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Fourier Plane-Wave Series Expansion for Holographic MIMO Communications

Andrea Pizzo, Luca Sanguinetti, Thomas L. Marzetta

arXiv:2105.01535v3cs.ITeess.SP

TL;DR

Electromagnetically large MIMO arrays require channel models that remain accurate beyond the far-field approximation. The paper develops a Fourier plane-wave series expansion from electromagnetic first principles, yielding a physically interpretable angular-domain model valid in the radiative near-field and under arbitrary scattering. Uniform discretization produces a low-rank semi-unitarily equivalent channel approximation used for ergodic-capacity analysis.

  • Problem

    Classical stochastic channel models rely on far-field planar-wave approximations, which break down for electromagnetically large arrays and radiative near-field propagation.

  • Method

    The paper derives a Fourier plane-wave series expansion whose angular response statistically describes coupling between source and receiver directions, then discretizes it uniformly into an angular-domain channel model.

  • Results

    Uniform discretization yields a low-rank semi-unitarily equivalent approximation of the electromagnetic channel, which the paper uses to compute ergodic capacity under different channel-state-information conditions.

  • Takeaways & Limitations

    The model provides a mathematically tractable, physics-consistent framework for studying electromagnetically large and dense arrays in near-field and arbitrary-scattering settings.

Abstract

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Imagine a MIMO communication system that fully exploits the propagation characteristics offered by an electromagnetic channel and ultimately approaches the limits imposed by wireless communications. This is the concept of Holographic MIMO communications. Accurate and tractable channel modeling is critical to understanding its full potential. Classical stochastic models used by communications theorists are derived under the electromagnetic far-field assumption, i.e. planar wave approximation over the array. However, such assumption breaks down when electromagnetically large (compared to the wavelength) antenna arrays are considered. In this paper, we start from the first principles of wave propagation and provide a Fourier plane-wave series expansion of the channel response, which fully captures the essence of electromagnetic propagation in arbitrary scattering and is also valid in the (radiative) near-field. The expansion is based on the Fourier spectral representation and has an intuitive physical interpretation, as it statistically describes the angular coupling between source and receiver. When discretized uniformly, it leads to a low-rank semi-unitarily equivalent approximation of the electromagnetic channel in the angular domain. The developed channel model is used to compute the ergodic capacity of a point-to-point Holographic MIMO system with different degrees of channel state information.

I. INTRODUCTION

Holographic MIMO seeks to exploit electromagnetic propagation with dense, electromagnetically large arrays, creating a need for accurate and tractable channel models beyond far-field approximations. The paper develops a Fourier plane-wave framework that separates array geometry from scattering and supports near-field, arbitrary-scattering analysis.

  • Holographic MIMO is designed to fully exploit the propagation characteristics of an electromagnetic channel.
  • Electromagnetically large arrays motivate channel models that accurately and tractably represent wave propagation.Deterministic models are accurate but site-specific because they rely on numerical electromagnetic solvers; stochastic models represent classes of environments.
  • Far-field plane-wave models break down in the radiative near-field because wavefront curvature causes magnitude and phase errors across the array.The far-field approximation treats wavefronts as locally planar over the entire array.
  • Exact plane-wave formulations can describe propagation at any source–receiver distance and under arbitrary scattering by mapping every transmit direction to every receive direction.Spherical waves can be decomposed into infinitely many plane waves, while scattering is embedded in an angular response.
  • The proposed model discretizes the Fourier plane-wave expansion into a low-rank semi-unitarily equivalent angular-domain channel representation.The representation separates deterministic array geometry from stochastic scattering and supports separate array and signal-processing design.
  • The paper applies the model to ergodic-capacity analysis under different degrees of channel state information.The framework also differs from prior work by covering both link ends and arbitrary scatterer configurations.

B. Fourier plane-wave representation of electromagnetic channels

The electromagnetic channel is represented as a spatially stationary random field using propagating plane waves and an angular response that couples source and receive directions. Its stochastic structure is governed by a spectral factor, yielding a tractable second-order model and a bandlimited representation.

