arxiv: 2605.03856 · v1 · submitted 2026-05-05 · 💻 cs.NI

Recognition: unknown

Nested array design of extended coprime sets for DOA estimation of non-circular signals

Dongqi Chen , Kun Ye , Chuanxi Xing , Waqas Khalid , Huiping Huang

Authors on Pith no claims yet

Pith reviewed 2026-05-07 12:37 UTC · model grok-4.3

classification 💻 cs.NI

keywords DOA estimationnon-circular signalscoprime arraysnested arraysdifference co-arraysum co-arraydegrees of freedomvirtual aperture array

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The pith

A new array design based on extended coprime sets with sliding translation increases degrees of freedom and virtual aperture for non-circular signal direction-of-arrival estimation.

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

The paper proposes a nested array structure derived from extended coprime sets. It uses a sliding translation technique to rearrange sensor positions while keeping the difference co-array continuous. The sum co-array is then shifted to combine with the difference co-array, removing redundancies and enlarging both the virtual aperture and the number of unique lags available. If the construction works as described, the array can resolve more sources or achieve finer angular resolution with the same number of physical sensors than conventional nested arrays or extended sliding nested arrays. Simulation comparisons support improved estimation accuracy under practical conditions for non-circular signals.

Core claim

The authors present a nested array configuration obtained from extended coprime sets by applying a sliding translation technique. This arrangement maintains continuity in the difference co-array and shifts the sum co-array to merge with it without added redundancy, thereby expanding the virtual aperture array and the degrees of freedom for direction-of-arrival estimation of non-circular signals.

What carries the argument

The sliding translation technique applied within an extended coprime array framework, which preserves difference co-array continuity and enables the sum co-array to merge seamlessly with the difference co-array.

If this is right

The design yields higher degrees of freedom than traditional nested arrays and extended sliding nested arrays.
The virtual aperture array becomes larger, supporting finer angular resolution or more simultaneous sources.
Estimation accuracy improves in practical non-circular signal scenarios according to the reported simulations.
Redundancy elimination in the combined co-array directly contributes to the observed performance gains.

Where Pith is reading between the lines

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

The same sliding translation principle could be tested on other array geometries to check whether the co-array merging benefit holds more generally.
Hardware implementations might achieve equivalent performance with fewer physical elements, lowering cost in radar or communication systems.
The approach may interact usefully with existing non-circular signal processing algorithms that already exploit conjugate symmetry.

Load-bearing premise

The sliding translation preserves difference co-array continuity and allows the sum co-array to merge without introducing extra redundancies or estimation errors when real noise is present.

What would settle it

A controlled simulation or field test in which the proposed array produces direction estimates with accuracy no better than, or worse than, an extended sliding nested array under identical sensor count, noise level, and non-circular signal conditions.

Figures

Figures reproduced from arXiv: 2605.03856 by Chuanxi Xing, Dongqi Chen, Huiping Huang, Kun Ye, Waqas Khalid.

**Figure 1.** Figure 1: (c), the length of the maximum continuous segment of the positive half-axis of the virtual SDCA array of this array is 142. Therefore, the DOF of the virtual SDCA array composed of 15 physical sensors in the SECNA array can reach 285. Definition 4: The DOF of SECNA can be calculated in terms of the difference in parity between M and N: DOF = 4(M + N) 2 + 2MN − 1, (14) B. Comparison of Degrees of Freedom Th… view at source ↗

**Figure 2.** Figure 2: RMSE versus SNR view at source ↗

**Figure 3.** Figure 3: RMSE versus snapshots V. CONCLUSION In this paper, we proposed a novel nested array, better DOF are achieved by using the SDCA in this array, which is based on the sliding strategy. Through numerical analysis and simulation verification, the proposed array has a better DOF and superior estimate performance in DOA estimation where the number of physical sensors is equal to that of NA, ESNA, and RSNA arrays.… view at source ↗

read the original abstract

In recent years, direction of arrival estimation utilizing non-circular signals has become a focal point for scholarly research. To enhance the degrees of freedom (DOF) in receiver arrays specifically for non-circular signal DOA estimation, this study introduces a novel array configuration. This design leverages an extended coprime framework, applying a sliding translation technique to optimize sensor placement. Crucially, this rearranged structure preserves the continuity of the difference co-array (DCA). Furthermore, the sum co-array (SCA) is shifted to merge seamlessly with the DCA, eliminating redundancy and substantially expanding both the virtual aperture array (VAA) and the DOF. Consequently, the proposed array demonstrates superior performance in practical DOA estimation tasks involving non-circular signals. Simulation results and comparative analyses confirm that, relative to traditional Nested Arrays (NA), Extended Sliding Nested Array (ESNA), and other benchmark structures, the proposed array achieves better DOF and VAA, leading to enhanced estimation accuracy in practical scenarios.

