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Fluid Antenna System: New Insights on Outage Probability and Diversity Gain

Wee Kiat New, Kai-Kit Wong, Hao Xu, Kin-Fai Tong, Chan-Byoung Chae

arXiv:2301.00073v2cs.IT

TL;DR

The fundamental limits and key performance factors of fluid antenna systems remain unclear, partly because their channels are strongly correlated. This paper develops analytical approximations and identifies an efficient suboptimal port configuration with favorable outage performance.

  • Problem

    The fundamental limits and key factors affecting FAS performance remain unclear because FAS channels are strongly correlated.

  • Method

    The paper derives a high-SNR approximation and analyzes FAS over a linear space of length Wλ.

  • Results

    Outage probability decreases significantly as N increases, while diversity gain is limited by min {N, N′}.

  • Takeaways & Limitations

    The proposed suboptimal FAS uses N∗ ports, substantially improves over N∗−1 ports, and makes N∗+1 ports yield negligible additional improvement.

Abstract

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To enable innovative applications and services, both industry and academia are exploring new technologies for sixth generation (6G) communications. One of the promising candidates is fluid antenna system (FAS). Unlike existing systems, FAS is a novel communication technology where its antenna can freely change its position and shape within a given space. Compared to the traditional systems, this unique capability has the potential of providing higher diversity and interference-free communications. Nevertheless, the performance limits of FAS remain unclear as its system properties are difficult to analyze. To address this, we approximate the outage probability and diversity gain of FAS in closed-form expressions. We then propose a suboptimal FAS with $N^{*}$ ports, where a significant gain can be obtained over FAS with $N^{*}-1$ ports whilst FAS with $N^{*}+1$ ports only yields marginal improvement over the proposed suboptimal FAS. In this paper, we also provide analytical and simulation results to unfold the key factors that affect the performance of FAS. Limited to systems with one active radio frequency (RF)-chain, we show that the proposed suboptimal FAS outperforms single-antenna (SISO) system and selection combining (SC) system in terms of outage probability. Interestingly, when the given space is $\fracλ{2}$, the outage probability of the proposed suboptimal FAS with one active RF-chain achieves near to that of the maximal ratio combining (MRC) system with multiple active RF-chains.

I. INTRODUCTION

FAS allows an antenna to change position and shape within a confined space, but its correlated channels make performance limits and practical port counts difficult to determine. The paper addresses these gaps with tractable approximations and a suboptimal port design.

  • Motivation: FAS is a promising 6G technology whose antenna can freely change position and shape within a given space.Its dynamic position and shape are intended to provide diversity and multiplexing gains.
  • Open problems: Strong spatial correlation among closely placed ports makes FAS channel PDF and CDF derivations intractable.Consequently, outage probability and diversity gain lack closed-form expressions.
  • Open problems: With one active RF-chain, increasing the number of ports has diminishing gains, but the satisfactory suboptimal port count remains unknown.Identifying that count matters because it can reduce implementation challenges and analysis complexity.
  • Distinctive features: Unlike traditional selection combining, FAS can provide infinitely many ports in limited space and freely switch its radiating element among them.The switching capability can be exploited to mitigate multi-user interference.
  • Paper contributions: The paper approximates FAS outage probability and diversity gain in closed form using a simple, accurate channel model that follows Jake’s spatial correlation.It also proposes a suboptimal FAS with N* ports and an algorithm to approximate N*.
  • Paper contributions: The proposed N* -port FAS is intended to achieve near-optimal performance with a minimal number of ports.A tolerance parameter ε_tol adjusts suboptimality; smaller ε_tol gives quantifiably near-optimal performance at higher cost.

II. SYSTEM MODEL

The system model considers a point-to-point FAS receiver with one RF-chain and N preset ports distributed along a linear space. Port channels are spatially correlated according to Jake’s model, and the receiver selects the port with the strongest channel magnitude.

  • System configuration: The point-to-point system uses a conventional transmitter and a fluid-antenna receiver with one RF-chain and N preset locations called ports.The ports are evenly distributed along a linear dimension of length Wλ.
  • Channel model: Closely packed ports exhibit strong spatial correlation modeled using Jake’s model and represented by correlation matrix J.The matrix can be decomposed as J = UΛU^H, with eigenvectors in U and eigenvalues in Λ.
  • Signal model: Only one port can be activated for communication, and the received signal at port n is modeled as y_n = h_n x + w_n.The channel coefficient is h_n, while w_n is additive white Gaussian noise.
  • Port selection: The globally optimal FAS port has the maximum channel magnitude, |h_FAS| = max{|h_1|,...,|h_N|}.The instantaneous received SNR and outage probability are then defined from this selected channel.

