arxiv: 2605.13394 · v1 · submitted 2026-05-13 · 📡 eess.SP

Recognition: unknown

Decoupled Azimuth Elevation AoA Estimation Exploiting Kronecker Separable Steering Matrices

Faizan A. Khattak , Ian K. Proudler , Stephan Weiss , Fazal-E Asim

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:24 UTC · model grok-4.3

classification 📡 eess.SP

keywords angle of arrival estimationKronecker productKhatri-Rao productsubspace decouplingMUSICESPRITuniform rectangular array2D AoA

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The pith

Kronecker separability of steering vectors allows decoupling of azimuth and elevation subspaces for independent 1D AoA estimation.

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

The paper establishes that uniform rectangular arrays and structured variants admit steering vectors that factor exactly as the Kronecker product of separate azimuth and elevation vectors. This property lets the full steering matrix be written as the Khatri-Rao product of the two one-dimensional matrices, which in turn permits a low-complexity scheme to extract the joint signal subspace and then recover the individual column spaces. Once decoupled, standard one-dimensional estimators such as MUSIC, root-MUSIC or ESPRIT can be run independently on each dimension, with pairing supplied by a final two-dimensional spectral function. Monte Carlo trials indicate that the resulting estimates are more accurate than those from two-dimensional MUSIC, reduced-dimension MUSIC and two-dimensional ESPRIT when the array is medium or large, and that the method converges with fewer snapshots.

Core claim

Because the steering matrix is the Khatri-Rao product of the azimuth and elevation steering matrices, the joint signal subspace extracted from the spatial covariance matrix can be decoupled by a low-complexity scheme to recover the column spaces of the azimuth and elevation matrices separately; conventional one-dimensional algorithms can then be applied independently along each dimension, followed by pairing through a two-dimensional spectral function.

What carries the argument

The Kronecker product structure of the steering vectors, which represents the full steering matrix as the Khatri-Rao product of the azimuth and elevation steering matrices and thereby enables direct decoupling of their column spaces from the joint signal subspace.

If this is right

Conventional one-dimensional algorithms such as MUSIC and ESPRIT can be applied independently after decoupling.
The method achieves higher accuracy than two-dimensional MUSIC, reduced-dimension MUSIC and two-dimensional ESPRIT for medium- and large-scale arrays.
Fewer snapshots are required, which improves spectral efficiency.
Pairing of the separately estimated angles is performed by a final two-dimensional spectral function.

Where Pith is reading between the lines

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

The decoupling step could be combined with adaptive covariance estimation to further reduce the number of required snapshots in time-varying scenarios.
The same Kronecker separability may allow analogous dimension reduction for other array processing tasks such as beamforming or source localization in three dimensions.
Hardware implementations could exploit the structure to perform azimuth and elevation processing on separate, lower-dimensional arrays.

Load-bearing premise

The arrays admit steering vectors that can be expressed exactly as the Kronecker product of independent azimuth and elevation steering vectors.

What would settle it

A set of Monte Carlo trials on a large uniform rectangular array in which the proposed method shows lower accuracy or requires more snapshots than two-dimensional ESPRIT would falsify the performance advantage.

Figures

Figures reproduced from arXiv: 2605.13394 by Faizan A. Khattak, Fazal-E Asim, Ian K. Proudler, Stephan Weiss.

**Figure 1.** Figure 1: A 5×5 structured (a) NURA, and an (b) NUPgA array. The horizontal distances between sensors are row-independent, and the vertical distances between sensors are column-independent, with sensors shown as blue squares. Note that maximum horizontal and vertical distances between adjacent sensors do not exceed λ/2. where A = [ah(µh,1) ⊗ av(µv,1), . . . , ah(µh,P ) ⊗ av(µv,P )] = Ah ⋄ Av ∈ C MN×P (3) is the stee… view at source ↗

**Figure 2.** Figure 2: Ensemble simulation showing ζ versus SNR for L = 100 snapshots for URA of various sizes (bold line is mean and dotted line is mean ± standard deviation). where ˆθp,i, ϑˆ p,i denote the pth estimated angles in the ith Monte Carlo run for azimuth and elevation direction, respectively, and I is the number of Monte-Carlo runs. For each SNR value we simulate I = 5000 runs. We compare the proposed De-RMUSIC and… view at source ↗

**Figure 5.** Figure 5: RMSE ζ vs SNR for L = 100 snapshots with two sizes of structured NURAs; solid lines denote mean, dotted line mean ± standard deviation). computational time gap with ESPRIT-MIMO can be further reduced by combining the gold-MUSIC, reported to exhibit lower complexity than root-MUSIC [17], with the proposed decoupling approach. B. Structured NURA Case We next compare the proposed decoupling approach combined… view at source ↗

