# Localization of a Power-Modulated Jammer

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## Abstract

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## 1. Introduction

**Contribution.**In this paper, we propose a new type of jammer, i.e., the power-modulated jammer. Our intuition is that a jammer that randomly changes its transmission power can make the localization process much harder (even impossible in some cases). In particular, after defining the power-modulated jammer model, we test a standard jammer (constant power) against a power-modulated jammer in the presence of a network of nodes running a classical localization technique, i.e., linear least square (LLS). Our findings confirm our intuition. In particular, we show that the power-modulated jammer—when compared against the constant-power jammer—makes the localization process particularly difficult, even in the presence of a high density of sensing nodes (10 nodes in a $40\times 40$ m area) and under conservative assumptions of the (shadow) fading process of the wireless channel ($\sigma <3$ dBm).

**Paper organization.**The remainder of this paper is organized as follows. In Section 2, we review background information and related work, while in Section 3, we introduce the system and the adversary model discussed in this work. In Section 4, we compare the new proposed jammer model (power modulated) with the classical one (constant-power) in terms of localization error. We provide a comprehensive discussion on the performance of the proposed jammer model and some potential mitigation technique in Section 5, while in Section 6, we draw conclusions and discuss future work.

## 2. Related Work

## 3. Scenario and Assumptions

**Path loss model.**We adopted the standard log-normal path loss model as depicted by Equation (1) where ${R}_{x}\left(t\right)$ is the received signal strength, ${T}_{x}\left(t\right)$ is the jamming power, $\gamma $ is the path loss exponent, and ${d}_{0}$ is a reference distance at which the path loss $PL\left({d}_{0}\right)$ is measured. Finally, ${X}_{\sigma}$ is a log-normal random variable with a mean $\mu =0$ and standard deviation $\sigma $ taking into account the shadowing effect of a flat-fading channel:

**Jamming power.**As previously mentioned, the jamming power is the result of a trade-off between the actual available transmission power of the jammer and the power requested to jam the whole area, i.e., the jammer wants to jam all the playground with a minimum amount of transmission power. Without loss of generality, we assume a node receiver sensitivity ${S}_{rx}$ is equal to ${S}_{rx}=-80$ dBm, and we dynamically set the minimum transmission power as a function of the deployment as depicted by Equation (2):

**Ranging.**Assuming the propagation model introduced by Equation (1), the approximated distance $\tilde{d}$ between the sensor and the jammer can be computed as described by Equation (3):

**Localization.**The estimated distances $\{{d}_{1},\dots ,{d}_{N}\}$ from the ranging process are combined to generate an approximated position for the jammer $[{x}_{J},{y}_{J}]$. We first compute a linearization of the problem by choosing one sensor (${x}_{N},{y}_{N}$) and its related distance to the jammer (${d}_{n}$) as a reference, and by subtracting it to the $n-1$ equations obtaining a system of $n-1$ equations in the form $Az=b$, yielding:

#### Constant and Power-Modulated Jammer

## 4. Jammer Localization

## 5. Discussion

**Mitigation and future work.**Locating a power-modulated jammer turns out to be a challenging problem. A naive solution might involve an ultra-dense sensing network, but this solution might be impractical in many scenarios and not scaling up with the area to be protected by the action of the jammer. All our considerations mainly focus on sensing devices featuring an isotropic antenna. While this set-up is very effective in the localization of a standard transmitter (such as a constant jammer), it turns out to be much less ineffective against a power-modulated jammer. Conversely, a directive antenna might be much more effective for determining the direction of the jamming signal, thus making multiple collaborating directive antennas able to effectively locate the power-modulated jammer. We observe how, in this case, the localization algorithm should not be based on multilateration but on triangulation. Another consideration might be concerned with detecting the actual presence of a power-modulated jammer in order to subsequently deploy an ad hoc strategy for its localization. Indeed, we can assume that the sensing nodes monitor the spectrum, and they can detect and identify the jamming signal. The sensing nodes might be able to model the random process behind the power-modulation, thus pre-conditioning the input to the localization algorithm. Finally, we would like to highlight that different localization algorithms might be affected in different ways by the modulated jammer—and in turn, the associated localization error. Our contribution aims at defining the theoretical framework for a new jammer strategy willing to evade the localization process. The choice of proper localization algorithm is only one of the parameters that can mitigate the evasion and we leave this wider discussion to future works.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A typical random deployment: 10 nodes and the jammer (black cross) at the center of the scene. The black dashed circle represents the jamming area.

**Figure 2.**Jammer localization by three-lateration: 3 sensors collaborating to estimate the jammer position.

**Figure 3.**Power-modulated jammer: the minimum transmission power is opportunistically chosen in order cover the whole area.

**Figure 4.**Localization of a constant jammer as a function of $\sigma $ (1 dBm $\le \sigma \le $ 6 dBm) and the number N of sensing nodes ($3\le N\le 10$).

**Figure 5.**Localization of a power-modulated jammer as a function of $\sigma $ (1 dBm $\le \sigma \le $ 6 dBm) and the number N of sensing nodes ($3\le N\le 10$).

**Figure 6.**Comparison between the modulated-power and constant-power jammer: localization error as a function of $\sigma $ while the number of nodes in the playground spans between $N=3$ and $N=9$.

**Figure 7.**Comparison between 100 and 5000 readings of the jamming signal as a function of $\sigma $ and the number of nodes.

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**MDPI and ACS Style**

Tedeschi, P.; Oligeri, G.; Di Pietro, R.
Localization of a Power-Modulated Jammer. *Sensors* **2022**, *22*, 646.
https://doi.org/10.3390/s22020646

**AMA Style**

Tedeschi P, Oligeri G, Di Pietro R.
Localization of a Power-Modulated Jammer. *Sensors*. 2022; 22(2):646.
https://doi.org/10.3390/s22020646

**Chicago/Turabian Style**

Tedeschi, Pietro, Gabriele Oligeri, and Roberto Di Pietro.
2022. "Localization of a Power-Modulated Jammer" *Sensors* 22, no. 2: 646.
https://doi.org/10.3390/s22020646