Joint Channel and Symbol Estimation for Communication Systems with Movable Antennas

Andr\'e L. F. de Almeida; Gilderlan T. de Ara\'ujo; Jose Carlos da Silva Filho; Josu\'e V. de Ara\'ujo; Paulo R. B. Gomes

REVIEW 2 major objections 7 minor 24 references

Joint Channel and Symbol Estimation for Communication Systems with Movable Antennas

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A movable-antenna uplink receiver can estimate the channel and decode all users' symbols at once, without training pilots, by fitting the received signal to a PARAFAC2 tensor model.

desk verdict Solid PARAFAC2 modeling for movable-antenna semi-blind reception, but the paper never resolves a per-user diagonal scaling ambiguity, so the reported NMSE/SER curves are not well-defined. read the letter →

arxiv 2506.07183 v1 pith:IKCWRCPJ submitted 2025-06-08 eess.SP

Josué V. de Araújo , Jose Carlos da Silva Filho , Gilderlan T. de Araújo , Paulo R. B. Gomes , André L. F. de Almeida This is my paper

classification eess.SP

keywords movableantennasPARAFAC2semi-blindreceiverjointchannelandsymbolestimationtensordecompositionmultiuseruplinkfluid

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 proposes that in an uplink multi-user system whose base station uses a movable antenna array, the received signal across blocks, time slots, and antennas forms a third-order tensor with PARAFAC2 structure. The authors show that this structure lets a single alternating least-squares algorithm jointly estimate the channel matrix H and the transmitted symbol matrix X, removing the need for pilot sequences. If the claim holds, future movable-antenna systems could decode data and acquire channel state information simultaneously, saving spectral resources that pilots consume. The paper's initial simulations indicate competitive channel and symbol error performance compared with a pilot-assisted reference.

What carries the argument

The machinery is the PARAFAC2 tensor decomposition, a variant of PARAFAC that allows the first-mode factor matrix to vary per frontal slice while constraining its cross-product to remain constant. Here the frontal slices are Y_p = S_p H D_p(C) X, with correspondences A_p = S_p H, B = X^T, C = C. This decomposition makes the received data a bilinear function of H and X, so the joint estimation problem splits into two linear least-squares subproblems solved alternately, a procedure known as bilinear alternating least squares (BALS).

What would settle it

Run Algorithm 1 on noise-free data generated from (9) with a known random H, X, C, and S_p, initializing X randomly. If the recovered pair does not satisfy \hat{H} = H \$Lambda^{{-1}}$ and \hat{X} = \Lambda X for a common diagonal \Lambda, or if the objective \sum_p \|Y_p - S_p \hat{H} D_p(C) \hat{X}\|$_F^{2}$ does not monotonically decrease, then the central claim of reliable joint estimation without pilots fails.

Watch

Extended reading notes

Core claim

The central claim is that the received signal model Y_p = S_p H D_p(C) X, where S_p is a port-switching matrix, H is the multi-user channel, C is a block coding matrix, and X is the symbol matrix, is a rank-R PARAFAC2 decomposition of the third-order tensor Y. Under this model, the pair (H, X) can be recovered semi-blindly by a bilinear alternating least squares (BALS) receiver that alternates between solving linear least-squares problems for H and X. The paper shows the identifiability conditions T M P ≥ N K and P M ≥ K that guarantee uniqueness in the least-squares sense, and verifies via Monte Carlo simulation that the receiver achieves low symbol error rates even when the number of antenna ports (and hence channel coefficients) is large.

Load-bearing premise

The load-bearing premise is that alternating least squares, started from a random initial symbol matrix, converges to a solution close enough to the true (H, X), and that the inherent scaling ambiguity - multiplying H by a diagonal matrix and X by its inverse leaves Y_p unchanged - is resolved or harmless in the evaluation; the paper does not prove convergence or specify a normalization, without which reported NMSE and SER are not well defined.

Editorial extensions

If this is right

Spectral efficiency improves because no pilot block is needed; data decoding can begin during the channel estimation stage.
The identifiability constraints K ≤ min(T M P / N, P M) give system designers a rule for choosing block count, antennas, and ports.
Channel estimation accuracy degrades as the number of ports N grows, because more coefficients are estimated, while symbol recovery stays nearly insensitive to N.
The method tolerates high spatial compression (M RF chains much smaller than N ports), supporting cost-efficient hardware.