  • After excluding reactive mechanisms near sources and scatterers, the channel response can be modeled as a spatially stationary electromagnetic random field.
  • The plane-wave representation decomposes the channel into source response, angular response, and receive response terms.The source and receive responses connect antenna locations to propagation directions, while the angular response maps source directions to receive directions.
  • The angular response is modeled as a non-negative spectral factor multiplied by independent, unit-variance circularly symmetric complex-Gaussian coefficients.This construction provides a stochastic representation with statistically independent complex-Gaussian random coefficients.
  • The power spectral density specifies average power transfer between every transmit and receive propagation-direction pair.Its spectral factor can be selected to represent a prescribed propagation environment, including isotropic or non-isotropic scattering.
  • The channel is bandlimited, with maximum circular bandwidth |D(κ)| = πκ^2 under isotropic scattering.
  • Under isotropic propagation, the model recovers Clarke’s isotropic sinc(2r/λ) correlation and is asymptotically valid.

III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS

The section derives a Fourier plane-wave series that approximates electromagnetic channel responses over finite planar arrays. Under electromagnetically large arrays, the approximation uses discretized plane waves and random Fourier coefficients, with accuracy improving as normalized array size increases.

  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: The expansion is motivated by the analogy between Fourier integrals and Fourier series, with plane waves corresponding to spatial-frequency Fourier harmonics.For fixed longitudinal coordinates, source and received plane waves are phase-shifted two-dimensional spatial-frequency harmonics.
  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: Electromagnetically large arrays are assumed, meaning their minimum normalized dimensions satisfy min(L/λ) ≫ 1.The assumption concerns size relative to wavelength rather than physical size alone.
  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: The discrete plane waves are sampled at array-dependent spatial frequencies and restricted to source and receive lattice ellipses.The corresponding index sets have cardinalities nS and nR.
  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: Theorem 2 approximates the channel response over finite source and receive array regions using a Fourier plane-wave series.The expansion applies for fixed source and receive longitudinal coordinates and represents the channel over the arrays’ fundamental periods.
  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: The Fourier coefficients are statistically independent, circularly symmetric, complex-Gaussian variables whose variances are determined by the array dimensions.These random coefficients weight the discretized plane waves in the channel expansion.
  • III. FOURIER PLANE-WAVE SERIES OF STOCHASTIC ELECTROMAGNETIC CHANNELS: The approximation error decreases as normalized array dimensions grow and vanishes asymptotically when the continuous and series representations coincide.The construction produces a periodic stationary random field and is accurate under the electromagnetically large-array assumption.

B. Physical considerations

The discretized expansion gives coupling coefficients a physical interpretation as angular power transfer between source and receive angular sets. Their joint pattern determines parallel-channel count and diversity, while finite array size limits the essential number of plane waves.

  • B. Physical considerations: The continuous source and receive plane waves become discretized angular waves, implying that finite arrays need only finitely many plane waves for essential channel information.The continuous angular response is replaced by a sequence indexed by receive and source angular sets.
  • B. Physical considerations: Each coupling coefficient represents the fraction of power transferred from a source angular set to a receive angular set.The coefficients form a discrete angular power distribution.
  • B. Physical considerations: Coupling strength depends jointly on array sizes and scattering, determining both the number of parallel channels and the level of diversity.Parallel channels follow from coupled source and receive angular sets; diversity varies across source angular sets.
  • B. Physical considerations: In the illustrated example, three source angular sets couple to six receive sets, producing three parallel channels with diversity levels three, two, and one.The source sets are represented by orange, blue, and green groups in the matrix arrangement.

C. Physical modeling of coupling coefficients

The coupling model expresses angular power transfer through array-defined source and receive regions and supports isotropic, clustered, and vMF-based non-isotropic scattering. Numerical examples show that array resolution and scattering jointly shape significant coupling.

