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Wavenumber-Division Multiplexing in Line-of-Sight Holographic MIMO Communications

Luca Sanguinetti, Antonio A. D'Amico, Merouane Debbah

arXiv:2106.12531v3eess.SPcs.ITmath.IT

TL;DR

The paper addresses how to represent and implement communication between spatially continuous volumes, particularly for LoS propagation where wave curvature affects MIMO behavior. It develops an electromagnetic MIMO model and Fourier-basis WDM, finding that receiver-only one-tap processing can match SVD with water-filling when receiver size is comparable to transmission range, although WDM modes become non-interfering only asymptotically.

  • Problem

    Classical LoS MIMO results rely on planar wavefronts, an approximation that becomes inadequate at high frequencies where wave curvature can make the channel high-rank.

    Source: §I · p. 1

  • Method

    The paper models electromagnetic communication between continuous volumes and applies Fourier basis functions to parallel LoS segments to create wavenumber-division multiplexing.

    Source: A. Motivation and Contribution · p. 2, §VI · p. 14

  • Results

    WDM with one-tap multipliers provides the same spectral efficiency as SVD with water-filling when receiver size is comparable to transmission range.

    Source: A. Motivation and Contribution · p. 2

  • Takeaways & Limitations

    WDM modes remain interferential at finite receiver size, with interference decreasing as the receiver grows and orthogonality reached only asymptotically.

    Source: A. Motivation and Contribution · p. 2

    Because the electromagnetic channel response has non-finite spatial support, WDM cannot provide non-interfering modes at finite receiver size.

    Source: A. Motivation and Contribution · p. 2

Abstract

from arXiv · show

Starting from first principles of wave propagation, we consider a multiple-input multiple-output (MIMO) representation of a communication system between two spatially-continuous volumes. This is the concept of holographic MIMO communications. The analysis takes into account the electromagnetic interference, generated by external sources, and the constraint on the physical radiated power. The electromagnetic MIMO model is particularized for a pair of parallel line segments in line-of-sight conditions. Inspired by orthogonal-frequency division-multiplexing, we assume that the spatially-continuous transmit currents and received fields are represented using the Fourier basis functions. In doing so, a wavenumber-division multiplexing (WDM) scheme is obtained, which is not optimal but can be efficiently implemented. The interplay among the different system parameters (e.g., transmission range, wavelength, and sizes of source and receiver) in terms of number of communication modes and level of interference among them is studied with conventional tools of linear systems theory. Due to the non-finite support (in the spatial domain) of the electromagnetic channel, WDM cannot provide non-interfering communication modes. The interference decreases as the receiver size grows, and goes to zero only asymptotically. Different digital processing architectures, operating in the wavenumber domain, are thus used to deal with the interference. The simplest implementation provides the same spectral efficiency of a singular-value decomposition architecture with water-filling when the receiver size is comparable to the transmission range. The developed framework is also used to represent a classical MIMO system and to make comparisons. It turns out that the latter achieves better performance only when a higher number of radio-frequency chains is used.

I. INTRODUCTION

At high frequencies, wavefront curvature can make line-of-sight channels high-rank, challenging the classical planar-wave account of LoS MIMO. The paper develops an electromagnetic MIMO model and applies Fourier bases to form an implementable wavenumber-division scheme.

I. INTRODUCTION

  • At high frequencies, reduced wavelengths and shorter ranges make wave curvature significant, so LoS MIMO can have high rank rather than the single mode predicted by planar-wave models.The introduction frames sub-THz spectrum as relevant to applications ranging from short-range interconnections to longer-range backhaul and satellite links.Source: §I · p. 1

A. Motivation and Contribution

  • The paper models communication between spatially continuous volumes and proposes Fourier-basis WDM for parallel line segments, while accounting for electromagnetic interference and radiated power.Unlike OFDM, WDM modes are not non-interfering; the simplest one-tap architecture matches SVD with water-filling when receiver size is comparable to transmission range.Source: A. Motivation and Contribution · pp. 1–2

B. Paper outline and notation

  • The paper proceeds from MIMO and LoS models to electromagnetic modeling, WDM design, spectral-efficiency evaluation, and conclusions.This outline describes the progression of the paper rather than a technical result.Source: B. Paper outline and notation · p. 2

II. MIMO COMMUNICATIONS

  • A conventional narrowband MIMO system maps transmitted signals to received signals through a channel matrix, with additive Gaussian noise and a total transmit-power constraint.Each channel-matrix entry represents the gain between one transmit antenna and one receive antenna.Source: §II · p. 2

