# A Novel Carrier Tracking Approach for GPS Signals Based on Gauss-Hermite Kalman Filter
###### Abstract
The EKF uses the linearization of a nonlinear system by Taylor series expansion. An EKF-based tracking approach was proposed in [11] to address the nonlinearity issue of the phase discriminator in LKF-based tracking system. In [12], a practical implementation and performance assessment of an EKF-based signal tracking loop were presented. In [13; 14], an EKF algorithm was proposed to track highly dynamic L1 and L2 GPS signals. However, the same problem exists in the above algorithms, such as loss of optimality, accuracy reduction, and filter divergence. That is, there is a bias of estimates and errors in the calculation of covariance matrices [6]. An approach [15] based on the extended strong tracking filter (ESTF) was developed to improve the carrier tracking estimation instead of the traditional EKF approach. However, it was concluded that the class of strong tracking filters such as the EKF will suffer the kinds of problems induced by the subjective fading factor. The UKF is one of the modifications of the sigma-point Kalman filter (SPKF) [6]. That is, the UKF algorithm is based on a set of symmetrically distributed sample points (so-called sigma points), which are used to determine the mean and the covariance [6]. This algorithm gives a more correct solution to nonlinear problems as compared with the EKF and forms suboptimal estimates of the state vector according to the criterion of the minimum mean-square error of the estimate. The authors of [16; 17] proposed an SPKF algorithm in the problem of GPS signal parameter estimation in coherent and noncoherent tracking modes in spacecraft autonomous navigation equipment. The obtained results suggested that the SPKF algorithm could form RNP estimates with appropriate accuracy under the conditions of space consumer functioning. The results also demonstrated that the accuracy of the noncoherent tracking mode was worse than that of the coherent tracking system. In [18], a new adaptive unscented Kalman filter (AUKF) was proposed to estimate the RNP of a GPSsignal tracking system. The experimental results showed that the proposed AUKF-based method improved the GPS tracking margin by approximately 8 and 3 dB as compared to the conventional algorithm and LKF-based tracking, respectively. The authors of [19] described an additive strong tracking filter using the UKF tracking algorithm for a GPS vector tracking loop. It should be noted that, from a practical point of view, the vector tracking approach requires large computational costs compared to the scalar tracking approach. However, implementation of the UKF algorithm requires the determination of several coefficients, which are called scaling parameters (\(\alpha\), \(\beta\), and \(\kappa\)) [6]. It should be noted that not much research effort has been spent on determining if there is a global optimal setting of the UKF scaling parameters [20]. It must also be noted that the choice of the scaling parameter (\(\alpha\)) noticeably affects the result of the UKF algorithm. Traditionally, the choice of this parameter is left to the developer. If we follow the recommendations of the authors [6], then \(\alpha\) should be chosen as a value other than 0, but as small as possible (\(\alpha\leq 0.01\)). In this paper, the value of this scale parameter was empirically selected as 0.7. With this choice of parameter \(\alpha\), it was possible to improve the quality characteristics of the UKF-tracking based system. This improvement occurred due to a wider spread of sigma points, which led to better consideration of nonlinearities in observations. However, it took several attempts. Thus, due to the tuning parameters, the rounding errors of numerical calculation for UKF may destroy the non-negative nature and asymmetry of the covariance matrix; thus, the convergence rate of the UKF approach is slow, and the system may also be unstable. Therefore, in order to avoid the influence of scaling parameters on the quality of the UKF algorithm, it is necessary to find other approaches to building a high-quality tracking system instead of UKF.