Polarization Aware Movable Antenna

The relentless pursuit of high-speed and reliable wireless communication has become a cornerstone of modern technological advancement. Emerging applications such as autonomous vehicles, the artificial intelligence of things, and augmented reality demand seamless connectivity with unprecedented data rates and minimal latency[1, 2]. These evolving requirements are pushing the boundaries of existing wireless communication systems, necessitating innovative solutions to address challenges like spectrum scarcity, signal degradation, and dynamic environmental conditions.

One promising technique that has emerged to meet these challenges is the use of movable antennas (MAs) [3, 4, 5, 6], also known as fluid antenna systems (FAS) [7, 8, 9, 10, 11]. By dynamically adjusting the position and orientation of antennas, it is possible to achieve enhanced signal-to-noise ratio (SNR), wider coverage, and adaptability to changing environments and user locations, without solely relying on traditional methods like increasing transmission power, beamforming, or deploying additional infrastructure[12, 13, 14, 15, 16, 17].

Previous research has explored the applications of MAs and FAS in radio frequency bands characterized by rich multipath environments [3, 7]. In these settings, antenna movement is leveraged to exploit spatial diversity arising from amplitude fluctuations caused by multipath propagation.

In [7], the authors propose FAS that allows the receiving antenna to move to positions to capture the strongest signal. Analysis of outage probability demonstrates that, within a movement range of 0.2$\lambda$, FAS can achieve performance comparable to a maximal ratio combining system with five antennas spanning 2$\lambda$. This flexibility in antenna positioning offers significant advantages, especially in scenarios requiring massive connectivity. It allows each user to find an optimal position where instantaneous interference is minimized, effectively creating deep nulls. This spatial multiplexing gain facilitates multi-user access, a concept introduced as Fluid Antenna Multiple Access (FAMA) in [8]. Building upon this concept, [9] extends FAS to a two-dimensional (2D) implementation and discusses reconfigurable pixel technologies as a practical approach for realization. Additionally, [18] introduces a continuous FAS (CFAS) architecture, demonstrating that CFAS outperforms discrete FAS with limited ports, particularly at medium to high threshold levels.

The MA system presented in [3] exhibits additional phase variations due to antenna translation, significantly surpassing fixed-position antennas in both deterministic and stochastic channels. The system’s performance further improves with increased positioning accuracy, larger movement ranges, and enhanced multipath effects. Building on 2D MA system, advancements in six-dimensional (6D) systems, such as those discussed in [4], enable simultaneous three-dimensional (3D) translation and rotation, enhancing the system’s directional capabilities. By imposing spatial constraints on surfaces equipped with 6D movable antennas (6DMA), user interference is reduced, improving the signal-to-interference-plus-noise ratio (SINR). In [5], the 6DMA model is simplified by integrating conventional fixed-position antenna arrays with 6DMA surfaces capable of rotating along a circular path. By limiting the degrees of freedom for surface movement to the azimuth angle along the circular track, computational complexity is significantly reduced while enhancing the system’s maximum capacity.

Despite these advancements, the impact of polarization changes due to antenna movement and rotation has been largely overlooked. Polarization is a fundamental characteristic of electromagnetic waves, describing the orientation of the electric field vector as the wave propagates through space [19, 20, 21, 22, 23]. In wireless communication systems, proper alignment of polarization between transmitting and receiving antennas is crucial for efficient signal transmission and reception. When polarizations are properly aligned, the receiving antenna can maximally capture the transmitted energy, resulting in stronger signal strength, improved SNR, and enhanced overall system performance. Conversely, polarization mismatch occurs when there is a misalignment between the polarization orientations of the transmitting and receiving antennas, leading to significant signal attenuation as the misaligned component of the electromagnetic wave is effectively lost. Neglecting polarization effects can lead to incomplete system models and suboptimal performance assessments, hindering a full understanding of the impact and advantages of MAs.

