DOA Estimation via Continuous Aperture Arrays: MUSIC and CRLB

Haonan Si; Jin Zhang; Xiansheng Guo; Yuanwei Liu; Zhaolin Wang

REVIEW 4 major objections 5 minor 32 references

DOA Estimation via Continuous Aperture Arrays: MUSIC and CRLB

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read CAPA-MUSIC estimates DOAs near the Cramér-Rao bound and proves continuous apertures beat discrete arrays.

desk verdict The CAPA-MUSIC algorithm is a plausible practical method, but the paper's continuous-limit derivation is internally inconsistent, so the CRLB and the CAPA-vs-SPDA gain claim are not established as written. read the letter →

arxiv 2507.21347 v1 pith:NKUP3HL4 submitted 2025-07-28 eess.SP

Haonan Si , Zhaolin Wang , Xiansheng Guo , Jin Zhang , Yuanwei Liu This is my paper · ORCID

classification eess.SP MSC 94A1265D32

keywords continuousaperturearrayCAPA-MUSICDOAestimationCramér-RaolowerboundMUSICalgorithmGauss-Legendrequadratureeigendecompositionspatialdegreesoffreedom

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper attempts to establish that direction-of-arrival (DOA) estimation can be performed directly on a continuous aperture array (CAPA), a receiving surface that measures the field at every point rather than at discrete antenna elements, and that such arrays are fundamentally more accurate than conventional discrete arrays. To make this practical, it proposes CAPA-MUSIC, a MUSIC-type algorithm that works with infinite-dimensional continuous measurements by reducing the covariance operator to a finite matrix and approximating integrals with Gauss-Legendre quadrature. It derives Cramér-Rao lower bounds for DOA estimation with CAPAs, both when the target snapshot signals are known and when they are unknown, and proves that CAPAs lower the bound by a factor on the order of $\lambda^2/(8 l_r^2)$ relative to a spatially discrete array of short dipoles. For a typical dipole length this is roughly two orders of magnitude, so the paper's message is that continuous apertures offer a concrete accuracy advantage along with a workable estimator. If the central claim holds, it supplies both a theoretical limit and a practical algorithm for the sensing side of 6G continuous-aperture systems.

What carries the argument

The load-bearing device is the continuous-to-discrete transformation in Section III-B: the eigendecomposition of the infinite-dimensional covariance operator is reduced to that of the finite $T \times T$ matrix $K = X^H X/T$, and the eigenvectors of the noise subspace are reconstructed by the Moore-Penrose pseudo-inverse in Eq. (38). The continuous inner products in $K$ and in the MUSIC spectrum are then evaluated by Gauss-Legendre quadrature (Eqs. (30)-(33)), giving a finite, tunable approximation and a complexity of $O(T^3 + K^4 T + N_S K^4)$. The CRLB analysis rests on a second device, the limit model $Z \to 0$ in Eqs. (16)-(21), in which the aperture is split into infinitesimal elements; that limit produces a finite non-vanishing covariance from which the Fisher information matrix is computed.

What would settle it

Generate a simulated CAPA with a large but finite number of very small antenna elements, run CAPA-MUSIC on a single target over many independent trials at a fixed SNR, and compare the empirical azimuth and elevation MSE with the claimed $CRLB_u$ from Eqs. (75)-(76); if the MSE remains clearly above the bound after increasing the quadrature order $K$ and the snapshot count $T$, the finite-rank noise-subspace reconstruction is not faithful and the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that CAPA-MUSIC (Eq. (40), Algorithm 1) estimates azimuth and elevation of multiple targets with mean squared error close to the CAPA Cramér-Rao lower bound, and that a CAPA reduces that bound by a factor on the order of $\lambda^2/(8 l_r^2)$ compared to a conventional spatially discrete array (Remark 5). The paper derives closed-form CRLBs for both known and unknown snapshot signals, proves that unknown snapshots never improve the bound, and shows that for a symmetric aperture with a single target the two bounds coincide (Proposition 1). It attributes the accuracy gain to the continuous aperture's large spatial degrees of freedom and demonstrates numerically that the algorithm's MSE approaches the bound as the quadrature order $K$ grows.

