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A FAS Channel Fitting Strategy Using Extreme Value Distributions for Accurate Outage Performance Evaluation
Rui Xu, Yinghui Ye, Guangyue Lu, Liqin Shi, Gan Zheng
TL;DR
Accurate outage-probability evaluation in FAS depends on characterizing the channel distribution, yet spatial correlation makes the exact distribution difficult to evaluate and global GEV fitting can miss the low-probability left tail. The paper combines GPD tail modeling with a GEV global fit and a left-tail-sensitive criterion. Simulations report improved left-tail fitting and OP accuracy in the low-outage region.
Problem
Spatial correlation makes the exact FAS channel distribution difficult to evaluate, while globally fitted GEV models may miss the low-probability left tail relevant to OP.
Source: §I · p. 1Method
The strategy combines a GPD model for the left tail with a GEV global model and a logarithmic-error MMSE criterion using uniform log-CDF sampling.
Source: §I · p. 2Results
The proposed strategy improves left-tail fitting and OP evaluation accuracy, and its errors are smaller than those of GEV ML in the reported comparisons.
Source: §V · pp. 5–6, §VI · p. 6Takeaways & Limitations
The results support fitting FAS channel distributions according to the target performance metric, with explicit attention to the left tail for OP evaluation.
Source: §VI · p. 6
Abstract
from arXiv · showhide
Modeling the channel in a single-antenna fluid antenna system (FAS) using extreme value distributions (EVDs) provides an accurate and tractable framework for FAS performance evaluation. When the objective of FAS channel fitting is outage probability (OP) evaluation, accurate characterization of the low-probability left-tail region becomes crucial, while existing fitting strategies that emphasize global fitting accuracy may fail to capture the critical tail behavior required for precise OP evaluation. In this paper, we propose an OP-oriented channel fitting strategy with a left-tail-sensitive target distribution and fitting criterion. Specifically, a combined EVD (CEVD) is introduced as the target distribution, where a generalized Pareto distribution (GPD) is employed to characterize the left tail and a generalized extreme value (GEV) distribution is used to model the global behavior. Furthermore, a modified mean-square-error (MMSE) criterion is developed, which employs logarithmic-domain errors to enhance sensitivity to left-tail discrepancies. Meanwhile, the evaluation points are constructed via uniform discretization on the logarithm of the cumulative distribution function, ensuring uniform sampling across all probability scales. This mitigates the under-representation of tail errors in the overall MMSE, which cannot be effectively addressed by error amplification alone due to the sparsity of tail samples. Simulation results demonstrate that the proposed fitting strategy significantly improves the OP evaluation accuracy in the ultra-low-OP regime.
I. INTRODUCTION
Accurate FAS outage-probability evaluation depends on the channel distribution, but spatial correlation makes the exact distribution difficult to handle. The paper proposes a left-tail-sensitive fitting strategy because globally accurate GEV fitting can miss low-OP behavior.
I. INTRODUCTION
- Spatial correlation among densely deployed FAS ports makes the exact channel distribution a multidimensional integral, complicating accurate, tractable OP evaluation.EVD-based fitting is attractive because it can provide relatively accurate OP expressions with low computational complexity.Source: §I · p. 1
- Below 10^-2, the ML-based GEV fit can deviate because global fitting and high-density samples underrepresent the left tail that dominates low-OP events.GEV captures asymptotic extreme-value behavior but does not necessarily model the left tail precisely.Source: §I · p. 1
- The proposed strategy combines a GPD left-tail target with a GEV global model and a logarithmic-error MMSE criterion using uniformly spaced log-CDF evaluation points.Uniform log-CDF sampling addresses sparse tail samples, which error amplification alone cannot adequately compensate for.Source: §I · p. 2
- Simulations show that the proposed OP-oriented strategy improves matching in the left tail and thereby improves OP evaluation accuracy.Source: §I · p. 2
II. SYSTEM MODEL AND CHANNEL FITTING FRAMEWORK
The system model selects the strongest channel envelope from a receiver's fluid antenna ports, and outage probability follows from that selected channel's distribution. Because port correlation makes the exact distribution intractable, the paper uses an EVD fitting framework and develops a left-tail-sensitive alternative to the prior GEV-ML strategy.
II. SYSTEM MODEL AND CHANNEL FITTING FRAMEWORK
- The receiver selects the maximum envelope across N preset antenna ports, and OP is the probability that this selected envelope does not exceed the normalized decoding threshold.The threshold is γth/γ̄, where γth is the decoding-SNR threshold and γ̄ is average transmit SNR.Source: §II · p. 2
- Dense port deployment creates spatial correlation across the channel vector, making the exact distribution of the selected envelope mathematically intractable.Source: §II · p. 2
- The EVD fitting procedure generates Monte Carlo samples, estimates their empirical distribution, and fits a target distribution's parameters using a selected criterion.Source: §II · p. 2
- Because the prior ML-based GEV fit may miss left-tail behavior and produce inaccurate low-OP estimates, this paper jointly designs a left-tail-sensitive target and criterion.Source: §II · p. 2
A. The CEVD
The CEVD uses complementary GEV and GPD components to fit the FAS channel envelope globally and in its low-probability left tail. Its threshold is set at a CDF probability of 10^-2 to target the practical low-OP region.
