Performance Bounds for Near-Field Velocity Estimation With Modular Linear Array

Ali. A. Nasir; Khalid A. Alshumayri; Mudassir Masood

arxiv: 2511.06383 · v4 · pith:TTELO3PWnew · submitted 2025-11-09 · 📡 eess.SP

Performance Bounds for Near-Field Velocity Estimation With Modular Linear Array

Khalid A. Alshumayri , Mudassir Masood , Ali. A. Nasir This is my paper

Pith reviewed 2026-05-21 18:47 UTC · model grok-4.3

classification 📡 eess.SP

keywords near-field velocity estimationmodular linear arrayCramer-Rao boundpredictive beamformingarray geometrytransverse velocityradial velocityantenna savings

0 comments

The pith

Modular linear arrays achieve uniform linear array velocity estimation accuracy with fewer antennas by increasing inter-module separation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives closed-form Cramer-Rao bounds for joint radial and transverse velocity estimation in a near-field predictive beamforming system that uses a modular linear array. The expressions show that wider gaps between modules enlarge the effective aperture and tighten the transverse-velocity bound while leaving the radial-velocity bound largely unchanged. The work also establishes that these modular designs reach the same accuracy as a single dense uniform linear array but require fewer total antennas, with the savings increasing as module spacing grows. A sympathetic reader would care because velocity estimates support predictive beamforming, so geometry-aware bounds can guide more efficient array hardware choices in sensing and communications. Simulations confirm the bounds align with the mean-squared error achieved by maximum-likelihood estimators.

Core claim

The paper establishes closed-form expressions for the Cramer-Rao bounds on joint radial and transverse velocity estimation using a modular linear array. These expressions reveal that increasing inter-module separation enlarges the effective aperture, thereby reducing the transverse-velocity CRB, while the radial-velocity CRB remains largely insensitive. Furthermore, an MLA achieves equivalent estimation accuracy to a collocated ULA with fewer antennas, and the relationship between inter-module spacing and the required number of antennas is quantified.

What carries the argument

Closed-form Cramer-Rao bound expressions for joint radial and transverse velocity estimation that explicitly capture how inter-module separation affects the effective array aperture.

If this is right

Increasing inter-module separation reduces the transverse-velocity CRB by enlarging the effective aperture.
The radial-velocity CRB stays largely insensitive to changes in inter-module separation.
An MLA reaches the same estimation accuracy as a collocated ULA while using fewer antennas overall.
Antenna savings grow as inter-module spacing increases, with the exact relation given by the derived expressions.
The bounds are tight under the modeled conditions because simulated MSE of the MLE matches the CRB values.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Designers could favor modular arrays in hardware-constrained settings where reducing total antenna count lowers cost without sacrificing velocity accuracy.
The radial-velocity bound's insensitivity suggests it depends more on parameters such as observation duration or carrier frequency than on array geometry.
The same geometry-CRB tradeoff could be tested in planar modular arrays to see whether the antenna-saving benefit scales to two-dimensional deployments.

Load-bearing premise

The analysis assumes a standard near-field array signal model with known geometry and Gaussian noise such that the maximum-likelihood estimator can approach the derived CRB without model mismatch or additional impairments.

What would settle it

A simulation or experiment in which the mean-squared error of the maximum-likelihood estimator deviates significantly from the closed-form CRBs when inter-module separation is varied, or when real-world data introduces geometry uncertainty or non-Gaussian noise.

Figures

Figures reproduced from arXiv: 2511.06383 by Ali. A. Nasir, Khalid A. Alshumayri, Mudassir Masood.

**Figure 3.** Figure 3: The MSE of MLE in [1] and the derived CRBs. The total number of antenna used is 240. For the modular array, the number of modules is 2, the number of antennas in each module is M = 120, and the modular spacing parameter L = 61. To understand the interplay between array geometry and estimation accuracy, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 2.** Figure 2: Effect of velocity mismatch on the array gain when [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Figure 4: CRB for radial velocity (top) and transverse velocity (bottom), with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

Velocity estimation is a cornerstone of the recently introduced near-field predictive beamforming. This paper derives the Cramer-Rao bounds (CRBs) for joint radial and transverse velocity estimation within a predictive beamforming framework employing a modular linear array (MLA). We obtain closed-form expressions that characterize the interplay between array geometry and estimation accuracy, showing that increasing the inter-module separation enlarges the effective aperture and reduces the transverse-velocity CRB, while the radial-velocity CRB remains largely insensitive to this separation. Furthermore, we show that an MLA can achieve the same accuracy as a collocated ULA with fewer antennas and quantify the relation between inter-module spacing and antenna savings. The derived expressions are validated through simulations by comparing them with the mean-squared error (MSE) of the maximum likelihood estimator (MLE) reported in the literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies closed-form CRBs for near-field radial and transverse velocity estimation on modular linear arrays, with the key new angle being how inter-module spacing trades off against antenna count for transverse accuracy.

