Secure Beamforming for Continuous Aperture Array (CAPA) Systems

Chongjun Ouyang; Mingjun Sun; Shaochuan Wu; Yuanwei Liu; Zhaolin Wang

REVIEW 3 major objections 4 minor 31 references

Secure Beamforming for Continuous Aperture Array (CAPA) Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Secure CAPA beamforming collapses to a single matrix inversion, and the paper derives the closed-form current pattern behind it.

desk verdict The FP-BCD optimality claim is undone by a false equivalence in Lemma 2, but the problem formulation and the ZF scheme have real merit. read the letter →

arxiv 2501.04924 v2 pith:CRGH6IKW submitted 2025-01-09 eess.SP

Mingjun Sun , Chongjun Ouyang , Zhaolin Wang , Shaochuan Wu , Yuanwei Liu This is my paper · ORCID

classification eess.SP

keywords continuousaperturearraysecurebeamformingweightedsecrecysum-ratefractionalprogrammingblockcoordinatedescentzero-forcingfunctioninversionphysicallayersecurity

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 tries to show that secure downlink beamforming for a continuous aperture array (CAPA) can be solved exactly, rather than approximated by discretizing the radiating surface. The goal is to maximize the weighted secrecy sum-rate (WSSR) for multiple legitimate users when several eavesdroppers are listening. The central claim is that the optimal source current pattern has a closed form: it is a linear combination of the channel spatial responses, so the search over continuous functions reduces to inverting an $N \times N$ matrix with $N = K + Q$ (users plus eavesdroppers). A complementary zero-forcing construction produces a zero-leakage current pattern with water-filling power allocation that approaches the optimization-based solution at high transmission power. If the argument is right, CAPA secure beamforming avoids the performance loss and high complexity of Fourier-basis discretization, with reported WSSR gains of 77\% and 117\% over discrete MIMO at $P = 10^2$ mA$^2$.

What carries the argument

The load-bearing object is the continuous inversion identity for kernels of the form $\Psi(s,s')=\delta(s-s')+\sum_{i=1}^{I}\psi_i^*(s)\psi_i(s')$. The paper proves that its inverse is $\delta(s-s')$ minus a finite-rank correction built from the matrix $(I+\Psi)^{-1}$, where $[\Psi]_{i,i'}=\int \psi_i(s)\psi_{i'}^*(s)\,ds$. This identity turns the infinite-dimensional task of inverting an operator into an $I \times I$ matrix inversion. It is applied to the kernel $W_{\lambda,k}$ that appears in the first-order optimality condition for the current pattern, with the functions $\psi_i$ taken as the normalized legitimate and eavesdropping channel responses; combined with fractional-programming auxiliary variables, it yields the closed-form current update in (51).

What would settle it

A direct numerical check settles whether the reformulation is equivalent: take $K=1$, pick any current $J_1(s)$ with $\gamma_1 < \Gamma_1$, and choose parameters so that $G_\Gamma > 2\Gamma_1$. Problem (15) then has optimum $b_1=0$ because the secrecy rate is negative, while problem (20) is positive at $b_1=\alpha_1$ since $\log_2((1+G_\Gamma-\Gamma_1)/(1+\Gamma_1))>0$; the two problems therefore have different optima on this instance.

Watch

Extended reading notes

Core claim

On the paper's own terms, the contribution is a continuous-domain counterpart to linear precoding. The transmit current density $J_k(s)$ for each legitimate user is not searched over all functions: the optimal pattern is shown to be a weighted sum of the conjugate channel spatial responses $H_i^*(s)$ of the legitimate users and $\bar{H}_q^*(s)$ of the eavesdroppers. The weights come from the inverse of a kernel $W_{\lambda,k}(s,s')$ built from the Dirac delta plus the Gram matrix of those responses. The paper's Theorem 2 supplies a continuous-function inversion identity that converts this operator inversion into an $(K+Q) \times (K+Q)$ matrix inversion, giving a closed-form current update at every step of a block coordinate descent algorithm based on fractional programming. The same span result drives the zero-forcing construction, whose coefficients are columns of the inverse channel-correlation matrix. The claimed consequence is that optimal secure beamforming for CAPA is achievable without Fourier truncation, with complexity $O(LKN^3)$ after the channel correlation matrix is precomputed.

Load-bearing premise

The load-bearing premise is that a term involving $G_\Gamma$ can be discarded as a constant during the reformulation, even though it is multiplied by the decision variable $b_k$, so the claimed equivalence of the transformed problems depends on that multiplication being harmless.

