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Certified Outage Dimensioning of Fluid Antenna Receivers: Required SNR, Aperture, and Port Count

Yuanjun Shen

arXiv:2610.13094v1eess.SPcs.ITmath.IT

TL;DR

Receiver dimensioning must determine the SNR or aperture–port resources needed for a target outage, while approximation error can affect whether a reliability target is met. The paper uses probability brackets and certified design decisions; the supplied results include narrow required-SNR intervals for low-port receivers, with a limitation for some lower models at very small thresholds.

  • Problem

    Approximation uncertainty can change whether a receiver configuration meets a target outage, especially at stringent targets such as 10^-5.

    Source: §I · p. 1

  • Method

    The paper uses probability brackets and certified classifications of feasible, infeasible, or unresolved candidates to support receiver dimensioning.

    Source: §I · p. 1, D. Certified Link-Design Quantities · p. 5

  • Results

    Low-port receivers have required-SNR widths of 0.10–0.20 dB, while dense receivers retain larger sampling costs.

    Source: §VI · p. 10

  • Takeaways & Limitations

    Pairwise correlation magnitudes alone do not specify physical-port outage, so the bounds retain covariance information through PSD covariance order.

    Source: B. Why Physical-Port Geometry Matters · p. 12

    If the stated ratio exceeds 1 + ϵ, the lower model cannot support that relative-error budget for all sufficiently small thresholds.

    Source: A. PSD Comparison and Its Tail… · p. 3

Abstract

from arXiv · show

We study the SNR, aperture, and port count needed to meet a target outage in a single-user downlink with a fixed transmit antenna and a single-RF fluid antenna receiver. For known spatially correlated Rayleigh channels and ideal port selection, a sampling-free residual certificate controls reduced-rank outage error over a continuous SNR interval. Positive Bessel features support large arrays, and bounded importance sampling converts the certificate into simultaneous design intervals. At 1 bit/s/Hz and outage $10^{-5}$, a four-wavelength, ten-port receiver requires 4.40--4.55 dB. Reducing its port count from 256 to ten costs 0.70--1.25 dB under a joint 95\% probability statement. At 3.5 wavelengths, ten ports require 4.90--5.05 dB. Integer-port screening across 3--8 dB shows how aperture and port requirements change with available SNR; an effective-SNR transformation extends these decisions to other rates. The reported intervals include model and sampling uncertainty under checked Gaussian covariance inputs. A constructive slow-growth rank bound and a 16--1024-port study support compact evaluation as port count grows.

I. INTRODUCTION

Receiver dimensioning asks what SNR or aperture–port resources meet a target outage, where approximation uncertainty can change feasibility. The paper develops certified outage intervals and a design map for ideal port selection in a spatially correlated single-user link.

II. SYSTEM MODEL AND APPROXIMATION OBJECTIVE

  • The link uses one fixed transmit antenna and one active RF chain, with ideal selection of the strongest among N spatially correlated ports across an aperture W.The joint port-fading law matters because outage occurs when all candidate ports are weak.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • At N = 12, W = 0.5, a block-model estimate and full-covariance reference fall on opposite sides of a 10^-3 target, motivating probability brackets for feasibility decisions.The estimates are about 5.2 × 10^-4 and 1.8 × 10^-3, respectively.Source: §I · p. 1

A. Question and Contribution

  • The paper frames the outcomes as required-SNR intervals and certified feasible, infeasible, or unresolved aperture–port candidates, with a resource frontier when relevant candidates are resolved.It also develops a residual certificate, positive Bessel features for large arrays, and fixed experiments relating aperture and port choices to power cost.Source: A. Question and Contribution · pp. 1–2, §I · p. 1

B. Organization and Notation

  • The paper’s roadmap assigns the channel and design objective to Section II, the residual certificate and design inversion to Section III, and numerical intervals and design results to Sections IV and V.It states that bold capitals and lowercase letters denote matrices and vectors, and defines A ⪯ B as B − A positive semidefinite.Source: B. Organization and Notation · p. 2

II. SYSTEM MODEL AND APPROXIMATION OBJECTIVE

  • The residual criterion controls relative outage error, unlike an average channel-energy rule; rescaling the reduced covariance to unit diagonal invalidates the stated covariance-order guarantee.The paper distinguishes computational rank from the physical port count.Source: §II · p. 2

A. Receiver-Dimensioning Problems

The design problems are to bracket required SNR for a specified geometry or classify finite aperture–port candidates at an available SNR. PSD comparison provides outage bounds, but its lower model can lose accuracy at sufficiently small thresholds.

