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 $\ddot{I}{}_G(\hat{I}´)=e^{-\ddot{I}{€}^2\hat{I}{´}^2}$. 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 $K_{\mathrm{eff}}^G\approx \ddot{I}€\sqrt{2}W$. 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 $\ddot{I}{}_{\min }$ and $\ddot{I}{}_{\max }$, 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 $P_{\mathrm{out}}\approx 1-e^{-x}(1+\ddot{I}€\sqrt{2}Wx)$ 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 $\hat{I}{»}_2=2\ddot{I}{€}^2$. A Pickands-constant refinement for the deep-outage regime and a threshold-dependent effective diversity $N_{\mathrm{eff}}\approx 1+\ddot{I}€\sqrt{2}Wx$ are further derived. Numerical results confirm that the Gaussian approximation incurs less than 10\% relative outage error for $W\le 2$ and that the continuous-aperture formula converges with as few as $N\approx 10W$ ports.