Movable Antennas in Wireless Systems: A Tool for Connectivity or a New Security Threat?

As shown in Fig. 1, a BS equipped with a ULA of size $N$ serves $K\leq N$ single-antenna users in downlink transmission at frequency $f=\frac{c}{\lambda_{\text{BS}}}$. Meanwhile, a jammer attempts to disrupt this communication using a device with $M$ movable antennas, as illustrated in Fig. 2. Though structurally similar to a ULA, the jammer’s antennas are mobile and can adjust positions according to the jammer’s objectives. Each antenna has defined movement boundaries in the $(X,Z)$ plane, ranging from $[x_{\text{min}},x_{\text{max}}]\times[z_{\text{min}},z_{\text{max}}]$. Along the $y$-axis, the antennas are unrestricted provided that the spacing between any two consecutive antennas remains at least $D=2\lambda_{\text{J}}$ to avoid coupling effects [9], where $\lambda_{\text{J}}$ denotes the jammer’s transmitted signal wavelength. The direct channel from the BS to user $k$ is represented by $\bm{h}_{\text{BS},k}\in\mathbb{C}^{N\times 1}$, and the channel from the jammer to user $k$ is denoted by $\bm{h}_{\text{J},k}\in\mathbb{C}^{M\times 1}$ $\forall k\in\mathcal{K}\triangleq\{1,\ldots,K\}$. We denote $n_{k}\sim\mathcal{CN}(0,\sigma^{2}_{k})\hskip 2.84544pt\forall k\in\mathcal{K}$, as the additive white Gaussian noise (AWGN).

Unaware of the jammer’s presence, the BS optimizes its beamforming vectors based solely on the BS-user channels. Consequently, the BS’s transmitted baseband signal can be expressed as $x=\sum_{k\in\mathcal{K}}\bm{w}_{k}s_{k}$, where $\bm{w}_{k}\in\mathbb{C}^{N\times 1}$ and $s_{k}$ are the BS beamforming vector and the information symbol for user $k$ with zero mean and unit variance. The jammer aims to disrupt the communication between the BS and the users by transmitting a signal $z$ through its antenna array, expressed as $z=\sum_{k\in\mathcal{K}}\bm{v}_{k}q_{k}$, where $\bm{v}_{k}\in\mathbb{C}^{M\times 1}$ is the jammer’s beamforming vector, and $q_{k}$ is the jamming signal symbol directed toward user $k$. We suppose that the jammer transmits over the same frequency used by the BS, i.e $\lambda_{BS}=\lambda_{J}=\lambda$. Additionally, the jammer possesses channel state information (CSI) for all channels between itself and the users¹¹1We assume the jammer performs prior channel estimation, storing estimated channels for various user locations in a dictionary. During jamming, it selects channels based on users’ current locations. The effect of imperfect field-response information (FRI) due to minor user location misalignment is minimal, allowing the algorithm to maintain robust performance [2].; however, the jammer lacks the CSI for the BS-user links. Given the above, the actual received signal at user $k$ is expressed as²²2In this paper, we assume that $M\geq K$, allowing the jammer to design a distinct beamforming vector for each user. If $K>M$, the jammer can still effectively target a group of $M$ users, making the problem equivalent to the scenario discussed in this paper.:

The rate expression for each user depends on the multiple access technique employed by the BS. In our case, the BS employs SDMA, thus the achievable rate for each user $k$ is given by:

where the term $\sum_{\begin{subarray}{c}j\in\mathcal{K}\\ j\neq k\end{subarray}}|\bm{h}_{\text{BS},k}^{H}\bm{w}_{j}|^{2}$ represents the interference from other users, $\sum_{\begin{subarray}{c}j\in\mathcal{K}\end{subarray}}|\bm{h}_{\text{J},k}^{H}\bm{v}_{j}|^{2}$ is the interference from the jammer, and $\sigma_{k}^{2}$ is the noise power for user $k$.