A. Capacity and number of degrees-of-freedom

  • Whitening the noise and applying an SVD decomposes MIMO into independent single-input, single-output modes, with nonzero singular values identifying usable communication modes.The number of modes determines the channel degrees of freedom, and water-filling allocates power across them.Source: A. Capacity and number of… · p. 3

B. Line-of-sight channels

  • Under far-field LoS and the planar-wave approximation, the channel has rank one; when range is comparable to array size, spherical-wave modeling can yield multiple modes.For uniform linear arrays, the cited full-rank condition uses equal antenna counts and spacing Δ = λd/N.Source: B. Line-of-sight channels · p. 3

III. ELECTROMAGNETIC WAVE COMMUNICATION

The electromagnetic model describes propagation between spatially continuous volumes through Maxwell-based fields and a Green function. It adds external electromagnetic interference as a random field whose spatial samples can be correlated.

III. ELECTROMAGNETIC WAVE COMMUNICATION

  • The model considers monochromatic current sources in a bounded source volume and observes the sum of the information-carrying field and external electromagnetic interference in the receive volume.The setting is an infinite homogeneous medium, and the bounded source-current condition is linked to bounded radiated power.Source: §III · p. 3

A. The Green function and signal-carrying electric field

  • In a homogeneous isotropic medium, the electric field obeys a vector wave equation, and its unbounded-medium solution is given by the Green function.The passage also gives a free-space far-field approximation and identifies the wave vector associated with propagation.Source: A. The Green function and… · pp. 3–4

B. The EMI field

  • External interference is modeled as incoming plane waves with random amplitudes; under isotropic propagation, its spatial samples are generally correlated.For the stated isotropic condition, samples along a straight line are uncorrelated only at integer multiples of λ/2 spacing.Source: B. The EMI field · p. 4

C. The electromagnetic MIMO channel

The electromagnetic MIMO channel is formed by expanding transmit currents and projecting received fields onto chosen basis functions. Its input-output model incorporates hardware noise, while the current-energy constraint is related to physical radiated power.

C. The electromagnetic MIMO channel

  • The model expands the transmit current in orthonormal basis functions and projects the received field onto a receive basis to produce spatial samples with receiver hardware noise.The hardware-noise samples are modeled as independent circularly symmetric Gaussian variables.Source: C. The electromagnetic MIMO channel · p. 4
  • Each channel-matrix entry represents coupling between a source mode and a reception mode, and the capacity-achieving architecture uses an SVD of the noise-whitened channel.The architecture performs electromagnetic current and field operations in an inner analog stage and signal generation and processing in an outer digital stage.Source: C. The electromagnetic MIMO channel · p. 5

D. The electromagnetic radiated power

  • Constraining the source current's L2 norm limits radiated power, but bounding radiated power does not necessarily bound that norm.The cited upper bound on radiated power establishes the first implication; the converse need not hold.Source: D. The electromagnetic radiated power · p. 5

IV. WAVENUMBER-DIVISION MULTIPLEXING

WDM adapts the OFDM idea to electromagnetic MIMO, seeking a simple, physically interpretable scheme where optimal eigenfunction bases are computationally demanding and difficult to implement. The section reviews OFDM and sets up a one-dimensional model using Fourier basis functions for parallel source and receiver segments.

IV. WAVENUMBER-DIVISION MULTIPLEXING

  • WDM’s contribution is an end-to-end description and analysis of Fourier-based communication, rather than the use of Fourier processing itself.The scheme is proposed as a simple, physically interpretable alternative to computationally demanding eigenfunction bases.Source: §IV · p. 5

A. Review of orthogonal-frequency-division multiplexing

  • In OFDM, observing the received signal over an interval that includes the channel response’s support can yield orthogonal parallel transmissions without a cyclic prefix.The reviewed conditions use an extended observation interval, Tr ≥ T + Tg; the section notes these conditions inform WDM design.Source: A. Review of… · p. 6

B. System and signal model

  • The model uses a one-dimensional source segment and a parallel receiver segment, with the received field projected onto receiver basis functions.The source and receiver lie along the z-axis, with the receiver at distance d; the projection processes the field along that axis.Source: B. System and signal model · pp. 6–7
  • Fourier source modes radiate in angular directions, with beamwidth 2λ/Ls, inversely proportional to source length Ls.The section describes Fourier basis functions for the transmit current and relates each mode to a radiation direction.Source: B. System and signal model · pp. 6–7

C. The coupling coefficients

The spatial channel’s unbounded support prevents exact decoupling of the Fourier communication modes for a finite receiver. Increasing receiver size reduces interference, with near-orthogonality reported once its size is on the order of the source distance.

