# A Novel Carrier Tracking Approach for GPS Signals Based on Gauss-Hermite Kalman Filter
## 2 Problem Description  
The development of a tracking system based on the Bayesian theory of nonlinear filtering involves setting mathematical models of states and measurements. Figure 1 shows a block diagram of the proposed tracking system model in one channel of the receiver.  
This tracking system model can be described as a discrete linear state model with additive white Gaussian noise (AWGN) and a discrete nonlinear measurement model with AWGN [6; 13].  
\[\left\{\begin{array}{c}x_{k}=Fx_{k-1}+u_{x,k-1};\\ z_{k}=h(x_{k})+v_{z,k},\end{array}\right., \tag{1}\]  
where \(x_{k}=[A_{k},\varphi_{k},f_{d,k},f^{\prime}_{d,k}]\) is the state vector of the RNP of the GPS signal, \(F=\begin{bmatrix}0&0&0&1\\ 1&T&0.5T^{2}&0\\ 0&1&T&0\\ 0&0&1&0\end{bmatrix}\) is the state transition matrix, \(u_{x,k-1}=[n_{A,k-1},n_{\varphi,k-1},n_{f,k-1},n_{f^{\prime},k-1}]\) is a Gaussian distributed process noise with a spectral power density \([q_{A},q_{\varphi},q_{f},q_{f^{\prime}}]\) and a covariance matrix \(Q\), \(T\) is the coherent integration time, \(z_{k}=[I_{p,k},Q_{p,k}]\) is the measurement vector at the \(k^{th}\) moment, \(h_{k}\) is a nonlinear function of measurements, and \(v_{z,k}\) is a Gaussian-distributed measurement noise with a Gaussian probability density function \(N(0,R)\). The nonlinear function of measurements \(h\) takes the following form [2]:  
\[h(x_{k})=\left\{\begin{array}{c}I_{p,k}=A_{k}R_{code}(\delta\tau_{k})\sin c( f_{d,k}T)\cos\left(2\pi(\varphi_{k}+\frac{f_{d,k}T}{2}+\frac{f^{\prime}_{d,k}T ^{2}}{6})\right);\\ Q_{p,k}=A_{k}R_{code}(\delta\tau_{k})\sin c(f_{d,k}T)\sin\left(2\pi(\varphi_{k }+\frac{f_{d,k}T}{2}+\frac{f^{\prime}_{d,k}T^{2}}{6})\right),\end{array}\right. \tag{2}\]  
where \(A_{k}\) is the signal amplitude, \(R_{code}(\tau)=\left\{\begin{array}{c}1-\frac{|\tau|}{\tau_{chip}},\,|\tau|<T _{chip}\\ 0,\,|\tau|>=T_{chip}\end{array}\right\}\) is the autocorrelation function of the C/A code, \(T_{chip}\) is the C/A code chip length (\(T_{chip}\approx 1\)\(\mu\)s) [2], \(\delta\tau_{k}\) is the code phase error, \(\delta\varphi_{k}\) is the carrier phase error, \(f_{d,k}\) is the carrier Doppler shift, and \(f^{\prime}_{d,k}\) is the carrier Doppler shift rate. It should be noted that, \(\delta\tau_{k}\), in Equation (2), is assumed to be known. In this case, the estimate of \(\delta\tau_{k}\) can be taken at the output of the traditional delay lock loop subsystem [1]. The covariance matrix \(Q\) of the process noise can be described as follows [2; 3; 4]:  
\[Q=\left(\frac{f_{L1}}{c}\right)^{2}q_{f^{\prime}}\begin{bmatrix}0&0&0&0\\ 0&\frac{T^{5}}{20}&\frac{T^{4}}{8}&\frac{T^{3}}{6}\\ 0&\frac{T^{4}}{8}&\frac{T^{3}}{3}&\frac{T^{2}}{2}\\ 0&\frac{T^{3}}{6}&\frac{T^{2}}{2}&T\end{bmatrix}+\begin{bmatrix}q_{A}T&0&0&0 \\ 0&f_{L1}^{2}(q_{f}\frac{T^{3}}{3}+q_{\varphi}T)&f_{L1}^{2}q_{f}\frac{T^{2}}{2} &0\\ 0&f_{L1}^{2}q_{f}\frac{T^{2}}{2}&T\end{bmatrix}, \tag{3}\]  
where \(f_{L1}\) denotes the nominal carrier frequency in Hz (\(f_{L1}=1575.420\) MHz for GPS L1 signal), and \(c\) denotes the velocity of light. A more detailed description of the process noise  
Figure 1: A block diagram of a nonlinear KF-based GPS carrier tracking system.  
spectral power density vector \([q_{A},q_{\varphi},q_{f},q_{f^{\prime}}]\) is presented in [12; 13]. The measurement covariance matrix \(R\) can be written as follows [2; 3; 4]:  
\[R=\left[\begin{matrix}\sigma_{IQ}^{2}&0\\ 0&\sigma_{IQ}^{2}\end{matrix}\right];\;\sigma_{IQ}^{2}=\frac{N_{0}}{2T}, \tag{4}\]  
where \(N_{0}\) is the power spectral density of the receiver white noise.