In this paper, we consider narrow-band channels with slow fading. The BS-user links are modeled as Rayleigh fading, expressed as:

where $PL(d_{BS,k})$ is the distance dependent path loss modeled as $PL(d_{BS,k})=\rho_{o}d^{-\alpha_{BS}}$. Here, $\rho_{o}$ represents the path loss at the reference distance of 1 m, $d_{BS,k}$ signifies the BS-user distance, and $\alpha_{BS}$ denotes the path loss exponent for the communication link. $\bm{h}^{NLOS}_{BS,k}$ represents the random non-LoS component modelled using a distribution with zero mean and unitary variance.

The channel vector between the jammer and each user is determined by the propagation environment and the position of the MA at the jammer which is characterized by the field-response channel model [6]. In this model, because the movement regions for the jammer’s antennas are significantly smaller than the signal propagation distance, we assume the far-field condition holds. With this assumption, a plane-wave model can accurately represent the field response from the jammer’s MA region to each user. Specifically, the angle of departure (AoD), angle of arrival (AoA), and the amplitude of the complex coefficient for each channel path between the base station and each user remain constant regardless of the MA’s position within its movement region, while only the phase of each channel path changes with the MA’s position.

Let $L^{t}_{k}$ and $L^{r}_{k}$, for $1\leq k\leq K$, denote the total number of transmit and receive channel paths from the jammer to user $k$, respectively. The elevation and azimuth AoDs for the $j$-th transmit path from the jammer to user $k$ are denoted by $\theta^{t}_{k,j}$ and $\phi^{t}_{k,j}$ for $1\leq j\leq L^{t}_{k}$. Similarly, the elevation and azimuth AoAs for the $i$-th receive path to user $k$ are denoted by $\theta^{r}_{k,i}$ and $\phi^{r}_{k,i}$ for $1\leq i\leq L^{r}_{k}$. For simplicity, we define the virtual AoDs and AoAs as:

In light of the above, the transmit and receive field-response vectors (FRVs) for the channel from the jammer to user $k$ are obtained as follows.

where, $\bm{u}_{k}\in\mathbb{R}^{3\times 1}$ is the antenna position vector at user $k$ and $\bm{p}_{m}\in\mathbb{R}^{3\times 1}$ is the m-th antenna position vector at jammer. $\rho^{r}_{k,j}(\mathbf{u}_{k})=X_{k}\vartheta^{r}_{k,j}+Y_{k}\varphi^{r}_{k,j}+Z_{k}\omega^{r}_{k,j}$ for $1\leq j\leq L^{t}_{k}$ represents the difference in signal propagation distance for the $j$-th receive channel path between the user antenna position $\mathbf{u}_{k}$ and the origin of the local coordinate system at user $k$. This indicates that the phase difference of the coefficient of the $j$-th receive channel path for user $k$ between user position $\mathbf{u}_{k}$ and $O_{k}$ is given by $\frac{2\pi}{\lambda}\rho^{r}_{k,j}(\mathbf{u}_{k})$.

Similarly, the transmit FRV, $\mathbf{f}_{k}(\mathbf{p}_{m})$, characterizes the phase differences in all $L^{t}_{k}$ transmit paths to user $k$, where $\rho^{t}_{k,i}(\mathbf{p}_{m})=x_{m}\vartheta^{t}_{k,i}+y_{m}\varphi^{t}_{k,i}+z_{m}\omega^{t}_{k,i}$ for $1\leq i\leq L^{t}_{k}$ represents the difference of the signal propagation distance for the $i$-th transmit channel path between the m-th jammer-antenna position $\mathbf{p}_{m}$ and the origin of the local coordinate system at jammer $O_{0}$.