A. Receiver-Dimensioning Problems

  • For a specified (W, N), target outage p, and threshold θ, P1 brackets required SNR; P2 classifies finite candidates and identifies feasible candidates not dominated in both resources.For other rates, the effective-SNR transformation maps a decision to the specified rate and available SNR.Source: A. Receiver-Dimensioning Problems · p. 2
  • The feasible resource frontier is defined within the finite candidate set, and unresolved candidates must remain under consideration before declaring an optimum.Each geometry is evaluated with its own covariance and certificate, including nonnested port grids.Source: A. Receiver-Dimensioning Problems · p. 2, Figure 1 · p. 3

A. PSD Comparison and Its Tail Limitation

  • PSD comparison brackets the full outage using lower and upper models, with the bounds coinciding at full rank.Anderson’s inequality supplies the comparison; singular covariances are handled by adding δI and taking δ ↓ 0.Source: A. PSD Comparison and Its Tail… · p. 3
  • For positive definite R, the lower majorant can lose residual directional structure, and if its stated ratio exceeds 1 + ϵ it cannot support that relative-error budget for all sufficiently small thresholds.The cited small-ball behavior is FR(x) ∼ x^N / det R.Source: A. PSD Comparison and Its Tail… · p. 3

B. A Sampling-Free Multiplicative Certificate

A residual-variance certificate bounds relative outage error for a reduced-rank Gaussian KL model without probability integration. A witness at one threshold extends the model-error guarantee across a continuous SNR range for fixed covariance.

B. A Sampling-Free Multiplicative Certificate

  • Writing the channel as h = Arz + e makes the reduced outage probability depend on the retained KL component, while e is an independent Gaussian residual.The residual covariance is Er; the retained component uses z ∼ CN(0, Ir).Source: B. A Sampling-Free Multiplicative… · p. 3
  • Theorem 1 uses residual variances to certify a relative bound between reduced and full outage probabilities.Any witness t satisfying ar(t, x) ≥ (1 + ϵ)^-1 certifies (Ur − FR)/FR ≤ ϵ.Source: B. A Sampling-Free Multiplicative… · p. 3
  • The proof combines independent retained-component and residual events with a Gaussian tail union bound, alongside the upper bound from Anderson’s inequality.The lower-bound argument does not require independence across residual ports.Source: B. A Sampling-Free Multiplicative… · pp. 3–4
  • A feasible scalar witness need not be globally optimal; the implementation searches a fixed logarithmic/linear grid with local refinement and evaluates the coefficient at 80 decimal digits.Numerical-input controls are detailed in Section V-A.Source: B. A Sampling-Free Multiplicative… · p. 4
  • For fixed rank and witness, passing at xmin guarantees the same model-error budget for every x ≥ xmin, giving a continuous SNR-range guarantee at fixed covariance.The sufficient rank can increase because of the union bound and Gaussian dilation; evaluation cost also depends on the estimator.Source: B. A Sampling-Free Multiplicative… · p. 4

C. Large-Port Evaluation with Positive Features

Positive-feature approximations represent the Jakes covariance as a positive semidefinite retained component plus a controlled residual, enabling certificates without dense covariance eigenvalues. The construction also gives a sufficient rank growing as O(log log N) under fixed conditions.