Editorial extensions

If this is right

Because the optimal current pattern lies in the span of the $K+Q$ channel responses, each BCD iteration only needs the precomputed channel correlation matrix $\mathbf{H}$; no numerical integration over the aperture is required inside the iterations.
The complexity of the optimization-based method is $O(LKN^3)$ after computing $\mathbf{H}$ with $M$-point quadrature, compared with $O(LKN_F^3)$ for Fourier discretization, where the number of Fourier bases $N_F$ grows rapidly with aperture size and carrier frequency.
The zero-forcing CAPA scheme achieves zero inter-user interference and zero eavesdropper leakage, uses water-filling for power allocation, and approaches the FP-based optimum at high SNR while costing only $O(N^3)$ beyond computing $\mathbf{H}$.
Reported simulations put CAPA's WSSR gain over discrete MIMO at 77% for the optimization-based method and 117% for the zero-forcing method at transmit power $10^2$ mA$^2$, with the gap widening as the aperture size grows.
The optimization-based method flexibly trades residual leakage against power, whereas the ZF method nulls all leakage; this is why the paper reports that the ZF method degrades as the number of users grows, while the optimization method improves.

Reading between the lines

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

If the span result is correct, a testable extension is that secure CAPA beamforming needs only the Gram matrix of channel responses, not full continuous channel functions, which would make CSI acquisition for CAPA substantially lighter than a literal continuous-channel model suggests.
The same inversion identity would apply to other continuous-aperture functional programming problems, such as joint sensing and communication or energy-splitting designs, whenever the kernel has the structure 'Dirac delta plus finite-rank channel correlations'.
A natural prediction to test experimentally is that the gap between the ZF heuristic and the FP-based optimum shrinks with SNR, so a hybrid scheme could switch from ZF at high SNR to the optimization method only when the number of users or eavesdroppers makes interference management difficult.
The dependence of the ZF method on inverting the full $K+Q$ channel matrix implies each additional eavesdropper consumes one spatial degree of freedom; the optimization method, by contrast, spends power to suppress leakage softly, which explains its reported advantage when many users compete.

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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

The headline: the paper's main claim—that the FP-based BCD algorithm exactly solves the multi-user CAPA secure beamforming problem—is unsupported. Lemma 2 is genuinely wrong: the term log2(1+GΓ) that gets discarded in (22) is multiplied by b_k in the objective, so it is not a constant with respect to the optimization variables. The b-update (30) and the claimed optimality rest on this false equivalence. Separately, the algebraic identity in (22b) is also incorrect unless Γ_k=0.

That said, the paper is not a waste of time. The problem formulation itself—WSSR for multiple LUTs and Eves under a continuous aperture—is new relative to the single-user single-eve [27], and it is a natural extension. The continuous-function inversion theorem (Theorem 2) is a useful tool, and the resulting closed-form current pattern as a linear combination of channel responses is a real simplification. The ZF-based scheme in Section IV, with the water-filling power allocation, appears to stand independent of Lemma 2; the zero-forcing construction (Theorem 3) is simple and correct as far as I can tell. The complexity analysis, comparing against the Fourier discretization, is reasonable.

The soft spot is the load-bearing one. Because the equivalence in Lemma 2 fails, Algorithm 1 does not solve problem (13). The simulation claims about 'optimal' performance are therefore not grounded, though the qualitative WSSR gains over discrete MIMO might still hold if the algorithm behaves reasonably. The paper gives no code, so I cannot check the numerics independently. Also, the heavy reliance on the same group's prior work ([20], [21], [27]) is not a flaw by itself, but it means the novelty hinges on how much of the derivation is genuinely different; the CoV lemma from [21] is explicitly imported.

The ZF part and the complexity analysis could survive a careful revision. If the authors can fix Lemma 2—or else reframe the algorithm as a heuristic—the paper would be worth a second look. As is, the central contribution is not established.

Recommendation: I would not accept this in its current form, but I would send it to peer review rather than desk-reject, because the problem is important and a fix is plausible. The ZF section alone might be publishable as a shorter paper.

For your reading group: maybe. It is a good case study in how a small algebraic slip can undermine a whole optimization claim.

Referee Report

3 major / 4 minor

Summary. The paper considers a downlink secure transmission system in which a continuous aperture array (CAPA) serves multiple legitimate users in the presence of multiple eavesdroppers. The authors formulate the weighted secrecy sum-rate (WSSR) maximization as a functional programming problem over the source current density, subject to a transmit power constraint. They propose an FP-based block coordinate descent (BCD) algorithm that they claim obtains the globally optimal current pattern in closed form via a continuous-function inversion theory, and they also propose a zero-forcing (ZF) heuristic with water-filling power allocation. Numerical results compare the proposed schemes with discrete MIMO baselines and a Fourier-based discretization benchmark.