This oversight is particularly critical as communication systems move towards higher frequency bands, such as the millimeter-wave (mmWave) spectrum [24, 25]. At these higher frequencies, signal propagation becomes more directional with reduced diffraction around obstacles, often resulting in a single line-of-sight (LOS) link between transmitter and receiver [26]. Previously, it was believed that MAs offer no gains in such LOS scenarios due to the lack of multipath diversity [7, 3]. However, our formulation incorporating polarization effects indicates otherwise – MAs can achieve significant performance gains even in LOS scenarios by facilitating polarization matching. As such, there is a clear gap in current research: the need to incorporate polarization effects into the design and optimization of MA systems, especially as the wireless industry advances towards higher frequency bands to achieve greater bandwidths and data rates.

To fill the gap, this paper makes the following main contributions:

• •

  We introduce a polarization-aware movable antenna (PAMA) framework, which incorporates polarization effects into the design and optimization of antenna movement and rotation. PAMA uncovers a previously overlooked, but defining benefit of antenna movement – facilitating optimal polarization matching between the transmitter and receiver. By accounting for the impact of polarization changes due to movement and rotation, PAMA provides a more refined system model and deepens our understanding of the dynamic capabilities of MAs.

• •

  Building upon the electromagnetic theory, we develop a polarization-aware channel model that describes the impact of 3D movement and rotation of antennas on the channel gains. Grounded in Maxwell’s equations, this model allows for precise characterization of how antenna positioning and orientation affect polarization alignment and, consequently, signal reception quality. By integrating these factors into the system design, we enable the optimization of antenna movements to maintain polarization alignment, thereby maximizing the sum rate of the communication system.

• •

  Contrary to prior beliefs that MAs are more effective in low-frequency, multipath-rich environments, our PAMA framework demonstrates that MAs are highly effective in higher frequency bands such as the mmWave band, even with a single LOS link. The smaller size of antennas at higher frequencies makes them easier to move and rotate, facilitating practical implementation of PAMA systems in these bands. This new understanding extends the potential and application of MA to next-generation wireless communication systems that demand higher bandwidths.

Organization: The remainder of this paper is structured as follows. Section II introduces the system model and essential geometric concepts used throughout the work. Section III presents our PAMA framework and the polarization-aware channel model. Section IV explores the impacts of 3D antenna translation and rotation on the PAMA channel gains. Section V demonstrates the potential of PAMA through a sum-rate optimization problem. Finally, Section VI concludes the paper.

Notations: We use boldface lowercase letters to denote column vectors (e.g., $\bm{h}$, $\bm{w}$) and boldface uppercase letters to denote matrices (e.g., $\bm{H}$, $\bm{P}$). For a vector or matrix, $(\cdot)^{\top}$ denotes the transpose, $(\cdot)^{*}$ denotes the complex conjugate, and $(\cdot)^{H}$ denotes the conjugate transpose. $\mathbb{C}$ stand for the sets of complex values. The imaginary unit is represented by $j$. The Euclidean norm of a vector $\bm{p}$ is denoted by $|\bm{p}|$. For two vectors $\bm{a}$ and $\bm{b}$, $\langle\bm{a},\bm{b}\rangle$ denotes the angle between them, and $\bm{a}\cdot\bm{b}$ denotes their inner product.

We consider a mobile edge network within a 3D environment. The network comprises a base station (BS) and $K$ mobile users distributed within the 3D space. The BS is equipped with $L$ half-wave dipole antennas, and each mobile user is equipped with a single half-wave dipole antenna. We focus on the downlink transmission from the BS to the users, operating in the mmWave frequency band. Both the antennas at the BS and mobile users are movable.

Translation typically requires larger operational spaces, often spanning tens of wavelengths, whereas rotation can be accommodated within a more compact area, approximately the size of a half-wavelength sphere, as shown in Fig. 1. In this work, we assume that the antennas on mobile users are restricted to 3D rotation, due to the spatial constraint. In contrast, the $L$ antennas at the BS can perform both 3D rotation and translation independently within a designated region. The extension of this model to include translatable antennas for mobile users is straightforward.

In this work, we consider the far-field scenario, which implies the following:

• •

  Plane wave. The waves received by the users can be treated as plane waves because the propagation distance is much greater than the wavelength and the dimensions of the movable regions.

