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Spectral Approximation and Ergodic-Capacity Convergence of HMIMO Channels under Spatial-Wavenumber Domain Mismatch

Hangsong Yan, Hong Yang, Shu Sun

arXiv:2608.26802v1cs.IT

TL;DR

The paper studies finite-dimensional approximation of continuous HMIMO channels with square apertures and circular wavenumber support, a mismatch that creates a non-separable square–disk concentration problem. It projects the operator onto a tensor-product 1D PSWF subspace while preserving circular support, and establishes super-exponential spectral and ergodic-capacity convergence beyond an explicit truncation threshold. Numerical results indicate that conventional spatial-DoF truncation can miss relevant modes for compact apertures.

  • Problem

    Square apertures with circular wavenumber support create a non-separable concentration problem that classical separable PSWF constructions cannot directly handle.

  • Method

    The operator is projected onto a tensor-product subspace of 1D PSWFs while preserving circular support, producing a generally non-diagonal sparse representation.

  • Results

    Spectral approximation error and the optimized ergodic-capacity gap are controlled by a 1D PSWF tail and decay super-exponentially beyond an explicit truncation threshold.

  • Takeaways & Limitations

    For compact apertures, retaining modes beyond the conventional 1D spatial-DoF benchmark can materially improve evaluated spectral efficiency.

  • Takeaways & Limitations

    The framework does not yet cover general spatial geometries or near-field channels, where spatial wide-sense stationarity breaks down.

Abstract

from arXiv · show

We establish quantitative results on finite-dimensional spectral approximation and ergodic-capacity convergence for continuous Holographic Multiple-Input Multiple-Output (HMIMO) channels with square apertures and physically prescribed circular wavenumber support. The resulting spatial-wavenumber domain mismatch leads to a non-separable square-disk concentration problem for which the classical separable construction based on prolate spheroidal wave functions (PSWFs) cannot be directly applied. We project the continuous operator onto a tensor-product subspace of one-dimensional (1D) PSWFs while preserving the circular wavenumber support, yielding a generally non-diagonal but highly sparse finite-dimensional matrix. We show that the whole-spectrum approximation error, accounting for retained-eigenvalue perturbations and the residual spectral tail, remains controlled by a 1D PSWF eigenvalue-tail envelope despite the loss of separability and induced off-diagonal coupling. Beyond an explicit 1D truncation threshold, this error decays super-exponentially. This analysis further yields an asymptotic upper envelope for the eigenspectrum under the flattened two-dimensional eigenvalue ordering. We further establish a non-asymptotic upper bound on the gap between the actual ergodic capacities of the continuous and tensor-PSWF-truncated channels under their respective transmit-covariance optimizations. Combined with the spectral result, this capacity-gap bound inherits the same super-exponential dependence on the truncation order. Finally, quadrature rules with explicit radial and angular node thresholds are developed for evaluating the projected matrix. Numerical results show that conventional truncation based on spatial degrees of freedom can omit performance-relevant modes, particularly for compact apertures.

I. INTRODUCTION

The paper addresses finite-dimensional approximation and capacity analysis for continuous HMIMO operators, where square apertures and circular wavenumber support create a non-separable concentration problem. It uses tensor-product 1D PSWFs to obtain sparse numerical representations and quantitative spectral and capacity guarantees.

  • Continuous HMIMO channels are modeled as spatial-wavenumber integral operators rather than finite-dimensional matrices, making eigenspectrum characterization central to capacity analysis.
  • Conventional spatial DoF measures do not quantify how accurately a finite-dimensional truncation captures the operator eigenspectrum.
  • The framework projects the square-aperture operator onto tensor products of 1D PSWFs while preserving circular wavenumber support, producing a generally non-diagonal sparse matrix.
  • Whole-spectrum error accounts for retained-eigenvalue perturbations and the residual tail, yet remains controlled by a 1D PSWF eigenvalue-tail envelope with super-exponential decay beyond an explicit threshold.
  • The ergodic-capacity gap under transmit-covariance optimization is bounded by the same trace defect and therefore inherits super-exponential dependence on truncation order.
  • Quadrature rules specify radial and angular node thresholds, while numerical results show conventional DoF truncation can omit performance-relevant modes for compact apertures.

A. Eigenfunction Evaluation

The paper reviews 1D PSWF evaluation as the basis for a tensor construction, then explains why the square-aperture/circular-support geometry produces a non-diagonal concentration problem. The resulting representation preserves the physical disk support while exploiting basis structure for analysis and computation.