III. OUTAGE PROBABILITY AND DIVERSITY GAIN OF FAS

The paper derives tractable approximations for the correlated FAS channel distribution, outage probability, and diversity gain. These results expose how correlation and port count constrain diversity and motivate the proposed suboptimal design.

  • Distribution analysis: The correlated channel magnitude vector |h| is modeled as a correlated Rayleigh random vector whose PDF and CDF are approximated analytically.These approximations support subsequent outage-probability and diversity-gain derivations.
  • Asymptotic analysis: At high SNR, the paper gives a closed-form outage approximation and derives an approximate diversity gain for FAS.The high-SNR expression is further simplified using a Taylor-series approximation.
  • Outage probability: Theorem 1 expresses FAS outage probability as P{|h_FAS| < Ω} = F_|h|(Ω,...,Ω).The result evaluates the joint CDF of the channel magnitudes at a common threshold.
  • Interpretation: The closed-form interpretations reveal performance limits that cannot be directly obtained from the more complicated prior outage expression.This provides more direct insight into how port count and correlation affect FAS.
  • Diversity gain: FAS diversity gain is limited by min{N, N′}, so increasing N beyond N′ may not provide useful improvement.Here N′ is the numerical rank associated with the covariance matrix in the limiting fixed-W formulation.

IV. SUBOPTIMAL SOLUTION: FAS WITH N∗PORTS

The paper introduces N* as a reduced port count based on dominant eigenvalues of the correlation matrix, enabling tractable channel approximation while retaining near-optimal performance. The proposed N*-port FAS substantially improves on smaller configurations, whereas additional ports provide only marginal gains.

  • Channel approximation: The proposed suboptimal FAS uses N* ports to reduce the required number of ports while maintaining an approximation to the original channel distribution.The method is motivated by the numerical rank of the correlation matrix and its best low-rank approximation.
  • Port-number selection: N* is chosen as the smallest integer satisfying εtol ≥ εN*, using the dominant eigenvalues of the covariance or correlation matrix.The tolerance εtol controls sub-optimality: smaller values produce a closer-to-optimal design, while larger values permit less optimality.
  • Performance trade-off: For fixed W, FAS with N* ports yields considerable improvement over configurations with N < N* ports, while most configurations with N > N* ports provide marginal improvement over N*−1 ports.The paper therefore identifies N* as a practical operating point between performance and port count.
  • Correlation-matrix conditioning: When J is ill-conditioned, N* is usually smaller than the numerical rank, whereas N* equals the numerical rank when J is well-conditioned.This distinction reflects how eigenvalue concentration affects the number of ports needed for the approximation.
  • Near-singular correlation matrices: For near-singular J, channels with N ports can be approximated using N* ports for computation, although a small distributional gap may remain.The reduction removes nearly dependent entries and makes otherwise incalculable analytical results calculable.

V. RESULTS AND DISCUSSIONS

The results validate the outage-probability and distribution approximations, identify space, port count, and matrix conditioning as key performance factors, and show that a suboptimal port count can approach optimal performance. With one active RF-chain, suboptimal FAS outperforms SISO and SC, while MRC remains strongest but can have similar performance when W = 0.5.

  • Approximation accuracy: The analytical outage-probability expression is more accurate than the simplified approximation, though its complexity restricts computation to small N.The simplified expression remains useful despite inaccuracies from replacing powers of single integrals.
  • Performance factors: At high SNR, the approximation becomes accurate, and det(J) and the conditioning of J determine whether increasing N improves performance.A near-singular J prevents N from becoming important; increasing W or decreasing N can improve conditioning.
  • Port-count effects: The diversity gain is limited by min {N, N′}, so increasing ports beyond N′ provides no improvement in a point-to-point setting.The outage probabilities for N and N′ ports are the same when N > N′, while the earlier case is lower bounded by the latter.
  • Suboptimal FAS: FAS with N∗ ports significantly improves over N∗−1 ports, whereas N + 1 ports provide negligible improvement over N∗ ports.This supports using the proposed suboptimal FAS for efficient performance.
  • System comparisons: With one active RF-chain, suboptimal FAS outperforms SISO and SC, while MRC has the lowest outage probability; FAS and MRC achieve similar performance when W = 0.5.MRC’s advantage is attributed to using more active RF-chains, whereas FAS switches to the best port within a finite space.

