Application of semi-circle law and Wigner spiked-model in GPS jamming confronting

Abstract

Detection and reduction of global positioning system (GPS) jamming threats are inevitable considering importance of satellite navigation in different applications such as fixed and moving platforms. Different methods utilizing statistical/spectral signal processing, time/frequency transforms and antenna array have been introduced. In this paper, a new method based on random matrix theory is proposed for detection and reduction of GPS jamming threat. By using the concept of Wigner matrix and introducing spiked matrix model, our proposed method is based on Karhunen–Loeve transform (KLT) by distinguishing eigen-values through systematic (analytic) thresholding. The authentic (cleaned) signal is reconstructed by projecting received signal on the eigen-functions domain where jamming components can be better removed. The objective is to detect the presence of jamming in the early stage of jamming attack (early warning) and then trying to reduce the jamming effect (mitigation). Different parameters including acquisition metric, position deviation, cross ambiguity function and run-time have been evaluated in simulation part to provide a good preview. According to the simulation results, the proposed method has better performance in comparison with some reference algorithms in terms of detection and false alarm probabilities and acquired satellites. At least 2.5 dB improvement in detection is achieved for the proposed algorithm. The run-time is reduced about 30% compared with wavelet packet coefficients transforms. Also, the overhead computational complexity for covariance matrix estimation ($O\left( {m^{3} } \right)$) is reduced compared to conventional covariance matrix estimation-based eigen-decomposition methods.

Appendix 1

The test statistic of Chi2GoF is given by $x_{hist} \left( m \right) = \mathop \sum \nolimits_{i = 1}^{{N_{b} }} \left( {O_{i}^{\left( m \right)} - E_{i} } \right)^{2} /E_{i}$ where $N_{b}$ is the number of bins,$O_{i}^{\left( m \right)}$ is the value of the $i$ th bin of the measured histogram at snapshot $m$ (each snapshot contains $N{ }$ samples), and $E_{i}$ is the value of the ${ }i$ th bin of the reference histogram, evaluated under $H_{0}$. For large ${ }N$, the variable $x_{hist} \left( m \right)$ under $H_{0}$ is approximately chi-squared distributed with $N_{b} - 1{ }$ degrees of freedom, whereas under $H_{1}$ it departs from a central chi-square distribution. The KS test is based on the empirical distribution function (EDF). Given N ordered values of a sample $X$, the EDF is defined as $\hat{F}_{N} \left( x \right) = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} I\left( {X_{i} \le x} \right)$, where $I\left( . \right)$ is the indicator function and $X_{i}$ is the $i$ th sample element to be tested (sorted from low to high). The basis of MVDET method is vectors distribution equality law, which means that the PDF of the received signal samples in battlefield is compared with the PDF of the reference one in clean environment. If significant deviation occurs, the decision on the existence of interference is confirmed. The test statistic can is considered as $T_{{mn}}= ||\hat{\boldsymbol\mu }_{{D_{F} }}- \hat{\boldsymbol\mu }_{{D_{G} }}||$ in which, ${\hat{\boldsymbol\mu }}$ are estimated mean vectors. If $T_{mn}$ exceeds a predefined threshold, the null hypothesis is rejected.