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Spatially-Stationary Model for Holographic MIMO Small-Scale Fading

Andrea Pizzo, Thomas L. Marzetta, Luca Sanguinetti

arXiv:1911.04853v4cs.ITeess.SP

TL;DR

The paper addresses the lack of channel models for Holographic MIMO that are simultaneously physically meaningful, mathematically tractable, and numerically reproducible. It models far-field small-scale fading as a correlated Gaussian scalar random field constrained by the Helmholtz equation, yielding Fourier plane-wave representations and an efficient sampling method. The resulting formulation also supports degrees-of-freedom limits tied to aperture geometry and wavelength.

  • Problem

    Holographic MIMO lacks channel models that simultaneously retain physical realism, mathematical tractability, and numerical reproducibility for analyzing compact massive arrays.

  • Method

    The paper models far-field small-scale fading as a zero-mean spatially stationary correlated Gaussian scalar field consistent with the Helmholtz equation and represents it using Fourier plane waves.

  • Results

    The formulation provides an exact Fourier plane-wave spectral representation and a computationally efficient way to generate small-scale fading samples over spatially constrained compact spaces.

  • Takeaways & Limitations

    The model links physically meaningful spatial correlation with tractable field analysis and supports degrees-of-freedom limits for different aperture geometries.

Abstract

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Imagine an array with a massive (possibly uncountably infinite) number of antennas in a compact space. We refer to a system of this sort as Holographic MIMO. Given the impressive properties of Massive MIMO, one might expect a holographic array to realize extreme spatial resolution, incredible energy efficiency, and unprecedented spectral efficiency. At present, however, its fundamental limits have not been conclusively established. A major challenge for the analysis and understanding of such a paradigm shift is the lack of mathematically tractable and numerically reproducible channel models that retain some semblance to the physical reality. Detailed physical models are, in general, too complex for tractable analysis. This paper aims to take a closer look at this interdisciplinary challenge. Particularly, we consider the small-scale fading in the far-field, and we model it as a zero-mean, spatially-stationary, and correlated Gaussian scalar random field. Physically-meaningful correlation is obtained by requiring that the random field be consistent with the scalar Helmholtz equation. This formulation leads directly to a rather simple and exact description of the three-dimensional small-scale fading as a Fourier plane-wave spectral representation. Suitably discretized, this leads to a discrete representation for the field as a Fourier plane-wave series expansion, from which a computationally efficient way to generate samples of the small-scale fading over spatially-constrained compact spaces is developed. The connections with the conventional tools of linear systems theory and Fourier transform are thoroughly discussed.

I. INTRODUCTION

Holographic MIMO extends massive-array ideas toward compact, potentially continuous apertures, but requires channel models that remain physically meaningful, tractable, and reproducible. The paper motivates a spatially stationary small-scale fading model grounded in wave physics and represented through Fourier plane waves.

  • Holographic MIMO: Massive MIMO’s theoretical channel capacity increases unboundedly as N →∞ with fixed K, but practical base-station form factors limit antenna counts.Massive MIMO’s recognized advantages include spectral efficiency, energy efficiency, and power control.
  • Holographic MIMO: Holographic MIMO approaches the N →∞ limit by integrating a massive, possibly infinite number of antennas into compact spaces.Its asymptotic form is a spatially continuous electromagnetic aperture with uncountably infinite antennas separated by infinitesimal distances.
  • Channel-modeling challenge: Deterministic channel models are scenario-specific, whereas stochastic models can support broader conclusions but may omit significant physical phenomena.The paper positions stochastic modeling as a way to retain environmental generality while addressing physical realism.
  • Channel-modeling challenge: The i.i.d. Rayleigh model becomes inadequate as antenna spacing decreases and spatial correlation arises.Its convenient closed-form spectral-efficiency expressions make it useful, but its assumptions obscure physical propagation properties in densely spaced arrays.
  • Channel-modeling challenge: Clarke’s far-field NLoS model yields spatial correlation even without spatial directivity, with autocorrelation J0(2πr/λ) in 2D and sinc(2r/λ) in 3D.The model assumes scalar waves propagating in an isotropic random-scattering environment.
  • Paper objective: The paper generalizes Clarke’s model to non-isotropic scattering using a physically meaningful, mathematically tractable, and numerically reproducible 3D spatially stationary model.A continuous formulation supports a simple small-scale-fading expression and connects propagation with linear-systems theory and Fourier transforms.