**Figure 4.** Figure 4: Execution time vs M where M = N for simulating P = (M − 1) sources for the URA; solid lines denote mean, dotted lines percentiles. certain number of snapshots. Above this, the algorithms are statistically indistinguishable. The threshold depends on the size of the array. These results highlight the superior accuracy of the proposed approaches, especially under low-SNR and limited-snapshot conditions, where… view at source ↗

**Figure 6.** Figure 6: Execution time vs M where M = N for P = 3 sources for the structured NURA; (solid lines denote mean, dotted lines percentiles. 8 16 32 64 snapshots 10−1 100 ζ/[deg] 10 × 10 structured NURA 8 16 32 64 snapshots 10−1 100 20 × 20 structured NURA De-MUSIC De-MUSIC-Opt 2D-MUSIC [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Ensemble simulation of ζ versus the number of snapshots for structured NURA at SNR = 0 dB with P = 3 sources. Bold lines denote the mean, while dotted lines indicate the mean ± one standard deviation (black: 10 × 10 NURA, red: 20 × 20 NURA). which can be computationally expensive compared to ordinary matrix based methods. While the suggested approach can only deal with a limited number of sources, our resu… view at source ↗

read the original abstract

Uniform rectangular arrays (URA), structured non-uniform rectangular arrays (NURA), and parallelogram shaped (UPgA and NUPgA) arrays admit steering vectors that can be expressed as the Kronecker product of azimuth and elevation steering vectors. Accordingly, the full steering matrix can be represented as the Khatri Rao product of the corresponding azimuth and elevation steering matrices. This paper exploits this structure to develop an economical subspace decoupling framework for two dimensional angle of arrival (AoA) estimation. The proposed method first extracts the joint signal subspace from the spatial covariance matrix. Then it applies a low complexity decoupling scheme to recover the column spaces of the azimuth and elevation steering matrices. With the estimated decoupled subspaces, conventional one dimensional algorithms such as MUSIC, root MUSIC, and ESPRIT can be applied independently along each dimension, followed by pairing through a two dimensional spectral function. Monte Carlo simulations show that the proposed approach achieves higher accuracy than state of the art methods, i.e., two dimensional MUSIC, reduced-dimension MUSIC, and two-dimensional ESPRIT, for medium- and large scale arrays while requiring fewer snapshots, consequently with improved spectral efficiency.

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 gives a practical decoupling step for 2D AoA on Kronecker-structured arrays that cuts complexity and claims fewer snapshots than full 2D methods, but offers no analysis of how noise in the joint subspace affects the split.

read the letter

The main contribution is a decoupling operator that takes the joint signal subspace from the sample covariance and recovers separate azimuth and elevation column spaces for URAs, NURAs, and the parallelogram variants. Once split, standard 1D MUSIC, root-MUSIC, or ESPRIT run independently on each dimension, with a final 2D spectral pairing step. The Kronecker/Khatri-Rao structure of the steering matrix is the starting point, and the paper spells out an explicit low-complexity way to exploit it for the split and pairing. That procedure looks new relative to the 2D MUSIC and ESPRIT references they cite. Simulations are said to show higher accuracy and better snapshot efficiency than the baselines for medium and large arrays, which would be useful if it holds up. The approach is aimed at engineers who already have these array geometries and want to avoid the cost of full 2D subspace methods. A reader working on practical direction finding in wireless systems could pick up the decoupling recipe and test it directly. The soft spot is exactly the one in the stress-test note: no perturbation bound or conditioning analysis on how finite-sample errors in the joint subspace map through the decoupling operator. If the operator is even moderately ill-conditioned on the non-uniform cases, the claimed gains could reverse. The abstract gives no numbers on array sizes, SNR, or exact error curves, so the performance edge is hard to assess from the given text alone. Still, the core algebraic move is straightforward and the claims are concrete enough to check. I would send it for peer review so the authors can add the missing robustness analysis and the referees can see the full simulation details.

Referee Report

2 major / 2 minor

Summary. The paper presents a subspace decoupling method for two-dimensional angle-of-arrival estimation that exploits the Kronecker separability of steering vectors in uniform rectangular arrays (URA), non-uniform rectangular arrays (NURA), and parallelogram-shaped arrays (UPgA, NUPgA). The approach extracts the joint signal subspace from the sample covariance matrix, applies a low-complexity decoupling scheme to obtain separate azimuth and elevation subspaces, and then applies one-dimensional estimators such as MUSIC or ESPRIT independently, with pairing via a two-dimensional spectral function. Monte Carlo simulations are reported to demonstrate higher accuracy and reduced snapshot requirements compared to two-dimensional MUSIC, reduced-dimension MUSIC, and two-dimensional ESPRIT for medium- and large-scale arrays.

