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Pattern-Division Multiplexing for Multi-User Continuous-Aperture MIMO

Zijian Zhang, Linglong Dai

arXiv:2111.08630v3cs.ITcs.SYeess.SPeess.SYmath.IT

TL;DR

Flexible pattern design for multi-user CAP-MIMO is difficult because continuous patterns make optimization challenging. The paper develops PDM to replace that functional design with finite-basis projection optimization and uses it in a BCD scheme. The proposed scheme achieves higher sum-rate than benchmark schemes, including a reported 24% gain over MF in one setting.

  • Problem

    Existing CAP-MIMO pattern methods mainly use fixed special functions in restricted scenarios, while flexible design for general multi-user settings remains understudied and entails difficult functional programming.

    Source: A. Prior Works · p. 2

  • Method

    PDM projects continuous patterns onto orthogonal bases for finite projection-length optimization, and a BCD scheme uses PDM to solve the multi-user sum-rate design problem.

    Source: B. Our Contributions · p. 2

  • Results

    At AT = 1 m^2, the proposed scheme achieves 17.90 bps/Hz, about 24% higher than the 13.69 bps/Hz achieved by MF.

    Source: B. Sum-Rate Against the Aperture Area AT · p. 11

  • Takeaways & Limitations

    The authors present PDM as a signal-processing framework for CAP-MIMO pattern design and related problems such as channel estimation and energy-efficiency optimization.

    Source: §VI · p. 14

Abstract

from arXiv · show

In recent years, thanks to the advances in meta-materials, the concept of continuous-aperture MIMO (CAP-MIMO) is reinvestigated to achieve improved communication performance with limited antenna apertures. Unlike the classical MIMO composed of discrete antennas, CAP-MIMO has a quasi-continuous antenna surface, which is expected to generate any current distribution (i.e., pattern) and induce controllable spatial electromagnetic (EM) waves. In this way, the information is directly modulated on the EM waves, which makes it promising to approach the ultimate capacity of finite apertures. The pattern design is the key factor to determine the communication performance of CAP-MIMO, but it has not been well studied in the literature. In this paper, we develop pattern-division multiplexing (PDM) to design the patterns for CAP-MIMO. Specifically, we first study and model a typical multi-user CAP-MIMO system, which allows us to formulate the sum-rate maximization problem. Then, we develop a general PDM technique to transform the design of the continuous pattern functions to the design of their projection lengths on finite orthogonal bases, which can overcome the challenge of functional programming. Utilizing PDM, we further propose a block coordinate descent (BCD) based pattern design scheme to solve the formulated sum-rate maximization problem. Simulation results show that, the sum-rate achieved by the proposed scheme is higher than that achieved by benchmark schemes, which demonstrates the effectiveness of the developed PDM for CAP-MIMO.

I. INTRODUCTION

CAP-MIMO uses a quasi-continuous aperture to generate configurable current patterns and radiated EM waves, making pattern design central to communication performance. The paper introduces a multi-user design framework and PDM to address the challenge of optimizing continuous patterns.

I. INTRODUCTION

  • CAP-MIMO replaces discrete half-wavelength-spaced antennas with a quasi-continuous EM surface intended to generate arbitrary current distributions.The paper notes that practical hardware currently has difficulty generating arbitrary patterns on a continuous aperture.Source: §I · p. 1

A. Prior Works

  • Earlier CAP-MIMO pattern schemes used fixed special functions and were effective mainly in restricted settings such as single-user, linear-aperture, or near-field transmission.The paper identifies flexible design for general scenarios with complex propagation and distributed receivers as an understudied problem.Source: A. Prior Works · p. 2

B. Our Contributions

  • The paper models a multi-user CAP-MIMO system and formulates pattern design as a sum-rate maximization problem.The model considers one planar-aperture transmitter serving multiple users coherently.Source: B. Our Contributions · p. 2, §II · p. 3
  • PDM projects continuous patterns onto orthogonal bases, replacing functional optimization with the design of finite projection lengths.The technique is designed to flexibly use knowledge of continuous channel functions.Source: B. Our Contributions · p. 2
  • A block coordinate descent scheme uses PDM to design multi-user patterns, which simulations report are almost mutually orthogonal and achieve higher sum-rate than benchmarks.The supplied passages do not identify the benchmark schemes or give numerical comparisons.Source: B. Our Contributions · p. 2

A. CAP-MIMO Transmitter

  • The transmitter superposes user-specific patterns as current distributions carrying their respective symbols, with patterns intended to make symbols as orthogonal at users as possible.The system model assumes narrowband operation and linear superposition for coherent transmission.Source: A. CAP-MIMO Transmitter · p. 3

B. Electromagnetic Channels

The system model links aperture currents to users’ received electric fields through environment-dependent EM channels. It formulates joint pattern design as maximizing multi-user sum-rate under a transmit-power constraint.

