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On the Performance of Randomly Generated Codebooks in Near-Field Communications

Akarsh Yadav, Ly V. Nguyen, James V. Krogmeier, David J. Love

arXiv:2610.12588v1eess.SP

TL;DR

Large arrays extend near-field operation, where spherical-wave channels depend on both user angle and distance. This paper analyzes four randomly generated codebooks and their gain distributions, finding that RNF has the most favorable high-gain scaling while RFF is bounded by phase mismatch.

  • Problem

    Near-field channels have distance-dependent spherical-wave phase, and the gain distributions of random codebooks in this setting need characterization.

    Source: §I · pp. 1–2

  • Method

    The paper characterizes gain distributions for RVQ, RP, RFF, and RNF, deriving approximations for RP and RFF and high-gain tail behavior for RNF.

    Source: §I · p. 2

  • Results

    RNF has the most favorable high-gain scaling, RVQ and RP deteriorate as array size grows, and RFF can outperform them at finite codebook sizes but saturates below one for finite-distance near-field users.

    Source: §V · p. 10

  • Takeaways & Limitations

    Random-codebook performance depends on codebook structure, array size, and codebook size in near-field beamforming.

    Source: §V · p. 10

Abstract

from arXiv · show

Near-field communication with large antenna arrays requires beamforming techniques that account for spherical wavefront effects. While predefined codebooks are widely used in multiple-antenna wireless systems, the performance of randomly generated codebooks in the near-field region has not been investigated. In this work, we study four randomly generated codebooks for near-field communications: random vector quantization (RVQ), random phase (RP), random far-field (RFF), and random near-field (RNF) codebooks. We use the known RVQ gain distribution as a benchmark. For RP, we derive a large-array bulk approximation and characterize the high-gain tail behavior of the gain distribution. For RNF, we characterize the high-gain tail behavior of the gain distribution. For RFF, we derive an analytical approximation for the maximum achievable beamforming gain when far-field beams are used to serve near-field users, along with the corresponding maximizing far-field direction. We also analyze the maximum-gain scaling with codebook size. Monte Carlo simulations verify the analytical results and show that RVQ and RP degrade significantly as the array size increases, while RNF achieves the most favorable high-gain scaling. RFF can outperform RVQ and RP for finite codebook sizes, but its maximum gain saturates below one due to the phase mismatch between the far-field beams and the near-field channel. These results show how codebook structure, array size, and codebook size affect random beamforming performance in the near-field.

I. INTRODUCTION

Near-field propagation adds distance-dependent phase curvature, making random-codebook performance uncertain and requiring beamforming that accounts for both angle and distance. The paper compares four random codebooks and finds RNF has the most favorable high-gain scaling, while RFF is limited by residual phase mismatch.

I. INTRODUCTION

  • Near-field channels depend on both user angle and distance, so codebook design must account for spherical wavefronts rather than only angular coverage.The paper studies RVQ, RP, RFF, and RNF; RNF samples an angle-distance pair to form a near-field steering vector.Source: §I · pp. 1–2
  • The study characterizes near-field gain distributions for RVQ, RP, RFF, and RNF, including limiting behavior as codebook size increases.For RP, it derives a central-limit-theorem-based large-array bulk approximation and a separate approximation for gains close to one.Source: §I · p. 2
  • For RFF, the paper approximates the maximum gain and its maximizing far-field direction when serving a near-field user.The maximum remains below one as codebook size increases because of residual phase mismatch.Source: §I · p. 2
  • For RNF, the probability of obtaining gain near one decreases only linearly as the target gain approaches one.The paper characterizes this high-gain tail behavior.Source: §I · p. 2
  • RNF has the most favorable maximum-gain scaling, followed by RP and RVQ, while RFF is limited by phase mismatch.RVQ and RP deteriorate significantly as array size grows; RFF can outperform them for finite codebook sizes but saturates below one for finite-distance near-field users.Source: §I · p. 2

A. System Model

  • The system model considers a single-cell downlink with a single-antenna user and a ULA base station, using a spherical-wave channel and a near-field steering vector.The channel response depends on the user’s normalized direction and distance.Source: A. System Model · pp. 2–3

B. Codebook Design

The codebook design section defines maximum beamforming gain as the performance measure and relates its behavior to the distribution of a randomly generated codeword. It uses RVQ as a benchmark and connects high-gain tail behavior to the codebook size needed for near-unit gain.

