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Benchmarking Curvature-Domain Signaling for Continuous-Aperture Wireless Communications: Capacity, Robustness, Detection, and Conditioning Against Legacy Modal Bases

Yasser Al-Eryani

arXiv:2609.28353v1eess.SPeess.SY

TL;DR

Continuous-aperture communication needs benchmarks that compare aperture signaling coordinates under shared physical constraints. This paper provides such a benchmark for curvature-domain signaling and legacy bases; curvature is the flattest tested basis under phase-sampling noise, while its practical value remains a system question.

  • Problem

    The paper addresses the missing external benchmark comparing curvature coordinates with established aperture and wavefront bases under common assumptions.

    Source: §I · p. 1

  • Method

    It benchmarks curvature-regularized modes and legacy bases under shared aperture, propagation, phase-only, power, noise, quantization, and sampling assumptions.

    Source: §I · p. 2, A. Two transmit constraints, and why… · p. 9, §X · p. 11

  • Results

    Curvature is the flattest tested basis under phase-sampling noise, at 2.55 against 8.17 for the energy-optimal reference, while staying within 0.856 of the capacity ceiling.

    Source: F. Verdict · p. 11

  • Takeaways & Limitations

    Curvature is a stability-oriented option in some regimes, not a universal winner, and whether its trade-off is worthwhile remains a system question.

    Source: F. Verdict · p. 11

    The benchmark is scalar, narrowband, and one-dimensional, and does not report a two-dimensional aperture benchmark.

    Source: D. Limitations · p. 11

Abstract

from arXiv · show

Continuous-aperture and holographic MIMO systems motivate signaling that operates directly on large electromagnetic apertures rather than on a few antenna ports. This paper is the empirical companion to the operator-theoretic curvature-domain framework: a reproducible benchmark of curvature-domain signaling under a common scalar aperture-channel model, stress-testing the theory's dual-budget generalized water-filling law against the modal bases used in near-field and holographic MIMO. All methods share the same apertures, quadrature, Fresnel or Green-function propagation, phase-only constraints, power, phase-energy and curvature-energy budgets, receiver noise, phase quantization and training assumptions. The compared coordinates are curvature-regularized eigenmodes, raw phase coefficients, Fourier phase modes, polynomial and Zernike-like wavefront modes, near-field matched-focus profiles, random and optimized RIS phase codebooks, and SVD water-filling upper bounds. The claim is deliberately limited: curvature-domain signaling is a gauge-invariant, physically realizable coordinate system that can approach the phase-space SVD water-filling bound with fewer stable modes where derivative noise, phase quantization, sampling density or ill conditioning limit conventional bases. It is not a claim of new electromagnetic physics, nor that curvature modes dominate every baseline. We prove the SVD upper-bound relation for the discretized phase-control tangent space, derive pairwise-error and perturbation bounds, and report capacity, retained modes, symbol error, robustness, quantization, sampling, regularization, conditioning and cost, with 95% bootstrap confidence intervals on Monte Carlo results. A verdict table locates the regimes where curvature-domain signaling is engineering-relevant: moderate-to-high phase noise, coarse phase quantization, and non-Fourier-diagonal channels.

I. INTRODUCTION

The paper benchmarks curvature-domain signaling against established aperture and wavefront coordinates under common physical constraints, asking when curvature regularization improves stability or efficiency. It treats curvature as a gauge-invariant coordinate for wavefront shape, not a universal winner or new electromagnetic physics.

I. INTRODUCTION

  • The benchmark compares aperture-signaling methods under matched apertures, propagation, phase-only constraints, noise, power, quantization, and sampling, with the discretized SVD as an upper bound.Its stated aim is to identify when curvature regularization improves stability or efficiency, not to show that curvature always wins.Source: §I · pp. 1–2

A. MIMO Capacity and Physical Degrees of Freedom

  • Classical MIMO capacity and physical degrees of freedom are framed through Gaussian channel theory and aperture- and geometry-constrained signal spaces [10]–,,.The cited work links singular functions and coupling strengths to wave communication limits.Source: A. MIMO Capacity and Physical Degrees… · p. 2

B. Holographic and Continuous-Aperture MIMO

  • Holographic and continuous-aperture MIMO replace sparse antenna-port abstractions with dense or nearly continuous surfaces, using plane-wave and near-field channel models, –.Continuous-aperture architectures also treat current distributions and beamforming variables continuously [7].Source: B. Holographic and Continuous-Aperture… · p. 2