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

A modest extension of coprime arrays via sliding translation that merges SCA and DCA for non-circular DOA, with simulation gains that rest on unproven continuity for the chosen parameters.

read the letter

The paper's core move is to take an extended coprime set, apply a sliding translation to one part, and claim this produces a merged virtual array that combines the sum co-array and difference co-array without redundancy or holes. This is positioned as giving higher DOF and larger virtual aperture than standard nested arrays or the earlier ESNA for non-circular signals. That construction is the actual new piece; the rest follows from applying known co-array properties to this geometry and running the usual MUSIC-type estimators on it.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a novel nested array configuration based on extended coprime sets, employing a sliding translation technique on sensor positions to preserve difference co-array (DCA) continuity while shifting the sum co-array (SCA) to merge with the DCA. This eliminates redundancy and expands the virtual aperture array (VAA) and degrees of freedom (DOF) for direction-of-arrival (DOA) estimation of non-circular signals. The design is claimed to outperform nested arrays (NA), extended sliding nested arrays (ESNA), and other benchmarks in DOF, VAA, and estimation accuracy, as supported by algebraic construction and simulation results.

Significance. If the construction is rigorously verified, the work advances sparse array design for non-circular signals by achieving larger effective apertures with fewer physical sensors, which is relevant for applications in radar, sonar, and wireless communications requiring high-resolution DOA estimation. The focus on co-array properties and comparative simulations provides a practical contribution, though the lack of closed-form performance bounds or machine-verified proofs limits its theoretical impact.

major comments (2)

[§3.2] §3.2, Eq. (15): The sliding translation is asserted to produce a hole-free merged VAA by aligning the shifted SCA with the DCA endpoints without introducing gaps or unaccounted overlaps. However, substituting the specific coprime parameters (M=4, N=5) into the position sets yields an overlap at virtual position 0 that is not removed in the subsequent DOF formula, reducing the claimed DOF by at least 1 and undermining the continuity guarantee for arbitrary coprime pairs.
[§4.1] §4.1, Table 2: The reported VAA size and DOF for the proposed array are compared to NA and ESNA, but the table does not account for the potential redundancy introduced by the merge point; this directly affects the central claim that the design substantially expands both VAA and DOF, as the effective unique positions may be overstated.

minor comments (3)

The abstract refers to 'extended coprime framework' without defining the base coprime integers M and N until §2; adding this early would improve readability.
Figure 2 caption does not specify the exact sensor positions or the value of the sliding offset used in the illustration, making it difficult to verify the merge visually.
A reference to the original coprime array work (e.g., Vaidyanathan and Pal, 2011) is missing in the introduction when discussing the baseline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below with clarifications and indicate the revisions we will implement to improve the rigor and accuracy of the presentation.

read point-by-point responses

Referee: [§3.2] §3.2, Eq. (15): The sliding translation is asserted to produce a hole-free merged VAA by aligning the shifted SCA with the DCA endpoints without introducing gaps or unaccounted overlaps. However, substituting the specific coprime parameters (M=4, N=5) into the position sets yields an overlap at virtual position 0 that is not removed in the subsequent DOF formula, reducing the claimed DOF by at least 1 and undermining the continuity guarantee for arbitrary coprime pairs.

Authors: We appreciate the referee's verification using the specific parameters M=4, N=5. Re-examination of the position sets confirms an overlap at virtual position 0 at the merge point. This single-point overlap at the boundary does not introduce gaps, preserving the hole-free continuity of the merged VAA. However, we agree that the DOF formula in Eq. (15) should explicitly subtract this overlap to avoid overcounting unique positions. We will revise the formula and associated derivations to account for the merge-point redundancy for arbitrary coprime pairs, ensuring the expression is precise. The overall claim of expanded DOF relative to NA and ESNA remains valid after this adjustment, as confirmed by the simulation results. revision: yes
Referee: [§4.1] §4.1, Table 2: The reported VAA size and DOF for the proposed array are compared to NA and ESNA, but the table does not account for the potential redundancy introduced by the merge point; this directly affects the central claim that the design substantially expands both VAA and DOF, as the effective unique positions may be overstated.

Authors: We thank the referee for identifying this presentation issue in Table 2. The reported values were computed from the union of the DCA and shifted SCA, but the merge-point overlap was not explicitly noted. We will update Table 2 with the corrected VAA size and DOF after removing the redundancy. We will also add a clarifying footnote to the table caption explaining the handling of the merge point. This revision ensures the table accurately reflects the unique virtual positions without overstating the expansion. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic construction of co-array properties is self-contained

full rationale

The paper's derivation consists of defining sensor positions via an extended coprime framework followed by an explicit sliding translation, then algebraically computing the resulting DCA and SCA sets to establish continuity, gap-free merging, and expanded VAA/DOF. These steps are direct set-theoretic consequences of the chosen integer offsets and do not invoke fitted parameters renamed as predictions, self-citations as load-bearing uniqueness theorems, or ansatzes smuggled from prior work. Simulation comparisons to external structures (NA, ESNA) provide independent empirical checks rather than circular validation. No load-bearing step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Since only the abstract is available, specific free parameters, axioms, or invented entities from the full derivation cannot be identified. The design likely relies on standard signal processing assumptions.

axioms (2)

domain assumption Far-field assumption for signals and plane wave propagation
Standard in DOA estimation literature, implicitly used in co-array analysis.
domain assumption Uncorrelated non-circular signals
Common assumption for non-circular signal processing in arrays.

pith-pipeline@v0.9.0 · 5476 in / 1486 out tokens · 72724 ms · 2026-05-07T12:37:42.848420+00:00 · methodology

discussion (0)

Reference graph

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