B. Codebook Design

  • The codebook’s performance metric is its maximum beamforming gain, formed from the gains of its codewords.Each codeword is unit norm, and the codeword gains are identically distributed realizations of a common random variable.Source: B. Codebook Design · p. 3
  • The high-gain tail of the single-codeword distribution determines how large a codebook must be to obtain near-unit maximum gain.For a target gap ε, the probability that the maximum exceeds 1 − ε approaches one when Kp_M(ε) → ∞.Source: B. Codebook Design · p. 3

A. Random Vector Quantization (RVQ)

  • For RVQ, the single-codeword gain distribution is independent of the near-field response because each codeword is isotropically distributed.The gain is defined by the squared inner product between the user’s near-field response and the RVQ codeword.Source: A. Random Vector Quantization (RVQ) · p. 3
  • For fixed array size, the maximum RVQ beamforming gain converges in probability to one as codebook size increases.RVQ codewords are independently and isotropically generated on the complex unit sphere, making RVQ a commonly used benchmark.Source: A. Random Vector Quantization (RVQ) · pp. 3–4

B. Random Phase (RP)

The RP analysis characterizes both typical gains and rare near-maximum gains, while the RFF analysis examines how random far-field beams serve near-field users despite phase-curvature mismatch.

B. Random Phase (RP)

  • RP codewords use independently uniform antenna phases, making the gain distribution independent of the user’s near-field phase profile.The gain is determined by the coherent sum of M independent unit-magnitude phasors.Source: B. Random Phase (RP) · p. 4
  • For large M, the scaled RP gain Mνk converges in distribution to a unit-rate exponential, approximating the distribution’s bulk behavior.This approximation is less accurate near the upper endpoint because it does not enforce the gain’s bounded support.Source: B. Random Phase (RP) · p. 4
  • Near γ = 1, Lemma 2 characterizes the RP gain’s right tail with a polynomial approximation that respects [0, 1] and does not require large M.This high-gain regime matters for large codebooks, whose maximum gains depend on rare high-gain events.Source: B. Random Phase (RP) · p. 4
  • For fixed M, the maximum RP beamforming gain converges in probability to one as codebook size K increases.Each additional independently generated codeword gives another chance to exceed any fixed threshold below one.Source: B. Random Phase (RP) · p. 4

C. Random Far-Field (RFF) Codebook

  • RFF codewords are randomly directed far-field steering vectors, but their linear phase progression cannot remove the near-field channel’s quadratic phase mismatch.The mismatch is associated with finite-distance propagation.Source: C. Random Far-Field (RFF) Codebook · p. 5
  • The dimensionless parameter A captures near-field quadratic phase curvature: larger values indicate stronger curvature, while A → 0 recovers the far-field phase model.A is largest near broadside and decreases as the user moves toward the array-axis direction.Source: C. Random Far-Field (RFF) Codebook · p. 5

1) Maximum achievable gain:

The best far-field beam for a near-field user is limited by residual phase mismatch, and the paper approximates its maximum gain using symmetry-based candidate directions. As array size grows, this mismatch affects more user locations; increasing the RFF codebook size cannot remove the ceiling.

1) Maximum achievable gain:

  • A symmetry-based family of approximate stationary directions makes the nonconvex far-field gain maximization tractable, and its approximation closely agrees with direct numerical maximization.Directions indexed by b and −b achieve the same gain, so only nonnegative b need be considered.Source: 1) Maximum achievable gain · pp. 5–6
  • The candidate-direction interval rule can select the wrong maximizer near the b⋆ = 2 to b⋆ = 3 transition, but the gain loss is at most ≈2%.The exceptional range has width O(1/M^2) in A(θt, rt), so it shrinks as the array grows.Source: 1) Maximum achievable gain · p. 6
  • As A(θt, rt) increases, the maximum far-field gain decreases and generally remains below one; larger RFF codebooks approach this location-dependent ceiling.For small A(θt, rt), the quadratic phase component and residual mismatch are small, so the maximum gain is close to one.Source: 1) Maximum achievable gain · p. 6
  • As M increases, degradation begins at smaller A(θt, rt) and spans more user locations; for M = 256, noticeable loss extends to distances around 100 m.The quadratic phase scales approximately as A(θt, rt)M^2, increasing mismatch for larger arrays.Source: 1) Maximum achievable gain · p. 6, Figure 4 · p. 7
  • Unlike RVQ and RP, the RFF maximum-gain CDF retains the probability that a user's best possible far-field gain is at most γ, even as codebook size grows.Some user locations cannot exceed the threshold with any far-field beam, regardless of codebook size.Source: 1) Maximum achievable gain · pp. 7–8

D. Random Near-Field (RNF) Codebook

RNF codewords are sampled from the near-field steering-vector set, incorporating both linear and quadratic phase components. The resulting high-gain tail is governed by how close a sampled angle-distance pair is to the user's location.