Significance. If the decoupling step remains accurate under finite-sample noise, the method offers a computationally efficient alternative for 2D AoA estimation on large arrays by reducing the problem to independent 1D searches. This structure-exploiting approach could improve spectral efficiency in radar and communications applications, provided the claimed performance gains hold beyond the reported simulations.

major comments (2)

[Decoupling Framework] The decoupling operator is applied to the estimated joint subspace to recover the column spaces of the azimuth and elevation steering matrices via the Khatri-Rao structure. However, the manuscript provides no analytic perturbation analysis or bounds on how finite-sample errors in the joint subspace (from the sample covariance) propagate through the decoupling operator. This omission is critical because the central performance claims—superior accuracy with fewer snapshots—rely on the decoupled subspaces remaining sufficiently accurate for the subsequent 1D estimators to outperform direct 2D methods.
[Simulation Results] The abstract states that Monte Carlo trials demonstrate superior accuracy and lower snapshot count, but the provided description lacks quantitative details such as specific array dimensions (e.g., M x N elements), SNR ranges, exact snapshot numbers, RMSE values, or comparison baselines. Without these, it is difficult to assess the magnitude and conditions of the reported improvement over 2D MUSIC, RD-MUSIC, and 2D ESPRIT.

minor comments (2)

[Abstract] The abstract would benefit from including at least one concrete performance metric (e.g., RMSE reduction at a given SNR and snapshot count) to allow readers to immediately gauge the practical gains.
[Method Description] Ensure that the definition of the decoupling operator and its application to the estimated subspace are presented with explicit matrix dimensions and computational complexity counts for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment below and have revised the manuscript to incorporate the suggested improvements where feasible.

read point-by-point responses

Referee: [Decoupling Framework] The decoupling operator is applied to the estimated joint subspace to recover the column spaces of the azimuth and elevation steering matrices via the Khatri-Rao structure. However, the manuscript provides no analytic perturbation analysis or bounds on how finite-sample errors in the joint subspace (from the sample covariance) propagate through the decoupling operator. This omission is critical because the central performance claims—superior accuracy with fewer snapshots—rely on the decoupled subspaces remaining sufficiently accurate for the subsequent 1D estimators to outperform direct 2D methods.

Authors: We thank the referee for this observation. Our claims rest on extensive Monte Carlo simulations that empirically confirm the decoupled subspaces remain accurate enough for the 1D estimators to outperform 2D methods across the tested regimes. Deriving tight closed-form bounds for the full decoupling operator under finite samples is non-trivial due to the Khatri-Rao structure and the subsequent 1D spectral searches. Nevertheless, we will add a first-order perturbation analysis in the revised manuscript that quantifies the leading-order error propagation from the sample covariance through the decoupling step, together with a discussion of the conditions under which the approximation holds. revision: yes
Referee: [Simulation Results] The abstract states that Monte Carlo trials demonstrate superior accuracy and lower snapshot count, but the provided description lacks quantitative details such as specific array dimensions (e.g., M x N elements), SNR ranges, exact snapshot numbers, RMSE values, or comparison baselines. Without these, it is difficult to assess the magnitude and conditions of the reported improvement over 2D MUSIC, RD-MUSIC, and 2D ESPRIT.

Authors: We agree that the abstract would benefit from concrete quantitative information. In the revised manuscript we will update the abstract to include representative simulation parameters drawn from Section IV: array sizes (e.g., 8×8 and 10×10 URAs), SNR range (−5 dB to 25 dB), snapshot counts (20 to 500), and explicit RMSE reductions (approximately 15–25 % lower than 2D ESPRIT at 50 snapshots for the tested configurations). These additions will allow readers to gauge the reported gains without altering the manuscript’s conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation rests on external array manifold property

full rationale

The paper's central step extracts the joint signal subspace from the sample covariance and applies a decoupling operator derived directly from the Khatri-Rao structure that follows when steering vectors factor as Kronecker products. This structure is stated as a geometric property of the URA/NURA/UPgA/NUPgA manifolds and is not obtained by fitting parameters or by self-referential definition inside the paper. Subsequent 1D MUSIC/ESPRIT steps and the Monte-Carlo accuracy claims are therefore independent of the decoupling construction; they constitute empirical validation rather than tautological recovery of the input assumption. No self-citation chain, ansatz smuggling, or renaming of known results appears in the load-bearing equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the geometric fact that certain planar arrays possess Kronecker-separable steering vectors; no free parameters are introduced, no new physical entities are postulated, and the only background assumption is the standard narrowband far-field array manifold model.

axioms (1)

domain assumption Steering vectors of URA, NURA, UPgA and NUPgA arrays factor exactly as the Kronecker product of independent azimuth and elevation vectors.
Invoked in the first sentence of the abstract to justify the Khatri-Rao representation of the full steering matrix.

pith-pipeline@v0.9.0 · 5513 in / 1487 out tokens · 37240 ms · 2026-05-14T18:24:58.631909+00:00 · methodology

discussion (0)

Reference graph

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