• •

  Point representation of antennas. The physical sizes of the transmitting and receiving antennas are negligible compared to the propagation distance, allowing us to represent their positions as points specified by their center coordinates.

To mathematically represent the network and the positions of the antennas, we establish two coordinate systems: the Cartesian Coordinate System (CCS) and the Spherical Coordinate System (SCS), as depicted in Fig. 2.

Table I provides the conversion formulas between the unit vectors of the two coordinate systems. For instance, the radial unit vector $\bm{\nu}$ in the SCS can be expressed in terms of the CCS unit vectors as $\bm{\nu}=\sin(\theta)\cos(\phi)\bm{i}+\sin(\theta)\sin(\phi)\bm{j}+\cos(\theta)\bm{k}$.

Given the two coordinate systems, we introduce several definitions regarding the antennas’ positions and orientations.

The polar and azimuthal angles are illustrated in Fig. 3. A summary of the definitions and their geometric relationships is given in Tab. II.

The CCS representation facilitates mathematical operations such as dot products, providing computational simplicity. In contrast, the SCS representation offers a clearer depiction of the relationship between channel gains and antenna orientation.

Building upon the system model, this section presents our PAMA framework. At the core of wireless communication is the interaction between transmitting and receiving antennas through electromagnetic fields. An alternating current in the transmitting antenna generates electromagnetic waves, creating an electromagnetic field that propagates through space to the receiver. The receiving antenna interacts with this field, specifically sensing the electric field component, which induces a current within the antenna used to recover the transmitted signal. This fundamental interaction is governed by Maxwell’s equations, which describe how electric and magnetic fields are generated by charges and currents, and how they propagate in space.

In this section, we begin by modeling the radiation of the electric field from movable transmitting antennas, characterizing how movements – 3D translations and rotations – affect the spatial distribution and polarization of the emitted electromagnetic waves. We then explore how movable receiving antennas detect and interact with the electric field. Ultimately, the analysis leads to a polarization-aware channel representation for movable antennas.

This section analyzes the PAMA channel model and examines how each variable influences channel quality, providing insights into the key characteristics of PAMA and identifying the sources of its performance gains. Fig. 5 summarizes the relationships between antenna movement and their corresponding impacts on the channel. It is essential to highlight that these properties are derived under the conditions of a far-field scenario and a dominant LOS path. For clarity, the indices $\ell$ and $\kappa$ in the subscripts are omitted in this section.

In the far-field LOS scenario, the electromagnetic waves can be approximated as plane waves, and the spatial fading characteristics are negligible. The relative orientation of the electric field vectors between the transmitter and receiver remains unchanged despite antenna translation. This means that antenna translation does not affect the amplitude or polarization alignment of the received signal – only the phase changes due to the difference in path length, leading to Proposition 1.

In practical systems, digital precoding is often easier to implement than mechanical antenna movements. Therefore, when studying MAs, it is essential to consider the unique advantages that MAs introduce beyond what precoding can achieve. Proposition 1 indicates that the benefits of antenna translation are more significant in near-field or rich multi-path environments, in which case antenna translation can impact the amplitude and spatial characteristics of the channel gain, offering advantages that precoding alone cannot replicate.

In the context of this paper, which focuses on far-field and mmWave dominant LOS paths, the unique gains that MAs can offer over precoding are primarily obtained through antenna rotation rather than translation.

When the transmitting antenna rotates, it changes the emission angle $\theta^{(e)}$ of the electromagnetic wave. This alteration affects the antenna’s radiation pattern – the distribution of the electric field w.r.t. the CCS – which in turn influences the electric field intensity received. As illustrated in Fig. 5 (a) and (b), different emission angles lead to variations in the radiation pattern, resulting in changes to the received signal’s strength. According to Corollary 3, the magnitude or intensity of the electric field reaches its maximum when the emission angle is $\theta^{(e)}=\frac{\pi}{2}$. To achieve this, the transmitting antenna should be positioned within the plane that is perpendicular to the direction of wave propagation.

Furthermore, the rotation of the transmitting antenna also modifies its polarization orientation, thereby changing the polarization matching angle $\alpha$ between the transmitter and receiver, and hence the matching efficiency $\mathcal{M}\left(\theta^{(i)},\alpha\right)$ in the PAMA channel gain.

