Hybrid Precoding for Multi-Group Multicasting in mmWave Systems

Gek Hong (Allyson) Sim; Luis F. Abanto-Leon; Matthias Hollick

REVIEW 4 major objections 4 minor 20 references

Hybrid Precoding for Multi-Group Multicasting in mmWave Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A joint SDR-based design for hybrid transmit precoders and receive combiners brings mmWave multi-group multicast to near-fully-digital performance with fewer RF chains, and two receive antennas raise successful reception by 60%.

desk verdict Plausible first joint hybrid multicast precoder/combiner design, but the headline gains may be counting slack-satisfied users as decoded, so those numbers need a fix before the claims stand. read the letter →

arxiv 1908.02678 v2 pith:MZXUEXOW submitted 2019-08-07 eess.SP cs.ITmath.IT

Luis F. Abanto-Leon , Matthias Hollick , Gek Hong (Allyson) Sim This is my paper · ORCID

classification eess.SPcs.ITmath.IT

keywords hybridprecodingmillimeter-wavemulticastmulti-groupsemidefiniterelaxationalternatingoptimizationfinite-resolutionphaseshiftersreceivecombiningmulti-antennareceivers

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 tackles a problem in millimeter-wave (mmWave) wireless: one base station sends different multicast streams to several groups of users using a hybrid precoder built from a small number of digital streams plus a network of cheap analog phase shifters with only a few allowed phase values. The authors claim that, despite the combinatorial phase constraints, their alternating semidefinite-relaxation (SDR) design can match the performance of a fully digital precoder using only 8 RF chains instead of 12. They further claim that giving each receiver just two antennas, and letting the design choose the receive combiner, is enough to cut required transmit power and lift the number of successfully decoded messages by 60% relative to single-antenna receivers. If true, this would make high-capacity multicast practical with simpler and cheaper transceivers.

What carries the argument

The load-bearing mechanism is a three-block alternating optimizer. First, the analog precoder $F$ is optimized by rewriting $F m_i = J_i f$, forming the rank-one matrix $D = f f^H$, relaxing the rank and phase constraints to a semidefinite program whose diagonal entries are pinned to $\delta$, and then recovering a phase vector by Cholesky factorization $\hat{D} = \hat{Q}^T \hat{Q}^*$, least-squares fitting of a common vector $\hat{u}$ and per-entry phases so that $\hat{q}_n \approx [\hat{f}]_n \hat{u}$, and randomization over candidate $\hat{u}$. Second, the digital precoders $m_i$ are optimized by a further SDR over $M_i = m_i m_i^H$. Third, the receive combiners $w_k$ are optimized by a per-user SDR over $W_k = w_k w_k^H$. Slack variables $x_k$ penalized in the objective keep every subproblem feasible, so the alternation does not stall. The central identity is $D = Q^T Q^*$ with $q_n \approx [f]_n u$: it converts the hard constant-modulus finite-alphabet constraint on $F$ into a solvable least-squares problem.

What would settle it

Run the relaxed program for the analog precoder ($P^{\mathrm{hyb}}_{\mathrm{SDR},1}$) on a channel realization and compute the two largest eigenvalues of $\hat{D}$. If the second eigenvalue is comparable to the first, the least-squares recovery cannot produce $\hat{q}_n \approx [\hat{f}]_n \hat{u}$ for all $n$, and the decoded-packet count should fall below the fully-digital baseline; observing that drop on a realistic realization would settle the limit of the claim.

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Extended reading notes

Core claim

The central claim is that joint design of the hybrid transmit precoder and the per-user receive combiners, rather than treating them separately, is what unlocks near-digital performance in mmWave multi-group multicast. With receivers limited to one omnidirectional antenna, the scheme saturates; with two antennas, the algorithm makes each receiver reject interference, raising decoded packets by 60% and improving average transmit power per received message by 16.8% for the hybrid precoder. At the transmitter, the SDR-based alternating scheme with finite-resolution phase shifts reaches the same number of decoded packets as the fully digital precoder, which needs 12 RF chains, using only 8 RF chains, at the cost of a few dB more transmit power. The design is stated to be the first to handle an arbitrary number of finite-resolution phase shifts together with multi-antenna receivers in this setting.

Load-bearing premise

The algorithm's success hinges on the relaxed SDP solution being close to rank one: after the rank constraint is dropped, the Cholesky factors $\hat{q}_n$ must be approximable as $[\hat{f}]_n \hat{u}$ for one common vector $\hat{u}$ and valid phase values, and the paper concedes this cannot be guaranteed.