B. Contribution

The paper develops a physically consistent, analytically tractable model of far-field small-scale fading for compact Holographic MIMO arrays. It connects Helmholtz-constrained spatial statistics with Fourier plane-wave representations and efficient numerical sampling.

  • The fading is modeled as a zero-mean, spatially stationary, correlated Gaussian scalar random field satisfying the Helmholtz equation.
  • The Helmholtz constraint yields a physically meaningful spatial correlation whose power spectral density is supported on a sphere of radius κ = 2π/λ.A spectral factor specifies scattering directionality and the propagation environment.
  • The resulting three-dimensional field has an exact Fourier plane-wave spectral representation as a superposition of statistically independent Gaussian propagating waves.
  • Under isotropic propagation, the model is presented as the closest physically tenable model to i.i.d. Rayleigh fading and recovers Clarke’s model.
  • For compact spaces, spatial discretization produces a Fourier plane-wave series expansion with a countably finite set of Gaussian coefficients.
  • The series representation enables efficient small-scale-fading sample generation through the inverse fast Fourier transform, with numerical validation by Monte Carlo simulations.The paper also interprets the model as an asymptotic Karhunen–Loève expansion and provides Matlab code for reproducing numerical results.

A. Plane-Wave Solution

The paper derives plane-wave solutions to the scalar Helmholtz equation and uses them to constrain the spatial statistics of far-field small-scale fading. Propagating waves occupy a disk in transverse wavenumber space, while the resulting spectral density is concentrated on a sphere.

  • Each realization of the electromagnetic small-scale fading is required to satisfy the scalar Helmholtz equation.
  • A plane-wave solution has wavenumber components constrained by the Helmholtz equation, with (kx, ky) supported on a disk of radius κ.
  • The analysis excludes evanescent waves and retains only propagating waves from the upgoing and downgoing half-spaces.Evanescent waves decay exponentially and contribute to near-field propagation.
  • A statistical model for the plane-wave amplitudes is chosen instead of scenario-dependent ray tracing or numerical partial-differential-equation methods.This choice leads to an analytically tractable model.
  • The spatial autocorrelation also satisfies the Helmholtz equation, forcing the power spectral density onto an impulsive sphere of radius κ.Because the support has zero measure, the spectrum is interpreted as a singular Delta distribution.
  • The resulting spectral constraint applies to physically meaningful fading, whereas i.i.d. Rayleigh fading is strictly incompatible with it in continuous three-dimensional space.The i.i.d. model produces an impulsive autocorrelation and constant power spectral density instead.

III. FOURIER DESCRIPTION OF PHYSICS-BASED CHANNELS

The paper converts the sphere-supported three-dimensional spectrum into a two-dimensional Fourier plane-wave representation by separating upgoing and downgoing components. This representation supports compact-space series expansions and efficient sample generation.

  • Fourier inversion in kz converts the sphere-supported spectrum into two two-dimensional plane-wave spectra over the disk D(κ).
  • The plane-wave spectra can exhibit large values near the boundary of D(κ) because of the spherical-parametrization Jacobian.If the spectral factor is bounded, a change of integration variables removes the associated singularity while preserving singular integrability.
  • The three-dimensional fading field is represented as the sum of upgoing and downgoing components with independent Gaussian random amplitudes.
  • The Fourier transform acts as a continuous plane-wave decomposition whose spatial harmonics correspond physically to propagating waves.
  • Spatial discretization over compact apertures yields a Fourier plane-wave series expansion and an efficient procedure for generating fading samples.

C. Connection with the Fourier Spectral Representation

The paper connects the Helmholtz-consistent fading field to Fourier spectral representations and linear space-invariant filtering. Migration filters then propagate the field between infinite z-planes.

  • Fourier spectral representation: The 3D field admits a Fourier spectral representation driven by white-noise Gaussian fields and the field power spectral density.Direct evaluation is inadequate for physics-based channels because the required square root of the impulsive spectrum is not defined, even distributionally.
  • Linear-systems interpretation: The filtering operation follows from the linear, space-invariant Helmholtz operator, so shifted and linearly combined solutions remain valid solutions.This permits wave propagation to be treated with linear-systems theory.
  • Migration filters: For any nonzero infinite z-plane, two phase-shift filters with responses e^±iγz generate the propagated components, whose sum yields h(x, y, z).These migration filters describe lossless propagation through the left and right half-spaces.