C. Reconfigurable Intelligent Surfaces and Near-Field Focusing

  • Near-field focus-aware and RIS-style codebooks are included as baselines, with the source work motivating range-angle codebooks and quantized phase optimization, –[18].The cited comparison treats them as baselines rather than competitors with identical implementation assumptions.Source: C. Reconfigurable Intelligent Surfaces… · p. 2

D. Fourier, Polynomial, and Zernike Wavefront Bases

  • Fourier modes suit shift-invariant paraxial propagation, while Zernike-like modes describe optical aberrations; aperture-norm orthogonality alone does not ensure good conditioning under propagation or implementation effects –.The stated conditioning factors include phase quantization, derivative noise, and finite-grid sampling.Source: D. Fourier, Polynomial, and Zernike… · p. 2

E. Electromagnetic Information Theory

  • The paper uses a scalar reproducible benchmark as a controlled first step, leaving vector, polarization-aware, and mutual-coupling-aware extensions for later work, [22].The broader field seeks models connecting wave propagation, antenna physics, channel modeling, and information theory.Source: E. Electromagnetic Information Theory · p. 2

F. Learning-Based Codebooks and Beamformers

  • Learning-based codebooks are excluded because fair comparison requires amortized accounting of offline training and online optimization costs, which this benchmark defers –.The adopted protocol has each baseline read channel state once and optimize once.Source: F. Learning-Based Codebooks and… · p. 2

C. Weighted Discretization

The channel is discretized with quadrature weights so vector norms approximate aperture L2 norms, and phase-only control is evaluated through a shared linearized tangent model and exact nonlinear checks.

C. Weighted Discretization

  • Weighted discretization is chosen so Euclidean vector norms approximate aperture L2 norms.The passage describes this as the purpose of the weighting convention.Source: C. Weighted Discretization · p. 3

D. Phase-Only Aperture Control

  • The phase-only model separates a shared carrier, focusing, or steering phase from the data-bearing phase perturbation before evaluating the received perturbation.The shared carrier response is subtracted first.Source: D. Phase-Only Aperture Control · p. 3
  • Modal capacity and pairwise-error comparisons use the same linearized tangent operator, while nonlinear Monte Carlo checks retain the exact phase-only map.The tangent model is a controlled comparison layer, not a replacement for electromagnetic validation.Source: D. Phase-Only Aperture Control · p. 3

E. Gauge Fixing and Curvature Coordinates

The benchmark fixes affine gauge choices before comparing curvature-bearing phase shapes, applies shared physical budgets, and aligns its finite-mode capacity allocation with the companion theory's dual-budget water-filling law.

E. Gauge Fixing and Curvature Coordinates

  • Gauge fixing excludes piston and coordinate-reference choices from being counted as data-bearing curvature information.A shared steering or focusing phase may instead be handled in the carrier phase.Source: E. Gauge Fixing and Curvature… · p. 3

F. Physical Budgets

  • Methods without a natural curvature-energy constraint are normalized to a common phase-energy or peak-phase budget, and active curvature constraints are reported with both phase- and curvature-energy normalization.This keeps the physical-budget comparison explicit across methods.Source: F. Physical Budgets · p. 4

G. Consistency with the Companion Theory Paper

  • The curvature capacity allocation follows the theory paper's dual-multiplier generalized water-filling law, with multipliers set by the phase and curvature budgets.In the flat limit, the allocation reduces to standard single-budget Gaussian water filling; the second multiplier captures the geometric constraint.Source: G. Consistency with the Companion… · p. 4

IV. COMPARED SIGNALING COORDINATES

The compared methods map coefficient vectors or symbol indices to phase profiles, with the curvature basis regularizing channel-aware modes and the raw-phase and Fourier bases serving as contrasting coordinates.