B. Electromagnetic Channels

  • The channel function G(r,s) maps transmitter current distributions to the receiver electric field and depends on the wireless environment.It is described mathematically as a system impulse response or Green function; the paper notes homogeneous unbounded and scattering-environment models.Source: B. Electromagnetic Channels · p. 3

C. Electromagnetic-Wave Receivers

  • Each far-field user is modeled as a point with an ideal tri-polarization antenna that combines the received field’s three components to decode its symbol.The model assumes mutually independent additive white Gaussian noise with covariance σ^2I3.Source: C. Electromagnetic-Wave Receivers · p. 4

D. Sum-Rate Maximization Problem Formulation

  • The design objective is to maximize the sum of user mutual information rates subject to an upper bound on total transmit power.The allowable power parameter PT is associated with physical radiation power and measured in A^2 (or mA^2).Source: D. Sum-Rate Maximization Problem… · pp. 4–5
  • The optimization is non-convex functional programming because continuous patterns appear inside both the objective and constraint, making classical discrete-array methods difficult to apply.The paper specifically notes difficulty obtaining partial derivatives with respect to the pattern functions.Source: D. Sum-Rate Maximization Problem… · p. 5

III. DEVELOPED PATTERN-DIVISION MULTIPLEXING (PDM) FOR CAP-MIMO

PDM converts continuous pattern design into optimization over projections on orthogonal bases. The paper uses Fourier series and a finite low-wavenumber approximation to make the design computationally feasible.

III. DEVELOPED PATTERN-DIVISION MULTIPLEXING (PDM) FOR CAP-MIMO

  • PDM transforms continuous pattern-function design into optimization over projection lengths on finite orthogonal bases.The method is presented as a way to address the functional-programming challenge in CAP-MIMO pattern design.Source: §III · p. 5

A. A General Pattern-Division Multiplexing (PDM) Technique

  • Fourier bases represent continuous patterns in wavenumber space and provide functional degrees of freedom rather than physical plane-wave channels.The bases are orthogonal, and the projection lengths are the quantities optimized.Source: A. A General Pattern-Division… · p. 5
  • The continuous-discrete transforms express the electric field and pattern power through discrete wavenumber-domain quantities while preserving system energy.The electric field uses channel transforms and pattern projections; Parseval’s theorem relates the continuous and discrete power representations.Source: A. A General Pattern-Division… · p. 6
  • Although channel gain falls off at high wavenumbers, the paper reports this makes their omission have very limited impact on pattern designs.The rationale is that CAP-MIMO channel gain is concentrated in the low-wavenumber band.Source: A. A General Pattern-Division… · p. 6
  • For the considered setups, channel gain dominates within κx ∈ [−κ0, κ0]; outside that band it is about -30 dB relative to κx/κ0 = 0.Fig. 3 plots normalized channel gain against normalized wavenumber, with frequency fixed at 2.4 GHz in one setup and distance fixed at 0.1 m in another.Source: A. A General Pattern-Division… · pp. 6–7, Figure 3 · p. 6
  • The finite-item approximation retains low-wavenumber, high-power terms and discards high-wavenumber, low-power terms to make computation practical.The approximation is stated to be valid when evanescent waves are negligible, so users are not too close to the aperture.Source: A. A General Pattern-Division… · p. 7

B. Performance Analysis of Developed PDM Technique

The performance analysis studies the loss from PDM’s finite-item approximation and expects that loss to vanish asymptotically as more expansion items are reserved.

B. Performance Analysis of Developed PDM Technique

  • The finite-item approximation can cause performance loss, which the analysis expects to approach zero asymptotically as the number of reserved expansion items increases.The analysis simplifies to a single-user system for tractability; the general multi-user case is studied in Section IV.Source: B. Performance Analysis of Developed… · p. 7
  • The analysis isolates loss due to truncating the expansion rather than evaluating the general multi-user case in this subsection.The paper states that the multi-user case is treated later in Section IV.Source: B. Performance Analysis of Developed… · p. 7
  • The authors use the simplified setting to seek analytically tractable and insightful results on the finite-item approximation..Source: B. Performance Analysis of Developed… · p. 7

1) Achievable user signal-to-noise ratio (SNR):

For the ideal single-user CAP-MIMO case, the paper derives a closed-form optimal pattern design and maximum achievable SNR. The SNR is approximately proportional to transmit SNR and aperture area, with a stated far-field approximation.