A. The CEVD
- For values below u, the lower-tail probability is formed from the probability of falling below u and the conditional probability of the lower-tail excess.The excess is Y = u − X given X < u, and its conditional distribution is approximated by a GPD.Source: A. The CEVD · p. 2
- Because the selected FAS channel is a maximum-type random variable, GEV provides a baseline for its global distribution while GPD targets the tail relevant to ultra-low outages.Source: A. The CEVD · p. 2
- The CEVD retains GEV as the baseline for the overall maximum channel envelope and uses GPD to model lower-tail excess below threshold u.It is an EVT-inspired fitted distribution for the FAS channel, not a strict asymptotic EVT law applied directly to correlated port channels.Source: A. The CEVD · pp. 2–3
- The CEVD joins the GPD-based left-tail expression to the GEV distribution at u, and continuity follows because the GPD CDF at zero is zero.Source: A. The CEVD · p. 3
- The threshold u is chosen where the GEV CDF equals 10^-2, matching the practical focus on OP below 10^-2 while retaining GEV fitting above it.The paper notes that threshold sensitivity is analyzed later and that classical threshold-selection methods add numerical and modeling complexity.Source: A. The CEVD · p. 3
B. The MMSE Criterion
The MMSE criterion is designed to make FAS channel fitting more sensitive to the left tail that matters for low-OP evaluation. It combines logarithmic-domain errors with evaluation points uniformly discretized on the log-CDF axis.
B. The MMSE Criterion
- Unlike sample-driven ML, whose abundant high-density samples dominate fitting, MMSE explicitly targets outage-relevant low-probability regions.The paper identifies ML's implicit emphasis on the sample distribution as a challenge for fitting the low-probability region.Source: B. The MMSE Criterion · p. 3
- Standard MSE can underweight low-OP discrepancies because tail CDF differences are small and tail samples sparse; amplifying errors alone may not overcome sparse sampling.The proposed criterion addresses both the magnitude of tail errors and the effective participation of tail samples.Source: B. The MMSE Criterion · p. 3
- MMSE combines logarithmic-domain CDF errors with uniformly discretized log-CDF evaluation points to increase the left tail's representation in fitting.The criterion uses logarithmic errors to amplify tail discrepancies and log-CDF sampling to represent probability scales comparably.Source: B. The MMSE Criterion · pp. 3–4
- Uniform discretization on the log-CDF axis gives each probability order of magnitude a comparable number of evaluation points.This construction increases the ultra-low-probability region's contribution without letting a few rare samples dominate parameter fitting.Source: B. The MMSE Criterion · pp. 3–4
IV. THE FAS CHANNEL DISTRIBUTION FITTING AND OP EVALUATION
The section combines a GPD fit for the left tail with an existing GEV fit for the global distribution. It then uses the fitted channel distribution to evaluate outage probability.
IV. THE FAS CHANNEL DISTRIBUTION FITTING AND OP EVALUATION
- The CEVD uses the GEV for global behavior and a MMSE-fitted GPD for the left tail below threshold u.The section reuses the previously obtained ML-based GEV fit and focuses on fitting the GPD parameters.Source: §IV · p. 4
- The GPD fitting procedure uniformly discretizes logarithmic CDF levels, maps them to sample-domain evaluation points, and minimizes MMSE over those points.It uses Monte Carlo samples and an empirical distribution to support the fitting procedure.Source: §IV · p. 4
- The GPD parameters are generalized as linear combinations of monomials over probabilities 10^-6 to 10^-2, W from 0.5 to 5, and ρ from 0.05 to 0.5.Here, ρ = W/(N − 1) is the physical spacing between adjacent ports in wavelength units.Source: §IV · p. 4
- Parameter generalization introduces minor approximation errors with negligible OP impact within the fitted parameter range; precise lookup tables can support discrete configurations.The lookup-table option is described for standardized W and N values.Source: §IV · p. 4
- OP is evaluated from the fitted channel distribution by setting x = γ̂, and the approximation is assessed through OP analysis because small parameter deviations may affect ultra-low-probability OP.The fitted parameter expressions assume unit-average-power small-scale fading, while the distribution remains applicable to arbitrary fading powers through scaling.Source: §IV · p. 5
V. NUMERICAL RESULTS
Across the reported settings, CEVD MMSE improves left-tail fitting and OP evaluation relative to GEV ML, including at high SNR and low decoding thresholds.
V. NUMERICAL RESULTS
- The MMSE remains relatively low and stable around FGEV(u) = 10^-2, supporting that threshold choice for fitting.The reported MMSE first decreases and then increases as FGEV(u) grows.Source: §V · p. 5
- CEVD MMSE mitigates GEV ML deviations in the ultra-low-probability CDF region across the reported W values and N = 10 and N = 20.The reported parameter-generalization approximation errors have negligible impact on fitting accuracy.Source: §V · p. 5
- For both moderate- and dense-density port deployments, CEVD MMSE eliminates GEV ML inaccuracies in the high-SNR region.The comparison uses γth = 10 dB.Source: §V · pp. 5–6
- CEVD MMSE has smaller log-domain OP errors than GEV ML in the specific and averaged settings reported for Fig. 4.The averages cover W ∈ [0.5, 5] and ρ ∈ [0.05, 0.5].Source: §V · p. 6, Figure 4 · p. 6
- At low decoding thresholds, CEVD MMSE achieves more accurate OP evaluation than the compared fitting strategy.Fig. 5 reports approximate OPs and log-domain absolute error versus γth with average transmit SNR ¯γ = 10 dB.Source: §V · p. 6, Figure 5 · p. 6
VI. CONCLUSIONS
The paper proposes a left-tail-focused FAS channel fitting strategy and reports improved fitting and OP evaluation accuracy in the low-outage region.
VI. CONCLUSIONS
- The proposed strategy combines a CEVD target distribution with an MMSE criterion, both tailored to emphasize the left-tail region.Source: §VI · p. 6
- The strategy improves FAS channel fitting accuracy in the left tail and OP evaluation accuracy in the low-outage region.Source: §VI · p. 6
- The conclusion highlights fitting strategies designed around the target performance metric as important for performance-oriented channel modeling.Source: §VI · p. 6