IV. COMPARED SIGNALING COORDINATES

  • Each signaling method maps a finite coefficient vector or symbol index to a phase profile, which is then represented through a modal channel.Linear-basis columns are normalized under a shared convention unless otherwise stated.Source: §IV · p. 4

A. Curvature-Regularized Eigenbasis

  • The curvature-regularized generalized eigenbasis is channel-aware, but penalizes modes with high received gain when they require excessive curvature energy.Its modes are ordered by eigenvalues and converted to gauge-fixed, phase-normalized patterns.Source: A. Curvature-Regularized Eigenbasis · p. 4

B. Raw Phase Coefficients

  • Raw phase coefficients provide maximal local control, but neighboring-pixel oscillations can have low phase energy and high curvature energy, yielding poor derivative conditioning.The baseline uses localized phase samples or local hat functions followed by gauge projection and normalization.Source: B. Raw Phase Coefficients · p. 4

C. Fourier and DFT Phase Modes

  • Fourier and DFT phase modes are gauge-projected and orthonormalized after sorting by spatial frequency, and are expected to be strong for shift-invariant paraxial channels.The benchmark uses real sine and cosine components.Source: C. Fourier and DFT Phase Modes · p. 4

D. Polynomial and Zernike-Like Modes

The section establishes polynomial and matched-focus profiles as distinct baselines, then separates RIS-style codebooks by how they are generated and optimized.

D. Polynomial and Zernike-Like Modes

  • Rectangular apertures use tensor-product Legendre modes, while circular or masked apertures use Zernike-like disk modes weighted-QR orthonormalized on the aperture and sorted by total degree.Source: D. Polynomial and Zernike-Like Modes · p. 4

E. Matched-Focus and Beamfocusing Profiles

  • Matched-focus profiles remove common carrier phase, optionally gauge-project, peak-phase normalize, and sample range-angle focal points for sparse near-field point-to-point settings.Source: E. Matched-Focus and Beamfocusing… · p. 5

F. RIS-Style Codebooks: Random, Greedy, and Optimized

  • Random profiles require no channel state or propagation knowledge and serve as a lower reference, whereas greedy farthest-point selection uses received-space distances and has a basis-level form.Source: F. RIS-Style Codebooks: Random,… · p. 5
  • The optimized codebook uses projected gradient ascent on minimum received-space pairwise distance under gauge, peak-phase, and quantization constraints, and has no basis-level analogue.Source: F. RIS-Style Codebooks: Random,… · p. 5
  • Halving the step after non-improving iterations handles the quantized feasible lattice, while retaining the best true minimum distance prevents a surrogate overshoot from discarding a good iterate.Source: F. RIS-Style Codebooks: Random,… · p. 5
  • Seeded and unseeded optimizer runs answer different questions, and the fair curvature comparison is against the seeded run rather than only an independent optimization run.Source: F. RIS-Style Codebooks: Random,… · p. 5

G. SVD Upper Bounds

The section treats phase-space SVD water-filling as an upper bound for restricted phase-coordinate bases and specifies shared assumptions for fair comparisons.

G. SVD Upper Bounds

  • The phase-space SVD is the upper bound for phase-coordinate bases under the same discretized phase-energy budget, while arbitrary complex-field SVD is an outer bound.Source: G. SVD Upper Bounds · p. 5

V. FAIRNESS PROTOCOL

  • Comparisons hold apertures, propagation, power, phase and curvature budgets, noise, mode count, quantization, training assumptions, and recorded random seeds constant.Source: §V · p. 6

A. Constrained Modal Capacity

B. Pairwise Error Probability

The section gives an AWGN pairwise-error framework for finite phase codebooks, describes derivative-noise regularization, and reports quantization degradation empirically.

B. Pairwise Error Probability

C. Phase Noise and Derivative Noise

  • Because D2 amplifies high spatial frequencies, regularized curvature reconstruction with ρ>0 has finite worst-case noise gain, unlike potentially singular unregularized inversion near gauge and high-frequency nullspaces.Source: C. Phase Noise and Derivative Noise · p. 6

D. Quantization

  • The quantization analysis gives a componentwise rounding bound but treats it as conservative, reporting empirical rate and SER degradation as a function of phase-bit count.Source: D. Quantization · p. 7

E. Conditioning and Computational Cost

The benchmark pairs conditioning and computational-cost diagnostics with a reproducible suite of capacity, detection, robustness, quantization, and sampling comparisons. Its figures specify what to measure and compare, but the supplied passages do not report numerical outcomes.