1) Achievable user signal-to-noise ratio (SNR):

  • The ideal-case analysis temporarily omits the user index by setting K = 1 and reformulates the sum-rate maximization as SNR maximization.This setup is used to analyze performance loss from the finite-item approximation.Source: 1) Achievable user signal-to-noise… · p. 7
  • The single-user CAP-MIMO system has a closed-form optimal pattern design and maximum achievable SNR characterized using the maximum eigenvalue and its corresponding eigenvector.The result is presented for the ideal case as part of analyzing performance loss from finite-item approximation.Source: 1) Achievable user signal-to-noise… · pp. 7–8
  • In the far-field region, the squared channel function has similar amplitude across the aperture, and the achievable SNR has a stated approximation.The approximation is introduced for the case where the user is in the far-field region.Source: 1) Achievable user signal-to-noise… · p. 8
  • The achievable user SNR is approximately proportional to transmit SNR and the CAP-MIMO aperture area.The supplied passage identifies transmit SNR as PT/σ2 and aperture area as AT.Source: 1) Achievable user signal-to-noise… · p. 8

2) Performance loss caused by finite-item approximation:

The finite-item approximation introduces a controllable SNR loss, while PDM converts continuous pattern design into finite-dimensional optimization that can be solved by alternating updates.

2) Performance loss caused by finite-item approximation:

  • SNR loss is defined as the difference between ideal and finite-item achievable SNR, and Lemma 4 upper-bounds it for a given set of reserved Fourier items N.The bound is presented as a threshold for the degree of finite-item approximation.Source: 2) Performance loss caused by… · p. 8
  • The SNR-loss upper bound grows approximately with transmit SNR and aperture area, but approaches zero as the reserved expansion-item count increases and η approaches 1.The authors say this makes the PDM-based design capable of approaching the ideal solution asymptotically.Source: 2) Performance loss caused by… · p. 8

A. Proposed Pattern Design Scheme for Sum-Rate Maximization

  • The proposed BCD scheme alternates auxiliary variables, user combiners, and continuous patterns, then uses finite Fourier projections to recast pattern optimization as a QCQP.The projection lengths are optimized subject to the power constraint, with a Lagrange multiplier found by one-dimensional binary search.Source: A. Proposed Pattern Design Scheme for… · p. 9, 3) Fix ρ and ψ, then optimize θ(s) · pp. 9–10
  • The authors focus on theoretical performance; online implementation is challenging because arbitrary current distributions are difficult to generate and the iterative procedure is high-complexity.They suggest low-mobility, slow-fading use or offline codebooks as possible future applications.Source: 3) Fix ρ and ψ, then optimize θ(s) · p. 10

1) Convergence:

The BCD scheme converges asymptotically when the finite-item approximation gap is made sufficiently small, and its complexity is reported as similar to WMMSE. Simulations use one transmitter serving eight users in a specified three-dimensional setup.

1) Convergence:

  • As the reserved Fourier-item count NF increases, the finite-item approximation gap can approach zero, ensuring strict convergence of Algorithm 1 when NF is sufficiently large.The auxiliary-variable and combiner updates are monotonic.Source: 1) Convergence · p. 10

2) Complexity:

  • The proposed scheme’s computational complexity is similar to that of weighted mean-square error minimization (WMMSE).The analysis attributes the main work to updating ρ, ψ, projection lengths w, and patterns θ(s).Source: 2) Complexity · p. 10

1) Simulation scenario:

  • The simulation scenario places one CAP-MIMO transmitter at the origin of the xy-plane and serves eight users in a 3-D setup.Four users are at (±1 m, ±1 m, 30 m), and four are at (±5 m, ±5 m, 30 m); the square aperture area is 0.25 m².Source: 1) Simulation scenario · p. 10, Figure 4 · p. 10

2) Simulation parameters:

  • Unless otherwise specified, simulations use f = 2.4 GHz, maximum transmit power PT = 100 mA² for all compared schemes, noise power σ² = 5.6 × 10^-3, and Is = 1024 integral samples.The finite-item setups use NF = 225, 81, and 9; for the considered aperture, the NF = 81 setup exactly matches the N setting in (19).Source: 2) Simulation parameters · p. 11

3) Simulation benchmarks:

Across the simulations, PDM outperforms the matched-filter benchmark, while its finite-item performance depends on the number of retained Fourier items; the paper also reports CAP-MIMO outperforming fully-digital MIMO.