Reading between the lines

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

A natural extension the paper leaves implicit is using the estimated channel's sparse structure (few dominant paths in movable-antenna channels) to reduce the number of parameters and improve NMSE at high port counts.
The PARAFAC2 formulation suggests a direct link to blind source separation: with C designed as a truncated DFT, the block-varying switching matrices S_p act as a spatial multiplexing code, so other PARAFAC2 fitting algorithms could replace BALS for faster convergence.
If the diagonal scaling ambiguity is resolved by a deterministic initialization or a normalization step, the receiver could be extended to online tracking of time-varying channels, since each new block updates only S_p and C.
The reported insensitivity of symbol error rate to N suggests a trade-off worth quantifying: increase ports for downlink beamforming freedom while paying a channel-estimation cost.

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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 paper's core contribution is legitimate: it organizes the uplink received signal from a movable-antenna base station into a PARAFAC2 tensor model, with the switching matrices and Khatri-Rao coding fitting naturally into that structure. The algebraic derivations in (13), (18), and (21) check out, and the PARAFAC2 correspondence in (10) is correct. That is a genuine new application of tensor signal processing to a current 6G topic, and the authors are right that prior PARAFAC2 receivers for OSTBC and prior tensor-based movable-antenna estimators did not do exactly this.

The soft spot, as the stress-test note explains, is not cosmetic: the model Y_p = S_p H D_p(C) X is invariant under replacing (H, X) with (H Lambda, Lambda^{-1} X) for any nonsingular diagonal Lambda. The paper never mentions this ambiguity, never imposes a normalization, and uses no pilot symbols or other disambiguation mechanism. Algorithm 1's alternating least squares updates preserve the entire scaling family as fixed points, so the NMSE for H and SER for X reported in Figures 3–6 have no well-defined reference. A random initialization does not fix this; it just picks an arbitrary member of the family. Without a normalization convention, the simulation results cannot be reproduced or interpreted.

Other issues are real but less severe. The identifiability section only checks that the two LS subproblems in (19) and (21) are overdetermined; it does not address joint uniqueness of the PARAFAC2 model, though here the PARAFAC2 constraint A_p^T A_p = H^T H holds automatically and does not resolve the scaling. There is also no comparison against the closest prior tensor-based movable-antenna estimator [3], only a pilot-assisted LS baseline. No code or data is provided. These are all fixable, and the paper itself says spectral efficiency results will appear in a full version.

This is a preliminary conference paper with a sound modeling idea and a correct algebraic core, but the missing scaling ambiguity makes the current numerical claims unsubstantiated. The fix is straightforward: adopt a convention such as normalizing each column of H to unit norm and choosing the phase from a known pilot symbol, or aligning the estimates to the constellation. I would encourage revision rather than rejection, and the idea deserves room to mature.

Who is this for? Researchers working on tensor-based receivers or movable-antenna estimation who want to see a novel PARAFAC2 formulation, and who can look past the missing normalization. It deserves a serious referee because the modeling is novel and the math is mostly solid, but the ambiguity must be addressed before any performance claims can be accepted.

Referee Report

2 major / 7 minor

Summary. This manuscript addresses joint channel and symbol estimation for the uplink of a multiuser system whose base station is equipped with a movable-antenna array: M RF chains select among N ports via a block-varying binary switching matrix S_p, and K single-antenna users spread their T-symbol blocks using known block-coding coefficients C. The received signal is written as frontal slices Y_p = S_p H D_p(C) X + Z_p and is shown to follow a PARAFAC2 decomposition, leading to the correspondences in (10). The proposed receiver (Algorithm 1) alternates two least-squares updates, (18)-(19) for the channel matrix H and (20)-(21) for the symbol matrix X, and the paper reports simulated NMSE and SER in Figures 3-6. The algebraic core of the paper is correct, but the model admits an unreported per-user diagonal scaling ambiguity, so the reported error metrics and the claimed joint estimation results are not well-defined as stated.

Significance. If the identifiability gap described in the major comments is closed, the paper makes a moderate but useful contribution: it brings the PARAFAC2 machinery to movable-antenna systems through a two-time-scale switching structure, and the resulting receiver is simple and amenable to the usual tensor-based analysis. On the credit side, the derivation of the LS updates (13), (18), and (21) is correct, the PARAFAC2 correspondence in (10) is valid, the per-iteration complexity is stated clearly, and the qualitative finding that channel NMSE degrades as the number of ports N grows (Figures 3 and 6) is plausible. The main weakness is that the model's scaling invariance is not addressed, which currently blocks reproduction of the numerical claims; the results are also preliminary, with only a pilot-assisted LS baseline for comparison.

major comments (2)