Load-bearing premise

The load-bearing premise is that the finite snapshot matrix $K = X^H X/T$ captures the continuous noise subspace faithfully enough that the pseudo-inverse reconstruction of eigenvectors yields a valid MUSIC spectrum; the paper acknowledges information loss in Remark 2 and supports the assumption only numerically.

Editorial extensions

If this is right

CAPA-MUSIC achieves estimation performance close to the Cramér-Rao bound for multiple targets, with computational complexity $O(T^3 + K^4 T + N_S K^4)$.
CAPAs lower the CRLB by a factor on the order of $\lambda^2/(8 l_r^2)$ versus half-wavelength-spaced short-dipole arrays, implying an accuracy gain of roughly $8\pi^2$ for $l_r = \lambda/(8\pi)$.
The CRLB with known snapshots is always no larger than the CRLB with unknown snapshots, and the two are equal for a symmetric aperture with a single target.
Enlarging the CAPA dimensions $L_x$ and $L_y$ sharpens the MUSIC peaks and reduces the CRLB through terms like $L_x^3 L_y$ and $L_x L_y^3$.
With multiple targets, the algorithm's MSE approaches the CAPA CRLB and worsens gracefully as the number of closely spaced targets grows.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

For real hardware, the continuous limit is a mathematical idealization; at finite element densities the gain over discrete arrays will be smaller, and a quantitative model of discretization loss would sharpen the practical prediction.
The same continuous-discrete transformation could carry other subspace methods (e.g., ESPRIT or tensor-based estimators) over to CAPAs, and the CRLB framework could extend to near-field DOA or joint position-and-attitude estimation.
The paper's own Remark 2 suggests a testable design rule: the number of snapshots $T$ needed for a faithful noise subspace may scale with the number of targets $M$, and a systematic study of that scaling would tell practitioners how large $T$ must be.
Because the CRLB expressions depend explicitly on aperture geometry, optimizing the aperture shape (not just its size) could yield further accuracy gains beyond the square CAPAs tested here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper has a genuine new piece: a MUSIC variant for continuous-aperture arrays, built on the standard X^H X versus XX^H eigendecomposition trick and Gauss-Legendre quadrature, with numerical experiments showing MSE tracking the CRLB. That part is worth reading. The complexity analysis is sensible, and the CAPA-MUSIC spectrum is clearly specified.

The problem is the continuous-limit theory. Equation (16) defines x_n(t) = Z E(r_n,t) with noise variance Z sigma^2, so the covariance of x is Z^2 A R_s A^H + Z sigma^2 I. Equation (21) instead writes A R_s A^H + sigma^2 I, dropping the Z factors. Equation (29) then claims Z^2 sum_n E*E tends to the integral, but for a Riemann sum with N proportional to 1/Z, Z^2 sum_n is about Z times the integral, which goes to 0. So the K matrix used in Algorithm 1 is not the limit stated; it is the right finite object only under a different normalization, which is not given. The same inconsistency enters the likelihood in (44) and the FIM in (46): they use sigma_nu^2 = Z sigma^2 without the corresponding Z^2 in the signal, so the CRLBs in (89)–(90) are not the limits of the stated model. The CAPA-vs-SPDA factor lambda^2/(8 l_r^2) in Remark 5 is computed from those CRLBs, so the central theoretical claim is unsupported. This is an internal contradiction, not a matter of interpretation.

There is also a smaller contradiction: Proposition 1 says known and unknown snapshot CRLBs are identical for a single symmetric target, which follows from the FIM block-diagonal structure. Remark 4 then says uncertainty in the signal still affects performance “via other statistical dependencies,” which directly contradicts the proposition. As written, that remark is wrong.

What is good: the algorithm itself may work; the simulations show near-CRLB behavior, and the Gauss-Legendre approximation is sensible. But the paper does not ship code or data, so the empirical claim is not independently reproducible from the text.

Who is this for? People working in CAPA sensing / holographic MIMO. They will want to cite it if it survives revision. My recommendation: send it to review, but it needs major revision. The authors must fix the Z->0 normalization, redo the CRLB derivation, and qualify or remove the CAPA superiority factor. If they do, it could be a solid contribution. As is, I would not accept.