read the letter

The central result is a set of closed-form CRB expressions that tie array geometry directly to velocity estimation accuracy. Larger inter-module separation widens the effective aperture and lowers the transverse-velocity bound while leaving the radial-velocity bound nearly unchanged. The authors also quantify how many fewer antennas an MLA needs to reach the same accuracy as a collocated ULA. These relations are not present in the earlier uniform-array CRB papers they cite, so the geometry-specific claims are genuinely new for this setting. The derivations follow the standard near-field manifold and Gaussian-noise model, then apply the usual Fisher-information formula; the algebra appears clean and the final expressions are explicit enough to be used for design rules. Simulations compare the bounds to MLE mean-squared error taken from the literature, which at least provides a consistency check under the reported conditions. The model assumptions are standard and stated up front, with no obvious circularity or post-hoc fitting. The main soft spot is the validation step. The stress-test note flags that the referenced MLE may come from a slightly different parameterization or far-field approximation; if the paper does not re-derive or re-simulate the MLE on its exact MLA observation model, the numerical match is weaker evidence than it first appears. That concern is real but not fatal, because the closed-form expressions themselves can still be checked analytically or with fresh Monte Carlo runs. Minor additional points include the lack of explicit error bars on the simulation curves and the usual caveat that the CRB is a lower bound that the MLE may not always attain at low SNR. Overall the work is technically competent and the claims are proportionate to the evidence shown. Readers working on near-field sensing, predictive beamforming, or array-geometry optimization will find the explicit trade-off curves useful. It is narrow but well-executed, so it belongs in a specialized journal or conference track rather than a broad venue. I would send it to peer review with a request that the authors confirm the MLE model match or add their own simulation code for the exact setup.

Referee Report

2 major / 2 minor

Summary. This paper derives closed-form Cramer-Rao bounds (CRBs) for joint radial and transverse velocity estimation in a near-field predictive beamforming framework that employs a modular linear array (MLA). The authors obtain analytical expressions characterizing the effect of array geometry, specifically showing that larger inter-module separation enlarges the effective aperture and thereby reduces the transverse-velocity CRB while leaving the radial-velocity CRB largely insensitive. They further claim that an MLA can match the estimation accuracy of a collocated uniform linear array (ULA) using fewer antennas and quantify the relation between inter-module spacing and the resulting antenna savings. The expressions are validated by comparing them to the mean-squared error of the maximum-likelihood estimator reported in the literature.

Significance. If the closed-form CRB derivations are correct under the stated near-field MLA model, the results supply direct analytical insight into geometry-accuracy trade-offs that can guide hardware-efficient array design for near-field velocity estimation. The explicit dependence on inter-module separation and the antenna-savings quantification are potentially useful for system optimization in radar or ISAC applications. The closed-form character itself is a positive feature, as it permits immediate evaluation of design parameters without repeated numerical optimization.

major comments (2)

[Abstract] Abstract (validation statement): The paper validates the derived CRBs by comparing them to the MSE of the MLE 'reported in the literature.' Because the abstract provides no explicit confirmation that the referenced MLE employs the identical near-field MLA observation model, the same radial/transverse velocity parameterization, and the same near-field manifold, this comparison does not necessarily verify that the closed-form expressions correctly capture the claimed interplay between inter-module separation and the two CRBs. This verification step is load-bearing for the central geometry-accuracy claims.
[Results] Results section (radial-velocity CRB claim): The statement that the radial-velocity CRB 'remains largely insensitive' to inter-module separation requires an explicit demonstration from the closed-form expression (e.g., showing that the relevant partial derivatives or Fisher-information terms are independent of the separation parameter). Without this step, the insensitivity assertion rests on simulation observation rather than analytic structure and weakens the contrast drawn with the transverse-velocity result.

minor comments (2)

[Abstract] The abstract and introduction should include a brief statement of the precise near-field array signal model (e.g., the exact form of the steering vector and the definition of radial versus transverse velocity components) so that readers can immediately assess the scope of the derived bounds.
[System Model] Notation for the modular-array geometry (module positions, inter-module spacing, total aperture) should be introduced once and used consistently; a small diagram or table summarizing the geometric parameters would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful comments, which help to strengthen the presentation of our results. Below we address each major comment in turn.

read point-by-point responses

Referee: [Abstract] Abstract (validation statement): The paper validates the derived CRBs by comparing them to the MSE of the MLE 'reported in the literature.' Because the abstract provides no explicit confirmation that the referenced MLE employs the identical near-field MLA observation model, the same radial/transverse velocity parameterization, and the same near-field manifold, this comparison does not necessarily verify that the closed-form expressions correctly capture the claimed interplay between inter-module separation and the two CRBs. This verification step is load-bearing for the central geometry-accuracy claims.

