# A Novel Carrier Tracking Approach for GPS Signals Based on Gauss-Hermite Kalman Filter
## 3 Problem Solution  
In probability theory, Bayesian estimation is a general probabilistic approach to recursive estimation of an unknown probability density function (PDF) of the state using a mathematical model of the process and a set of receiving measurements. Many problems require an estimation of the state at each receiving measurement. In this case, a recursive filter is a convenient solution. Such a filter consists of mainly two stages: prediction and update. In modern GPS technologies, statistical signal processing methods based on Bayesian filtering theory are widely used. When constructing algorithms for a GPS signal tracking system, it is necessary to describe mathematical models of the system state and measurements. The mathematical model for discrete system state and measurements of a Gaussian process can be described as follows [6]:  
\[\left\{ \begin{array}{l}x_{k}=f_{k}(x_{k-1})+u_{x,k-1},\\ z_{k}=h_{k}(x_{k})+v_{z,k}.\end{array} \right. \tag{5}\]  
Here, \(f_{k}\) is a possibly nonlinear function of the state \(x_{k-1}\), \(h_{k}\) is a possibly nonlinear function of measurements \(z_{k}\), \(u_{x,k-1}\) is an independent and identically Gaussian distributed process noise sequence, and \(v_{z,k}\) is an independent and identically Gaussian distributed measurement noise sequence. According to the Bayes method, it is required to form a posteriori PDF \(p(x_{k}|z_{1:k})\) at the moment \(k\), if there is a sequence of all measurements up to the moment \(k\). In this case, it is assumed that the priori PDF \(p(x_{0}|z_{0})=p(x_{0})\) is known. Then, the PDF \(p(x_{k}|z_{1:k})\) can be obtained recursively in two stages: prediction and update. It is assumed that the required PDF \(p(x_{k-1}|z_{1:k-1})\) is known. Then, the prediction stage uses the system model in Equation (5) to obtain the priori pdf of the state at time \(k\) using the Chapman-Kolmogorov equation [6]. \[p(x_{k}|z_{1:k-1})=\int p(x_{k}|x_{k-1})p(x_{k-1}|z_{1:k-1})dx_{k-1}. \tag{6}\]  
At the timestep \(k\), a new measurement is received, which is used to predict the posterior PDF and implement the update stage using Bayes' rule. \[p(x_{k}|z_{1:k})=\frac{p(z_{k}|x_{k})p(x_{k}|z_{1:k-1})}{p(z_{k}|z_{1:k-1})}. \tag{7}\]  
Here, the normalizing constant \(p(z_{k}|z_{1:k-1})\) in Equation (7) has the following form:  
\[p(z_{k}|z_{1:k-1})=\int p(z_{k}|x_{k})p(x_{k}|z_{1:k-1})dx_{k}. \tag{8}\]  
It can be seen from Equation (8) that \(p(z_{k}|z_{1:k-1})\) depends on the observation likelihood function \(p(z_{k}|x_{k})\) determined by the observation model in Equation (5) and the known statistics of \(v_{z,k}\). The recurrence relations in Equations (6) and (7) form the basis for the optimal Bayesian solution of the estimating problem. In practice, solving the Bayesian problem is possible in a number of cases using the Kalman filtering approach. As is known, [PERSON] obtained a recursive algorithm for generating linear estimates of the state by the criterion of the minimum mean square error for linear models. In this case, the optimal Bayesian estimate \(\hat{x}_{k,opt}\) of the state is the posteriori mean value of \(x_{k}\)[6]. \[\hat{x}_{k,opt}=E[x_{k}|z_{1:k}]=\int x_{k}p(x_{k}|z_{1:k})dx_{k}. \tag{9}\]  
This algorithm is called the LKF and can be applied for estimating linear dynamical systems by forming Gaussian approximations to the joint distribution of the state \(x_{k}\) and measurement \(z_{k}\). Therefore, the first and second moments (mean values and covariances) of posteriori estimates are sufficient. In nonlinear systems, other modifications of the traditional KF were considered to find Bayesian estimates of \(x_{k}\) in Equation (9). The most common algorithms for nonlinear filtering problems are EKF and UKF. The EKF extends the scope of KF to nonlinear optimal filtering problems by forming a Gaussian approximation to the nonlinear model using a Taylor series-based transformation. The drawback of this method is the filter divergence in some circumstances [6].