C. Large-Port Evaluation with Positive Features

  • A positive feature approximation R = AAH + E, with E ⪰ 0, enables the residual certificate without computing tiny eigenvalues of a dense covariance matrix.The theorem uses UA = FAAH and residual diagonal entries vi = Eii.Source: C. Large-Port Evaluation with Positive… · p. 4
  • For centered line positions, Neumann’s addition formula yields Bessel features whose omitted covariance is positive semidefinite and has a uniform diagonal-tail bound.The bound follows from a Bessel-function bound and a geometric tail condition when m + 1 > b.Source: C. Large-Port Evaluation with Positive… · p. 4
  • The thin SVD of the finite feature matrix supplies retained modes, and the residual bound adds retained-mode residuals to the omitted-feature term.Both components are nonnegative, avoiding subtraction of nearly unit retained energy from Rii = 1.Source: C. Large-Port Evaluation with Positive… · p. 4
  • An explicit sufficient test sets tϵ and ηϵ from the target error ϵ and rank r, then certifies qr ≤ ηϵ and ar ≥ (1 + ϵ)^-1.Direct retention can use dr(W) as a residual bound; experiments use numerical residuals from (18).Source: C. Large-Port Evaluation with Positive… · p. 4
  • The marginal union-bound certificate can become vacuous at fixed rank and x as N grows when a fixed rank leaves positive residual variance at a linear number of ports.An N-uniform certificate would require control of the continuous residual process.Source: C. Large-Port Evaluation with Positive… · p. 4
  • At fixed W, xmin, and ϵ under the exact Jakes model, direct Bessel-feature retention admits a sufficient rank of order O(log log N) for every x ≥ xmin.The constants depend on W, xmin, and ϵ.Source: C. Large-Port Evaluation with Positive… · p. 4
  • The experiments use SVD, while direct-feature and SVD constructions can be certified at the same threshold and tolerance before choosing the one with fewer latent coordinates.For m ≪ N, feature SVD costs O(Nm^2), residual preprocessing O(Nm), and one witness evaluation O(N).Source: C. Large-Port Evaluation with Positive… · p. 4

D. Certified Link-Design Quantities

Monotonicity turns simultaneous outage-probability brackets into certified SNR intervals and feasibility classifications over a declared receiver-design map. Unresolved candidates remain relevant when identifying the resource frontier.

D. Certified Link-Design Quantities

  • When both threshold sets are nonempty, monotonicity bounds the least SNR meeting the outage target using the first endpoint that fails and the second that passes.An empty set leaves its corresponding endpoint unresolved.Source: D. Certified Link-Design Quantities · p. 5
  • Monotonicity converts grid evaluations into bounds on the continuous design quantity, while model error, sampling uncertainty, and grid spacing affect interval width.A local dB-width estimate additionally assumes an approximately constant outage slope d.Source: D. Certified Link-Design Quantities · p. 5
  • For any rate threshold θ > 0, the outage quantile gives γreq(θ) = θγreq(1), shifting every SNR bracket by 10 log10 θ dB.The same effective-SNR design decisions therefore extend to other rates.Source: D. Certified Link-Design Quantities · p. 5
  • A grid of simultaneous brackets supports certified feasible, certified infeasible, or unresolved classifications for each aperture–port candidate at available SNR.The classifications are ud ≤ p, ℓd > p, and ℓd ≤ p < ud, respectively.Source: D. Certified Link-Design Quantities · p. 5
  • Nondominated certified-feasible candidates form a safe resource frontier; it is the true frontier within D only if every candidate is resolved.Unresolved candidates must be considered before declaring an optimum.Source: D. Certified Link-Design Quantities · p. 5

IV. RELIABLE EVALUATION AND NUMERICAL CERTIFICATION

Importance sampling and bounded finite-sample intervals provide numerical outage brackets, with simultaneous coverage across declared estimates and checks on added sampling uncertainty. The procedure distinguishes these sampling bounds from the sampling-free model-error certificate.

