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Broadcasting in Centimeter-Wave Gigantic MIMO Systems: Beam Shaping is All You Need

Boyu Ning, Yike Xie, Jaebum Park, Songjie Yang, Weidong Mei, Emil Björnson, Robert W. Heath Jr

arXiv:2610.29542v1cs.ITmath.IT

TL;DR

Broadcasting in centimeter-wave G-MIMO faces a coverage-versus-sweeping trade-off: more beams narrow beamwidths and raise gain but increase initial-access sweep time. The paper uses geometric and propagation models to optimize a prostrate beam for ground coverage; at a −70 dB demodulation threshold, the traditional wide beam covers 40 m to 100 m, less than half the prostrate beam’s coverage capability.

  • Problem

    Broadcasting must balance coverage and complexity: more beams provide narrower beamwidths and higher gain but increase initial-access beam-sweeping time.

    Source: §I · p. 2

  • Method

    The paper combines a geometric broadcasting model and signal-propagation model to optimize a prostrate beam for ground-level coverage.

    Source: §I · p. 2

  • Results

    At a −70 dB demodulation threshold, the traditional wide beam covers 40 m to 100 m, less than 50% of the prostrate beam’s coverage capability.

    Source: §V · p. 12

  • Takeaways & Limitations

    The approach can achieve arbitrary desired broadcast coverage patterns when the array has sufficient spatial degrees of freedom.

    Source: §VI · p. 12

Abstract

from arXiv · show

Beamforming is a key enabling technology for centimeter-wave Gigantic MIMO (G-MIMO) systems, using antenna arrays to generate wide beams for cell-wide broadcasting or narrow beams for data transmission. This paper presents a novel \emph{beam shaping technology} for designing environment-aware broadcast beams (i.e., beams for common control information delivery, as in 5G SSB). As a case study, we shape a \emph{prostrate beam} in the vertical plane to approximate uniform electromagnetic power distribution across the ground, thereby enhancing broadcast coverage quality. We first propose a geometric broadcasting model that, through coordinate transformation, relates base station height, beam spread, maximum service distance, and downtilt angle. Building on this model and the signal propagation model, we formulate the beam shaping problem as a non-convex optimization problem that aims to maximize ground-level coverage. To solve it, we develop a generalized optimization framework. We reformulate the problem into a matrix representation, equivalently transforming it into a target pursuit optimization (TPO) problem. Using variable substitution, the objective is convexified into a quadratically constrained quadratic program, and the non-convex rank constraint is handled via successive convex approximation, decomposing the problem into iteratively solved subproblems. We provide complexity and convergence analyses for the proposed algorithm. Numerical simulations confirm that the optimization effectively shapes the intended beam pattern and that the proposed prostrate beam yields significant performance gains, including substantially extended coverage distance, over conventional wide-beam benchmarks.

I. INTRODUCTION

The introduction proposes environment-aware broadcast beam shaping, using geometry and propagation models to design a prostrate beam for ground-level coverage. The system model describes a downlink base station and a planar array that supports three-dimensional beam control.

I. INTRODUCTION

  • The prostrate beam redistributes radiation to provide uniform power along the ground, unlike conventional wide beams with uniform power across coverage directions.The authors describe this redistribution as enhancing broadcast performance.Source: §I · p. 2
  • A coordinate-transformed geometric model relates base-station height, beam spread or coverage width, maximum service distance, and downtilt angle.Source: §I · p. 2
  • Using the geometric and propagation models, the authors formulate a generally non-convex problem to maximize ground-level coverage with a prostrate beam.The design objective is for radiated power to “crawl” along the ground.Source: §I · p. 2
  • The framework recasts the problem as TPO, convexifies the objective into QCQP, and iteratively applies successive convex approximation to handle the non-convex rank constraint.Source: §I · pp. 2–3
  • The prostrate beam effectively shapes the intended pattern and significantly extends coverage distance compared with traditional wide beams.The paper also provides complexity and convergence analyses for its algorithm.Source: §I · p. 3

II. SYSTEM MODEL

  • The downlink system has a base station broadcasting a public message to system users.Source: §II · p. 3

A. Array Model

  • The UPA controls horizontal and vertical radiation for 3D beam control; its beam power separates into horizontal and vertical patterns.The two-dimensional beamforming vectors can be combined with a Kronecker product to form the 3D beamforming vector.Source: A. Array Model · p. 3

B. Geometric Broadcasting Model

The geometric model links base-station height, service distance, beam spread, and downtilt to describe broadcast coverage. It supports planning adjustments to downtilt or beam spread under the stated ample-transmit-power assumption.

