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

Tuo Wu

Abstract

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.

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

Abstract

Fluid antenna systems (FAS) achieve spatial diversity by dynamically switching among 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 -dimensional correlated FAS channel into independent eigenmodes and performs a controlled rank- truncation, reducing the outage analysis to a -dimensional integration with . 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 . 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 eigenmodes are needed regardless of , where 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.
Paper Structure (25 sections, 8 theorems, 62 equations, 7 figures, 2 tables)

This paper contains 25 sections, 8 theorems, 62 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Signal Model
  4. FAS Channel Correlation Model
  5. Outage Probability
  6. Karhunen-Loève Expansion of the FAS Channel
  7. Eigendecomposition of the Correlation Matrix
  8. KL Representation of the Channel
  9. Low-Rank Truncation
  10. Outage Probability via KL Expansion
  11. General K-Dimensional Integral Formulation
  12. Gauss-Hermite Quadrature Evaluation
  13. Special Case: Rank-1 Approximation (K=1)
  14. Special Case: Rank-2 Approximation (K=2)
  15. Ergodic Rate Under KL Expansion
  16. ...and 10 more sections

Key Result

Proposition 1

Let $F_{\max}(x)$ and $\tilde{F}_{\max}^{(K)}(x)$ denote the CDFs of $\max_n |g_n|^2/\eta$ under the full and $K$-truncated KL expansions, respectively. The fraction of total channel power lost due to truncation is bounded as

Figures (7)

  • Figure 1: Entropy fraction $h(\tilde{\mathbf{g}}^{(K)})/h(\mathbf{g})$ (solid) vs. power fraction $1 - \varepsilon_K$ (dashed) as a function of the number of retained modes $K$, for $N = 40$ and different apertures $W$. Vertical lines mark the theoretical prediction $K^* = 2\lceil W \rceil + 1$ from Theorem \ref{['thm:dof']}.
  • Figure 2: Ergodic capacity $\bar{C}$ vs. average SNR $\bar{\gamma}$. Markers: Monte Carlo simulation ($2\times 10^5$ trials); solid gray line: rank-1 closed-form \ref{['eq:erg_rank1']}. The analytical curve passes through the $K = 1$ MC markers, validating the theoretical result. $N = 20$, $W = 3$.
  • Figure 3: Gaussian rate-distortion curves for the true channel (solid), BCM ($D = 4$, dash-dot), and i.i.d. (dotted) models. Blue squares mark the KL truncation operating points $(\theta = \lambda_{K+1})$, which lie exactly on the true R-D bound. $N = 20$, $W = 3$.
  • Figure 4: Outage probability $P_{\mathrm{out}}$ vs. average SNR $\bar{\gamma}$ for FAS with $N = 20$, $W = 3$, and $\gamma_{\mathrm{th}} = 0$ dB. The percentage in parentheses indicates the fraction of total channel power captured by the $K$-mode KL truncation.
  • Figure 5: KL truncation error $\varepsilon_K = 1 - \sum_{k=1}^{K}\lambda_k/N$ vs. the number of retained modes $K$ for different apertures $W$ and port numbers $N$. The red dashed line indicates $\varepsilon_K = 1\%$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Remark 1: Analytical Challenge
  • Remark 2: Physical Interpretation of Eigenmodes
  • Proposition 1: Truncation Error Bound
  • Remark 3: Choosing $K$
  • Remark 4: Complexity Comparison
  • Remark 5: Interpretation of Rank-1
  • Theorem 1: Conservative Outage Bound
  • Corollary 1: Monotone Convergence
  • Corollary 2: Ergodic Capacity Lower Bound
  • Remark 6: Design Implication
  • ...and 9 more