  • C. Physical modeling of coupling coefficients: The angular power-transfer function specifies the fraction of power transmitted over a source angular region and received over a receive angular region.This establishes that power transfer is generally coupled between the two arrays.
  • C. Physical modeling of coupling coefficients: Under isotropic propagation, scattering decouples and source and receive coupling powers can be computed in closed form as solid angles.The receiver-side quantity is determined by the array’s xy-dimensions.
  • C. Physical modeling of coupling coefficients: A clustered non-isotropic model makes coupling nonzero only when source and receive angular sets are at least partly subtended by scattering clusters.Unlike isotropic propagation, coupling then depends jointly on spatial resolution and scattering.
  • C. Physical modeling of coupling coefficients: A mixture of 3D von Mises-Fisher angular densities offers a tractable accuracy trade-off for modeling non-isotropic propagation.The mixture uses modal directions, concentration parameters, and positive component weights.
  • C. Physical modeling of coupling coefficients: For isotropic scattering with L/λ = {10, 30}, nonzero coupling coefficients number nR = {344, 2928} and lie within the lattice ellipse.Their bowl-shaped, unequal strengths imply spatial correlation even under isotropic scattering.
  • C. Physical modeling of coupling coefficients: For the non-isotropic example, significant coefficients concentrate near modal directions, with n′R = 229 sufficient to capture 99.7% of channel power.The value combines 145 significant sets for one array size and 84 for the other figure panel.

D. Connection to Karhunen-Loeve expansion

The electromagnetic channel admits an asymptotic Karhunen–Loève expansion in spatial Fourier eigenfunctions, with lower dimensionality imposed by the double-sphere spectral support. Nyquist-rate sampling preserves the channel information needed to exploit its spatial degrees of freedom.

  • Spatial expansion: The spatially stationary electromagnetic field expands over source and receiver Fourier eigenfunctions with non-negative coupling eigenvalues.This is identified as the asymptotic Hilbert–Schmidt and Karhunen–Loève decomposition of the correlation kernel.
  • Spatial expansion: Unlike temporal sampling, the spatial expansion integrates the singularly integrable spectral density over frequency neighborhoods rather than sampling only discrete frequencies.Direct sampling at integer multiples of the fundamental spatial frequencies would produce a divergent series.
  • Spatial expansion: The electromagnetic field has lower dimensionality because its six-dimensional power spectral density is impulsive and supported on a double sphere of radius κ.The resulting basis describes the physically admissible spatial modes rather than a full unconstrained six-dimensional spectrum.
  • Sampling: Uniform spatial sampling must satisfy the Nyquist condition because the channel is circularly bandlimited with maximum bandwidth πκ^2.The sampling requirement applies for any scattering environment.
  • Sampling: Sampling at Nyquist rate or higher ensures no channel information is lost, enabling exploitation of the electromagnetic channel’s available degrees of freedom.The paper connects this property directly to the scope of Holographic MIMO systems.

A. Karhunen-Loeve expansion

The channel is represented in fixed angular basis functions through an angular random matrix whose entries encode source–receiver coupling strengths. This yields a physics-based low-rank approximation that preserves the dominant singular values of the electromagnetic channel.

  • Basis construction: The source and receiver basis vectors are formed from sampled Fourier harmonics and assemble into semi-unitary matrices Φ_S and Φ_R.Under uniform sampling, these matrices are collected from 2D inverse discrete Fourier transform bases.
  • Angular representation: The fixed angular bases do not depend on channel statistics, and each coupling coefficient specifies the average energy transferred between one source and receive basis function.The average channel power is preserved by the coupling coefficients.
  • Angular representation: The angular matrix H_a contains independent complex-Gaussian coupling coefficients scaled by the source–receiver standard deviations.Its dimensions are n_R × n_S, while the scaling collects n_Rn_S coupling strengths.
  • Low-rank structure: The angular matrix is semi-unitarily equivalent to H, so their most significant min(n_R,n_S) singular values are identical.This provides a low-rank approximation using geometry-defined spatial bases rather than full antenna-dimensional unitary factors.
  • Low-rank structure: The number of channel singular values equals the number of nonzero rows or columns of H_a and corresponds to the channel’s degrees of freedom.The degrees of freedom depend jointly on scattering conditions and array sizes, not solely on antenna count.
  • Model structure: The general model is not Kronecker structured, although a Kronecker channel follows when the propagation condition is separable.The separable model generalizes a prior far-field linear-array model to planar arrays and near-field propagation.