E. Conditioning and Computational Cost

  • The benchmark records costs for basis construction, channel projection, water filling, codebook optimization, and Monte Carlo detection; SVD has the highest global channel-decomposition cost.Conditioning and computational cost are reported separately by basis.Source: E. Conditioning and Computational Cost · p. 7

VII. REPRODUCIBLE EXPERIMENT SUITE

  • The reproducible suite uses a shared aperture geometry, transmit and receive grids, wavelength, separation, phase-only aperture, and equal-budget constraints.All figures and machine-readable result files are generated by the benchmark suite.Source: A. Fig. 1: Geometry and Constraints · p. 7, §VII · p. 7

B. Fig. 2: Modal Spectra and Conditioning

  • Modal diagnostics compare singular and generalized curvature spectra, Gram conditioning, and curvature-energy growth across bases.The SVD spectrum is the phase-space upper-bound reference; curvature modes are assessed by generalized eigenvalue decay and derivative-noise conditioning.Source: B. Fig. 2: Modal Spectra and… · p. 7, Figure 2 · p. 7

C. Fig. 3: Capacity Versus SNR

  • Capacity-versus-SNR curves compare curvature, Fourier, polynomial/Zernike-like, raw-phase, matched-focus, RIS-style, and SVD baselines under equal physical budgets.Curves should remain below the phase-space SVD upper bound; a crossing signals a normalization or implementation error.Source: C. Fig. 3: Capacity Versus SNR · p. 7, Figure 3 · p. 7

F. Fig. 6: Robustness to Phase-Sampling Noise

  • The phase-sampling-noise experiment reports achievable-rate and symbol-error-rate degradation as sampling noise increases.The figure notes that curvature regularization is expected to help when derivative-noise amplification dominates the raw phase representation; SER below 10^-3 is treated as below the Monte Carlo sensitivity floor.Source: F. Fig. 6: Robustness to… · p. 7, Figure 6 · p. 8

D. Fig. 4: Capacity Versus Retained Modes

  • Capacity versus retained-mode count tests whether curvature signaling approaches a relevant fraction of the SVD reference with fewer well-conditioned modes under the same budget.The figure measures mode efficiency using K, the number of retained modes.Source: D. Fig. 4: Capacity Versus Retained… · p. 7, Figure 4 · p. 8

E. Fig. 5: Pairwise Error and SER

  • Pairwise error probability and symbol error rate are compared against SNR for matched codebook sizes, receiver noise, and phase constraints.The figure compares analytical pairwise-error and union-bound predictions with Monte Carlo symbol-error rates.Source: E. Fig. 5: Pairwise Error and SER · p. 7, Figure 5 · p. 8

G. Fig. 7: Phase Quantization

  • Quantization, aperture-grid refinement, and regularization are tested through performance versus phase bits, sampling-density sensitivity, and a sweep of curvature regularization.Quantized profiles share peak-phase range and wrapping convention; stable coordinates should converge with grid refinement, while larger regularization suppresses derivative-noise amplification but may reduce rate.Source: G. Fig. 7: Phase Quantization · p. 8, Figure 7 · p. 8, Figure 8 · p. 8

H. Fig. 8: Aperture Sampling Density

The results compare curvature signaling with modal and SVD baselines across capacity, detection, robustness, sampling, regularization, and conditioning. The comparisons depend on keeping RMS phase-energy and peak phase-excursion constraints distinct.

H. Fig. 8: Aperture Sampling Density

I. Fig. 9: Regularization Sweep

  • Fig. 9 reports the rate–stability tradeoff as curvature regularization ρ varies.The supplied passage does not state which regularization setting is preferable.Source: §I · p. 8

J. Fig. 10: Cost and Conditioning

  • At K = 24, SVD’s condition number is 8.75 versus curvature’s 32.43, while curvature construction takes 0.1250 s versus SVD’s 0.8812 s.The timing is machine-dependent; the text recommends reading it as an order-of-magnitude comparison.Source: Figure 10 · p. 9, B. Capacity and conditioning · p. 9

VIII. RESULTS

  • The benchmark tables are generated from recorded results and regenerated whenever experiments are rerun.The two tables draw on the same recorded results.Source: §VIII · p. 9

A. Two transmit constraints, and why they must not be mixed

  • Capacity uses an RMS phase-energy budget, whereas detection uses a peak phase-excursion budget and shared noise anchored to realized symbol energy.Mixing the conventions creates a 40 dB error.Source: A. Two transmit constraints, and why… · p. 9

B. Capacity and conditioning

  • At 15 dB, curvature reaches 25.153 bits/use versus 29.386 for the complex SVD bound and 18.358 for Fourier; its capacity is 0.856 of the SVD ceiling.The paper states that curvature cannot exceed the discretized-channel SVD optimum.Source: B. Capacity and conditioning · p. 9