The path-response matrix (PRM), $\Sigma_{k}\in\mathbb{C}^{L^{t}_{k}\times L^{r}_{k}}$, represents the response between all the transmit and receive channel paths from $O_{0}$ to $O_{k}$, $\forall k\in\mathcal{K}$. Specifically, the entry in the $i$-th row and $j$-th column of $\Sigma_{k}$ is the response coefficient between the $i$-th transmit path and the $j$-th receive path for user $k$. Subsequently, the channel vector from the jammer to user $k$ can be expressed as:

where $\mathbf{F}_{k}=\begin{bmatrix}\mathbf{f}_{k}(\mathbf{p}_{1}),\mathbf{f}_{k}(\mathbf{p}_{2}),\dots,\mathbf{f}_{k}(\mathbf{p}_{M})\end{bmatrix}\in\mathbb{C}^{L^{t}_{k}\times M}$ is the field-response matrix (FRM) at the jammer targeting user $k$, and $\bm{P}=[\bm{p}_{1},\bm{p}_{2},\ldots\bm{p}_{M}]\in\mathbb{R}^{3\times M}$ is the antenna position matrix at the jammer. Since the positions of the antenna at each user are fixed, the field receive vectors $g_{k},\forall k\in\mathcal{K}$ are constant.

In this paper, the aim of the jammer is to jointly optimize the MA location matrix $\bm{P}$, while designing the jamming beamforming matrix $\bm{V}=[\bm{v}_{1},\bm{v}_{2},\ldots,\bm{v}_{K}]\in\mathbb{C}^{M\times K}$ to minimize the network sum rate, while adhering to available power budget $P_{J}$ and the antenna array hardware constraints, which can be formulated as:

Constraint (9b) represents the jammer’s power budget limit. Constraints (9c) and (9d) define the permitted movement region for the MAs within the $(X,Z)$ plane. The typical size of the antenna moving region is in the order of several wavelengths [6], however the most common value in research is a distance of $\pm\lambda$ around the $x$ and $z$ axes. As previously mentioned, the MAs in this architecture are free to move along the entire length $2L$ of the array on the $y$-axis, as specified in (9e), provided they maintain a minimum spacing of $2\lambda$ between each other, as ensured by (9f). Since the jammer lacks information about the BS-user links or the BS beamforming vectors, its objective is to align its beamforming with each jammer-user channel to maximize interference. Therefore, the problem is formulated as follows.

The problem $\mathcal{P}$ is a non-convex optimization problem for two main reasons. First, the dependence on antenna location matrix $\bm{P}$ in the jammer-user channel $\bm{h}_{J,k}^{H}$ is non linear, making the objective function non-convex. Additionally, constraints (9c)-(9e) are obviously non-convex.

Given the non-convexity of problem $\mathcal{P}$ and the coupling between variables $\bm{P}$ and $\bm{V}$ in the objective function, we divide the problem into two sub-problems that will be solved iteratively until convergence. In each sub-problem, we focus on optimizing a set of variables while fixing the rest, which is usually referred to as block coordinate descent (BCD) or alternating optimization (AO). The detailed algorithm is described in Algorithm 1.

The simulation setup is as follows: the BS is positioned at $(0,0)$, while users are uniformly distributed within a circular area of radius $40$ m centered at $(50,50)$. The jammer is located at $(100,0)$. All coordinates are given in meters. We adopt the PRM matrix model used in [2], where the PRM for each user $k$, $\Sigma_{k}=\operatorname{diag}\{\sigma_{1},\sigma_{2},\dots,\sigma_{L}\}$, is a diagonal matrix where each diagonal element $\sigma_{i}$ follows a circularly symmetric complex Gaussian (CSCG) distribution, $\mathcal{CN}(0,\frac{c_{k}^{2}}{L})$. Here, $c_{k}^{2}=g_{0}d_{k}^{-\alpha_{J}}$ represents the expected channel power gain for user $k$, with $g_{0}$ denoting the average channel power gain at a reference distance of 1 meter, and $\alpha_{J}$ being the path-loss exponent. The elevation and azimuth AoDs and AoAs for the channel paths of each user are modeled as random variables with a joint probability density function (PDF) given by $f_{\theta_{k},\phi_{k}}(\theta_{k},\phi_{k})=\frac{\cos\theta_{k}}{2\pi}$, where $\theta_{k}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ and $\phi_{k}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. Key simulation parameters are summarized in Table I and remain consistent across all simulations unless specified otherwise.