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

Abstract

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.

Figures (10)

• Figure 1: (a) Spatial correlation functions: Jakes model $J_0(2\pi\delta)$ versus the Gaussian approximation $e^{-\pi^2\delta^2}$. (b) Pointwise approximation error and the fourth-order bound $\pi^4\delta^4/4$ from Proposition \ref{['prop:error_bound']}.
• Figure 2: Power spectral density comparison. The Jakes PSD has compact support on $[-1,1]$ with singularities at $|f|=1$, while the Gaussian PSD $S_G(f) = \frac{1}{\pi}e^{-f^2}$ extends to all frequencies with approximately 15.7% spectral leakage beyond $|f|=1$.
• Figure 3: (a) Normalized eigenvalues $\lambda_k/N$ of the Jakes and Gaussian correlation matrices ($N=50$, $W=3$). The Gaussian eigenvalues exhibit smoother decay while the Jakes eigenvalues show a sharper cutoff. (b) Cumulative energy captured by the first $K$ eigenmodes.
• Figure 4: Outage probability versus average SNR for $N=10$ ports and aperture $W=1$ with $\gamma_{\mathrm{th}}=0$ dB. Monte Carlo simulations (markers) validate the KL rank-1 closed form \ref{['eq:pout_rank1_G']}, rank-2 semi-closed form \ref{['eq:pout_rank2_G']}, Slepian upper/lower bounds \ref{['eq:outage_sandwich']}, and the continuous-aperture formula \ref{['eq:pout_closed']}.
• Figure 5: Effective degrees of freedom $K_{\mathrm{eff}}$ versus normalized aperture $W$ ($N=200$). The numerical participation ratios (markers) closely match the asymptotic predictions: $2W+1$ for Jakes and $\pi\sqrt{2}\,W \approx 4.44W$ for Gaussian, confirming Theorem \ref{['thm:dof_gauss']}.

Theorems & Definitions (39)

• Proposition 1: Second-Order Matching
• proof
• Proposition 2: Approximation Error
• proof
• Remark 1: Practical Implication
• Proposition 3: Spectral Densities
• proof
• Remark 2: Spectral Leakage
• Remark 3: Eigenvalue Decay Rate
• Remark 4: Comparison with Jakes Rank-1