A. Gaussian Exponential Tilting

  • Ordinary Monte Carlo poorly resolves probabilities near 1/n, so Gaussian exponential tilting is used for rare-outage evaluation.The production tilt is frozen after a pilot; conditional on that pilot, production samples are iid and use an analytical weight bound.Source: A. Gaussian Exponential Tilting · p. 5
  • A deterministic eigenvalue-based tilt places typical proposed energy near the event boundary, while a second rule minimizes the analytical cap to reduce the range contribution to interval width.The implementation also balances variance and range terms using an independent pilot.Source: A. Gaussian Exponential Tilting · p. 5

B. Simultaneous Finite-Sample Brackets

  • Empirical Bernstein bounds applied to normalized bounded weights produce probability intervals, and a union bound gives simultaneous coverage for a prespecified family without requiring independence between family members.Samples within each estimate must satisfy the stated iid assumption.Source: B. Simultaneous Finite-Sample Brackets · p. 5
  • Allocating failure probability across thresholds and declared sample-count looks covers potential looks and permits stopping by a declared width or feasibility rule.Pilot-selected thresholds and tilts are allowed by conditioning on the pilot.Source: B. Simultaneous Finite-Sample Brackets · pp. 5–6
  • Sampling uncertainty can exceed the sampling-free model-error budget, and the guarantee assumes exact weight evaluation.A zero lower endpoint leaves relative accuracy unresolved but may still allow a feasibility decision if u ≤ p.Source: B. Simultaneous Finite-Sample Brackets · p. 6
  • Refinement must remain within the declared coverage family or receive an additional failure budget; more samples alone need not overcome model-error or threshold-grid limits.If no covered refinement remains, the decision stays unresolved.Source: B. Simultaneous Finite-Sample Brackets · p. 6

V. NUMERICAL RESULTS

The numerical study combines simultaneous design intervals with compact certified models for large port counts. Its rank comparisons use distinct outage-error, channel-energy, and effective-mode criteria.

A. Protocol, Controls, and Reproducibility

  • The scalability protocol tests 490 decisions across N = 16–1024, W = 0.5–8, seven SNRs, and 5% and 20% model budgets.Positive Bessel features use m = max{40, ⌈πW⌉+40}; their omitted diagonal bound is at most 8.73 × 10^-41, and the added residual-variance cushion changes none of the selected ranks.Source: A. Protocol, Controls, and… · p. 6
  • The reported intervals share one 95% event across four families, including the broad study, dense-array refinements, and low-port designs.The last completed look is covered by a union over prescribed looks; selected witnesses and numerical checks use high-precision recomputations, while rigorous interval-arithmetic certification of covariance and SVD inputs remains open.Source: A. Protocol, Controls, and… · p. 6
  • At W = 4, N = 256, the 20%, 5%, and 1% model budgets select ranks 15, 16, and 17 at 12 dB; for (4, 10), all select rank ten.These budgets are numerical tolerances, not reliability specifications.Source: A. Protocol, Controls, and… · p. 6

B. Large-Port Rank and Residual Scaling

  • At fixed aperture, added modes rapidly reduce residual variance while port density has a smaller effect on this grid; Corollary 1 separately guarantees slowly growing sufficient ranks.At N = 1024 and 30 dB, rank-to-K* ratios are 8/3 for W = 1 and 26/17 for W = 8.Source: B. Large-Port Rank and Residual Scaling · p. 6, Figure 2 · p. 7
  • At N = 1024, W = 8, the 30-dB sufficient rank is 26, versus 18 by the 99%-energy criterion and 17 by the effective-mode count.These counts target relative outage error, retained channel energy, and simplified diversity description, respectively.Source: B. Large-Port Rank and Residual Scaling · p. 6, Figure 2 · p. 7

C. Required SNR and the Cost of Reducing Port Count

The results quantify required SNR for low-port receivers, map aperture and port choices against available SNR, and compare outage accuracy with energy-based rank selection. The design map also shows that feasibility depends on operating point and rate.