Referee Report

4 major / 5 minor

Summary. The paper studies direction-of-arrival (DOA) estimation with continuous aperture arrays (CAPAs). It proposes CAPA-MUSIC, which avoids eigendecomposition of an infinite-dimensional covariance operator by forming a T-by-T matrix K from field snapshots, reconstructing noise-subspace eigenvectors through Gauss-Legendre quadrature and a pseudo-inverse, and then searching a MUSIC spectrum. It derives Cramér-Rao lower bounds (CRLBs) for DOA estimation in both known-snapshot and unknown-snapshot cases, and claims a theoretical accuracy advantage of CAPA over conventional spatially discrete arrays (SPDAs), quantified in Remark 5 as a reduction by roughly lambda^2/(8 l_r^2). Numerical experiments show the proposed algorithm's MSE closely tracking the derived CRLB for single- and multi-target scenarios.

Significance. If the central claims are correct, the paper would provide a practical, low-complexity MUSIC algorithm for continuous apertures and a useful CRLB benchmark, together with a theoretical argument that CAPAs offer a substantial accuracy gain over SPDAs. The algorithm is clearly specified with a complexity analysis, and the numerical validation is extensive. However, the theoretical comparison and the CRLB formulas rest on a limiting discretization that is internally inconsistent. The claimed CAPA-over-SPDA gain is not established by the derivation as written. With a repaired normalization of the continuous-limit model, the contribution could be significant for the CAPA sensing literature.

major comments (4)

[§II-B and §III-B, Eqs. (16), (20), (21), (29)] The limiting discretization is internally inconsistent. From Eq. (16), x_n(t)=Z(a^T(r_n)s(t)+n(r_n,t)) with noise variance sigma_nu^2=Z sigma^2, so Eq. (20) is a vector whose entries scale as Z; taken literally the limit is the zero vector, and the covariance in Eq. (21) should be Z^2 A R_s A^H + Z sigma^2 I_N, not A R_s A^H + sigma^2 I_N. Consequently, in Eq. (29), Z^2 sum_n E^*(r_n,i)E(r_n,j) tends to 0 as Z->0, not to the surface integral; the displayed equality is dimensionally wrong. The same scaling issue affects the eigenvector reconstruction in Eqs. (34)-(38), where the components of u_i acquire a Z-dependent normalization. The paper needs to define the limiting continuous observation model explicitly (for example, by normalizing voltages per unit area or by working directly with the integral covariance operator) and redo Eqs. (21)-(38) under that model.
[§IV-A, Eqs. (43)-(46)] The likelihood limit in Eq. (44) has the same normalization error. With x(r_n,t)=Z E(r_n,t) and sigma_nu^2=Z sigma^2, the exponent in Eq. (43) equals -(1/(Z sigma^2)) sum_t sum_n Z^2 |E-a^H s|^2 = -(Z/sigma^2) sum_t sum_n |E-a^H s|^2, whose limit is -(1/sigma^2) sum_t int_S |E-a^H s|^2 dr, not -(1/sigma_nu^2) sum_t int_S |E-a^H s|^2 dr. Thus the FIM in Eq. (46) should be proportional to 1/sigma^2 rather than 1/sigma_nu^2 = 1/(Z sigma^2). As written, sigma_nu^2->0 makes the CRLBs in Eqs. (89)-(90) tend to zero, i.e., infinite precision from a finite aperture, which is unphysical. The CRLB derivation must be redone with a consistent noise normalization.
[§IV-C, Remark 5] The claimed CAPA-over-SPDA gain factor lambda^2/(8 l_r^2) is not obtained from the displayed formulas under any consistent assignment of noise variances. If one evaluates the elevation CRLB ratio from Eqs. (90) and (98) with P=2 L_x/lambda, Q=2 L_y/lambda, and a common noise variance, the ratio is approximately 2 l_r^2/lambda^2 (i.e., CAPA improves by lambda^2/(2 l_r^2)), not lambda^2/(8 l_r^2). If instead the infinitesimal-element variance sigma_nu^2=Z sigma^2 is retained, the CAPA CRLB vanishes with Z and the comparison is meaningless. A corrected comparison after the normalization fix is required before the theoretical superiority claim can be sustained.
[§III-B, Eqs. (34)-(38) and Remark 2] The reduction of the infinite-dimensional eigendecomposition to the T-by-T matrix K is not fully justified. The exact algebraic relation between eigenvectors of X X^H and X^H X is u_i = (1/lambda_i) X e_i; the paper instead reconstructs u_i from e_i via the pseudo-inverse of the quadrature matrix E^H Omega in Eq. (38). It is not shown that this pseudo-inverse recovers the noise-subspace eigenvectors, nor that the Gauss-Legendre quadrature error remains controlled after the pseudo-inverse. Remark 2 concedes information loss but does not quantify it. Please provide a consistency argument or error bound for this step, or replace it by the exact algebraic reconstruction.

