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REVIEW 3 major objections 6 minor 41 references

Joint Spatial Division and Multiplexing with Customized Orthogonal Group Channels in Multi-RIS-Assisted Systems

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read RISs placed on the BS array's DFT directions let a fixed DFT pre-beamformer block-diagonalize the multiuser channel, cutting JSDM pre-beamforming design from matrix decomposition to O(1).

desk verdict A plausible low-complexity RIS-assisted JSDM scheme whose central block-diagonalization claim is real but conditional on exact DFT placement and strong LoS, and whose numerical evidence would be stronger with baseline sum-SE comparisons and a quantified offset threshold. read the letter →

arxiv 2507.01641 v2 pith:PNMFCKSQ submitted 2025-07-02 cs.IT eess.SPmath.IT

Weicong Chen , Chao-Kai Wen , Wankai Tang , Xiao Li , Shi Jin This is my paper · ORCID
classification cs.ITeess.SPmath.IT MSC 94A05
keywords reconfigurableintelligentsurfacejointspatialdivisionandmultiplexingblockdiagonalizationpre-beamformingDFTbeamformingchannelcustomizationspectralefficiencyusergrouping
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 a way to make joint spatial division and multiplexing (JSDM) much cheaper to implement in RIS-assisted FDD systems. It claims that placing each reconfigurable intelligent surface on one discrete-Fourier-transform (DFT) direction of the base-station array makes the BS–RIS links mutually orthogonal line-of-sight channels, so a fixed DFT pre-beamformer separates the user groups without any matrix decomposition. The RIS reflection patterns are then optimized from long-term statistical channel information to steer each surface's beam toward its own user group and suppress interference from other groups. The result is an approximately block-diagonal effective channel whose pre-beamforming design costs O(1) instead of the O($M_B^{3}$) of classical JSDM. Numerical results show the scheme raises sum spectral efficiency and that the block structure survives across Rician factors, RIS sizes, and practical phase and deployment offsets.

What carries the argument

The load-bearing object is the Dirichlet-type array-response inner product $D(\Theta,\Phi)=\sin(M^{\mathrm{R},v}\Theta)\sin(M^{\mathrm{R},h}\Phi)/(\sin\Theta\sin\Phi)$ (with an analogous BS-side form), whose zeros create orthogonality: placing RISs on the DFT angular grid makes $b_{k,0}^{\mathrm{H}}b_{m,0}=0$, and for a large RIS the same factor drives out-of-group UE responses to zero. On top of it are two closed-form designs: the pre-beamformer $F$ whose columns are the DFT array responses $f_i=b_{i,0}$, fixed for as long as the RIS positions are fixed; and the RIS reflection vector $\gamma_k = M^{\mathrm{R}}(a(\Delta^v_k,\Delta^h_k)\odot r^*_{k,0})^*$, which turns that inner product into the controllable factor $D(\Theta_{k,n},\Phi_{k,n})$ in the mean effective channel. The beam-bias pair $(\delta^{\mathrm{bias}}_{v,k},\delta^{\mathrm{bias}}_{h,k})$ is the tuning knob: in the large-RIS regime it aligns the beam exactly on a user, in the interference-limited regime it minimizes the worst interference-to-signal ratio, and in the noise-limited regime it minimizes interference while keeping every served user inside the half-power beamwidth.

What would settle it

In the paper's own scenario, run the same sum-spectral-efficiency and effective-channel evaluation with RISs deliberately placed halfway between DFT grid directions (or with a moderate direct BS–UE link left unblocked) while keeping the fixed DFT pre-beamformer; if the off-diagonal block power of H^H F no longer remains at least an order of magnitude below the diagonal block power across channel realizations, the claimed approximate block diagonalization fails in that regime.

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Extended reading notes

Core claim

The paper's central discovery is that the RISs' ability to customize the propagation environment can replace the heavy covariance-eigenvector machinery that classical JSDM uses for block diagonalization. With K RISs positioned on the DFT angles of the BS array, their line-of-sight array responses at the BS are mutually orthogonal, $b_{k,0}^{\mathrm{H}}b_{m,0}=0$ for $k\neq m$, so choosing the DFT vectors as pre-beamformers produces an effective channel $H^{\mathrm{H}}F \approx [\beta^{B}_{1,0}U^{\mathrm{H}}_1\Gamma^{\mathrm{H}}_1r_{1,0}, \ldots, \beta^{B}_{K,0}U^{\mathrm{H}}_K\Gamma^{\mathrm{H}}_Kr_{K,0}]$: the BS–RIS side of inter-group interference is already gone. The residual RIS–UE interference is attacked by designing each RIS reflection vector as a beam pointed at its group's coverage center, with beam-bias choices for the interference-limited and noise-limited regimes, and by grouping users and RISs with the help of a closed-form approximation of the effective-channel cross-correlation. The upshot is an approximately block-diagonal $H^{\mathrm{H}}F$ whose rank is roughly $K$, so $K$ users can be served simultaneously even when all of them lie along one angular direction from the BS, a case where conventional JSDM fails.