B. Geometric Broadcasting Model

  • The service region is the ground area the array is designed to cover, extending to maximum distance dmax; its downtilt and beam spread are mutually constrained by base-station height and service distance.The model assumes a level ground plane and a beam centered on the array boresight.Source: B. Geometric Broadcasting Model · p. 4
  • Given base-station height, beam spread, and maximum service distance, Proposition 1 specifies downtilt, with the stated condition sin θspread + 2/N ≤ 1.The proposition also gives beam spread in terms of height, downtilt, and maximum service distance.Source: B. Geometric Broadcasting Model · p. 4
  • When coverage is short, engineers can adjust downtilt, or adjust beam spread if downtilt cannot be changed, to meet the same coverage objective.These adjustments assume ample base-station transmission power; coverage deficiencies are attributed to array orientation and beam design.Source: B. Geometric Broadcasting Model · p. 4
  • The intended main-lobe service region spans dmin to dmax, while nearby unplanned regions are served by sidelobes or reflected waves.The array covers a 120° horizontal sector; the service region is the intended main-lobe area.Source: Figure 3 · p. 4, B. Geometric Broadcasting Model · p. 4
  • Near-field effects are treated as negligible because the design targets far-field cell-edge users and near-field distances are only a few meters in most configurations considered.The passage states that near-field beam misalignment does not cause significant energy loss under this setup.Source: B. Geometric Broadcasting Model · p. 4

C. Propagation Model

The propagation model maps directions from the tilted array to ground-user distance and loss. This makes the loss across the service region dependent on both azimuth and elevation.

C. Propagation Model

  • The model maps each direction from the tilted array to propagation loss, enabling environment-aware beam design to compensate for non-uniform path loss across the coverage area.A target user's received signal power differs from radiated power because of propagation loss.Source: C. Propagation Model · p. 4
  • For the urban-macro path-loss model, fc is center frequency normalized by 1 GHz and d3D is target distance normalized by 1 m.The passage says loss parameters should be adjusted for other wireless scenarios using statistical evaluation.Source: C. Propagation Model · p. 5
  • The ground-reaching propagation distance depends on base-station height, downtilt, azimuth, and elevation, so loss varies with both angular coordinates.Proposition 2 relates d3D to the array height and target direction; the figure defines the tilted-array geometry.Source: C. Propagation Model · p. 5, Figure 4 · p. 5

III. MOTIVATION AND BEAM-SHAPING PROBLEM

Uniform wide beams distribute radiated power evenly, but non-uniform propagation can leave received power imbalanced across the service region. The paper motivates a prostrate beam that redistributes gain and formulates its design as a target-pattern optimization.

A. Motivation for Optimizing the Benchmark

  • Conventional uniform wide beams can produce imbalanced received power because close-range propagation loss is lower than at far range.The passage reports that this imbalance was confirmed by SS-RSRP measurements in a real-world base-station scenario.Source: A. Motivation for Optimizing the… · p. 5, §III · p. 5
  • With uniformly distributed blockage points, the stated design objective is to distribute received power evenly across the region while maximizing service-region size.The passage presents this as the most effective method for ensuring all blocking points can access the broadcast signal.Source: A. Motivation for Optimizing the… · pp. 5–6
  • The proposed prostrate beam redistributes gain from near the base station toward distant areas to make received power more even and extend the service region.For the same service region, if the proposed approach cannot serve all blocked users, the benchmark approach will also fail to do so.Source: A. Motivation for Optimizing the… · p. 6

B. Problem Formulation

  • The case study shapes a 2D prostrate beam whose gain compensates for propagation loss, targeting uniform received power across the service region.The general beam-shaping approach is not tied to a specific target pattern, and 3D shaping can be reduced to 2D by separately designing vertical or horizontal weights.Source: B. Problem Formulation · p. 6
  • The prostrate-beam formulation specifies target magnitude separately across coverage, transition, and stop ranges, with an auxiliary function governing the transition range.For the 2D case, the target accounts for propagation loss without azimuth dependence; the 3D case also has azimuth-dependent loss.Source: B. Problem Formulation · pp. 6–7

IV. BEAM SHAPING OPTIMIZATION

The optimization discretizes the beam-shaping objective and rewrites it in matrix form as an equivalent target pursuit problem. Variable transformation and successive convex approximation yield iteratively solved convex subproblems.