minor comments (5)

[Eq. (18)] The dimension of A(alpha, phi) is stated as C^{M x N}, but the construction [a(r_1), ..., a(r_N)]^T with a(r_n) in C^M gives an N x M matrix; the product A s(t) in Eq. (20) also requires A to be N x M.
[Eqs. (34)-(35)] There are notation slips: Eq. (34) integrates E^*(r, k) where the index should be t, and Eq. (35) states omega = diag{omega_1,...,omega_K} in R^{K^2 x K^2}; the intended object is likely a K x K diagonal matrix before the Kronecker product.
[§V-C] In the M=3 target list, the third target location z3 = [200, 50, 15] is identical to the second target's location; this appears to be a typo and should be corrected for reproducibility.
[Proposition 1 and Remark 4] Proposition 1 proves that CRLB_k equals CRLB_u for a symmetric single-target aperture, but Remark 4 then states that uncertainty in the signal still affects estimation performance; the remark appears to contradict the proposition and should be clarified or removed.
[Eq. (12)] The delta-correlated noise in Eq. (12) is not pointwise well-defined; the derivation should integrate the noise over each cell before taking the limit, rather than writing n(r_n, t) as a pointwise random variable.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity is found: the CAPA-MUSIC estimator and the CRLBs are derived from the stated signal model, the near-CRLB claim is benchmarked against the paper's own bound in standard practice, and the self-citations are background only.

full rationale

The claimed derivation chain is self-contained, so no circular step is identified. Algorithm 1 estimates angles by searching peaks of a MUSIC spectrum built from the data-dependent matrix K = X^H X/T (Eqs. (29)-(40)); no parameter is fitted to targets, and the eigenvector reconstruction u_i is approximated by (E^H Omega)^dagger e_i (Eq. (38)), which is the exact algebraic relation e_i = X^H u_i of the finite sample model combined with a Gauss-Legendre quadrature whose convergence is checked externally in Fig. 3. Remark 2 candidly admits the information loss of this finite-rank reduction, but that stated limitation does not make the near-CRLB result circular, because the CRLB is derived independently from the Gaussian likelihood in Section IV, with the unknown-snapshot FIM block attributed to the non-self reference [22]. The CAPA-versus-SPDA comparison in Remark 5 is a closed-form ratio of two independently derived CRLBs, hence an algebraic consequence of the model rather than a value fitted after the fact; whether the ratio is physically meaningful depends on the noise normalization and on the consistency of the Z-to-0 limiting model in Eqs. (16)-(29) (the x_n = |S_n|E(r_n) normalization carries implicit Z factors that are dropped), but those are modeling-consistency and correctness concerns, not a derivation loop, since the CAPA-superiority conclusion never appears as an input to the derivation. Self-citations ([3], [14], [15], [21], [23], and the cited position paper defining CAPA) motivate the architecture only and are not load-bearing for the MUSIC derivation or the CRLB formulas, and no uniqueness theorem from the authors' prior work is invoked. Benchmarking the estimator's MSE against the paper's own CRLB is standard practice for optimality claims and does not constitute renaming or fitting. Verdict: no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard linear algebra (eigendecomposition of X^H X versus XX^H, Gauss-Legendre quadrature, FIM formulas) and on two domain assumptions (far-field planar waves, white Gaussian noise). The paper's own ad-hoc assumption is the Z to 0 limiting model, which is not mathematically consistent as written. The only freely chosen parameter that affects the headline comparison is the SPDA dipole length l_r.