Load-bearing premise

The construction rests on two geometric assumptions: the BS–RIS paths are almost pure line-of-sight and every RIS sits exactly on one of the DFT angular directions of the BS array; if the direct BS–UE link is not almost fully blocked or the RIS positions drift from those directions, the effective channel is no longer approximately block diagonal and the rank bound rank(H)≈K no longer holds.

Editorial extensions

If this is right

  • With $K$ RISs on the DFT grid, up to $K$ users can be served simultaneously even when all users lie along the same angular direction from the BS, a scenario where classical JSDM has no angular separation to exploit.
  • Because the pre-beamformer is a fixed DFT codebook, it never needs to be recomputed while the RISs stay in place, and it can run on a hybrid antenna array with one RF chain per RIS instead of a fully digital array.
  • Larger RIS arrays sharpen the approximate block diagonalization, because the Dirichlet factor for out-of-group users shrinks and the gap between desired and interfering channel coefficients widens.
  • When the number of users exceeds the number of RISs, the derived cross-correlation approximation yields a concrete association rule: cluster users by their LoS direction vectors, group RISs by path loss, and select one user per RIS to minimize inter-group correlation.
  • The paper's simulations indicate graceful degradation under deployment offsets and phase noise, with 1-bit phase quantization causing the largest loss and deployment offset hurting the interference-limited configuration more than the noise-limited one.

Reading between the lines

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

  • A natural extension the paper leaves implicit is to keep a partially blocked direct BS–UE link in the model; the DFT-grid argument alone would no longer block-diagonalize, but treating the direct link as an extra rank component with its own channel estimate could restore some of the multiplexing gain at the cost of extra CSI.
  • Because the pre-beamformer is fixed and only RIS reflection beams are retuned, the scheme is a plausible fit for slowly moving users: the LoS direction vectors that drive grouping change slowly, and the cross-correlation approximation could double as a reassociation metric.
  • The rank bound rank(H)≈K means the number of served users scales with the number of deployed RISs, not the BS array size; a testable design consequence is that more users require more surfaces rather than more BS antennas.
  • The same Dirichlet-kernel orthogonality could be transferred to near-field or wideband settings by replacing DFT beams with near-field-focused or frequency-dependent codebooks, although the paper's analysis is restricted to far-field narrowband planar arrays.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a low-complexity joint spatial division and multiplexing (JSDM) design for multi-RIS-assisted FDD MIMO systems. The authors place RISs on the discrete Fourier transform (DFT) angular directions of the base station, use the corresponding DFT vectors as a fixed pre-beamformer, and optimize RIS reflection beams to approximately block-diagonalize the effective multiuser channel. They derive a closed-form approximation of the channel cross-correlation coefficient, use it for UE/RIS grouping and association, and evaluate the scheme through simulations of effective-channel structure, sum spectral efficiency, deployment offset, phase quantization, and Rician factor.

Significance. If the central claim holds, the paper offers a noteworthy simplification of JSDM: instead of eigen-decomposition and multiple SVDs, the pre-beamformer is constructed from array response vectors with O(1) design complexity, and the channel rank is enhanced by distributed RISs placed at DFT directions. The analytical expressions in Proposition 1 and Lemma 1 are derived from the channel model rather than fitted to the final performance curves, and the paper explicitly studies robustness to deployment offset, phase noise, and Rician factor. These are genuine strengths. However, the significance is tempered by the fact that the block-diagonalization claim is demonstrated only in an idealized exact-DFT-placement, LoS-dominated regime, and the numerical validation does not include a sum-spectral-efficiency comparison against the cited RIS-JSDM baselines.