C. Required SNR and the Cost of Reducing Port Count

  • At 10^-5 outage, reducing W = 4 from 256 to ten ports changes required SNR from 3.30–3.70 dB to 4.40–4.55 dB, a 0.70–1.25-dB power cost.At W = 3.5, ten ports require 4.90–5.05 dB, a 1.20–1.75-dB cost relative to the same dense four-wavelength reference.Source: C. Required SNR and the Cost of… · p. 7

D. Aperture-to-SNR Design Curve

  • For p = 10^-5, the fixed-density curve gives required-SNR intervals of [17.6,17.8] dB at W = 0.5, [12.3,12.5] at W = 1, [7.4,7.6] at W = 2, and [3.4,3.7] at W = 4.The curve uses N = ⌈10W⌉+1, a dense reference rather than a minimum-port prescription; W = 8 has a 1.10-dB-wide interval at 10^-5 after reaching the sample limit.Source: D. Aperture-to-SNR Design Curve · pp. 7–8

E. Diversity References and the SNR–Aperture–Port Map

  • At 5 dB, ten is the certified minimum at each tested aperture from 3.5 to 8; at 8 dB, seven suffice from 2.5 to 8.The screening tests integer port counts 1–24 across ten apertures and available SNRs 3, 4, 5, 6, and 8 dB; the cap is a finite search budget, and each reported minimum excludes every smaller integer individually.Source: E. Diversity References and the… · p. 8
  • At 3 dB, all tested counts fail for W ≤ 4, while W = 6 needs fourteen ports and W = 8 needs fourteen or fifteen.At 5 dB the nondominated certified-feasible candidates are (3.125, 19), (3.25, 14), and (3.5, 10), though unresolved lower counts at the first two apertures may improve that subset frontier.Source: E. Diversity References and the… · p. 8
  • At R0 = 2 bit/s/Hz, 5 dB is equivalent to 0.229 dB at the reference rate, and all tested port counts fail at each tested aperture for p = 10^-5.Changing rate shifts P1 required-SNR endpoints by 10 log10(2^R0 −1) dB.Source: E. Diversity References and the… · p. 8

F. Energy-Based Rank Selection and Outage Accuracy

  • For N = 12, W = 0.5, the 99%-energy rule selects rank three; energy-model/full-outage ratios are 1.015, 1.051, and 1.169 at 5, 10, and 15 dB.For N = 64, W = 2, the six-mode energy approximation has a ratio of 1.177 at 5 dB; energy retention alone does not quantify the outage-error margin.Source: F. Energy-Based Rank Selection and… · p. 8

G. Independent-Mode and Block-Model Comparisons

The section compares reduced-model and block-model estimates, tests finite-angle scattering and computational controls, and clarifies the assumptions and boundaries of the design conclusions.

G. Independent-Mode and Block-Model Comparisons

  • Fig. 5 plots EDoF/WIM ratios against reduced-model point estimates for (N=64, W=2) and (N=256, W=4), with ratios describing agreement with the reference.The residual certificate separately quantifies uncertainty in that reference.Source: Figure 5 · p. 9, G. Independent-Mode and Block-Model… · p. 9
  • For N=12, W=0.5, BCM/full ratios were 0.928, 0.290, and 6.66 × 10^-6 at 0, 10, and 20 dB under the stated calibration.The comparison fixes eigenvalue threshold one and µ2=0.97; the reported discrepancies apply to that calibration.Source: G. Independent-Mode and Block-Model… · p. 9

H. Scattering Sensitivity and Consistency Controls

  • For N=256, W=4, certified ranks for a 20% model budget ranged from (14,14,15,16) at κ=0 to (12,13,14,15) at κ=8 across 0–30 dB.These are specified finite-angle Gaussian models; a 3 × 3 isotropic square needs rank eight at 0 and 20 dB.Source: H. Scattering Sensitivity and… · p. 9
  • At N=12, W=0.5, the common 20% output-accuracy experiment took 0.332 s for direct sampling and 0.359 s for residual screening plus sampling.The timings exclude shared factorization and I/O; savings depend on workload and estimator.Source: H. Scattering Sensitivity and… · p. 9

