How Many Independent Modes Does a Fluid Antenna Have? A Closed-Form Outage Analysis via Equivalent Degrees of Freedom

In a fluid antenna system (FAS), a single reconfigurable antenna is able to activate one of $N$ correlated ports to exploit spatial diversity. However, outage analysis is challenging because exact evaluation requires an $N$-dimensional multivariate integral, while existing closed-form approximations based on block-correlation models tend to underestimate the true outage probability. This paper shows that the spatial correlation matrix of a FAS with a normalized linear aperture length $W$ has at most $K^{*}=2\lceil W\rceil+1$ significant eigenmodes, regardless of the number of deployed ports. This is a spatial counterpart of the Slepian-Landau-Pollak spectral concentration theorem and reveals that the spatial degrees of freedom are determined by aperture size rather than port count. Motivated by this result, we derive an \emph{equivalent degree of freedom} (EDoF) approximation, under which the outage probability can be expressed in closed form as that of selection combining over $K^{*}$ independent branches. We propose a refined \emph{weighted independent modes} (WIM) approximation, to incorporate eigenvalue-dependent branch weights $\{β_k\}$ and yield a product-form closed-form expression with improved accuracy at moderate signal-to-noise ratio (SNR). Both approximations achieve the exact diversity order, become asymptotically exact at high SNR, and provably never underestimate the true outage probability by Anderson's inequality. The proposed framework is further extended to obtain closed-form expressions for ergodic capacity, characterize multi-user fluid antenna multiple access (FAMA) with explicit interference-limited outage floors. Besides, we analyze two-dimensional planar FAS, for which the diversity order scales multiplicatively with the aperture dimensions.

Sparse Fluid Antenna Arrays: Continuous Position Design Beyond Classical DOF Limits

Fluid antenna system (FAS), which continuously repositions a single physical element across a deployment region $[0, D]$, breaks this limit by freeing antenna positions from the discrete grid entirely. This paper establishes the theoretical foundations of sparse FAS design for direction-of-arrival (DOA) estimation and shows that continuous position freedom unlocks three compounding advantages over the classical designs. \emph{First}, we derive a universal dual DOF bound and prove that FAS-optimized positions can approach it, growing the DOF linearly with $D/λ$ , where $λ$ is the signal wavelength, rather than saturating at $O(N^2)$. \emph{Second}, the CRB scales as $O(1/D^{2L})$ for $L$ sources, a $(D/(N^2 d_0))^{2L}$ improvement over the best grid design, with $d_0 = λ/2$ and D-optimal positions admitting closed-form solution for single sources and efficient Frank-Wolfe algorithm for multiple sources. \emph{Third}, we propose a two-stage FAS-MUSIC approach that combines coarray MUSIC disambiguation with full-aperture local maximum likelihood (ML) refinement to track the CRB, overcoming the grating-lobe ambiguity inherent in large-aperture non-uniform arrays. Robustness to minimum spacing constraints, mutual coupling, and finite position accuracy is also analyzed. Extensive simulations show that FAS-MUSIC achieves $17.5\times$ lower root mean squared error (RMSE) than uniform linear array (ULA) MUSIC and that FAS with $4$ antennas outperforms MRA with $8$ antennas, gains that are unattainable by any grid-constrained design.

Hybrid Architecture Gets Fluid: A New Paradigm for Direction-of-arrival Estimation in 6G Networks

High-precision direction-of-arrival (DOA) estimation, as a key sensing capability for 6G-enabled applications such as autonomous driving and extended reality, is increasingly dependent on the effective exploitation of spatial degrees of freedom (DOFs). This paper integrates two frontier DOFs-oriented paradigms and proposes a fluid antenna-enabled hybrid analog-digital (FA-HAD) architecture, which features an extremely lightweight front-end configuration mechanism and efficient spatial DOFs exploitation. Within this architecture, a collaborative spatial-phase sampling strategy is first developed to enable real-time 2-D DOA estimation under compressive observations, and a single-source CRLB analysis is provided to quantify the achievable performance limit, offering quantitative guidance for accuracy-overhead trade-offs. Furthermore, an efficient virtual-array spatial covariance matrix reconstruction method is proposed to recover a physically meaningful covariance representation, thereby providing a covariance-domain interface that is directly reusable by a broad class of existing covariance-based array processing and array design techniques, which strengthens the scalability and transferability of the proposed architecture. Building upon the reconstructed SCM, a Jacobi-Anger expansion based dimension-reduced MUSIC estimator is further derived for arbitrary planar arrays with a favorable computational cost. Simulation results demonstrate that the proposed FA-HAD framework attains DOA accuracy close to fully digital systems while substantially reducing RF hardware complexity and training overhead.