B. Channel Statistics

The channel covariance has a compact eigenstructure determined by angular coupling strengths, while its effective rank depends on array dimensions and scattering richness. The model explains why electromagnetic channels remain spatially correlated and can approximate Clarke’s model with few modes.

  • Covariance structure: The covariance eigendecomposition uses fixed Kronecker products of source and receiver basis vectors as eigenvectors, with NSNRσ^2 coupling powers as eigenvalues.Only n_Rn_S eigenvalues are physically nonzero, reducing covariance computation relative to direct antenna-dimensional calculation.
  • Covariance structure: The covariance matrix is generally coupled rather than Kronecker separable; separability makes transmit or receive correlations independent of the opposite link end.The general coupled model does not have this invariance across the opposite antenna index.
  • Degrees of freedom: The number of significant coupling coefficients depends on scattering directionality, array size, and antenna spacing, rather than antenna count alone.For the illustrated L/λ = 10 arrays, Fig. 5 compares λ/2 spacing with λ/4 spacing and Fourier, Clarke, and i.i.d. Rayleigh models.
  • Degrees of freedom: More uneven coupling coefficients produce steeper eigenvalue decay, concentrating channel energy into fewer dominant modes.This links propagation anisotropy to the effective rank observed in the covariance spectrum.
  • Model comparison: For isotropic propagation, the Fourier model gives an n_R-order low-rank approximation of Clarke’s model, discarding approximately 4.6% of total channel power at L/λ = 10.The discarded-power error falls to 2.3% at L/λ = 30 and approaches zero asymptotically.
  • Model comparison: The electromagnetic channel cannot be modeled as i.i.d. Rayleigh fading because its coupling strengths are unequal and therefore induce spatial correlation.Clarke’s isotropic model is identified as the closest physically tenable approximation to i.i.d. Rayleigh fading.

C. Channel generation and migration filters

The channel can be generated from angular coupling strengths, Gaussian coefficients, fixed Fourier transforms, and deterministic migration filters. These filters account for z-axis propagation, while uniform sampling and pilot design determine practical representation and estimation requirements.

  • Channel generation: Channel generation samples independent Gaussian coefficients, forms the angular matrix H_a, applies migration filters, and transforms it through the fixed array bases.The coupling matrix is determined by the coupling strengths collected in Σ.
  • Channel generation: With uniform sampling, the spatial basis matrices reduce to 2D IDFT transforms, making the final spatial-domain conversion relatively low complexity through the FFT.Changing the source and receiver z-positions requires only updated propagation filtering and transformation steps.
  • Channel generation: For capacity-type metrics, the channel can instead be generated from the covariance eigendecomposition vec(H) = UΛ^1/2vec(W), without applying propagator matrices.This alternative relies only on the statistical equivalence between the migrated and angular representations.
  • Model assumptions: The Fourier expansion generates a periodic stationary random field, so arrays must be sufficiently large for the aperiodic channel’s correlation properties to remain preserved.The exact repetition periods are determined separately at the source and receiver.
  • Channel estimation: Approximately n_Rn_S pilot signals suffice for estimating the coupling coefficients by transmitting along the known deterministic basis vectors.The required pilot resources increase with normalized array length because n_R and n_S increase accordingly.
  • Channel estimation: The coupling strengths vary slowly relative to H and can be estimated from repeated projections onto receive basis vectors after transmitting source basis functions.Only the n_Rn_S real-valued coupling powers are needed to specify the correlation matrix.

V. CAPACITY EVALUATION

The paper evaluates ergodic capacity in the angular-domain channel model under different channel-state-information assumptions. Its Fourier plane-wave model closely matches the capacity approximation and captures the effects of sub-wavelength antenna spacing.

