Cooperative Speed Estimation of an RF Jammer in Wireless Vehicular Networks

Abstract

In this paper, we are concerned with the problem of estimating the speed of an RF jammer that moves towards a group/platoon of moving wireless communicating nodes. In our system model, the group of nodes receives an information signal from a master node, that they want to decode, while the Radio Frequency (RF) jammer desires to disrupt this communication as it approaches them. For this system model, we propose first a transmission scheme where the master node remains silent for a time period while it transmits in a subsequent slot. Second, we develop a joint data and jamming estimation algorithm that uses Linear Minimum Mean Square Error (LMMSE) estimation. We develop analytical closed-form expressions that characterize the Mean Square Error (MSE) of the data and jamming signal estimates. Third, we propose a cooperative jammer speed estimation algorithm based on the jamming signal estimates at each node of the network. Our numerical and simulation results for different system configurations prove the ability of our overall system to estimate with high accuracy and the RF jamming signals and the speed of the jammer.

A Appendix

Recall that the $N-1$ receivers are assumed to be close to each other, resulting in a constant value for the free space propagation loss $po_{j,i}$ and the random variable $\gamma _{i}$ for the observation interval in the above equations. Under these assumptions the set of Eq. (14) is simplified to:

$$\begin{aligned} \frac{\hat{z}_{1}}{\hat{z}_{2}}&=\frac{e^{j\frac{2\pi }{\lambda } \varDelta {u}_{1} \frac{f_c}{c} \cos {\phi _{1}} \tau _{1}}}{e^{j\frac{2\pi }{\lambda } \varDelta {u}_{2} \frac{f_c}{c} \cos {\phi _{2}} \tau _{2}} } \\&...\\ \frac{\hat{z}_{N-2}}{\hat{z}_{N-1}}&=\frac{e^{j\frac{2\pi }{\lambda } \varDelta {u}_{N-2} \frac{f_c}{c} \cos {\phi _{N-2}} \tau _{N-2}}}{e^{j\frac{2\pi }{\lambda } \varDelta {u}_{N-1} \frac{f_c}{c} \cos {\phi _{N-1}} \tau _{N-1}} } \end{aligned}$$

By taking the natural logarithm of the expressions on the left and right we have:

$$\begin{aligned} \ln {(\frac{\hat{z}_{1}}{\hat{z}_{2}})}&=\ln {(\frac{e^{\omega |u_{r,1}-u_{j}| \cos {\phi _{1}}\tau _{1}}}{ e^{\omega |u_{r,2}-u_{j}| \cos {\phi _{2}}\tau _{2}}} ) } \\&...\\ \ln {(\frac{\hat{z}_{N-2}}{\hat{z}_{N-1}})}&=\ln {(\frac{e^{\omega |u_{r,N-2}-u_{j}| \cos {\phi _{N-2}}\tau _{N-2}}}{ e^{\omega |u_{r,N-1}-u_{j}| \cos {\phi _{N-1}}\tau _{N-1}}} ) } \end{aligned}$$

where $u_{r,i}$ is the speed of every receiver, $u_{j}$ the speed of the jammer in the area and the variables $f_{cx}=\frac{f_c}{c}$, $\omega =j\frac{2\pi }{\lambda } f_{cx}$. If we assume that the jammer approaches the i-th receiver at a speed lower than its own speed the relative speed between jammer and receiver is positive and so $|u_{r,i}-u_{j}|= u_{r,i}-u_{j}$. By simplifying the previous logarithmic equations we have:

$$\begin{aligned} \small \ln {(\frac{\hat{z}_{1}}{\hat{z}_{2}})}&=\omega [(u_{r,1}-u_{j}) \cos {\phi _{1}}\tau _{1} -(u_{r,2}-u_{j}) \cos {\phi _{2}}\tau _{2}]\\&...\\ \ln {(\frac{\hat{z}_{N-2}}{\hat{z}_{N-1}})}&=\omega [(u_{r,N-2}-u_{j}) \cos {\phi _{N-2}}\tau _{N-2}-(u_{r,N-1}-u_{j}) \cos {\phi _{N-1}}\tau _{N-1}] \end{aligned}$$

In the above equations the estimated jamming signal values on the left-hand side are complex numbers of the form: $(\hat{a}_{1}+\hat{b}_{1}j),...,\hat{a}_{N-2}+\hat{b}_{N-2}j$. We observe that the real part of the above equations on the right side is equal to zero. So all the real parts, that is the $\hat{a}$’s, are equal to zero. We also assume that the receivers move at similar speeds ($u_{r,1}\simeq u_{r,2} \simeq ... \simeq u_{r,N-1}=u_r$) as they are members of the platoon. By replacing the AOD values with the order of Eq. (1) and the time delays as $\tau _{1}=\frac{dist_{1}}{c}, \tau _{2}=\frac{dist_{2}}{c}, ..., \tau _{N-1}=\frac{dist_{N-1}}{c}$ we have:

$$\begin{aligned} \tiny \hat{b}_{1}&=\omega [(u_{r}-u_{j}) \frac{x_{dist}}{dist_{1}}*\frac{dist_{1}}{c} - (u_{r}-u_{j}) \frac{x_{dist}+d}{dist_{2}}*\frac{dist_{2}}{c}]\\&...\\ \tiny \hat{b}_{N-2}&=\omega [(u_{r}-u_{j}) \frac{x_{dist}+(N-3)*d}{dist_{N-2}}*\frac{dist_{N-2}}{c} - (u_{r}-u_{j}) \frac{x_{dist}+(N-2)*d}{dist_{N-1}}*\frac{dist_{N-1}}{c}] \end{aligned}$$