{f}\) and integration time \(T\)[21].