Authors: We thank the referee for highlighting this point. The maximum-likelihood estimator referenced in the literature is derived for the near-field modular linear array model with the same radial and transverse velocity parameterization and near-field manifold as employed in our CRB analysis. To address the concern and make the validation more transparent, we will revise the abstract to explicitly note that the MLE uses the identical observation model. We will also add a clarifying sentence in the simulation section confirming the model match. revision: yes
Referee: [Results] Results section (radial-velocity CRB claim): The statement that the radial-velocity CRB 'remains largely insensitive' to inter-module separation requires an explicit demonstration from the closed-form expression (e.g., showing that the relevant partial derivatives or Fisher-information terms are independent of the separation parameter). Without this step, the insensitivity assertion rests on simulation observation rather than analytic structure and weakens the contrast drawn with the transverse-velocity result.

Authors: We agree that an explicit analytic demonstration would enhance the rigor of our claim. In the revised manuscript, we will include a brief derivation in the results or appendix section that shows the relevant entries of the Fisher information matrix for the radial velocity are independent of the inter-module separation. This follows directly from the structure of the near-field steering vector, where the radial component depends on the range and angle but not on the transverse module spacing in the first-order approximation used. revision: yes

Circularity Check

0 steps flagged

Standard CRB derivation on MLA model shows no circular reduction

full rationale

The paper derives closed-form CRB expressions by direct application of the standard Cramer-Rao formula to the near-field modular linear array observation model with known geometry and Gaussian noise. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the geometry-CRB interplay (inter-module separation enlarging effective aperture for transverse velocity while leaving radial largely unchanged) follows from the model equations themselves. Validation against literature MLE MSE is external benchmarking rather than a load-bearing internal loop. The derivation remains self-contained and independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard array signal processing assumptions for CRB derivation and the near-field propagation model; no free parameters or invented entities are explicitly introduced in the summary.

axioms (2)

standard math Standard assumptions underlying the Cramer-Rao bound (Gaussian noise, known deterministic signal model, regularity conditions for the likelihood function).
Invoked implicitly when deriving closed-form CRB expressions for velocity parameters.
domain assumption Near-field spherical-wave propagation model with known array geometry and constant velocity during observation.
Required for the joint radial-transverse velocity estimation framework described.

pith-pipeline@v0.9.0 · 5674 in / 1507 out tokens · 81531 ms · 2026-05-21T18:47:32.428221+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We obtain closed-form expressions that characterize the interplay between array geometry and estimation accuracy, showing that increasing the inter-module separation enlarges the effective aperture and reduces the transverse-velocity CRB, while the radial-velocity CRB remains largely insensitive to this separation.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

CRB(vr)Mod = 12 / γ(MK)² (12 − (δ/r)²(U²(K²−1) + M²−1)), CRB(vt)Mod = 12(r/δ)² / γ(MK)²(U²(K²−1) + M²−1) sin²θ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Near-field velocity sensing and predictive beamforming,

Z. Wang, X. Mu, and Y . Liu, “Near-field velocity sensing and predictive beamforming,”IEEE Transactions on V ehicular Technology, 2024

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C. Giovannetti, N. Decarli, and D. Dardari, “Performance bounds for velocity estimation with extremely large aperture arrays,”IEEE Wireless Communications Letters, vol. 13, no. 12, pp. 3513–3517, 2024

work page 2024
[3]

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A. Kosasih, ¨O. T. Demir, and E. Bj¨ornson, “Achieving beamfocusing via two separated uniform linear arrays,” in2024 58th Asilomar Conference on Signals, Systems, and Computers, 2024, pp. 168–172

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W.-C. Kao, J.-Y . Wu, S.-H. Tsai, and T.-Y . Wang, “Fast ambiguity- free subspace-based multiple aoa estimation for hybrid linear arrays,” in 2023 IEEE 34th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), 2023, pp. 1–5

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X. Li, Z. Dong, Y . Zeng, S. Jin, and R. Zhang, “Multi-user modular xl-mimo communications: Near-field beam focusing pattern and user grouping,”IEEE Transactions on Wireless Communications, vol. 23, no. 10, pp. 13 766–13 781, 2024

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X. Li, H. Lu, Y . Zeng, S. Jin, and R. Zhang, “Near-field modeling and performance analysis of modular extremely large-scale array communi- cations,”IEEE Communications Letters, vol. 26, no. 7, pp. 1529–1533, 2022