major comments (3)
  1. [Section III-A, Eq. (12), Appendix A] The rank bound rank(H)≈K and the approximate block-diagonal form in Eq. (16) both rest on the two idealizations B_k≈β^B_{k,0} b_{k,0} r^H_{k,0} and b^H_{k,0} b_{m,0}=0 for k≠m, which require exact placement of RISs on DFT angular directions and a large BS–RIS Rician factor. The robustness evidence in Figs. 10 and 12 reports only sum SE, an indirect aggregate metric. Since the central claim is that the effective channel is block-diagonalized, the paper should additionally quantify the block-diagonal quality directly, e.g., ||H_e^H F_c||/||H_c^H F_c|| for e≠c or the effective rank of Eq. (11), as a function of deployment offset and Rician factor, and state a quantitative tolerance threshold. Without such a metric, the claim remains conditional on deployment precision and strong LoS in a way that is not bounded by the current simulations.
  2. [Section V-A, Fig. 4] The reported intra-group signal-to-interference ratio of UE 4 is 0.70 in Configuration 2 (and 0.45 and 0.29 in Configurations 4 and 3). If this quantity is E{[HHF]4,4}/E{[HHF]4,3}, a value below 1 means the desired coefficient is weaker than an intra-group interference coefficient, which is contrary to the claimed block-diagonal structure. Please clarify the definition of this ratio; if the metric is correct, explain why the effective channel should still be regarded as block-diagonalized. The panels in Fig. 4 also appear block-shaped rather than truly block-diagonal, so a norm-based quantification would remove the ambiguity.
  3. [Section V-B, Fig. 6] The sum-spectral-efficiency comparisons are made only against Configuration 1, where the RIS behaves passively, and not against the cited RIS-JSDM schemes [14]–[16] that the paper positions itself against. The effective-rank comparison in Fig. 3 does not substitute for sum-SE, because rank alone does not determine achievable spectral efficiency under interference. Please add sum-SE baselines for the RIS-JSDM schemes of [14], [15], and [16] using the same channel model and simulation parameters, so that the claimed 'significant enhancement' is supported by a direct comparison.
minor comments (6)
  1. [Eq. (10)] In the denominator of the sum-spectral-efficiency expression, the sum runs over m≠n, but the outer index is s; this should be m≠s.
  2. [Section II-A] The text states each RIS comprises M_R = M_{R,v} × M_{S,h} elements; the subscript S,h should be R,h.
  3. [Section V-D] The sentence 'Figure 10 illustrates the performance of Configuration 3 versus varying PN variance' refers to the phase-noise results shown in Fig. 11, not Fig. 10; the cross-reference should be corrected.
  4. [Section IV-C, Eq. (35)] The UE selection and RIS association minimize the approximate cross-correlation Q_{n,m}, and the block-diagonalization evaluation in Fig. 4 partly uses the same type of correlation metric; to avoid a circularity concern, include a baseline with random association to show that the sum-SE improvement is not solely driven by minimizing Q_{n,m}.
  5. [Algorithm 1] Line 1 uses C both as the number of groups and as the set of group indices; using a different symbol, e.g., G, for the set would improve readability.
  6. [Section V-D] There is a typo in 'with bing σ2_PN' that should read 'with being σ2_PN'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: core BD/rank/SE claims are derived from the stated channel model and tested against independent sum-SE and channel-matrix metrics.

full rationale

The paper's load-bearing chain is not circular. Section III-A derives rank(H)≈K from the channel model B_k≈β^B_{k,0}b_{k,0}r^H_{k,0} and the condition b^H_{k,0}b_{m,0}=0; Appendix A proves the DFT-placement condition using the standard UPA inner-product formula, so the orthogonality is a derived property of the assumed model rather than an input fitted to the final SE curves. Section III-B obtains the approximate block-diagonal effective channel (16) by substituting F=[b_{1,0},...,b_{K,0}] into H^H F and using the same orthogonality; this is a construction, not a tautology. Proposition 1 and Lemma 1 are proven in Appendices B and C from the Rician channel model and the reflection form (18); their approximations are stated as assumptions (LoS dominance, large Rician factor) and are tested under finite Rician factors in Fig. 12. The only self-citations appear in Appendices A and B: [41] is used for the elementary ARV inner product, and [35] is cited for the DFT-direction placement construction. Both are parameter-free mathematical facts re-derived in the text, and neither is the paper's target result, so they do not make the argument circular. The UE-RIS association minimizes the approximate cross-correlation Q_{n,m} of Lemma 1, but the paper's reported evidence is sum SE and realized effective-channel matrices (Figs. 4–12), not the optimized objective itself; thus the evaluation is not statistically forced by the fit. Overall, the central claims are conditional on exact DFT placement and LoS-dominated links, which the paper explicitly flags, but conditionality is not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on four modeling assumptions (blocked direct link, LoS-dominated BS-RIS links, exact DFT alignment, and a simplified reflection vector for correlation analysis) and three hand-chosen design parameters (search grid, τ, C). No new physical entities are postulated.

