# Student's t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter for GPS Navigation
## 4 Results and Discussion  
To evaluate the performance of the proposed algorithm, two different environments are designed for testing. Environment 1 focuses on the filters' ability to mitigate noise induced by multipath effects, where satellite signals are reflected by surrounding surfaces and cause interference. In this case, outliers with varying magnitudes are inserted at specific time segments, allowing examination of the filters' sensitivity to changes in outlier strength and timing. Environment 2 is constructed to evaluate the filters' robustness under time-varying observations with heavy-tailed noise and randomly occurring outliers. This setting is intended to assess how well each filter handles non-Gaussian noise characteristics and abrupt disturbances, providing insight into their ability to maintain estimation accuracy under dynamic and unpredictable noise conditions. To facilitate comparison, the filters are grouped according to their underlying design. One set of comparisons involves the EKF, MCCEEK-1, MCCEEK-2, and STMCCEKF (with fixed kernel bandwidth), aiming to evaluate the influence of different kernel functions and robust modeling strategies under various noise scenarios. Another comparison includes the EKF, STMCCEKF, and STMCCEKF-AKB to highlight improvements introduced by integrating the Student's t-distribution, with fixed and adaptive kernel bandwidths. These structured tests provide a comprehensive assessment of each filter's effectiveness in handling distinct types of noise and interference. The testing environment used Matlab 2022a, with the Satellite Navigation (Satnav) Toolbox 3.0 [29; 30] to generate the necessary satellite data for GPS positioning, including satellite positions, velocities, and pseudoranges. The simulation will involve navigating a vehicle in motion. Figure 1 represents the motion trajectory of the simulated vehicle. Figure 2 illustrates a skylplot showing the spatial distribution of visible satellites during the simulation period, with each satellite labeled by its PRN (Pseudorandom Noise number) identifier to provide a clear understanding of satellite geometry and visibility conditions. The distribution of visible satellites is illustrated in the satellite skylplot shown in Figure 2, which visualizes the azimuth and elevation of the observed satellites. The plot consists of concentric circles, each representing a \(10^{\circ}\) elevation increment, ranging from \(0^{\circ}\) at the outermost ring (horizon) to \(90^{\circ}\) at the center (zenith). The multipath effect, along with other stochastic errors, is captured in the noise term as \(\epsilon_{i}\) is commonly used in GNSS observation models when exact modeling is impractical or environment-dependent. After introducing various types of noise interference, the equation representing the pseudorange between a satellite and a receiver in a satellite-based navigation system, such as GPS, can be expressed as [24]  
\[\rho_{i}=c(t_{u}-t_{si})=\sqrt{\left(x_{i}-x_{u}\right)^{2}+\left(y_{i}-y_{u} \right)^{2}+\left(z_{i}-z_{u}\right)^{2}}+ct_{b}+\
u_{pi}+d_{iono,i}+d_{tropo,i }+\epsilon_{i} \tag{45}\]  
where the pseudorange error is represented by \(\
u_{pi}\), and the receiver thermal noise, multipath, ionospheric delay, and tropospheric delay are the error sources which affect the accuracy of GPS measurements. These delays must be estimated and corrected to improve the positioning accuracy. The majority of inaccuracies are subsequently addressed by using the differential GPS (DGPS) mode, although the effects of multipath and the receiver's thermal noise cannot be completely avoided in the current investigation, and thus, the remaining error becomes \(\delta_{mp}+\delta_{noise}\), where \(\delta_{noise}\) and \(\delta_{mp}\) represent the additional multipath error, which can be in the form of outliers. Atmospheric delays are mitigated using the Klobuchar and Saastamoinen models for ionospheric and tropospheric corrections, respectively.