Rotating the receiving antennas affects both the incident angle $\theta^{(i)}$ of the incoming electromagnetic wave and the polarization matching angle $\alpha$, both of which impacts the polarization matching efficiency.

1. 1.

   The change of the incident angle $\theta^{(i)}$ alters the energy reflection coefficients $\Gamma_{\parallel}$ and $\Gamma_{\perp}$, as defined in (14) and (15). These reflection coefficients characterize the proportion of electromagnetic energy that enters the antenna. A smaller reflection coefficient indicates that more energy is transmitted into the antenna, enhancing the received signal strength.

2. 2.

   The orientation of the receiving antenna relative to the electric field vector of the incoming wave determines the polarization matching angle $\alpha$, which describes the ratio of the horizontal to vertical components of the electromagnetic wave’s polarization relative to the antenna’s orientation. A better alignment (i.e., a smaller mismatch angle) leads to higher polarization matching efficiency.

Figs. 5 (c) and (d) illustrate how the polarization matching efficiency varies with the incident angle $\theta^{(i)}$ and the polarization matching angle $\alpha$ for antennas with different relative dielectric constants $\varepsilon_{r}$. As can be seen, without proper optimization of the receiving antenna’s rotation, the matching efficiency can decrease significantly, potentially dropping to zero. In such cases, the antenna fails to effectively receive the incoming signal, resulting in a loss of information. Therefore, careful control of the receiving antenna’s orientation is essential for maximizing the channel gain and ensuring reliable communication in the PAMA framework.

Propositions 2 and 3 indicate that the amplitude of the channel gain is influenced by the orientations of both the transmitting and receiving antennas. To illustrate the extent to which antenna rotations affect the channel amplitude, we simulate two scenarios:

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  Scenario 1: Rotatable transmitting antenna with fixed receiving antenna;

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  Scenario 2: Fixed transmitting antenna with rotatable receiving antenna.

In the simulations, the positions of the antennas are set as $\bm{p}_{t}=[0,0,0]$, $\bm{p}_{r}=[75,-40,50]$. The fixed antennas are oriented vertically upward, with a direction vector of $[0,0,1]$. For visualization purposes, we vary the polar and azimuthal angles of the rotatable antennas over two periods, even though their actual values lie within a single period. The channel amplitude variations for the two scenarios are presented in Figs. 6(a) and 7(a), respectively.

For scenario 1 with a fixed receiving antenna, Fig. 6(a) shows that a randomly oriented transmitting antenna has only a $67.5\%$ probability of delivering at least half of the maximum possible energy to the user. Each period can yield two sets of maximum values due to the symmetry of the dipole antennas, where the collinear reverse sets of polar and azimuthal angles provide equivalent performance.

For scenario 2 with a fixed transmitting antenna, Fig. 7(a) shows that a randomly oriented receiving antenna has a $99.0\%$ probability of capturing at least half of the maximum energy. However, the remaining $1.0\%$ of orientations present a significant risk of receiving virtually no energy. This outcome highlights the critical importance of antenna orientation in ensuring reliable communication.

Figs. 6(b) and 7(b) depict the positions and orientations of the transmitting and receiving antennas for both scenarios. In these illustrations, fixed Antennas are shown with vertical black arrows; the optimal orientations of the rotatable antennas are shown with solid red arrows, indicating the directions that maximize the channel gain; the worst-case orientations of the rotatable antennas are shown with dashed green arrows, representing the directions that minimize the channel gain.

When the electromagnetic field reaches the receiving antenna, it can be decomposed into three orthogonal components along: the direction of the receiving antenna $\bm{n}_{r}$; the normal direction of the incident signal $\bm{n}_{r}^{\perp}$; a direction perpendicular to both $\bm{n}_{r}$ and $\bm{n}_{r}^{\perp}$, denoted by $\bm{n}_{3}$. This decomposition is illustrated in Fig. 8. The component along $\bm{n}_{r}$ represents the horizontal polarization component, while the components along $\bm{n}_{r}^{\perp}$ and $\bm{n}_{3}$ represent the vertical polarization components.