[§IV–§VI, Eqs. (9), (18)–(21)] The bilinear model Y_p = S_p H D_p(C) X in (9) is invariant under the transformation (H, X) → (HΛ, Λ^{-1}X) for every nonsingular diagonal Λ ∈ ℂ^{K×K}, because diagonal matrices commute and D_p(C) is diagonal. Both LS subproblems are equivariant under this family: if X = Λ^{-1}X_true, the H-update in (19) returns H_trueΛ when W in (17) has full column rank, and the X-update in (21) returns Λ^{-1}X_true when Z has full column rank. Hence every member of this family is a fixed point of Algorithm 1. The inequalities in §V (TMP ≥ NK and PM ≥ K) only make the two linear subproblems overdetermined; they do not remove the scaling ambiguity. Because §VI reports NMSE for Ĥ (Figures 3, 5, 6) and SER for X̂ (Figure 4) without stating any normalization or pilot-based disambiguation (and neither exists in Algorithm 1), the reported error metrics are not well-defined and the numerical results cannot be reproduced as described. This directly affects the paper's central claim of joint channel and symbol estimation.
[§II and §V] The identifiability discussion is incomplete and in tension with the model. Section II recalls that PARAFAC2 uniqueness holds under the constraint A_p^T A_p = Φ for all p, but with the correspondence A_p = S_p H from (10) this requires H^T S_p^T S_p H to be constant across p, which is not verified and generically fails because S_p^T S_p is the N×N orthogonal projector onto the row space of S_p and changes from block to block. The standard PARAFAC2 uniqueness results therefore do not carry over to the proposed model. Section V instead equates identifiability with the overdeterminedness conditions TMP ≥ NK and PM ≥ K; these conditions concern unique solvability of the linear subproblems (18) and (20), not identifiability of the bilinear pair (H, X), as the diagonal scaling invariance makes plain. The concluding sentence of §V, which states that the parameter choices ensure uniqueness of the estimated parameters, is consequently not justified. The authors should either prove identifiability under explicit constraints (for example, a small number of pilots or unit-norm/phase constraints on the factors) or explicitly characterize the remaining ambiguities and how the receiver resolves them.

minor comments (7)

[§III, Eq. (4)] For M < N the binary matrix S ∈ ℝ^{M×N} cannot satisfy exactly one unity element in each row and column, since that would force M = N; the intended condition is one unit per row and at most one unit per column (a selection matrix), and the paper should state it that way.
[§V] The section heading 'Indentifiability' should read 'Identifiability'.
[§VI and Algorithm 1] Several experimental details are missing: the SNR ranges on the horizontal axes of Figures 3–6, the convergence threshold δ and the maximum number of iterations of Algorithm 1, the random initialization procedure for X, and the definition of the NMSE/SER averaging (in particular whether per-user scaling/phase alignment was applied before computing the metrics).
[Introduction and References] The introduction cites the same reference twice as '[9], [9], [10]', and reference [13] duplicates reference [10] in the bibliography; reference [19] also has an incomplete author list.
[§VI] The anticipated spectral-efficiency gain from avoiding pilots is mentioned twice but is never quantified; a preliminary quantitative statement would support the claimed advantage.
[Abstract and §VI] The abstract's 'remarkable results' and the conclusion's 'competitive performance' are supported only by comparisons with a pilot-assisted LS channel estimator and a fixed-antenna case; a comparison with at least one existing semi-blind or tensor-based receiver would make the performance claims more persuasive.
[§II and §III] The same symbol X is used for the PARAFAC2 tensor in (3) and for the K×T symbol matrix in Section III, which may confuse readers; the tensor in (3) should use a different symbol.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: model, LS updates, and simulations are self-contained; the scaling ambiguity is an identifiability caveat, not a circular step.

full rationale

The derivation chain is self-contained and non-circular. The received-signal model in (5)-(9) is assembled from standard physical assumptions (block fading, port switching matrix S_p, and per-block diagonal coding D_p(C)); the PARAFAC2 identification in (10) is a direct algebraic relabeling of (9) and does not presuppose the estimates. The BALS receiver solves two ordinary least-squares problems (18)-(21) whose sensing matrices are formed from the data and the currently fixed factor; this is a standard fitting procedure, not a fitted parameter dressed as a prediction. The identifiability discussion in Section V is parametric (TMP >= NK and PM >= K) and cites external tensor literature [20]-[23], with self-citations [9]-[17] appearing only as background on tensor receivers, not as load-bearing support for the central claim. A real identifiability caveat exists: the model has the exact invariance Y_p = S_p H D_p(C) X = S_p (H Lambda) D_p(C) (Lambda^{-1} X) for any diagonal Lambda, and Algorithm 1 and Section V do not specify a normalization, so the reported NMSE/SER values are not uniquely defined unless an implicit scaling convention is used. That is a reproducibility and identifiability flaw, not a circular derivation: no equation in the paper reduces a prediction to an input fit, and no load-bearing argument reduces to the authors' prior work by citation. Therefore no circular step is identified.