C. The coupling coefficients

  • The channel spectrum is band-limited to |κz| ≤ 2π/λ, and its approximate 3-dB bandwidth is 0.6 times that limit.A lower wavelength gives a larger spatial-frequency bandwidth.Source: C. The coupling coefficients · p. 8
  • At d = 5 m, Lr = 0.4 m distorts the received spectrum and destroys orthogonality among three modes, while Lr = 5 m leaves them barely affected.The larger receiver captures the electric field almost undistorted over the considered wavenumber interval.Source: C. The coupling coefficients · p. 8, Figure 5 · p. 8
  • At d = 5 m, the number of coefficients in the 3-dB bandwidth is 3, 17, and 25 for Lr = 1, 5, and 10 m; at d = 10 m, it is 1, 9, and 16.The number of significant coefficients decreases as distance increases; the approximation from (1) is good only when d ≥ Lr.Source: C. The coupling coefficients · p. 9, Figure 7 · p. 9
  • As receiver length increases, each normalized coupling coefficient tends toward a sampled version of the channel spectrum.This convergence is stated for a fixed distance and holds asymptotically as receiver length grows unboundedly.Source: C. The coupling coefficients · pp. 8–9
  • Exact decoupling occurs only as receiver length Lr tends to infinity, because the spatial channel response has unbounded support.A finite receiver can approximate orthogonality when its extra length beyond the source is large enough to capture the response’s significant support.Source: C. The coupling coefficients · p. 9
  • As distance grows with array sizes and wavelength fixed, the number of significant coefficients tends to one, matching the single degree of freedom under planar-wavefront propagation.In the far field, the channel is approximated by a planar wavefront.Source: C. The coupling coefficients · p. 9

D. The EMI samples

The spatial correlation of electromagnetic-interference (EMI) samples depends on propagation conditions and receiver size. Finite receiver size leaves even isotropic EMI samples correlated, while non-isotropic propagation can produce stronger correlation.

D. The EMI samples

  • The spatial correlation of EMI samples depends on the power angular density; under isotropic propagation, the spatial correlation function is sinc-shaped.The isotropic condition is characterized by the stated angular propagation model.Source: D. The EMI samples · p. 10
  • Finite Lr leaves EMI samples correlated even under isotropic propagation because the receiver limits the spatial observation interval.As Lr grows, the isotropic-case correlation diminishes.Source: D. The EMI samples · p. 10
  • As Lr→∞, covariance R becomes diagonal with entries Lr/(2κ), so the samples become uncorrelated but their variance grows without bound.The paper notes that this unbounded variance is not physically meaningful asymptotically, while the model is considered safe for practical Lr≥d.Source: D. The EMI samples · p. 10
  • Under non-isotropic propagation, large eigenvalue variations across the considered Lr indicate that EMI samples may be highly correlated.Fig. 8 considers Ls = 0.2 m and λ = 0.01 m under isotropic and non-isotropic conditions.Source: D. The EMI samples · p. 10, Figure 8 · p. 10

V. SPECTRAL EFFICIENCY

The spectral-efficiency analysis compares WDM processing architectures under finite receiver size, where communication modes are not orthogonal. It imposes a transmit-power constraint and assumes isotropic EMI.

V. SPECTRAL EFFICIENCY

  • For finite Lr, WDM modes are not guaranteed to be orthogonal, so wavenumber-domain digital processing is needed to handle interference and compare spectral efficiency.Orthogonality is guaranteed only as Lr→∞.Source: §V · p. 10
  • The system SNR is defined for the spectral-efficiency analysis, but the supplied passage does not state its expression.The SNR definition follows the stated power constraint.Source: §V · p. 10
  • The analysis constrains transmit power by PN_n=1 |ξn|²/Ls ≤ Ps and assumes isotropic EMI.It defines P = (κZ0)²Ps, with Ps measured in A² and P in V²/m².Source: §V · p. 10

A. With Singular-Value Decomposition

The SVD architecture achieves WDM's maximum spectral efficiency by decomposing a noise-whitened channel. Its channel and noise-covariance matrices depend on system parameters, and computational complexity can become high at large sizes or high frequencies.