VI. DISCUSSION AND RESEARCH SCOPE

  • The analysis assumes specified centered Gaussian covariance and ideal instantaneous selection; implementation patterns and coupling require effective covariance, and low rank does not determine exact high-SNR diversity order.Integer-port minima apply to tested apertures and uniform placement; application-specific placement constraints require screening another candidate set.Source: §VI · pp. 9–10
  • Low-port required-SNR intervals are 0.10–0.20 dB wide, while dense receivers retain larger sampling costs and eight-wavelength intervals motivate tighter bounds.The constructive rank result supports compact latent models as port count grows at fixed aperture; the operating map separately quantifies physical resources.Source: §VI · p. 10

VII. CONCLUSION

  • At outage 10^-5 and 1 bit/s/Hz, ten ports at W=4 require 4.40–4.55 dB, costing 0.70–1.25 dB versus 256 ports.These are specified points on the SNR–aperture–port map for known Gaussian covariance and ideal selection.Source: §VII · p. 10

APPENDIX A POSITIVE DEFINITENESS OF THE FINITE JAKES MATRIX

The appendix establishes positive definiteness for finite distinct-port Jakes matrices and develops small-threshold, low-dimensional, and residual-rank bounds for outage evaluation.

APPENDIX A POSITIVE DEFINITENESS OF THE FINITE JAKES MATRIX

  • For any fixed finite set of distinct ports, the finite Jakes matrix is positive definite even when its smallest eigenvalues are extremely small.The proof uses continuity, analytic continuation, and a Vandermonde system formed from derivatives of the exponential polynomial.Source: Appendix A · p. 10

APPENDIX B SMALL-THRESHOLD BEHAVIOR OF THE TILTED

  • For fixed N and positive definite covariance, the tilted estimator has bounded relative error as x decreases to zero, with limiting factor approximately 1.544^N.The per-coordinate factor is minimized at c* ≃ 1.5936; the limiting factor depends exponentially on N.Source: Appendix B · p. 10

A. A Low-Dimensional Integral

  • The low-dimensional integral evaluates the outage bound over r complex variables, or 2r real variables, and eliminating latent phases without an invariance argument changes the event.At r=0, the bound reduces to (1 − e^(-x/λ1))^N when λ1>0, whereas U0=1; diagonal residual majorants may tighten the isotropic choice.Source: A. A Low-Dimensional Integral · p. 10

B. Outage-Relative Rank Rule

  • A sufficient rank can be conservative when τrI overstates structured residuals, and it is evaluated at the operating threshold x.The residual-amplitude bound holds for correlated residual ports and can be combined with Lr by taking their maximum.Source: B. Outage-Relative Rank Rule · p. 11

C. Outage-Capacity Conversion

The capacity conversion uses outage-probability endpoints to bound ideal-selection outage capacity. The appendix gives a finite-threshold small-ball identity and estimator, while noting that polydisc sampling can be inefficient outside the small-threshold regime.

C. Outage-Capacity Conversion

D. Exact Polydisc Identity

  • Uniform polydisc sampling of the finite-threshold identity has relative variance tending to zero as x decreases to zero for fixed R, but can be inefficient before that regime.The identity integrates the complex Gaussian density over the scaled unit polydisc; its finite-threshold form matters when small eigenvalues make the asymptotic condition demand very large SNR.Source: D. Exact Polydisc Identity · p. 11

APPENDIX D RANK, DIVERSITY, AND PHYSICAL-PORT GEOMETRY

The appendix distinguishes diversity behavior across rank and threshold limits and shows why outage depends on physical-port geometry, not covariance spectrum alone.

A. Diversity and the Order of Limits

  • For Rδ=Rr+δI, every δ>0 gives diversity N, whereas δ=0 gives diversity r, so rank and diversity-order limits do not commute.For fixed positive definite covariance, outage scales as x^N at small x; a rank-r covariance instead has x^r scaling.Source: A. Diversity and the Order of Limits · p. 11

B. Why Physical-Port Geometry Matters

  • Covariances with matched eigenvalues can have distinguishable physical-port outage because a unitary basis change preserves Euclidean norm but not the port maximum.Even positive definite covariances with identical pairwise correlation magnitudes can have different determinants and small-ball constants.Source: B. Why Physical-Port Geometry Matters · pp. 11–12
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