free parameters (2)

dipole length l_r in SPDA comparison = lambda/(4 pi) approximately 0.00796 m in simulations
The claimed CAPA accuracy gain over SPDA is the factor lambda^2/(8 l_r^2) (Remark 5). The numerical value in Section V-B is set to lambda/(4 pi), giving about 79x (19 dB). This is a modeling choice, not a measured quantity; a different l_r changes the headline conclusion.
Gauss-Legendre dimension K = 30
Chosen from the convergence test in Fig. 3; determines quadrature accuracy and runtime. It is an approximation parameter, not a physical constant.

assumptions (4)

domain assumption Far-field planar wave approximation (Eqs. (8)-(9)): the received phase at point r is e^{jk r dot d(alpha, phi)}, neglecting the second-order Fresnel term.
Required for the steering vector model. Fails for near-field targets; the paper restricts to z0 greater than 2D^2/lambda.
domain assumption Spatially white Gaussian noise with correlation sigma^2 delta(r1 - r2) across the aperture (Eq. (12)).
Underpins the covariance form and the likelihood in Eq. (43); idealized, and the paper notes extension to colored noise follows reference [20].
domain assumption Non-coherent point sources with T >= M snapshots so that the sample covariance has rank M signal-plus-noise structure (Section III-A).
Required for the MUSIC subspace separation; coherent sources would break the rank argument.
ad hoc to paper The Z to 0 discretization limit in Eqs. (16)-(21) yields a nonvanishing covariance R_x = A R_s A^H + sigma^2 I.
The paper lets element area Z go to 0 while keeping E[x x^H] nonzero; as written, E[x x^H] = Z^2(...) + Z sigma^2 I goes to 0, so the limit is not rigorously defined. This underpins both the algorithm's K matrix and the CRLB.

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Cite this review

Pith. "Pith review of DOA Estimation via Continuous Aperture Arrays: MUSIC and CRLB." pith.science (2026). https://pith.science/paper/NKUP3HL4

@misc{pith2026250721347,
  author       = {Pith},
  title        = {Pith review of: DOA Estimation via Continuous Aperture Arrays: MUSIC and CRLB},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKUP3HL4}},
  note         = {Machine review of arXiv:2507.21347}
}

read the original abstract

Direction-of-arrival (DOA) estimation using continuous aperture array (CAPA) is studied. Compared to the conventional spatially discrete array (SPDA), CAPA significantly enhances the spatial degrees-of-freedoms (DoFs) for DOA estimation, but its infinite-dimensional continuous signals render the conventional estimation algorithm non-applicable. To address this challenge, a new multiple signal classification (MUSIC) algorithm is proposed for CAPAs. In particular, an equivalent continuous-discrete transformation is proposed to facilitate the eigendecomposition of continuous operators. Subsequently, the MUSIC spectrum is accurately approximated using the Gauss-Legendre quadrature, effectively reducing the computational complexity. Furthermore, the Cram\'er-Rao lower bounds (CRLBs) for DOA estimation using CAPAs are analyzed for both cases with and without priori knowledge of snapshot signals. It is theoretically proved that CAPAs significantly improve the DOA estimation accuracy compared to traditional SPDAs. Numerical results further validate this insight and demonstrate the effectiveness of the proposed MUSIC algorithm for CAPA. The proposed method achieves near-optimal estimation performance while maintaining a low computational complexity.

Figures

Figures reproduced from arXiv: 2507.21347 by the authors.

**Figure 2.** An explanation of geometry of targets. r Td(αm, ϕm). Then, substituting Eq. (8) into Eq. (7) and neglecting the 2-order term results in: h(zm) ≈ jkη0e −jk||qm||2 4π||qm||2 e jkr Td(αm,ϕm) , (9) where αm and ϕm denote the azimuth and elevation angles, respectively. Then, due to the presence of noise, the excited electric filed at time t at the point r ∈ S can be written as follows: E(r, t) = X M m=1 J(qm, t)h(zm) + n… view at source ↗

**Figure 3.** An illustration of the MSE and running time versus the number of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: An illustration of the achieved MSE of CAPA-MUSIC along with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: An illustration of achieved CRLB. results highlighted in Remark 5 and validates the remarkable superiority of CAPA systems. Furthermore, since a symmetric CAPA array is employed in the simulation, the snapshot signals do not influence the CRLB of the DOA estimation (as…

**Figure 8.** Figure 8: MUSIC spectrum for CAPA-based DOA estimation under different CAPA sizes. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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