D. Random Near-Field (RNF) Codebook

  • RNF codewords sample an angle-distance pair and form the corresponding near-field steering vector, capturing both linear and quadratic phase components.This makes RNF a near-field benchmark distinct from codebooks restricted to far-field steering vectors.Source: D. Random Near-Field (RNF) Codebook · p. 8
  • An RNF codeword has high gain when its sampled angle-distance pair lies near the user's location, so local geometry in the joint domain governs the high-gain behavior.The sampling distribution and local steering-vector geometry determine the constant in the tail characterization.Source: D. Random Near-Field (RNF) Codebook · p. 8
  • Near unit gain, the RNF gain's right-tail probability scales linearly with 1 − γ because codewords and channels share the same two-parameter near-field model.A high-gain event occurs when the sampled pair falls within a sufficiently small neighborhood of the true user location.Source: D. Random Near-Field (RNF) Codebook · p. 8

A. Limiting Behavior of the Maximum Beamforming Gain

The maximum-gain limits and codebook-size scaling distinguish RNF, RP, RVQ, and RFF: RNF has the most favorable high-gain scaling, while RFF saturates below unit gain for finite-distance users. Simulations likewise show that performance depends on array size and codebook structure.

A. Limiting Behavior of the Maximum Beamforming Gain

  • As K → ∞, RVQ, RP, and RNF maximum gains converge in probability to one; RFF instead saturates below one for finite-distance near-field users.The distinction follows from the high-gain behavior of the three near-field-capable codebooks versus RFF’s restriction to far-field steering vectors.Source: A. Limiting Behavior of the Maximum… · p. 8
  • RNF has the most favorable high-gain scaling, while RVQ and RP require exponential growth in M to maintain a fixed target gap from unit gain.RP’s exponent is smaller than RVQ’s; RNF’s right-tail probability is linear in the gap from unit gain.Source: A. Limiting Behavior of the Maximum… · p. 8

B. Performance of Different Codebooks

  • As M increases, RVQ and RP degrade significantly, while RP consistently outperforms RVQ by preserving constant modulus across antennas.RVQ randomizes both amplitudes and phases, whereas RP randomizes phases only; amplitude randomness can reduce antenna contributions even when phases align.Source: B. Performance of Different Codebooks · p. 9
  • For RFF, increasing K improves maximum gain, but increasing M degrades it, and even large K does not remove its below-unit ceiling.Larger arrays produce narrower far-field beams and more significant near-field quadratic phase variation.Source: B. Performance of Different Codebooks · p. 9, Figure 6 · p. 9
  • At M = 128, increasing RNF codebook size shifts the maximum-gain CDF toward one, consistent with convergence in probability to unit gain.The CDF rises sharply near γ = 1 as K increases.Source: B. Performance of Different Codebooks · p. 9, Figure 7 · p. 9
  • At M = 128 and K = 1024, RNF ranks first, followed by RFF, RP, and RVQ in maximum-gain CDF performance.Better performance here means the CDF lies farther to the right, or equivalently gives a smaller probability that maximum gain is at or below a threshold.Source: B. Performance of Different Codebooks · p. 9, Figure 8 · p. 9

V. CONCLUSIONS

  • The analysis characterizes gain distributions and codebook-size limits; simulations show RVQ and RP deteriorating with array growth, RNF scaling best, and RFF saturating below one.RFF can outperform RVQ and RP at finite codebook sizes.Source: §V · pp. 9–10

APPENDIX A PROOF OF LEMMA 1

The appendix establishes the RP gain’s large-array bulk approximation and develops a separate near-unit-gain tail argument for phase-only beams. The tail proof uses the gain’s invariance to a common phase rotation.