Fundamental Analysis of Scalable Fluid Antenna Systems: Identifiability Limits, Information Theory, and Joint Processing

Unlike fixed-position arrays with static observation entropy, the scalable fluid antenna system (S-FAS) can dynamically adjust its aperture to form different observation spaces with configuration-dependent entropy budgets. This reconfigurability requires an information-theoretic framework beyond traditional algebraic identifiability analysis. This paper establishes an observation entropy framework for S-FAS, which unifies the derivation of identifiability limits, the diagnosis of processing bottlenecks, and system design optimization. For an S-FAS with mutual coupling suppression, we derive a complete capacity hierarchy among compressed, extended, and jointly stacked configurations. The entropy framework reveals that sequential two-stage processing suffers from an information bottleneck that restricts achievable capacity, while the noise entropy ratio can be used to distinguish fundamental performance limits from algorithmic deficiencies. A joint MUSIC algorithm is proposed to approach the theoretical joint capacity bound. Extensive Monte Carlo simulations, validated by both algebraic and information-theoretic criteria, verify the derived capacity hierarchy and identifiability boundaries.

DOA Estimation for Low-Altitude Networks: HAD Architectures, Methods, and Challenges

With the rapid expansion of low-altitude economy (LAE) services and the growing demand for integrated sensing and communication (ISAC) in air-ground networks, reliable direction-of-arrival (DOA) estimation has become essential for both directional communication and sensing functions. DOA underpins beam alignment, spatial-reuse scheduling, and ISAC-critical tasks such as airspace situational awareness and multi-target monitoring. Hybrid analog-digital (HAD) architectures have emerged as a practical solution for large-aperture directional operation under stringent radio frequency (RF), analog-to-digital converter (ADC), and size, weight, and power (SWaP) constraints. However, HAD compresses antenna-domain observations through analog combining, fundamentally reshaping the measurement model and introducing new algorithmic and system-level challenges for DOA estimation. This article first reviews the principles and representative architectures of HAD, highlighting their advantages for scalable beam-centric and ISAC-oriented operation in LAE scenarios. We then provide a structured overview of HAD-enabled DOA estimation methodologies, including spatial covariance matrix (SCM) reconstruction, multi-combiner scan-based acquisition, and pilot-aided estimation, along with key design tradeoffs. Finally, we discuss open challenges and outline reliability-driven research directions toward robust, deployable HAD-enabled DOA solutions for practical ISAC-enabled low-altitude environments.

A Gaussian Process Framework for Outage Analysis in Continuous-Aperture Fluid Antenna Systems

This paper develops a comprehensive analytical framework for the outage probability of fluid antenna system (FAS)-aided communications by modeling the antenna as a continuous aperture and approximating the Jakes (Bessel) spatial correlation with a Gaussian kernel $ρ_G(δ) = e^{-π^2δ^2}$. Three complementary analytical strategies are pursued. First, the Karhunen--Loève (KL) expansion under the Gaussian kernel is derived, yielding closed-form outage expressions for the rank-1 and rank-2 truncations and a Gauss--Hermite formula for arbitrary rank~$K$, with effective degrees of freedom $K_{\mathrm{eff}}^G \approx π\sqrt{2}\, W$. Second, rigorous two-sided outage bounds are established via Slepian's inequality and the Gaussian comparison theorem: by sandwiching the true correlation between equi-correlated models with $ρ_{\min}$ and $ρ_{\max}$, closed-form upper and lower bounds that avoid the optimistic bias of block-correlation models are obtained. Third, a continuous-aperture extreme value theory is developed using the Adler--Taylor expected Euler characteristic method and Piterbarg's theorem. The resulting outage expression $P_{\mathrm{out}} \approx 1 - e^{-x}(1 + π\sqrt{2}\, W\, x)$ depends only on the aperture~$W$ and threshold~$x$, is independent of the port count~$N$, and is identical for the Jakes and Gaussian models since both share the second spectral moment $λ_2 = 2π^2$. A Pickands-constant refinement for the deep-outage regime and a threshold-dependent effective diversity $N_{\mathrm{eff}} \approx 1 + π\sqrt{2}\, W\, x$ are further derived. Numerical results confirm that the Gaussian approximation incurs less than 10\% relative outage error for $W \leq 2$ and that the continuous-aperture formula converges with as few as $N \approx 10W$ ports.

Karhunen-Loève Expansion for Fluid Antenna Systems: Information-Theoretic Optimal Channel Compression and Outage Analysis

Fluid antenna systems (FAS) achieve spatial diversity by dynamically switching among $N$ densely packed ports, but the resulting spatially correlated Rayleigh channels render exact outage analysis intractable. Existing block-correlation models (BCM) impose structural approximations on the channel covariance matrix that can introduce optimistic performance bias. This paper proposes a principled Karhunen-Loève (KL) expansion framework that decomposes the $N$-dimensional correlated FAS channel into independent eigenmodes and performs a controlled rank-$K$ truncation, reducing the outage analysis to a $K$-dimensional integration with $K \ll N$. Closed-form outage expressions are derived for the rank-1 and rank-2 cases, and a general Gauss-Hermite quadrature formula is provided for arbitrary $K$. On the theoretical front, it is proved via Anderson's inequality that the KL approximation \emph{always} overestimates the outage probability, providing a conservative guarantee essential for secure system design. Leveraging the Slepian--Landau--Pollak concentration theorem, it is established that only $K^* = 2\lceil W \rceil + 1$ eigenmodes are needed regardless of $N$, where $W$ is the normalized aperture. It is further shown that the KL truncation achieves the Gaussian rate-distortion bound, certifying it as the information-theoretically optimal channel compression. Extensive numerical results confirm that (i) theoretical predictions match Monte Carlo simulations, (ii) the entropy fraction converges faster than the power fraction, (iii) the KL framework uniformly outperforms BCM in approximation accuracy while avoiding the optimistic bias inherent in block-diagonal models, and (iv) the effective degrees of freedom scale with the aperture rather than the number of ports.