Editorial extensions

If this is right

With 8 RF chains and 12 transmit antennas, the hybrid precoder matches the fully digital precoder's decoded-packet count over the simulated SINR targets of 4, 6, and 8 dB.
Adding a second receive antenna improves decoded packets by 60% for the hybrid precoder and 9% for the fully digital precoder, so receive-side combining is a cheap performance multiplier.
The transmit power per successfully received message falls by 16.8% for the hybrid precoder and 12.9% for the fully digital precoder when receivers use two antennas.
The optimizer works with naive initialization and without any knowledge of the optimal fully digital precoder, so the design does not depend on an oracle solution.
Because the framework allows an arbitrary number of phase levels, the same algorithm covers both coarse and finely quantized analog phase shifters.

Reading between the lines

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

Beyond the paper, the same SDR-plus-Cholesky-plus-least-squares phase recovery is a generic recipe for any constant-modulus finite-alphabet constraint, so it could transfer to reconfigurable-intelligent-surface phase profiles or discrete-phase analog beamforming in other frequency bands.
Beyond the paper, the reported 60% gain is measured in one correlated scenario (60 users, four groups, 30-degree angular spread) and is likely scenario-dependent, shrinking when inter-group channels are already well separated or when the number of receive antennas exceeds two.
Beyond the paper, the algorithm's dependence on random candidate vectors suggests a testable engineering tradeoff: when the relaxed SDP solution is nearly rank one, far fewer random draws should be needed, potentially cutting complexity by an order of magnitude.
Beyond the paper, since the fully digital baseline also fails to reach 100% decoded packets in the hardest scenarios, the practical bottleneck is multicast interference itself rather than the hybrid architecture.

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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

Two things to know. First, this is the first paper I know of that jointly designs hybrid transmit precoders with arbitrary finite-resolution phase shifts and receive combiners for multi-group multicast. Refs [14]–[17] handle only single-group, single-antenna, or severely restricted phase sets, so the problem formulation itself is a real step forward. Second, the central quantitative claims are not yet trustworthy because the success metric may be padded by the slack variables the authors introduce to keep the alternating subproblems feasible.

What the paper does well: the algorithmic machinery is standard but sensibly assembled — SDR for each subproblem, Cholesky factorization and least-squares for phase recovery, randomization for rank-one candidates. The authors are refreshingly honest about the heuristic parts: they explicitly say in Section VI that there is no convergence proof, and in Stage A2 they admit the common-vector phase recovery 'cannot be guaranteed.' The simulation setup is challenging, with co-located users in different multicast groups, and the observation that NRF=8 gets close to a fully-digital NRF=12 is plausible. Multi-antenna receivers do help in mmWave multicast; that qualitative point is not new, but the quantification in this hybrid setting is a fair contribution.

Now the soft spots, in proportion. The biggest is the slack-augmented QoS constraint. Algorithm 1 defines K(t) as the number of users satisfying (5b), which is the inequality with x_k added to the left-hand-side numerator. If x_k > 0 at the final solution, that user's true SINR is below the target γ. The paper never reports final slack values, and never states that Npackets is recomputed with x_k = 0 before counting successes. So the abstract's 60% improvement in successful reception and the 16.8% transmit-power-per-message gain may both be inflated. This is fixable — just report slack levels or re-evaluate with the original constraint — but without that, the headline numbers are not established. This is more load-bearing than the rank-one recovery heuristic, which is a known SDR risk and is at least admitted.

Other gaps are moderate. Averages over 100 channel realizations are given without error bars or confidence intervals. The 'average transmit power per received message' metric is not crisply defined. No comparison is run against prior hybrid multicast designs (e.g., [15]), so we cannot tell whether the proposed method beats existing hybrid approaches or just beats its own fully-digital variant. Code is not released. None of these are fatal, but they add up to a paper whose qualitative message is sound and whose quantitative message is currently over-claimed.

Who is this for? Researchers working on mmWave hybrid beamforming or physical-layer multicast. It is a competent engineering contribution with a load-bearing reporting flaw. I would send it to peer review, but with a strong referee instruction: resolve the slack-variable accounting, add confidence intervals, and add at least one prior hybrid multicast baseline. If those are addressed, the paper is a decent IEEE TSP or TWC candidate. As it stands, cite cautiously for the problem formulation, not for the 60% number.