IV. BEAM SHAPING OPTIMIZATION

  • The proposed design transforms the non-convex beam-shaping problem into iteratively solved convex subproblems using successive convex approximation.The method lifts the vector variable to a positive semidefinite matrix and introduces rank-one equivalent constraints.Source: §IV · p. 7

A. Matrix Representation

  • The matrix representation is shown to be equivalent to a target pursuit optimization that optimizes phases of the target beam pattern.The formulation uses target-pattern values compared with the array response at sampled angles.Source: §IV · p. 7, A. Matrix Representation · p. 7
  • The objective is discretized at O elevation-angle samples across [−π/2, π/2] to represent the beam pattern in matrix form.The sample spacing is Δφ = π/(O − 1).Source: A. Matrix Representation · p. 7
  • The discretized objective contains two modulus operations, which the formulation identifies as difficult to handle directly with existing optimization methods.The derivation introduces a matrix whose columns contain array responses at the sampled angles.Source: A. Matrix Representation · p. 7

B. Equivalent TPO Problem

After obtaining a closed-form beamforming vector, the remaining design task is to choose target-pattern phases across all ranges and magnitudes in the transition range. The authors recast this parameter optimization as a target pursuit optimization (TPO) problem.

B. Equivalent TPO Problem

  • Substituting the closed-form wv solution leaves the phases {κo} across all ranges and the transition-range magnitudes {µo | φo ∈Πt} to optimize.The derivation assumes O > N so that VVH is invertible.Source: B. Equivalent TPO Problem · p. 7
  • The authors name this remaining target-pattern parameter design a TPO problem.Its variables are the target-pattern parameters {κo, µo}O_o=1.Source: B. Equivalent TPO Problem · pp. 7–8
  • Although the objective is convex in the target-pattern function vector, it is non-convex in the parameters being optimized.The proposed formulation seeks to impose the vector elements’ constant-modulus property as constraints.Source: B. Equivalent TPO Problem · p. 8

C. Convexification Attempts

The TPO formulation is recast as a QCQP and lifted to a positive semidefinite matrix problem. Successive convex approximation (SCA) then handles the remaining non-convex rank-one condition through iterative convex subproblems.

C. Convexification Attempts

  • Replacing the target-pattern function vector with variable vector t turns its structure into quadratic constraints and yields a QCQP.The original P4 variables are implicit in t.Source: C. Convexification Attempts · p. 8
  • Lifting t to a positive semidefinite matrix reformulates the constraints, but the rank-one condition remains non-convex.The rank-one constraint is equivalent to the condition referenced in the formulation.Source: C. Convexification Attempts · p. 8
  • The rank penalty uses Tr(T) − ∥T∥2, which is nonnegative for positive semidefinite T and penalizes solutions that are not rank one.This leaves minT −∥T∥2 as the non-convex component of the objective.Source: C. Convexification Attempts · p. 8

D. SCA Decomposition

  • SCA replaces the non-convex objective with a convex approximation at each optimization step.The approximation matches the function value at the current iterate and is an upper bound in its neighborhood.Source: D. SCA Decomposition · p. 8
  • The approximation uses a first-order Taylor expansion of ∥T∥2 around the current matrix Tk, with qk the top singular vector of Tk.The resulting subproblem is solved repeatedly, using each solution in the next step until a convergence criterion is met.Source: D. SCA Decomposition · p. 8
  • Algorithm 1 for prostrate-beam design takes inputs N, O, lf, ld, fc, hBS, dmax, θtilt, ρ, T0, forms V from sampled directions, and calculates ML(φo) using hBS and θtilt.Source: D. SCA Decomposition · p. 9

E. Overall Algorithm and Convergence Analysis

The overall procedure computes geometry-dependent beam targets, iteratively optimizes them with SCA, and recovers the beam from the final matrix solution. The objective is non-increasing across iterations, and the stated complexity is O(RN^5.5 + O^3).

E. Overall Algorithm and Convergence Analysis

  • The algorithm sets sampling directions and steering vectors, calculates the beam spread and angular ranges, then computes required beam gains before SCA optimization.The iteration count R can be empirical or stopped when the objective increase falls below ϵ.Source: E. Overall Algorithm and Convergence… · p. 9
  • At iteration R, eigen-decomposition of TR yields the dominant eigenvector used to recover the prostrate beam pattern wv.The beamforming vector is recovered from the final matrix solution.Source: E. Overall Algorithm and Convergence… · p. 9
  • Each optimization iteration is non-increasing, so convergence is guaranteed, and computational complexity is O(RN^5.5 + O^3).The authors say this complexity has minimal design impact because codewords are predefined offline and only selected online.Source: E. Overall Algorithm and Convergence… · p. 9

V. NUMERICAL RESULTS

The numerical results compare the proposed prostrate beam with a conventional wide beam and examine geometry, propagation, and beam-pattern behavior. Under the stated demodulation threshold, the prostrate beam covers the full 40–180 m range, while the wide beam covers only 40–100 m.