Significance. Should the main optimality claim hold, the paper would establish that a nonconvex functional WSSR problem is exactly solvable with low computational complexity, and it would quantify substantial secrecy gains of CAPA over discrete MIMO. The continuous-function inversion theorem and the observation that the optimal current is a linear combination of channel spatial responses are elegant and potentially useful. However, the central equivalence on which the FP-BCD algorithm rests is flawed, so the main contribution is not established. The ZF heuristic is a reasonable baseline, but its conceptual novelty is limited.

major comments (3)

[Section III-A, Lemma 2 (Eqs. (15a), (19)-(22))] The equivalence claimed in Lemma 2 is false. Starting from Eq. (15a), one has sum_k b_k [log2(1+gamma_k) - log2(1+Gamma_k)] = sum_k b_k [log2(1+gamma_k) + log2((1+G_Gamma)/(1+Gamma_k))] - log2(1+G_Gamma) sum_k b_k. The term -log2(1+G_Gamma) sum_k b_k depends on the optimization variables b_k, so discarding it does not preserve equivalence. In addition, Eq. (22b) is algebraically incorrect: (1+G_Gamma)/(1+Gamma_k) equals 1 + (G_Gamma - Gamma_k)/(1+Gamma_k), not (1+G_Gamma - Gamma_k)/(1+Gamma_k); the two quantities differ unless Gamma_k = 0. Thus problem (20) is not equivalent to problem (15), and the subsequent FP transformations and the optimality claim for Algorithm 1 are unsupported.
[Section III-B and III-C, Eq. (30) and Eq. (61)] The b_k update rule in Eq. (30) is the optimal solution for problem (15) for fixed J_k(s), but in the transformed problem (20a) the coefficient of b_k is different, namely log2(1+gamma_k) + log2((1+G_Gamma - Gamma_k)/(1+Gamma_k)) as written, or log2((1+G_Gamma)/(1+Gamma_k)) after the algebraic correction. The threshold condition gamma_k >= Gamma_k therefore does not maximize the transformed objective. Moreover, the convergence proof in Eq. (61) omits the b_k block and states that b_k is a non-negative constant, which contradicts its role as an optimization variable in Algorithm 1. Consequently, the monotonicity chain does not establish convergence of Algorithm 1 to a stationary point of the WSSR problem (13).
[Appendix A, proof of Theorem 1] The proof of Theorem 1 invokes "Lemma 3 in [21]" to conclude that V_k(s) = 0 from the condition that the integral of U_k^*(s) V_k(s) over S_T vanishes for all smooth U_k vanishing on the boundary. Since [21] is an unpublished preprint and the lemma is not stated in the manuscript, the derivation of the optimal current structure in Eq. (41) is not self-contained. The authors should state and prove the needed variational lemma or provide a complete derivation.

minor comments (4)

[Lemma 3 and Eqs. (23a)-(26)] The logarithm base switches between log2, used in the problem statement, and log, apparently the natural logarithm, in the FP transformations; this should be made consistent.
[Section III-C] The sentence "We omit b_k since it is a non-negative constant" is inaccurate because b_k is an optimization variable updated in Algorithm 1 and must be treated as a block in the BCD analysis.
[Theorem 3] Theorem 3 requires the N x N channel correlation matrix H to be invertible, but the paper does not discuss rank deficiency when K + Q exceeds the available spatial degrees of freedom; a regularity condition should be stated.
[Eq. (21b)] The definition of G_Gamma as P times the sum over q of the integral of |bar_H_q(s)|^2 ds lacks parentheses; the intended grouping should be clarified.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity (score 1): the FP-BCD derivation is self-contained and parameter-free, its only same-group citation ([21]) supplies a textbook CoV lemma, and the benchmark [20] is external. The claimed optimality is nonetheless undercut by an algebraic error in Lemma 2 (Eqs. 22a–b; dropped Σb_k term) — a correctness defect, not a circular reduction.

full rationale

The derivation chain is self-contained rather than circular. The Green's-function model (Eqs. 1–2) is standard EM theory from the external reference [6]; the FP transformations of Lemmas 3–4 build on the external framework of [28], [29]; and the numerical benchmark is the external Fourier-based method of Zhang & Dai ([20]). No parameter is fitted to data and no simulation output is fed back into the derivation, so the 'predictions' — the closed-form current pattern (51) and the reported WSSR gains — do not reduce to their inputs by construction. The only same-group citation inside the derivation is [21] (Wang, Ouyang, Liu), invoked in Appendix A as 'Lemma 3 in [21]' for the fundamental lemma of calculus of variations; that is a textbook-standard, externally checkable result, so under the rules it is real evidence and does not raise the circularity score. Reference [27] is contextual prior work. The honest circularity finding is therefore near zero (score 1 for the cosmetic same-group citation). Separately, and per the reviewing rule to weigh flagged flaws explicitly, the central claim of 'obtaining the optimal solution' is not supported — but for a non-circularity reason. Lemma 2 breaks the claimed equivalence (15)↔(20): Eq. (22b) asserts log2((1+GΓ)/(1+Γ_k)) = log2((1+GΓ−Γ_k)/(1+Γ_k)), which fails unless Γ_k = 0, and the term log2(1+GΓ) dropped in passing from (15) to (20) actually multiplies Σ_k b_k, which is not constant because {b_k} are optimization variables in (15c) and are reset each iteration by (30). The b-subproblem (29) is itself solved against the original objective (15a), not the 'equivalent' (20a). The β-update (34), the J-update (51), and the asserted optimality of Algorithm 1 inherit the defect, as does the convergence argument in Section III-C, which treats b_k as 'a non-negative constant' although it is an iterated variable. These are validity defects in the claimed reduction (the conclusion is not assumed as input; the asserted equivalence is simply false), so they belong in correctness risk rather than in the circularity score. The ZF scheme of Section IV is an independent, standard construction and is not affected by the Lemma 2 defect.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data in the derivation; simulation inputs (P, noise, geometry) are user-set. The paper invents no new physical entities. The load on the ledger is carried by inherited channel and security assumptions and by a variational lemma imported from the authors' own [21]. The decisive problem is not an unacknowledged axiom but the false equivalence in Lemma 2.