Referee Report

4 major / 4 minor

Summary. The paper studies the joint design of hybrid transmit precoders (analog phase-shifter network plus digital precoder) and receive combiners for multi-group multicast in mmWave systems. It formulates an NP-hard QoS power-minimization problem, introduces slack variables to keep the alternating subproblems feasible, and solves the subproblems via semidefinite relaxation with randomization, alternating optimization, and a Cholesky-factorization-based phase-recovery step. Simulation results over 100 channel realizations claim that a hybrid precoder with NRF=8 performs close to its fully-digital counterpart and that two receive antennas yield roughly 60% improvement in decoded packets and 16.8% improvement in average transmit power per received message.

Significance. If the numerical claims are credible, the paper would be a useful extension of multicast beamforming to hybrid arrays with finite-resolution phase shifters and multi-antenna receivers, an area that prior work treated only with single-antenna receivers. The problem formulation is clearly structured, the algorithmic pipeline is described in detail, and the simulation study covers challenging co-located-user scenarios. However, the central performance claims are not yet sufficiently supported: the success metric can be inflated by slack variables, the rank-one recovery step has no guarantee, and no convergence proof or statistical dispersion measures are provided. These are load-bearing issues that can be fixed within the scope of the manuscript.

major comments (4)

[Section IV-A, Algorithm 1, Section V] Algorithm 1 defines K(t) as the number of users satisfying (5b), but (5b) is the slack-augmented inequality in which x_k is allowed to be positive. A user with x_k>0 does not necessarily satisfy the original SINR constraint (3b); its true SINR is γ_i - x_k/(sum_{j≠i}|w_k^H H_k F m_j|^2 + σ^2||w_k||^2), which is below γ_i. The paper reports Npackets and the 60% and 16.8% gains without stating that the final counts are recomputed with x_k=0 or reporting the final slack values. If any counted user has x_k>0, the headline results are not established. Please re-evaluate all reported Npackets using the un-relaxed constraint (3b), or report the slack magnitudes and verify they are zero for all users counted as successfully received.
[Section IV-A, Stage A2] The phase-recovery step assumes q_n ≈ [f]_n u after Cholesky factorization of the SDR solution. The paper itself states that a solution may not exist, and no diagnostic is provided for how close the SDR solution is to rank one in the simulated scenarios. Since the quality of the recovered analog precoder F depends directly on this approximation, the reader cannot assess whether the reported hybrid performance is robust or an artifact of favorable randomizations. Please add a quantitative measure of rank-one closeness of D (e.g., the eigenvalue ratio or normalized residual of the least-squares fit) and report its distribution over the 100 channel realizations, or provide a theoretical bound on the approximation error.
[Section VI, convergence discussion] The paper admits there is no theoretical evidence supporting convergence of Algorithm 1. Because the simulation study uses Niter as low as 1, and the acceptance rule in Algorithm 1 is based on K(t) and g(t), the operating point reported could be a poor local solution in some realizations. A monotonicity or local-convergence statement, or at least a numerical convergence check showing the objective and K(t) versus iteration for individual representative realizations, would support the claim that the plotted results reflect the algorithm's typical behavior rather than its stagnation.
[Section V, Figures 1-3] All numerical results are averages over 100 channel realizations, but no confidence intervals, standard deviations, or error bars are shown. The abstract reports precise percentage gains (16.8% and 60%) from these averages; without statistical dispersion, those numbers are not established with any stated uncertainty. Additionally, the metric 'average transmit power per received message' is not explicitly defined. Please specify its formula (e.g., Ptx/Npackets) and provide error bars or confidence intervals for the key comparisons, especially the Nrx=1 versus Nrx=2 contrast in Fig. 2.

minor comments (4)