  • Eigenfunction Evaluation: Classical 1D PSWFs solve the problem of maximizing bandlimited-function energy concentration within a finite spatial interval.
  • Eigenfunction Evaluation: PSWFs can be evaluated through a commuting differential operator and a Legendre-polynomial expansion that yields a symmetric pentadiagonal matrix eigenproblem.
  • Eigenfunction Evaluation: Parity-aware evaluation computes finite-Fourier eigenvalue magnitudes separately for even and odd indices, providing the 1D ingredients for the tensor-product construction.
  • Domain Mismatch: The square aperture observes the field over a square spatial region, whereas its physical wavenumber support is confined to a circular disk.
  • Domain Mismatch: The tensor-product basis remains complete on the square, allowing an infinite matrix representation that shares the continuous operator’s non-zero eigenvalues.
  • Domain Mismatch: Tensor-product PSWFs do not diagonalize the square–disk operator because their separable Fourier support is square rather than circular.

B. Tensor-PSWF Spectral Approximation

The square–disk concentration operator is non-separable, so the paper represents it in a tensor-product 1D PSWF basis and analyzes finite-dimensional spectral truncation. The resulting approximation error remains controlled by 1D PSWF tails and decays super-exponentially beyond an explicit threshold.

  • The square–disk problem seeks functions concentrated in a square spatial domain while remaining bandlimited to a circular wavenumber disk.
  • The continuous operator is represented by expanding its eigenfunctions in a complete tensor-product basis of normalized 1D PSWFs.Completeness makes the resulting infinite matrix an exact representation sharing the operator’s non-zero eigenvalues.
  • The finite matrix retains N1D PSWFs along each axis, giving N = N1D^2 basis functions for the square aperture.
  • Parity causes selected matrix elements to vanish, reducing the original four-dimensional integral calculations to sparse two-dimensional polar-form evaluations.The truncated matrix inherits sparsity and symmetry, reducing the number of required computations.
  • The total whole-spectrum error accounts for retained-eigenvalue perturbations and the residual tail through a trace defect controlled by a 1D PSWF eigenvalue-tail envelope.Once N1D > ec/4, the bound has an explicit non-asymptotic super-exponential form; analogous tail control extends to rectangular apertures.
  • The flattened two-dimensional eigenspectrum admits an explicit asymptotic upper envelope, which is a bound rather than an exact asymptotic equivalence.The actual eigenspectrum may decay faster than the derived envelope.

V. CHANNEL CAPACITY

The paper quantifies information loss when the continuous channel is restricted to the finite tensor-PSWF subspace. It analyzes actual ergodic capacity with channel averaging and optimized transmit covariance rather than relying on a Jensen-based upper bound.

  • The capacity analysis directly studies averaged instantaneous mutual information with transmit covariance optimized under a total-power constraint.
  • The analysis contrasts this actual capacity with an analytical upper bound obtained through Jensen’s inequality.
  • The section quantifies the information-theoretic loss caused by restricting the continuous channel to the finite tensor-PSWF subspace.

A. Ergodic Capacity of the Continuous Aperture Channel

The continuous HMIMO channel is modeled as a spatial integral operator for symmetric square apertures with circular wavenumber support. Its ergodic capacity is formulated under perfect CSIR, statistical CSIT, and optimized transmit covariance.

  • Continuous HMIMO uses propagating fields over spatially continuous apertures, so the physical channel is modeled as an integral operator rather than a finite-dimensional matrix.The operator’s eigenspectrum is required to characterize fundamental capacity limits.
  • The capacity setting assumes perfect instantaneous CSIR, statistical CSIT, and identical square transmit and receive apertures with circular wavenumber support.
  • Under separable scattering, receive and transmit covariance eigenfunctions coincide with those of the concentration operator, with eigenvalues scaled by γr and γt.The product γrγt is absorbed into the effective SNR, and the normalized model sets the scattering factors equally.
  • The continuous input-output relation is converted into a discrete representation using Mercer’s theorem and the Karhunen–Loève expansion.The resulting channel matrix has i.i.d. complex Gaussian entries in its whitened representation, with transmit power and noise defining the SNR.
  • The actual ergodic capacity uses a Fredholm determinant and optimizes a deterministic transmit covariance under non-negativity and a total-power constraint.The optimization can be restricted to diagonal covariance matrices without loss of optimality, and explicit optimal power allocation is unnecessary for the convergence analysis.

B. PSWF-Truncated Random Channel

The tensor-PSWF truncation induces a finite-dimensional Gaussian channel from the continuous channel, with transmit optimization reducible to the eigenbasis of M_N. Its capacity loss is bounded by the same trace defect governing spectral approximation and therefore converges super-exponentially beyond the truncation threshold.