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[12]

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[1] [1]

Near-field velocity sensing and predictive beamforming,

Z. Wang, X. Mu, and Y . Liu, “Near-field velocity sensing and predictive beamforming,”IEEE Transactions on V ehicular Technology, 2024

work page 2024

[2] [2]

Performance bounds for velocity estimation with extremely large aperture arrays,

C. Giovannetti, N. Decarli, and D. Dardari, “Performance bounds for velocity estimation with extremely large aperture arrays,”IEEE Wireless Communications Letters, vol. 13, no. 12, pp. 3513–3517, 2024

work page 2024

[3] [3]

Achieving beamfocusing via two separated uniform linear arrays,

A. Kosasih, ¨O. T. Demir, and E. Bj¨ornson, “Achieving beamfocusing via two separated uniform linear arrays,” in2024 58th Asilomar Conference on Signals, Systems, and Computers, 2024, pp. 168–172

work page 2024

[4] [4]

Fast ambiguity- free subspace-based multiple aoa estimation for hybrid linear arrays,

W.-C. Kao, J.-Y . Wu, S.-H. Tsai, and T.-Y . Wang, “Fast ambiguity- free subspace-based multiple aoa estimation for hybrid linear arrays,” in 2023 IEEE 34th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), 2023, pp. 1–5

work page 2023

[5] [5]

Multi-user modular xl-mimo communications: Near-field beam focusing pattern and user grouping,

X. Li, Z. Dong, Y . Zeng, S. Jin, and R. Zhang, “Multi-user modular xl-mimo communications: Near-field beam focusing pattern and user grouping,”IEEE Transactions on Wireless Communications, vol. 23, no. 10, pp. 13 766–13 781, 2024

work page 2024

[6] [6]

Near-field modeling and performance analysis of modular extremely large-scale array communi- cations,

X. Li, H. Lu, Y . Zeng, S. Jin, and R. Zhang, “Near-field modeling and performance analysis of modular extremely large-scale array communi- cations,”IEEE Communications Letters, vol. 26, no. 7, pp. 1529–1533, 2022

work page 2022

[7] [7]

Performance bounds for near-field localization with widely-spaced multi-subarray mmwave/thz mimo,

S. Yang, X. Chen, Y . Xiu, W. Lyu, Z. Zhang, and C. Yuen, “Performance bounds for near-field localization with widely-spaced multi-subarray mmwave/thz mimo,”IEEE Transactions on Wireless Communications, vol. 23, no. 9, pp. 10 757–10 772, Sep. 2024

work page 2024

[8] [8]

Cram ´er–rao bounds for near-field sensing: A generic modular architecture,

C. Meng, D. Ma, X. Chen, Z. Feng, and Y . Liu, “Cram ´er–rao bounds for near-field sensing: A generic modular architecture,”IEEE Wireless Communications Letters, vol. 13, no. 8, pp. 2205–2209, 2024

work page 2024

[9] [9]

In-band full-duplex wireless: Challenges and opportuni- ties,

A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportuni- ties,”IEEE Journal on Selected Areas in Communications, vol. 32, no. 9, pp. 1637–1652, 2014

work page 2014

[10] [10]

Near-field localization and sensing with large-aperture arrays: From signal modeling to pro- cessing,

Z. Wang, P. Ramezani, Y . Liu, and E. Bj ¨ornson, “Near-field localization and sensing with large-aperture arrays: From signal modeling to pro- cessing,”IEEE Signal Processing Magazine, vol. 42, no. 1, pp. 74–87, 2025

work page 2025

[11] [11]

Near- field communications: A tutorial review,

Y . Liu, Z. Wang, J. Xu, C. Ouyang, X. Mu, and R. Schober, “Near- field communications: A tutorial review,”IEEE Open Journal of the Communications Society, 2023

work page 2023

[12] [12]

Joint motion, angle, and range estimation in near-field under array calibration imper- fections,

A. Hussain, A. Abdallah, A. Celik, and A. M. Eltawil, “Joint motion, angle, and range estimation in near-field under array calibration imper- fections,”arXiv preprint arXiv:2507.13463, 2025

work page arXiv 2025

[13] [13]

Ofdm radar algorithms in mobile communication net- works,

K. M. Braun, “Ofdm radar algorithms in mobile communication net- works,” Ph.D. dissertation, Karlsruhe, Karlsruher Institut f¨ur Technologie (KIT), Diss., 2014, 2014

work page 2014

[14] [14]

S. M. Kay,Fundamentals of statistical signal processing: estimation theory. USA: Prentice-Hall, Inc., 1993

work page 1993