C. Detection

  • At 14 dB, symbol-error rates are 0.025590 for curvature, 0.000780 for phase-realizable SVD, and 0.002300 for Fourier.This is the highest SNR at which the compared methods remain above the stated Monte Carlo sensitivity floor.Source: C. Detection · p. 9
  • A seeded packing optimizer reaches dmin = 14.1391, improving 1.45516 over curvature; curvature is a useful starting point, not a stationary packing solution.The independent optimizer agrees with the phase-realizable SVD eigendecomposition to within 0.1% on dmin, Esym, and packing efficiency.Source: C. Detection · p. 10

D. Robustness: the one axis curvature wins

Curvature has its clearest measured advantage in phase-sampling-noise robustness, while its quantization and broader benchmark results remain mixed. The paper limits its claims to the benchmark’s scalar, narrowband, one-dimensional setting and identifies conditions where curvature may or may not help.

D. Robustness: the one axis curvature wins

  • Across the phase-sampling-noise sweep, curvature’s SER rises by a factor of 2.55 versus 8.17 for phase-realizable SVD; this is a flatter degradation curve, not a lower error rate.Curvature starts from a worse operating point and ends at a comparable one.Source: D. Robustness: the one axis curvature… · p. 10

E. Quantization

  • At one bit, curvature retains 0.839 of infinite-resolution capacity; its measured SER is below the eight-bit value because peak-normalized quantization raises RMS excursion and delivered energy.The one-bit SER is below the 0.000150 sensitivity floor; under the energy-constrained capacity convention, capacity falls monotonically with quantization.Source: E. Quantization · pp. 10–11

F. Verdict

  • On this geometry, curvature wins one of eight verdict rows; it is not a capacity improvement, does not beat SVD conditioning, and loses minimum-distance packing to seeded optimization.The benchmark leaves whether its flatter phase-noise degradation is worth the tradeoff to the system context.Source: F. Verdict · p. 11

A. When Curvature Should Help

  • Curvature is expected to be useful when phase measurements are noisy, high-frequency patterns are costly or undersampled, or constraints include smoothness, quantization, or curvature energy.The passage also identifies gauge-invariant representation and smooth channel subspaces poorly represented by initial Fourier or polynomial modes as relevant cases.Source: A. When Curvature Should Help · p. 11

B. When Curvature Should Not Be Expected to Win

  • Curvature need not dominate: Fourier may suit nearly shift-invariant paraxial channels, matched focus may suit simple near-field point links, and SVD remains the upper bound with arbitrary complex fields.Its stability advantage may matter less when phase and derivative noise are negligible.Source: B. When Curvature Should Not Be… · p. 11

C. No Claim of New Physics

  • Curvature coordinates leave the propagation operator and physical propagation laws unchanged; observed advantages are attributed to coordinate choice, regularization, conditioning, or related constraints.The paper makes no claim of new electromagnetic physics.Source: C. No Claim of New Physics · p. 11

D. Limitations

  • The benchmark is scalar, narrowband, and one-dimensional; it does not fully model polarization, mutual coupling, hardware nonlinearities, thermal constraints, beam squint, calibration drift, or closed-loop training overhead.A two-dimensional aperture benchmark is also not reported.Source: D. Limitations · p. 11

X. CONCLUSION

  • The conclusion positions curvature as a stable, gauge-invariant benchmarking coordinate, not a method that beats SVD or changes propagation physics.The discretized phase-space SVD upper-bounds finite phase-coordinate bases under the same energy model.Source: §X · p. 11

APPENDIX A WATER-FILLING COMPUTATION

The appendices describe computational procedures for water filling, weighted orthonormalization, and a nonlinear consistency check for phase-only codebooks.

APPENDIX A WATER-FILLING COMPUTATION

  • The water-filling computation sorts inverse gains to select an active set, checks its consistency, and assigns zero power to numerically negligible singular values.The passage also identifies the water level as the quantity used to select the active set.Source: Appendix A · pp. 11–12

APPENDIX B WEIGHTED QR ORTHONORMALIZATION

  • Weighted QR orthonormalization transforms a sampled basis into a weighted-orthonormal basis.The appendix names Euclidean QR as the factorization used in the construction.Source: Appendix B · p. 12

APPENDIX C NONLINEAR PHASE-ONLY CONSISTENCY CHECK

  • The consistency check compares nonlinear and linearized received distances; large relative error marks excursions outside the reliable tangent regime, where nonlinear SER should be emphasized.Source: Appendix C · p. 12
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