According to (14) and (15), we have $\Gamma_{\parallel}\geq\Gamma_{\perp}$, meaning that the horizontal polarization component has a smaller reflection coefficient, and thus a better matching efficiency, than the vertical component. Therefore, increasing the component along $\bm{n}_{r}$ by adjusting antenna orientations enhances the channel gain. However, achieving complete horizontal polarization requires the polarization matching angle $\alpha$ to be zero, which also corresponds to an incident angle $\theta^{(i)}$ of zero. Under this condition, the reflection coefficient $\Gamma_{\parallel}$ increases, reducing the matching efficiency. Therefore, simply minimizing $\alpha$ is suboptimal. Instead, we need to balance the horizontal and vertical polarization components to maximize the overall matching efficiency.

Since the polarization direction $\bm{n}_{\bm{\mathcal{E}}}$ is perpendicular to the propagation direction, and $\bm{p}_{r}$ is perpendicular to the antenna direction, the polarization matching angle satisfies $\alpha\allowbreak=\langle\bm{n}_{\bm{\mathcal{E}}},\allowbreak\bm{n}_{r}\rangle\allowbreak\geq\theta^{(i)}=\allowbreak\langle\bm{n}_{r}^{\perp},\allowbreak\bm{p}_{r}\rangle$. To increase the incident angle and reduce the reflection coefficient without introducing additional vertical polarization components along $\bm{n}_{3}$, it is optimal to maintain $\alpha=\theta^{(i)}$. For linearly polarized antennas, this configuration implies that the transmitting antenna, the receiving antenna, and the propagation path are all aligned within the same plane, confirming Proposition 4. It is important to note that lying in the same plane does not necessarily mean that the transmitting and receiving antennas are parallel, as depicted in Figs. 6(a) and 7(a).

Figs. 6(b) and 7(b) demonstrate that the worst-case orientation consistently occurs when one antenna is directed toward the other. This phenomenon is explained by:

• •

  For the transmitting antenna in Figs. 6(b), when the emission angle $\theta^{(e)}=0$, meaning the transmitting antenna points directly at the receiver, the electric field intensity in that direction is effectively zero due to the dipole antenna’s radiation pattern.

• •

  For the receiving antenna in Figs. 7(b), when the incident angle $\theta^{(i)}=\frac{\pi}{2}$, the reflection coefficients $\Gamma_{\parallel,1}=\Gamma_{\perp,1}=1$, indicating total reflection with no energy entering the antenna. Additionally, the polarization matching angle $\alpha=\frac{\pi}{2}$, indicating complete polarization mismatch and resulting in zero capturable energy.

By incorporating polarization, we have redefined movable antennas and established more refined, polarization-aware channel models. This enhancement serves as a foundation for further optimizations, specifically tailored to harness the dynamic adjustments of antenna orientations and positions to maximize system performance. In this paper, we focus on the fundamental sum-rate optimization [13] to reveal the potential of PAMA.

This paper presented the PAMA framework, which integrates polarization effects into the translation and rotation of antennas in wireless communication systems. By factoring in polarization, PAMA uncovers a defining advantage of MA: the ability to optimize polarization matching between the transmitter and receiver. This enhancement debunks the belief that MA is only effective in low-frequency multipath environments and extends the applicability of MA to higher frequency bands like the mmWave spectrum.

At mmWave frequencies, antennas are physically smaller, making them easier to move and rotate, which facilitates the practical implementation of PAMA systems. The dominance of LOS paths in the mmWave band simplifies channel estimation compared to traditional MA systems operating in environments with rich multipath. Additionally, PAMA achieves near-optimal performance with low rotation granularity. These factors collectively enhance the feasibility of deploying PAMA in real-world scenarios.

The insights from this work underscore the critical role of polarization in maximizing the performance of MA. Future research may focus on integrating PAMA with other advanced communication technologies, evaluating its performance in various high-frequency settings, and addressing practical implementation challenges such as antenna control mechanisms and mobility constraints. By advancing the understanding of how polarization can be harnessed in movable antenna systems, this work lays the foundation for developing more efficient and reliable wireless communication networks.