free parameters (3)
  • Exhaustive search grid step for beam biases = 2π/M_R,v and 2π/M_R,h
    Optimization problems (25) and (26) are solved by exhaustive search on a grid with step 2π/M_R in each dimension; the grid resolution is chosen, not derived, and limits the achievable interference suppression.
  • Noise-limited signal threshold τ = 1/√2
    Constraint D(Θ,Φ)≥τ M_R in (26) uses τ=1/√2 to keep users inside the half-power beamwidth; this hand-set threshold controls the noise-limited configuration's tradeoff.
  • Simulation group count C = 3
    The numerical study fixes C=3 coverage groups with radii 2, 3, and 4 meters; sum-SE results depend on this geometry and no sensitivity analysis over C is reported.
assumptions (5)
  • domain assumption Direct BS-UE and multi-RIS cascaded links are negligible, so the end-to-end channel is hn≈Σ_k B_k Γ_k u_{k,n}.
    Invoked in (2) and (11); footnote 6 justifies it by path-loss numbers for one 6.5 GHz geometry. The rank and BD analysis assumes the worst-case blocked-direct-link scenario.
  • domain assumption BS-RIS channel is LoS-dominated, B_k≈β^B_{k,0} b_{k,0} r^H_{k,0}.
    Used in (12), (16), Appendices B and C; Fig. 12 tests robustness, but the derivations require large Rician factor κB.
  • ad hoc to paper RISs are placed exactly on DFT angular directions of the BS array so b^H_{k,0} b_{m,0}=0 for k≠m.
    Appendix A constructs the vertical AoDs; Section V-D shows sum SE degrades with deployment offset, so exact alignment is load-bearing.
  • ad hoc to paper For correlation analysis, RIS k is assigned to UE k with γ_k=M_R(r^*_{k,k,0}⊙r_{k,0}).
    Lemma 1 and Appendix C use this simplified reflection vector to derive Qn,m, while the actual design (18) includes optimized beam biases; Section IV still uses Lemma 1 for grouping and association.
  • domain assumption NLoS path angles in the BS-RIS channel are uniformly distributed and NLoS gains are independent zero-mean.
    Appendix C uses this to drop the NLoS term in E{B^H_k FF^H B_k} and keep only the LoS contribution.

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Cite this review

Pith. "Pith review of Joint Spatial Division and Multiplexing with Customized Orthogonal Group Channels in Multi-RIS-Assisted Systems." pith.science (2026). https://pith.science/paper/PNMFCKSQ

@misc{pith2026250701641,
  author       = {Pith},
  title        = {Pith review of: Joint Spatial Division and Multiplexing with Customized Orthogonal Group Channels in Multi-RIS-Assisted Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNMFCKSQ}},
  note         = {Machine review of arXiv:2507.01641}
}
read the original abstract

Reconfigurable intelligent surfaces (RISs) offer the unique capability to reshape the radio environment, thereby simplifying transmission schemes traditionally contingent on channel conditions. Joint spatial division and multiplexing (JSDM) emerges as a low-overhead transmission scheme for multi-user equipment (UE) scenarios, typically requiring complex matrix decomposition to achieve block-diagonalization of the effective channel matrix. In this study, we introduce an innovative JSDM design that leverages RISs to customize channels, thereby streamlining the overall procedures. By strategically positioning RISs at the discrete Fourier transform (DFT) directions of the base station (BS), we establish orthogonal line-of-sight links within the BS-RIS channel, enabling a straightforward pre-beamforming design. Based on UE grouping, we devise reflected beams of the RIS with optimized directions to mitigate inter-group interference in the RISs-UEs channel. An approximation of the channel cross-correlation coefficient is derived and serves as a foundation for the RISs-UEs association, further diminishing inter-group interference. Numerical results substantiate the efficacy of our RIS-customized JSDM in not only achieving effective channel block-diagonalization but also in significantly enhancing the sum spectral efficiency for multi-UE transmissions.

Figures

Figures reproduced from arXiv: 2507.01641 by the authors.

Figure 1
Figure 1. Multi-UE MIMO system assisted by multiple RISs. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Overview of the multi-RIS assisted JSDM system. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. CDF of the effective rank. 1 2 3 4 5 6 1 2 3 4 5 6 Configuration 1 1.332e-07 1.697e-07 1.575e-07 1.275e-07 1.539e-07 1.196e-07 1.42e-07 1.548e-07 1.597e-07 1.415e-07 1.363e-07 1.501e-07 1.425e-07 1.413e-07 1.659e-07 1.65e-07 1.086e-07 9.107e-08 8.208e-08 8.396e-08 1.18e-07 1.115e-07 1.008e-07 1.135e-07 8.328e-08 1.055e-07 1.101e-07 6.301e-08 8.739e-08 9.588e-08 9.422e-08 5.25e-08 9.294e-08 7.683e-08 1.012e-07 9.409e… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Visualization of the averaged effective channel matrix across different RIS configurations. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Amplitude of the desired and interference coefficients with different RIS sizes across 100 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Sum SE versus transmit power with different RIS configurations for [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Sum SE comparison of Configurations 3 and 4 in noise- and interference-limited regimes with [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Sum SE versus the transmit power with different RIS configurations for [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: SE versus the transmit power for each UE with different RIS configurations. [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Sum SE versus the deployment offset of RISs with [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: Sum SE versus the phase shift offset of RISs with [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Sum SE versus the Rician factor of the BS–RISs channel. [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.