A. With Singular-Value Decomposition

  • The maximum WDM spectral efficiency is achieved using an SVD of the noise-whitened channel, with noise covariance obtained by Cholesky decomposition.The architecture applies the resulting singular-vector processing at the transmitter and receiver.Source: A. With Singular-Value Decomposition · p. 11
  • H and C are square N×N matrices assumed known at both link ends; H depends on wavelength, distance, and source and receiver sizes.Computing C also requires knowledge of the noise variance.Source: A. With Singular-Value Decomposition · p. 11
  • The preprocessing and postprocessing cost can become high for large sources or high frequencies; at λ = 0.001 m, Nmax reaches 401 for Ls = 0.2 m and 1001 for Ls = 0.5 m.The stated maximum complex-operation count scales with N².Source: A. With Singular-Value Decomposition · p. 11

B. With Linear Processing at the Receiver only

Receiver-only WDM uses linear receive processing without source preprocessing, with choices that trade interference handling against computational simplicity. As receiver size grows, the modes approach orthogonality, enabling a simple one-tap implementation.

B. With Linear Processing at the Receiver only

  • With no source preprocessing, receiver processing transforms the observation and noise, and decoding treats interference as Gaussian noise.The transformed noise entries are independent, identically distributed Gaussian variables.Source: B. With Linear Processing at the… · p. 11
  • MMSE combining maximizes each stream's SINR, whereas maximum-ratio combining avoids matrix inversion by maximizing desired-signal power while neglecting interference.The MMSE combiner requires an N×N matrix inversion, which may be cumbersome for large N.Source: B. With Linear Processing at the… · p. 11
  • As Lr grows, WDM modes become orthogonal and the asymptotically optimal receiver is a bank of one-tap complex-valued multipliers.This is identified as the simplest WDM implementation.Source: B. With Linear Processing at the… · p. 11
  • Iterative water-filling can fail to converge depending on interference; for sufficiently large Lr, the analysis instead neglects interference when computing optimal powers.The powers are computed with respect to the squared diagonal gains |eHnn|².Source: B. With Linear Processing at the… · p. 11

C. Numerical analysis

The numerical comparisons show how WDM performance depends on receiver size and processing: simple processing approaches more complex schemes as the receiver grows. Comparisons with EM-MF and conventional MIMO highlight trade-offs between spectral efficiency and implementation complexity.

C. Numerical analysis

D. Matched electromagnetic processing at the receiver

E. Comparisons with a fully-digital MIMO communication scheme

  • The simplest WDM implementation highly outperforms MIMO with MR combining for any Lr and MIMO with SVD processing when Lr > 3.MIMO achieves marginal gains over WDM with SVD only when Nr/N > 8, at much higher implementation complexity.Source: E. Comparisons with a fully-digital… · pp. 13–14

VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

The paper develops an electromagnetic MIMO model and applies Fourier basis functions to create WDM for parallel one-dimensional apertures in LoS. WDM modes become orthogonal only asymptotically as receiver size grows, while its simplest implementation performs sufficiently well when receiver size is of the same order as transmission distance.

VI. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

  • The electromagnetic MIMO representation models wave propagation between spatially continuous volumes and serves as a baseline for optimal or implementable holographic MIMO schemes.Source: §VI · p. 14
  • WDM's Fourier modes are not orthogonal for finite receiver size; for the investigated scenarios, simplest WDM performs sufficiently well when receiver size is of the same order as source distance.Source: §VI · p. 14
  • Future directions include multipath, LoS blockage, arbitrary source and receiver orientations, multiuser communication, and two-dimensional apertures.Source: §VI · p. 14

APPENDIX A

The appendices derive a radiated-power bound, characterize fields generated by basis functions, and establish Fourier-mode orthogonality under stated conditions.

APPENDIX A

  • The radiated-power derivation uses the far-field electric field and Poynting vector to obtain an upper bound involving the current distribution.Source: Appendix A · p. 14

APPENDIX B

  • The field and radiation-pattern derivations characterize the electric field generated by a source basis function and its normalized radiation pattern.Source: Appendix B · pp. 14–15

APPENDIX C

  • Under conditions (66) and (67), the appendix reduces the relevant integral to its i = 0 term, from which the Fourier-mode orthogonality condition follows.Source: Appendix C · p. 15
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