APPENDIX A PROOF OF LEMMA 1

  • For large M, the RP gain is approximately exponentially distributed with rate M, giving a bulk CDF approximation away from the upper endpoint.The real and imaginary parts of the random phasor sum are approximated as independent standard normal variables using the Berry–Esseen theorem.Source: Appendix A · p. 10

APPENDIX B PROOF OF LEMMA 2

  • Because gain is invariant to common phase rotation, the tail-volume argument counts the centered phase vector in an (M − 1)-dimensional hyperplane and accounts separately for the common phase.The common-phase degree of freedom contributes a factor proportional to the diagonal circle’s length.Source: Appendix B · p. 11
  • Near unit gain, the RP tail probability is asymptotic to C_Mϵ, obtained by restricting phase differences to a small neighborhood of a common phase.On the high-gain event, unwrapped phase differences are O(√ϵ) when M^2ϵ ≪ 1.Source: Appendix B · p. 11

APPENDIX C PROOF OF LEMMA 3

The proof rewrites the beamforming-gain sum by index difference, handles its negative, zero, and positive cases, and applies finite geometric-series identities to obtain the stated expression.

APPENDIX C PROOF OF LEMMA 3

  • Changing variables to the index difference ℓ rewrites the beamforming gain as a sum organized by offset.This sets up separate treatment of negative, zero, and positive offsets.Source: Appendix C · p. 11
  • For ℓ = 0, the exponential term is one, so the inner sum contributes M terms.The proof treats this zero-offset case separately from the nonzero offsets.Source: Appendix C · p. 11
  • Finite geometric-series identities simplify the inner sums for the nonzero offsets and yield equation (34).The proof also uses e^jθ + e^-jθ = 2 cos(θ).Source: Appendix C · p. 11

APPENDIX D PROOF OF LEMMA 4

The appendix makes the RP summand approximately symmetric about the array midpoint for large M, then uses that symmetry to identify approximately stationary far-field directions.

APPENDIX D PROOF OF LEMMA 4

  • For large M, the factor u(p) is approximately constant, while the other factor is symmetric under p → M − p, making f_p approximately even about M/2.The bound on A(θ_t, r_t) decreases with M, supporting the small-angle approximation used for u(p).Source: Appendix D · p. 12

APPENDIX E PROOF OF LEMMA 5

  • When g_p is odd about M/2, the approximately even f_p terms cancel in the sum, leaving only the residual contribution.The proof identifies conditions on the far-field direction that make g_p odd.Source: Appendix E · p. 12
  • The resulting far-field direction is approximately stationary for large M.Source: Appendix E · p. 12

APPENDIX F PROOF OF LEMMA 6

The proof scales near-field curvature as τ = AM^2 and identifies which candidate far-field direction maximizes gain over curvature intervals. Higher-order candidates become relevant at larger curvature, with a narrow exception to the interval rule.

APPENDIX F PROOF OF LEMMA 6

  • The natural curvature scale is τ = AM^2; at τ = 0, candidate q = 0 is optimal because its gain is 1 while the gains for q ≥ 1 are 0.Source: Appendix F · p. 12
  • The first local maximum shifts to larger τ as b increases, so higher-order candidates become relevant only at larger A(θt, rt).Source: Appendix F · p. 12
  • On β1 ≤ τ ≤ β2, b = 1 gives the largest gain, with b = 0 tying at the left endpoint and b = 2 tying at the right endpoint.The previous candidate decreases across the interval, while higher-order candidates remain below b = 1.Source: Appendix F · p. 13
  • For large M, when A lies between thresholds Ath,b⋆ and Ath,b⋆+1, candidate direction θ(b⋆) maximizes gain, except on a narrow interval where b⋆ = 0 dominates.The exception occurs for τ in (10.50374, 10.81127).Source: Appendix F · p. 13

APPENDIX G PROOF OF LEMMA 7

The proof approximates RNF gain near a target user by a quadratic form in the angular and range offsets. This makes the high-gain region an ellipse whose area, and thus high-gain probability, scales linearly with the gain deficit.

APPENDIX G PROOF OF LEMMA 7

  • For small angle and range offsets, the phase difference vector is approximated to first order by the phase-map Jacobian times the offset vector.Source: Appendix G · p. 13
  • The RNF high-gain event is locally an ellipse in angle-range offset space, with area scaling linearly in ε and a cubic-order boundary correction.Source: Appendix G · p. 13
  • The high-gain tail probability is determined by a positive coefficient that depends on local sampling density and the geometry of Q.Source: Appendix G · p. 13
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