  • Capacity formulation: The spatial-domain MIMO capacity is evaluated through the semi-unitarily equivalent angular-domain channel representation.The angular transformation depends only on array geometry, while signal processing operates on the angular channel coefficients.
  • Implementation implications: The transceiver can operate with N_S ≫ n_S and N_R ≫ n_R without increasing signal-processing complexity, since processing depends on the angular dimensions.This applies to channel estimation, optimal signaling, and coding.
  • Channel-state information: The model supports ergodic-capacity analysis with instantaneous receiver channel knowledge, including exact and large-dimensional formulations based on channel eigenvalues.Under large n_S and n_R, random-matrix tools provide an asymptotic approximation using fixed-point coefficients Γ_R and Γ_S.
  • Numerical evaluation: Fig. 7 varies ergodic capacity with antenna spacing from λ/2 to λ/8 for a squared array with L/λ = 10 and snr = 10 dB.The Fourier plane-wave model uses the settings of Fig. 3(a) and Fig. 4(a).
  • Numerical evaluation: The Fourier plane-wave model achieves a perfect match with the finite-dimensional capacity calculation, validating the low-rank approximation under isotropic propagation.Compared with i.i.d. Rayleigh fading, a large capacity gap appears below λ/2 spacing because antenna correlation increases as spacing decreases.

B. Perfect channel knowledge at source and receiver

With perfect channel knowledge at both transmitter and receiver, capacity-achieving transmission uses the singular modes and waterfilling power allocation. At high SNR, capacity growth reflects the available degrees of freedom, while spatial correlation reduces the growth slope and shifts capacity downward.

  • Perfect channel knowledge at source and receiver: Perfect channel knowledge enables singular-value decomposition of H_a and capacity-achieving transmission over its right singular vectors.The optimal powers are computed with the waterfilling algorithm.
  • Capacity versus SNR: Fig. 8 plots ergodic capacity C versus snr for L/λ = 10 and λ/4-spaced antenna elements, corresponding to N_R = 1600.The Fourier plane-wave model uses the setups of Fig. 3(a) and Fig. 4(a).
  • Capacity versus SNR: At large SNR, an increasing number of communication modes becomes active, and capacity scales linearly with log2(snr).The asymptotic slope equals the number of degrees of freedom in (48).
  • Capacity versus SNR: Spatial correlation reduces the high-SNR slope and introduces a negative capacity shift relative to i.i.d. Rayleigh fading and Clarke’s model.These effects arise in the capacity comparison reported for Fig. 8.
  • Statistical channel knowledge: With transmitter knowledge of coupling-strength statistics, the optimal angular covariance is diagonal and statistically independent information streams are sent angularly.The corresponding spatial-domain input symbols are correlated through the array transformation.
  • Scope: The model is valid in the near-field and under arbitrary scattering, extending the scope of the capacity analysis beyond far-field channel assumptions.The conclusion identifies electromagnetically large and dense arrays as a principal application area.

APPENDIX

The appendix constructs the discrete Fourier plane-wave expansion by partitioning transmit and receive angular domains, projecting the spectral field onto orthonormal functions, and numerically evaluating the resulting coefficient variances.

  • Discretization: The appendix discretizes the Fourier plane-wave representation over a finite spatial observation region by uniformly partitioning transmit and receive angular domains.The resulting angular sets are denoted W_S(m_x,m_y) and W_R(ℓ_x,ℓ_y).
  • Coefficient construction: Applying the first mean-value theorem to each partition yields the approximated Fourier series coefficients H_a(ℓ_x,ℓ_y,m_x,m_y).The coefficients are indexed by receive and source angular sets.
  • Coefficient construction: The coefficients are mutually independent circularly symmetric complex-Gaussian variables because they result from projecting a four-dimensional white-noise field onto orthonormal functions.Their variances are obtained from the average power expression in (23).
  • Variance evaluation: Unlike the isotropic-scattering case, the coefficient variances require a general numerical procedure for non-isotropic propagation conditions.A closed-form variance expression is not used for every possible non-isotropic environment.
  • Coordinate representation: The angular integration variables correspond to cosine directions, while spherical-coordinate mappings use k_x = sin θ_R cos φ_R and k_y = sin θ_R sin φ_R.The resulting relation at the receiver is γ(k_x,k_y) = cos θ_R.
  • Integration regions: The receive integration region is the intersection of each angular set with the projected upper hemisphere and is divided into three azimuthal subregions.The source region is handled similarly, with additional fourth-quadrant regions omitted from the appendix for space reasons.
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