assumptions (5)

domain assumption The EM channel is modeled by the free-space Green's function (2) with scalar, vertically polarized source current and point receivers (Section II-A).
The whole optimization depends on this channel model from [6]; mutual coupling, antenna size, and polarization mismatch are ignored.
domain assumption The eavesdroppers' CSI is known perfectly at the BS via TDD reciprocity because Eves behave as registered users (Section II).
Without this, secure beamforming against the assumed Eves is not implementable.
domain assumption The variational lemma 'if Re(integral U* V)=0 for all smooth U vanishing on the boundary, then V=0' is taken from Lemma 3 of [21] (Appendix A, Eqs. 81-83).
This unproved lemma from the authors' prior work is needed to pass from an integral vanishing condition to pointwise optimality.
standard math Standard fractional programming identities from [28] and [29] are used in Lemma 3 and Lemma 4.
These identities are established in the cited literature and used correctly modulo the b_k issue in Lemma 2.
standard math The kernel inverse in Theorem 2 exists because lambda I + Phi is invertible for lambda > 0 with Phi positive semidefinite.
Used implicitly when writing (47).

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Cite this review

Pith. "Pith review of Secure Beamforming for Continuous Aperture Array (CAPA) Systems." pith.science (2026). https://pith.science/paper/CRGH6IKW

@misc{pith2026250104924,
  author       = {Pith},
  title        = {Pith review of: Secure Beamforming for Continuous Aperture Array (CAPA) Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRGH6IKW}},
  note         = {Machine review of arXiv:2501.04924}
}

read the original abstract

Continuous aperture array (CAPA) is considered a promising technology for 6G networks, offering the potential to fully exploit spatial DoFs and achieve the theoretical limits of channel capacity. This paper investigates the performance gain of a CAPA-based downlink secure transmission system, where multiple legitimate user terminals (LUTs) coexist with multiple eavesdroppers (Eves). The system's secrecy performance is evaluated using a weighted secrecy sum-rate (WSSR) under a power constraint. We then propose two solutions for the secure current pattern design. The first solution is a block coordinate descent (BCD) optimization method based on fractional programming, which introduces a continuous-function inversion theory corresponding to matrix inversion in the discrete domain. This approach derives a closed-form expression for the optimal source current pattern. Based on this, it can be found that the optimal current pattern is essentially a linear combination of the channel spatial responses, thus eliminating the need for complex integration operations during the algorithm's optimization process. The second solution is a heuristic algorithm based on Zero-Forcing (ZF), which constructs a zero-leakage current pattern using the channel correlation matrix. It further employs a water-filling approach to design an optimal power allocation scheme that maximizes the WSSR. In high SNR regions, this solution gradually approaches the first solution, ensuring zero leakage while offering lower computational complexity. Simulation results demonstrate that: 1) CAPA-based systems achieve better WSSR compared to discrete multiple-input multiple-output systems. 2) The proposed methods, whether optimization-based or heuristic, provide significant performance improvements over existing state-of-the-art Fourier-based discretization methods, while considerably reducing computational complexity.

Figures

Figures reproduced from arXiv: 2501.04924 by the authors.

**Figure 2.** Convergence of the proposed FP-based BCD algorithm. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** WSSR for non-approximate beamforming schemes. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: WSSR versus transmit power. coefficients, which can then be solved using techniques commonly applied in conventional discretized MIMO systems. After obtaining the discrete Fourier coefficients of the source current, the continuous source current pattern can be reconstr…

**Figure 6.** Figure 6: WSSR versus number of LUTs [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: WSSR versus number of Eves. approach, as they eliminate the approximation of continuous functions through Fourier series discretization, enabling direct acquisition of the optimal solution. As shown in Section III C, the computational complexity of the proposed scheme …

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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