IV. ISOTROPIC PROPAGATION

The isotropic model provides a physically tenable reference for small-scale fading and connects arbitrary scattering channels to filtered isotropic fields. Its autocorrelation and independence properties agree with established Clarke-model results.

  • Connection to Rayleigh fading: The isotropic small-scale-fading model is identified as the closest physically tenable model to i.i.d. Rayleigh fading under the stated conditions.The model remains consistent with physics principles while matching the relevant independence behavior.
  • Isotropic model: An isotropic channel has a radially symmetric spectral factor invariant under rotations.The wavenumber coordinates can therefore be aligned conveniently with an axis.
  • Isotropic model: The isotropic spectrum is bandlimited and singularly integrable, which guarantees convergence of its spectral integral and permits closed-form autocorrelation.The autocorrelation depends on the distance between spatial points.
  • Isotropic model: Samples along a straight line separated by an integer multiple of λ/2 are independent, and isotropic samples become asymptotically independent as λ → 0.This result follows from zeros of the isotropic autocorrelation function.
  • Connection to prior models: The same autocorrelation functions arise from the 3D and 2D Clarke models, while the framework supports general non-isotropic fading through linear filtering.This places the proposed tools in agreement with previous models.

A. Linear System-Theoretic Interpretation of Scattering

The paper interprets arbitrary Helmholtz-consistent scattering fields as filtered isotropic fields. It combines shaping filters on z = 0 with migration filters to extend the representation across space and enable finite-aperture discretization.

  • A. Linear System-Theoretic Interpretation of Scattering: A non-isotropic field h(x, y, 0) with arbitrary spectrum is obtained by filtering an isotropic field with a 2D linear space-invariant response.The spectral factor acts as the shaping-filter response.
  • A. Linear System-Theoretic Interpretation of Scattering: The full field is bandlimited with wavenumber support D(κ), and its z-plane values follow from the z = 0 field through migration filters.The resulting procedure is used to generate samples over compact rectangular apertures.
  • A. Linear System-Theoretic Interpretation of Scattering: The finite-region channel energy is contained in a countably finite number of angular directions, each corresponding to a propagating plane wave.The resulting series is periodic over the spatial region with fundamental side lengths Lx and Ly.
  • A. Linear System-Theoretic Interpretation of Scattering: The 3D series extends across z using migration responses e^±iγℓmz, with accuracy increasing as min(Lx, Ly)/λ grows large.The stated approximation condition is |z| < min(Lx, Ly).
  • A. Linear System-Theoretic Interpretation of Scattering: The same discretization and filtering construction yields an efficient numerical procedure for linear, planar, and volumetric compact apertures.The procedure uses independent Gaussian lattice variables and inverse Fourier processing.

B. Planar and Volumetric Arrays

The discretized plane-wave model generates samples over compact arrays using Gaussian lattice fields, migration filters, and inverse Fourier transforms. Nyquist sampling and numerical experiments support its efficiency and accuracy for practical apertures.

  • B. Planar and Volumetric Arrays: The spatial grid contains N = NxNyNz points over a parallelepiped with side lengths Lx, Ly, and Lz < min(Lx, Ly).The grid counts are determined by the aperture dimensions and spacings along the three Cartesian axes.
  • B. Planar and Volumetric Arrays: The transverse channel is 2κ-bandlimited, so conventional half-wavelength spacing is generally adequate for spatial sampling.A half-wavelength array can be viewed as Nyquist samples of a spatially continuous aperture.
  • B. Planar and Volumetric Arrays: 3D channel samples are generated by drawing two independent Gaussian lattice fields, applying migration filters at each z sample, and using a 2D IFFT.The z sampling interval may be arbitrary, while the transverse grid follows the spectral bandwidth.
  • B. Planar and Volumetric Arrays: O(N log(N)) complexity applies to the 1D channel-generation procedure when IFFT algorithms are used.For a 2D plane aperture, the cost is O(NxNy log(NxNy)); for 3D samples, the procedure performs a 2D IFFT for each Nz sample.
  • B. Planar and Volumetric Arrays: The numerical procedure generates samples through spectral filtering and inverse fast Fourier transforms over compact apertures.The block diagram summarizes the process for practical array geometries.
  • B. Planar and Volumetric Arrays: For a linear aperture with Lx = 16λ and ∆ = λ/16, empirical autocorrelation matches its known closed form.Additional planar and propagated-plane experiments validate the 2D and 3D procedures and the Fourier plane-wave expansion.
  • B. Planar and Volumetric Arrays: The numerical results validate the procedure and the Fourier plane-wave expansion for modeling fields over compact rectangular arrays of practical size.The validation covers both 2D and 3D propagation models.