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

The central claim rests on the system model, standard linear algebra, and the PARAFAC2 decomposition theory. There are no fitted free parameters or invented physical entities. The main unstated assumptions are the block-fading protocol, perfect knowledge of the coding and switching matrices, and the algorithmic assumption that BALS converges from random initialization.

assumptions (5)

standard math Kronecker product identities (1)-(2) and the vec identity vec(ABC) = (C^T tensor A)vec(B) are valid.
Invoked in Section IV to derive the linear least-squares forms for channel and symbol updates.
domain assumption Block fading: the channel H is constant across P blocks, while the switching matrix S_p and the coding vector c_p vary per block.
Stated in Section III; the entire PARAFAC2 model depends on this time structure.
domain assumption The receiver has perfect knowledge of the coding matrix C and the switching matrices S_p for p = 1, ..., P.
Implicit in Section III and used to build W in (17) and Z in (23).
domain assumption The received tensor Y follows the exact PARAFAC2 structure with no model mismatch, and the joint decomposition is unique up to the stated ambiguities.
The paper checks only the tall-matrix conditions TMP >= NK and PM >= K in Section V; it does not establish joint uniqueness or discuss the scaling ambiguity.
ad hoc to paper The BALS algorithm converges from a random initialization to a stationary point close enough to the true factors.
No convergence proof or initialization strategy is provided; this is assumed for the reported performance in Algorithm 1.

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Cite this review

Pith. "Pith review of Joint Channel and Symbol Estimation for Communication Systems with Movable Antennas." pith.science (2026). https://pith.science/paper/IKCWRCPJ

@misc{pith2026250607183,
  author       = {Pith},
  title        = {Pith review of: Joint Channel and Symbol Estimation for Communication Systems with Movable Antennas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKCWRCPJ}},
  note         = {Machine review of arXiv:2506.07183}
}

read the original abstract

Communication systems aided by movable antennas have been the subject of recent research due to their potentially increased spatial degrees of freedom offered by optimizing the antenna positioning at the transmitter and/or receiver. In this context, a topic that deserves attention is channel estimation. Conventional methods reported recently rely on pilot-assisted strategies to estimate the channel coefficients. In this work, we address the joint channel and symbol estimation problem for an uplink multi-user communication system, where the base station is equipped with a movable antenna array. A semi-blind receiver based on the PARAFAC2 model is formulated to exploit the tensor decomposition structure for the received signals, from which channel and symbol estimates can be jointly obtained via an alternating estimation algorithm. Compared with reference schemes, our preliminary numerical simulations yield remarkable results for the proposed method.

Figures

Figures reproduced from arXiv: 2506.07183 by the authors.

**Figure 2.** Transmission time structure. chain and can connect to one of the N available ports, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** depicts the channel estimation performance regarding the normalized mean squared error (NMSE). In this scenario, we can observe that the accuracy of the estimated MU channel H decreases when the number of ports increases. This result [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: SER vs. SNR (dB) is expected since increasing the possible antenna positions implies estimating more channel coefficients. On the other hand, more spatial degrees of freedom are available for optimizing the transmit beamforming in the downlink transmission stage. Spec…

**Figure 5.** Figure 5: NMSE vs. SNR (dB) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: NMSE vs. Number of ports (N). In particular, the estimation accuracy reaches its upper limit (worst case) for configurations close to N = 40 ports, showing very similar results regardless of the SNR. These results are in agreement with those of [PITH_FULL_IMAGE:figure…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Channel estimation for movable-antenna MIMO systems via tensor decomposition,

R. Zhang, L. Cheng, W. Zhang, X. Guan, Y . Cai, W. Wu, and R. Zhang, “Channel estimation for movable-antenna MIMO systems via tensor decomposition,”IEEE Wireless Commun. Lett., vol. 13, no. 11, pp. 3089–3093, 2024

2024

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Fixed and movable antenna technology for 6G integrated sensing and communication,

Y . Zeng, Z. Dong, H. Wang, L. Zhu, Z. Hong, Q. Jiang, D. Wang, S. Jin, and R. Zhang, “Fixed and movable antenna technology for 6G integrated sensing and communication,” 2024. [Online]. Available: https://arxiv.org/abs/2407.04404

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