# A Novel Carrier Tracking Approach for GPS Signals Based on Gauss-Hermite Kalman Filter
## 3 Problem Solution
The UKF approach is based on a set of symmetrically distributed sample points used to form the mean and the covariance. There is another approach to selecting the set of sample points for obtaining the mean and covariance. This approach is based on so-called Gaussian filters. In [24], it was shown that the UKF can be considered as a special case of Gaussian filters, where the nonlinear filtering problem is solved using Gaussian-assumed density approximations. In this paper, we present the GHKF as a Gaussian filtering framework to be used in a GPS carrier tracking system. Next, we conduct a detailed analysis of this filter [6; 24; 25; 26]. In GHKF, the linear equations of KF can be adapted to the nonlinear state space model. Desired moments (mean and covariance) of the original distribution of \(x\sim N(m,P)\) can be obtained exactly by calculating the following integrals [6]:  
\[ \begin{array}{l}\mu_{U}=\int f(x)N(x|m,P)dx_{k};\\ S_{U}=\int\left(f(x)-\mu_{U}\right)\left(f(x)-\mu_{U}\right)^{T}N(x|m,P)dx_{k}.\end{array}  \tag{10}\]  
These integrals can be evaluated with practically any analytical or numerical integration method. The Gauss-Hermite quadrature rule is a method for solving special integrals of the previous form in a Gaussian kernel with an infinite domain. More specifically, the Gauss-Hermite quadrature can be applied to integrals of the following form:  
\[\int\limits_{-\infty}^{+\infty}f(x)e^{-x^{2}}dx\approx\sum\limits_{i=1}^{p}w_{ i}f(x_{i}), \tag{11}\]  
where \(x_{i}\) denotes the sample points (quadrature points), \(w_{i}\) denotes the associated weights to use for the approximation, and \(p\) is the number of distinct quadrature points. The quadrature points \(x_{i}\) are chosen to be the zeros of the \(p^{th}\)-order Hermite polynomial. The Hermite polynomial of degree \(p\) is defined as follows:  
\[H_{p}(x)=(-1)^{p}\exp(\frac{x^{2}}{2})\frac{d^{p}}{dx^{p}}\exp(-\frac{x^{2}}{ 2}). \tag{12}\]  
The first two Hermite polynomials are \(H_{0}(x)=1;\ H_{1}(x)=x.\) Further polynomials can be found from the following recursion:  
\[H_{p+1}(x)=xH_{p}(x)-pH_{p-1}(x). \tag{13}\]  
The weights \(w_{i}\) are given by  
\[w_{i}=\frac{2^{p-1}p!\sqrt{\pi}}{p^{2}\big{[}H_{p-1}(x_{i})\big{]}^{2}}. \tag{14}\]Therefore, one-dimensional Gauss-Hermite quadrature integration refers to the special case of Gaussian quadrature with a unit Gaussian weight function \(N(x|0,1)\), i.e., to approximations of the following form:  
\[\int\limits_{-\infty}^{+\infty}f(x)N(x|0,1)dx\approx\sum\limits_{i=1}^{p}w_{i}f (x_{i}). \tag{15}\]  
Now, consider a multidimensional random variable \(\mathbf{x}\) having a Gaussian density \(N(\mathbf{x}|\mathbf{0},\mathbf{I}_{n_{x}})\), where \(n_{x}\) is the identity matrix of dimensions \(n_{x}\times n_{x}\). The one-dimensional integration method can be extended to multidimensional integrals of the following form:  
\[\int\limits_{\mathbb{R}^{n_{x}}}f(\mathbf{x})N(\mathbf{x}|0,\mathbf{I}_{n_{x}} )d\mathbf{x}\approx\sum\limits_{l=1}^{k}w_{l}f(\xi_{l});\;k=p^{n_{x}}, \tag{16}\]  
where \(\xi_{l}=\left[\xi_{l}^{1},\xi_{l}^{2},\ldots,\xi_{l}^{n_{x}}\right]^{T}\) is the \(l^{th}\) column of the quadrature matrix of dimensions \(n_{x}\times k\), which can be described as follows:  
\[\xi=\left[\xi_{1}\;\ldots\;\xi_{k}\right]=\left( \begin{matrix}\xi_{1}^{1}& \ldots&\xi_{k}^{1}\\ \vdots&\ddots&\vdots\\ \xi_{1}^{n_{x}}&\ldots&\xi_{k}^{n_{x}}\end{matrix} \right). \tag{17}\]  
Additionally, the weight of the quadrature point vector \(\xi_{l}\) can be written as  
\[w_{l}=\prod\limits_{j=1}^{n_{x}}w_{l}^{j}=\prod\limits_{j=1}^{n_{x}}\frac{2^ {p-1}p!\sqrt{\pi}}{p^{2}\Big{[}H_{p-1}(\xi_{l}^{j})\Big{]}^{2}}. \tag{18}\]  
Therefore, if \(p\) points are required to calculate the one-dimensional integral, then \(k=p^{n_{x}}\) points are required for the \(n_{x}\)-dimensional case. This set of quadrature points are given by the Cartesian product of \(n_{x}\) vectors, each of which consists of \(p\) points. These points are the roots \(\xi_{1,2,\ldots,p}\) of the Hermite polynomial \(H_{p}(x)\).