VI. CONCLUSIONS AND OUTLOOK

The paper develops a physically grounded, exact stochastic model for far-field small-scale fading in compact Holographic MIMO arrays and derives efficient numerical sampling procedures. It also identifies spatial degrees-of-freedom limits and outlines extensions to broader system analyses.

  • Analytical framework: The fading field is modeled as a zero-mean, spatially stationary, complex-Gaussian scalar random field satisfying the Helmholtz equation.
  • Analytical framework: The physically meaningful power spectral density yields an exact 2D Fourier plane-wave spectral representation of the small-scale fading.
  • Numerical procedure: Spatial discretization produces a computationally efficient procedure for generating accurate fading samples over compact arrays of practical size.
  • Outlook: The framework supports theoretical analysis of frequency-flat fading, capacity computation, multi-user systems with single-antenna terminals, and extensions to vector electromagnetic fields.
  • Degrees of freedom: 2/λ DoF per meter for linear apertures and π/λ^2 DoF per square meter for planar apertures are asymptotic limits.
  • Degrees of freedom: Expanding a planar aperture into a volume aperture asymptotically yields only a two-fold increase in available DoF, limiting parallel communication channels.

APPENDIX I

Appendix I connects Fourier spectral representations of random processes with linear-systems theory and derives spatial autocorrelation expressions for isotropic channels.

  • Linear-systems interpretation: The output power spectral density satisfies Sy(ω) = Sx(ω)|F(ω)|2 for an input spectrum Sx(ω) and filter response F(ω).
  • Linear-systems interpretation: A stationary white-noise process passed through a linear time-invariant filter generates a random process with a prescribed power spectral density.
  • Linear-systems interpretation: White noise is interpreted as the stochastic counterpart of the Dirac delta function, linking random-process generation to deterministic Fourier-transform filtering.
  • Spatial autocorrelation: The autocorrelation of an isotropic 3D channel follows from a 3D Fourier inverse transform of a spherically symmetric spectrum and involves the spherical Bessel function j0(x) = sin(x)/x.
  • Spatial autocorrelation: The corresponding isotropic 2D autocorrelation is obtained similarly using polar coordinates and the integral representation of Bessel functions.

APPENDIX IV

Appendix IV constructs a finite Fourier-series approximation to a bandlimited stationary Gaussian process over a finite observation interval and relates it asymptotically to the Karhunen–Loève expansion.

  • Discrete approximation: A bandlimited stationary Gaussian process over t ∈ [0,T] is approximated by partitioning its frequency integral into discrete intervals.
  • Discrete approximation: The approximation error becomes negligible as ΩT → ∞, yielding a countably finite Fourier series with statistically independent Gaussian coefficients.
  • Discrete approximation: The resulting series approximates the process over a finite interval through a periodic stationary random process with fundamental period T = 2π/∆ω.
  • Limitation: Direct application of the Karhunen–Loève expansion can produce a divergent series because the singular spectrum is evaluated on its boundary.
  • Karhunen–Loève connection: As ΩT grows, Fourier coefficients approach the power spectral density and Fourier harmonics assume the roles of Karhunen–Loève eigenvalues and eigenfunctions.

C. Computation of variances of Fourier coefficients

This appendix computes variances of Fourier coefficients by integrating the spatial power spectra over wavenumber regions, using symmetry to reduce the required calculations.

  • 1D coefficients: The coefficient variances are obtained by substituting the 1D power spectrum and κ = 2π/λ into the Fourier-coefficient expression.
  • 1D coefficients: Because the integrand is symmetric about the origin, variances for negative indices follow from those for nonnegative indices.
  • 2D coefficients: For planar apertures, the 2D spectrum is integrated over the lattice ellipse, with rotational symmetry allowing attention to the first wavenumber quadrant.
  • 2D coefficients: The 2D integrations are evaluated after transforming to polar coordinates and partitioning angular intervals according to the ordered wavenumber parameters.
  • Scope: The appendix notes that some mathematical details are omitted because of space limitations.
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