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.