B. Sum-Rate Against the Aperture Area AT

  • At AT = 1 m², PDM achieves 17.90 bps/Hz versus 13.69 bps/Hz for matched filtering, about 24% higher.Fig. 5 plots sum-rate against aperture area.Source: B. Sum-Rate Against the Aperture Area AT · p. 11, Figure 5 · p. 12
  • NF = 9 incurs increasingly large performance loss, whereas NF = 225 offers almost negligible improvement over NF = 81 for the considered aperture.The paper attributes the smaller-NF loss to insufficient functional degrees of freedom and explains that the recommended truncation works because channel gain is concentrated in the low-wavenumber band.Source: B. Sum-Rate Against the Aperture Area AT · p. 12
  • The CAP-MIMO schemes outperform fully-digital MIMO in the reported comparison.The paper gives three explanations, including limited patch degrees of freedom and the continuous aperture’s coverage of the region.Source: B. Sum-Rate Against the Aperture Area AT · p. 12

C. Sum-Rate Against the Transmit Power PT

The proposed PDM scheme outperforms the benchmarks across the considered transmit-power range, and its optimized patterns distribute power across users’ non-overlapping regions. User distance and network radius also affect sum-rate.

C. Sum-Rate Against the Transmit Power PT

  • At 103 mA2, PDM achieves about 15.96 bps/Hz versus 12.69 bps/Hz for fully-digital MIMO, an improvement of about 26%.PDM outperforms the benchmark schemes throughout the considered transmit-power range.Source: C. Sum-Rate Against the Transmit Power… · p. 12
  • At about 316.2 mA2, fully-digital MIMO overtakes CAP-MIMO with MF; the paper attributes this to MF ignoring inter-user interference.The authors state that interference-suppressed patterns are necessary for effective multi-user CAP-MIMO.Source: C. Sum-Rate Against the Transmit Power… · pp. 12–13

D. Patterns θ(s) of CAP-MIMO

  • Optimized patterns concentrate power in mainly non-overlapping regions, while their phase values are symmetrically distributed toward the users’ angular directions.The paper relates this pattern structure to reducing inter-user interference and to spatially orthogonal MIMO beams.Source: D. Patterns θ(s) of CAP-MIMO · p. 13, Figure 7 · p. 13, Figure 8 · p. 13

E. The Impact of User Positions on Sum-Rate

  • At R=10 m, PDM achieves 29.32, 26.51, 22.90, and 9.18 bps/Hz for L=2 m, 5 m, 10 m, and 30 m, respectively.The paper attributes the decrease with vertical distance to larger-scale fading and limits on the aperture’s spatial resolution.Source: E. The Impact of User Positions on… · pp. 13–14
  • As network radius increases, each scheme’s sum-rate first rises and then falls: reduced inter-user interference gives way to stronger large-scale fading.Source: E. The Impact of User Positions on… · p. 14

VI. CONCLUSIONS AND FUTURE WORKS

  • The paper develops PDM to replace continuous-pattern-function design with finite-basis projection lengths and uses BCD for sum-rate maximization; lower-complexity design remains future work.Source: §VI · p. 14

APPENDIX A PROOF OF LEMMA 1

Appendix A extends a physical radiation-power bound to stochastic CAP-MIMO currents composed of multiple symbol-carrying patterns.

APPENDIX A PROOF OF LEMMA 1

  • The proof starts from the Poynting vector and evaluates expected radiation power for a stochastic current composed of multiple patterns.Source: Appendix A · p. 14
  • For sufficiently large distances, the channel function is approximated using plane-wave directions parameterized by elevation and azimuth angles.Source: Appendix A · p. 14
  • Applying Cauchy-Schwarz and taking the infinite-distance limit bounds physical radiation power by a component independent of distance.Source: Appendix A · p. 14

APPENDIX B PROOF OF LEMMA 3

Appendix B derives the combiner and pattern design used in Lemma 3, while Appendix C bounds the change in the objective used in Lemma 4.

APPENDIX B PROOF OF LEMMA 3

  • The power constraint is met at equality, and the combiner maximizing the Rayleigh quotient is the eigenvector for the matrix’s maximum eigenvalue.Source: Appendix B · pp. 14–15
  • Substituting the derived eigenvector into the optimization yields the pattern design scheme and the corresponding optimal objective value.Source: Appendix B · p. 15

APPENDIX C PROOF OF LEMMA 4

  • Appendix C derives the objective difference by substitution, norm inequalities, Cauchy-Schwarz, and Parseval’s theorem.Source: Appendix C · p. 15

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