  • Channel construction: The projected channel can be represented in the eigenbasis of M_N without changing its distribution under the separable Gaussian scattering model.The corresponding random matrix has i.i.d. CN(0, 1) entries in that representation.
  • Capacity characterization: The actual ergodic capacity of the projected channel is optimized over truncated transmit covariance matrices Q_N.The covariance constraint includes positive semidefiniteness, a transmit-power trace bound, and the stated channel model.
  • Capacity characterization: The capacity maximization can be restricted without loss of optimality to transmit covariance matrices diagonal in the eigenbasis of M_N.This reduces the covariance optimization to diagonal Q_N in that basis.
  • Channel construction: The truncated random channel is induced by projecting the continuous square–disk channel onto the tensor-PSWF subspace, rather than introduced as an independent surrogate.The retained spatial subspace is spanned by tensor products of the first N_1D one-dimensional PSWFs.
  • Capacity gap and convergence: The capacity loss under perfect instantaneous CSIR and statistical CSIT is controlled by the trace defect Tr(P)−Tr(M_N), which also controls whole-spectrum error.For N_1D > ec/4, the bound becomes explicit and, combined with the spectral result, decays super-exponentially with truncation order.
  • Capacity gap and convergence: The capacity-gap upper bound also applies to rectangular extensions after replacing the square-domain operators with their rectangular counterparts.The extension relies on spectral ordering and trace-defect structure rather than square geometry itself.

VI. QUADRATURE RULES AND NUMERICAL RESULTS

The section develops quadrature rules for the projected matrix and evaluates how truncation dimension affects eigenvalues and spectral efficiency. Results show that the analytical threshold avoids artificial truncation, with the largest benefits for compact apertures.

  • Quadrature Rules: Beyond the radial-node threshold Mr > e^4, Gauss–Legendre quadrature for the radial integral converges super-exponentially.
  • Quadrature Rules: Beyond the angular-node threshold Mθ > 2ec, the uniform trapezoidal rule for the angular integral converges exponentially.
  • Numerical Results: The eigenvalue curves benchmark conventional truncation N1D = 2L/λ against the analytical threshold N1D = ⌊ec/4⌋+1 and numerically converged references.
  • Numerical Results: Insufficient subspace dimension causes an artificial eigenvalue drop, while increasing N1D mitigates truncation and approaches the reference spectrum.
  • Numerical Results: Once N1D reaches ⌊ec/4⌋+1, spectral efficiency nearly saturates across aperture sizes and SNR regimes, while compact apertures benefit most from extra modes.
  • Numerical Results: At L/λ = 1 and ρ = 10, increasing N1D from 2L/λ to ⌊ec/4⌋+1 raises normalized spectral efficiency by nearly 40%, versus below 5% at L/λ = 10.

APPENDIX A

The appendix derives the tensor-PSWF matrix representation by reducing the continuous operator’s integrals and exploiting the circular wavenumber support. Symmetry and parity make the resulting matrix sparse through exact zero patterns.

  • PSWF Foundations: The 1D PSWF operator commutes with a second-order differential operator, making PSWFs eigenfunctions of a singular Sturm–Liouville problem.
  • PSWF Foundations: Legendre-polynomial expansions and recurrences convert the PSWF relations into an eigenvalue decomposition problem.
  • Matrix Representation: A quadruple integral is reduced to a double integral using band-limited Fourier-transform identities and Parseval’s identity.
  • Matrix Representation: The disk indicator restricts the wavenumber integration to the circular support, while the 1D transform terms remain equivalent to their band-limited versions on that domain.
  • Matrix Sparsity: Parity and axis symmetry force M(pq)(jℓ) to vanish when p+j or q+ℓ is odd, producing exact sparsity in the projected matrix.

APPENDIX D

The appendix proves that finite tensor-PSWF projection preserves nonnegative spectral ordering and controls the whole-spectrum error through a one-dimensional eigenvalue tail. Beyond the explicit threshold, the resulting envelope and trace defect decay super-exponentially.

  • Spectral Projection: The retained tensor-product PSWF subspace is a finite-dimensional principal truncation of the compact self-adjoint operator representation.
  • Spectral Projection: The max–min principle implies nonnegative retained spectral differences after projection.
  • Trace Defect: The trace defect equals the contribution of the discarded index complement and is bounded using the one-dimensional eigenvalue tail.
  • Eigenvalue Envelope: The resulting strict eigenvalue envelope enters the super-exponential decay regime beyond its stated threshold and yields the bound in Theorem 1.
  • Eigenvalue Envelope: When N1D > ec/4, the tail ratio satisfies 0 < q(N1D,c) < 1, enabling geometric control of the remainder.