VI. CONCLUSIONS

The paper derives closed-form approximations for FAS outage probability and diversity gain, identifies spatial correlation as a key performance factor, and proposes a port-selection scheme. The proposed suboptimal FAS improves over SISO and SC, approaches MRC under a half-wavelength space, but remains limited by its single active RF-chain.

  • The paper approximates FAS outage probability and diversity gain in closed-form expressions.
  • FAS performance strongly depends on the spatial correlation matrix J.
  • Increasing ports beyond N′ yields no diversity gain in a point-to-point setting and makes J ill-conditioned.For fixed N, increasing W or decreasing N can address the ill-conditioning.
  • The proposed scheme with N* ports provides a significant gain over FAS with N*−1 ports while nearly matching FAS with N*+1 ports.The approximation uses a fixed ε_tol and reflects diminishing gains from additional ports.
  • The proposed suboptimal FAS outperforms SISO and SC systems but falls behind MRC because it has a single active RF-chain.When W = 0.5, suboptimal FAS and MRC achieve similar performance.
  • The study leaves MIMO-FAS and MIMO performance for future work.

APPENDIX A: APPROXIMATED PDF OF |h|

Appendix A develops an approximated probability density function for the FAS channel magnitude by transforming amplitude-phase variables, introducing matrix G, and simplifying the resulting series.

  • The derivation introduces an N × N matrix G to approximate the PDF of |h|.The matrix is used with the angle representation to simplify integration.
  • The infinite series is truncated to a finite one for ease of computation.The appendix also invokes the binomial theorem and Cauchy product in the derivation.
  • The appendix derives the FAS channel magnitude PDF from amplitude-and-phase channel variables.
  • The appendix uses a mapping function and permutation-based terms to organize the series representation.
  • The derivation assumes t is sufficiently large and monotonically decreasing in each relevant variable.

APPENDIX B: APPROXIMATED CDF OF |h|

Appendix B obtains the cumulative distribution function of |h| by integrating the approximated joint magnitude density and organizing the resulting terms with auxiliary notation.

  • The CDF of |h| is obtained by integrating the approximated magnitude PDF over the channel magnitudes.
  • The derivation represents terms using indexed quantities such as s*_t, its sums, and entries of an auxiliary matrix.

APPENDIX C: OUTAGE PROBABILITY AT HIGH SNR

Appendix C derives the high-SNR outage probability from an asymptotic channel PDF, using amplitude-phase representations and Taylor-series approximations. It relates the resulting expression to the diversity behavior of FAS.

  • At high SNR, outage probability is derived from an approximation of the fading-channel PDF.
  • The FAS PDF is rewritten in amplitude and phase before applying the high-SNR approximation.
  • Taylor-series approximation around zero is used to solve an integral term in the outage-probability derivation.
  • Comparing the asymptotic expression with the general form gives M = N −1 and ξ = N.

APPENDIX D: DIVERSITY GAIN OF FAS

The appendix derives diversity gain behavior for FAS, showing gain N as W →∞ and an approximate limitation by min {N, N′} for finite-space channel correlations.

  • Unbounded space: When W →∞, the diversity gain of FAS is N.This result follows from M = N −1 in Appendix C.
  • High-SNR diversity gain: At high SNR, diversity gain can be obtained from the probability density function of the fading channels.The appendix assumes the high-SNR channel PDF can be approximated as in (37) and uses it to derive diversity gain.
  • Finite-space correlation: For finite W, the covariance matrix J may be nearly singular because adjacent ports can become highly correlated.The n-th and (n + 1)-th ports are considered within a finite space, and their joint CDF reduces to a singular case when the ports coincide in the limiting argument.
  • Finite-rank approximation: Using N′ finite ports to approximate an FAS with N ports yields diversity gain approximately limited by min {N, N′}.Here, N′ is the numerical rank of the covariance matrix J′ defined from the N →∞ fixed-W covariance structure.
  • Numerical treatment: Rank-revealing QR factorization or Gauss-Jordan elimination with a given tolerance can remove nearly dependent entries of J.These procedures are suggested for handling the covariance matrix's near-dependence.
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