V. NUMERICAL RESULTS

  • With downtilt fixed at π/4, maximum service distance increases with both base-station height and beam spread, approaching 1000 m for hBS = 30 m and θspread ≈86°.For a fixed height, the increase is especially rapid over beam spreads of 0°–60°; for fixed spread, increasing height extends coverage almost linearly.Source: Figure 7 · p. 10, §V · p. 11
  • At ground level, propagation loss rises from 84 dB at 0 m to 95 dB at 100 m; with the base station fixed at 30 m, moving the user from 0 m to 30 m adds about 3 dB.The heatmap plots propagation loss against user horizontal distance and height, with a 30 m base-station height and user distances from 0 to 100 m.Source: §V · p. 11
  • In the beam-pattern comparison, rotating the prostrate pattern clockwise by the 45-degree array tilt shows its energy advancing forward while skimming the ground.The simulation uses beam spread π/4 and normalized transmit power P = 1.Source: Figure 9 · p. 10, §V · p. 11
  • Within the coverage range, the prostrate beam’s gain gradually decreases toward the ground and has no significant dips, while the wide beam stays near −5 dB.Both beams’ coverage ranges, defined by gain above −20 dB, are essentially equal to the beam spread; the prostrate beam provides higher gain in farther directions.Source: Figure 10 · p. 10, §V · p. 11
  • The prostrate beam overcomes propagation loss across 40–180 m at a −70 dB demodulation threshold, while the traditional wide beam covers only 40–100 m.The comparison uses equal total EIRP for the two beams.Source: §V · pp. 11–12, Figure 11 · p. 11
  • The prostrate beam gives cell-edge users significantly higher spectral efficiency, with the gain more pronounced at greater distances.The evaluation uses 400 MHz bandwidth, a 3 dB noise figure, 4 dB receive array gain, and transmit powers of 20 dBm and 40 dBm.Source: Figure 12 · p. 11, §V · p. 12

VI. CONCLUSION

The paper proposes environment-aware beam shaping for broadcast beams, with desired coverage patterns achievable when the array has sufficient spatial degrees of freedom. It designs a ground-following vertical prostrate beam and identifies an energy threshold for full coverage.

VI. CONCLUSION

  • The beam-shaping approach can produce arbitrary desired coverage patterns when the array has sufficient spatial degrees of freedom.It also allows horizontal customization based on user distribution or obstacle layouts.Source: §VI · pp. 12–13
  • The vertical prostrate beam follows the ground surface and extends coverage range versus traditional wide beams, but needs energy above a full-coverage threshold.Below that threshold it can yield zero coverage, and no beam can fully cover the intended service area.Source: §VI · p. 13
  • The method is broadly applicable to centimeter-wave G-MIMO broadcast systems, while future work could address movable antenna positions and weights or constant-modulus constraints.The paper notes that codebook-based broadcast beams are often optimized offline, so computational complexity is not stringent.Source: §VI · p. 13

APPENDIX A PROOF OF PROPOSITION 1

The appendices derive geometric relationships and the angle between the base-station pole and a target direction. The latter uses coordinate transformations to express the target direction in the global frame.

APPENDIX A PROOF OF PROPOSITION 1

  • Proposition 1 follows from geometric relationships in Fig. 2, with the result derived through trigonometric functions and equation transformation.The proof uses the stated geometric relationships as its starting point.Source: Appendix A · p. 13

APPENDIX B PROOF OF PROPOSITION 2

  • For Proposition 2, the target direction is transformed from the array frame to the global frame through successive rotations about the x-axis, z-axis, and x-axis.The final rotation uses the array tilt angle.Source: Appendix B · p. 13
  • The angle between the base-station pole and target direction is calculated in the global coordinate system, yielding the stated result of Proposition 2.The displayed expression relates the angle to ψ, φ, and θtilt.Source: Appendix B · p. 13

APPENDIX C PROOF OF THE RELATION IN (19)

The appendix derives the radiation-power integration used in relation (19). It introduces an auxiliary function, expands an expression, and substitutes the result to complete the proof.

APPENDIX C PROOF OF THE RELATION IN (19)

  • The proof begins by expressing the integration of radiation power in the beam domain.An auxiliary function is then defined for the derivation.Source: Appendix C · p. 13
  • The integration is further calculated by expanding the expression in square brackets.The passage identifies this expansion as an intermediate step.Source: Appendix C · p. 13
  • Substituting the expanded expression into the preceding result completes the proof.The appendix identifies the substitution as the final step.Source: Appendix C · p. 14
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