Near-Field NLOS Localization via Position-Unknown HRIS:From Self-Localization to Target Positioning

Current reconfigurable intelligent surface (RIS)-aided near-field (NF) localization methods assume the RIS position is known a priori, and it has limited their practical applicability. This paper applies a hybrid RIS (HRIS) at an unknown position to locate non-line-of-sight (NLOS) NF targets. To this end, we first propose a two-stage gridless localization framework for achieving HRIS self-localization, and then determine the positions of the NF targets. In the first stage, we use the NF Fresnel approximation to convert the signal model into a virtual far-field model through delay-based cross-correlation of centrally symmetric HRIS elements. Such a conversion will naturally extend the aperture of the virtual array. A single-snapshot decoupled atomic norm minimization (DANM) algorithm is then proposed to locate an NF target relative to the HRIS, which includes a two-dimensional (2-D) direction of arrival (DOA) estimation with automatic pairing, the multiple signal classification (MUSIC) method for range estimation, and a total least squares (TLS) method to eliminate the Fresnel approximation error. In the second stage, we leverage the unique capability of HRIS in simultaneous sensing and reflection to estimate the HRIS-to-base station (BS) direction vectors using atomic norm minimization (ANM), and derive the three-dimensional (3-D) HRIS position with two BSs via the least squares (LS)-based geometric triangulation. Furthermore, we propose a semidefinite relaxation (SDR)-based HRIS phase optimization method to enhance the received signal power at the BSs, thereby improving the HRIS localization accuracy, which, in turn, enhances NF target positionings. The Cramer-Rao bound (CRB) for the NF target parameters and the position error bound (PEB) for the HRIS coordinates are derived as performance benchmarks.

Physical Layer Security for FAS-Aided Short-Packet Systems: A Variable Block-Correlation Approach

This paper presents a comprehensive physical layer security (PLS) framework for fluid antenna system (FAS)-aided short-packet communications under the variable block-correlation model (VBCM). We consider a downlink wiretap scenario in which a base station transmits confidential short packets to a legitimate receiver user (RU) in the presence of an eavesdropper user (EU), where both the RU and EU are equipped with fluid antennas. Unlike existing FAS security analyses that rely on constant block-correlation models or infinite-blocklength assumptions, we incorporate the VBCM to accurately capture the non-uniform spatial correlation structure inherent in practical FAS deployments. By employing a piecewise linear approximation of the decoding error probability and Gauss-Chebyshev quadrature, we derive closed-form and asymptotic expressions for the average achievable secrecy throughput (AAST). We further prove that the AAST is monotonically non-decreasing in the number of RU ports, which reduces the three-dimensional joint optimization of transmit power, blocklength, and port number to a two-dimensional grid search (GS). Numerical results demonstrate that the FAS-aided system achieves up to an order-of-magnitude secrecy throughput improvement over conventional fixed-position antenna systems, and reveal that blocklength selection is the most critical design parameter in the joint optimization.

Fluid Antennas Meet Intelligent Surfaces: Security Analysis of NOMA Systems Under Hardware Impairments

The revolutionary convergence of fluid antenna systems (FAS) and reconfigurable intelligent surfaces (RIS) creates unprecedented opportunities for secure wireless communications, yet the practical implications of hardware impairments on this promising combination remain largely unexplored. This paper investigates the security performance of non-orthogonal multiple access (NOMA) systems when fluid antennas (FAs) meet intelligent surfaces under realistic hardware constraints. We develop a comprehensive analytical framework that captures the complex interplay between adaptive spatial diversity, intelligent signal reflection, and hardware-induced distortions in short-packet communications. Through novel piecewise linear approximations and block-correlation models, we derive tractable expressions for average secure block error rate (BLER) that reveal fundamental performance limits imposed by hardware impairments. Our analysis demonstrates that while the synergy between FAs and intelligent surfaces offers remarkable degrees of freedom for security enhancement, practical hardware imperfections create performance ceilings that persist regardless of spatial diversity gains. The theoretical framework exposes critical design trade-offs between system complexity and achievable security performance, showing that hardware quality becomes a decisive factor in realizing the full potential of FAS-RIS architectures. Extensive simulations validate our analytical insights and provide practical design guidelines for implementing secure NOMA systems that effectively balance the benefits of fluid-intelligent cooperation against the constraints of realistic hardware limitations.