4) Conclusion:

The conclusion consolidates the continuous and projected HMIMO channel constructions, their covariance-optimized capacity analysis, and the resulting distributional equivalence. It emphasizes rigorous spectral and capacity control while identifying broader geometries and near-field channels as future scope.

  • Channel Representation: The channel fields are expanded in orthonormal spatial eigenfunctions, producing an equivalent Hilbert–Schmidt random-operator representation.
  • Capacity Optimization: The covariance optimization can be restricted to diagonal transmit covariances under statistical CSIT, using phase invariance and concavity.
  • Projected Channel: Projecting onto tensor-product PSWFs yields an N-dimensional random channel whose covariance is represented by the truncated matrix MN.
  • Projected Channel: Unitary invariance of the i.i.d. Gaussian matrix permits representation in the eigenbasis of MN without changing the channel distribution.

APPENDIX F

The appendix compares continuous and receive-truncated channels using common Gaussian coefficients and covariance allocations, then bounds the resulting rate difference through trace-class operator inequalities.

  • Common comparison setup: The comparison uses a common admissible power-allocation sequence and the same Gaussian array for continuous and truncated channels.The finite channel matrix is identified with the leading N × N block of the shared Gaussian array.
  • Operator comparison: The min–max principle orders the eigenvalues of comparable positive trace-class operators, while log2(1+x) concavity converts eigenvalue differences into rate bounds.These inequalities support both receive- and transmit-side comparisons.
  • Operator comparison: The continuous and receive-truncated Gram operators are positive trace-class operators almost surely.Hilbert–Schmidt well-posedness and shared Gaussian construction establish the required operator properties.
  • Rate bound: The intermediate receive-truncated ergodic rate is bounded through a trace difference involving Tr(P) and Tr(MN).The displayed bound has the form ln 2 [Tr(P) − Tr(MN)].

3) Transmit-Side Truncation:

The transmit-side analysis connects the fully truncated auxiliary operator to the finite-dimensional rate functional and transfers fixed-allocation bounds to optimized capacities.

  • Transmit-side truncation: The fully truncated operator KN is supported only on its leading N × N block and represents a zero-padded auxiliary embedding.Its leading block is the finite-dimensional weighted truncated channel.
  • Transmit-side truncation: The transmit-side perturbation operator is Hilbert–Schmidt almost surely, and its Gram operators satisfy an exact identity.The identity follows from the corresponding operator constructions and shared Gaussian coefficients.
  • Capacity optimization: Every admissible infinite power-allocation sequence induces a continuous covariance and finite-dimensional restriction, while every feasible N-dimensional diagonal covariance has a zero extension.This establishes the common feasible set needed to compare the two optimizations.
  • Capacity optimization: The continuous capacity is at least the truncated capacity, and their gap is controlled by Tr(P) − Tr(MN).The resulting upper bound applies under the shared admissible power-allocation parameterization, even when optimizers differ.
  • Capacity optimization: When N1D > ec/4, the capacity-gap bound inherits an explicit super-exponential envelope from the spectral trace-tail bound.The derivation uses a factorial envelope and a strict lower bound from Robbins’ version of Stirling’s approximation.

APPENDIX H

The appendix derives radial and angular quadrature thresholds by bounding the exponential types and Fourier-coefficient tails of the projected-matrix integrands.

  • Radial quadrature: The radial analysis maps 1D PSWF bandlimiting into polar coordinates to bound the integrand’s exponential type uniformly over angles.The product’s exponential type is bounded by the sum of the component types, while the polynomial factor r does not change it.
  • Radial quadrature: The radial quadrature threshold is obtained when the quadrature error envelope enters its super-exponential regime.The condition is expressed through the threshold derived from e^(4M_r) < 1 in the supplied derivation.
  • Angular quadrature: The angular integrand is treated as a 2π-periodic function whose phase amplitude is maximized over the expanded wavenumber domains Kx, Ky ∈ [−2c, 2c].The worst case occurs at r = c and Kx, Ky = ±2c, defining Amax.
  • Angular quadrature: Jacobi–Anger expansion and the Bessel-function envelope show that angular Fourier coefficients enter super-exponential decay beyond the stated order threshold.The supplied derivation gives conditions involving m > eAmax and the resulting threshold in terms of c.
  • Angular quadrature: The trapezoidal rule is sufficient for angular evaluation when its node count exceeds the threshold, because aliasing then samples only the super-exponentially decaying Fourier tail.This provides an explicit sufficient angular quadrature rule for evaluating the projected matrix.
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