[Section IV] The text states that P^hyb_SDR,1, P^hyb_SDR,2, and P^hyb_SDR,3 'can be recast as linear programs'; these are semidefinite programs, not linear programs. The statement should be corrected to say they are convex SDPs solvable in polynomial time.
[Algorithm 1] The randomization step for W_k uses w_k = W_k v_k with v_k uniformly distributed on the unit sphere, which is not the standard Gaussian randomization used in SDR. The text should justify why this spherical randomization is valid, or use the standard Gaussian approach.
[Section V] The units for P_max_rx and σ^2 are given in dBm, but the equations use linear power quantities. Please make the conversion explicit to avoid ambiguity.
[Section II, Eq. (2)] In the SINR expression, the denominator is written as ∑_{j≠i}|...|^2 + σ^2||w_k||^2_2; the subscript 2 on the norm appears only in the second term, which is correct but stylistically inconsistent. This is a minor notation issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reports a self-contained simulation study of an SDR-based alternating optimization, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claims are simulation outputs of an optimization procedure, not derivations from first principles or fits to data. The objective (4a) and QoS constraints (4b) define the problem, and the reported Npackets and Ptx are evaluations of the proposed algorithm, so no prediction reduces by construction to an input. The phase-recovery stage is explicitly acknowledged as heuristic: the authors state that a solution (f_hat, u_hat) for q_hat_n = [f_hat]_n u_hat 'may not exist,' and Section VI states 'There is no theoretical evidence supporting the convergence of Algorithm 1.' These are honest limitations, not hidden circularities. The only self-citation ([1], by author G. H. Sim) supports a background statement on multicast benefits and is not load-bearing. One correctness risk should be tracked separately: Algorithm 1 defines K(t) as the count of users satisfying the slack-augmented constraint (5b), and if Npackets were not recomputed with x_k = 0, the headline 60% reception gain could be inflated; however, the paper nowhere equates Npackets to K(t), so under the hard-evidence rule this is not a demonstrated circular step.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the efficacy of the SDR-plus-randomization heuristic, which is a domain-specific assumption for this class of QCQP problems, not a guaranteed mathematical fact. Hyperparameters beta, Nrand, and Niter are hand-set and materially affect results. No new physical entities are introduced.

free parameters (3)

beta (regularization weight) = G^3 * N_RF_tx * N_tx * N_rx
Hand-chosen in Eq. (4a); the paper says beta must be 'sufficiently large' but does not prove sufficiency. It trades transmit power against slack, and the reported performance depends on this choice.
Nrand (number of randomizations) = 500, or 400+300*(Ntx+Nrx-11) in Section V.C
Fig. 3 shows performance improves with Nrand; the headline numbers are reported for Nrand=500 (or scaled). This is a computational budget parameter that is essential to the claimed results.
Niter (alternating iterations) = 3-5 in simulations
Algorithm 1 has no convergence proof; Section VI admits this. Fig. 3 shows Niter affects performance, so the reported gains are conditional on the chosen Niter.

assumptions (4)

domain assumption Narrowband flat-fading channel model
The received signal model in Eq. (1) assumes narrowband flat-fading, standard in mmWave hybrid precoding but not validated against measurements in this paper.
domain assumption Geometric channel model with Mp=8 paths and specified angular spreads
Simulation setup in Section V fixes the number of paths and angle distributions. These are inputs chosen by the authors, not derived from real measurements.
ad hoc to paper SDR relaxation tightness
The method drops rank-one constraints (Eqs. 7d, 12d, 14e) and relies on randomization to recover rank-one solutions. No tightness guarantee is provided for this problem.
ad hoc to paper Convergence of alternating optimization
Section VI states 'There is no theoretical evidence supporting the convergence of Algorithm 1.' The performance claims assume the algorithm reaches a useful fixed point despite non-convexity.

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Cite this review

Pith. "Pith review of Hybrid Precoding for Multi-Group Multicasting in mmWave Systems." pith.science (2026). https://pith.science/paper/MZXUEXOW

@misc{pith2026190802678,
  author       = {Pith},
  title        = {Pith review of: Hybrid Precoding for Multi-Group Multicasting in mmWave Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZXUEXOW}},
  note         = {Machine review of arXiv:1908.02678}
}

abstract

Multicast beamforming is known to improve spectral efficiency. However, its benefits and challenges for hybrid precoders design in millimeter-wave (mmWave) systems remain understudied. To this end, this paper investigates the first joint design of hybrid transmit precoders (with an arbitrary number of finite-resolution phase shifts) and receive combiners for mmWave multi-group multicasting. Our proposed design leverages semidefinite relaxation (SDR), alternating optimization and Cholesky matrix factorization to sequentially optimize the digital/analog precoders at the transmitter and the combiners at each receiver. By considering receivers with multiple-antenna architecture, our design remarkably improves the overall system performance. Specifically, with only two receive antennas the average transmit power per received message improves by $ 16.8\% $ while the successful information reception is boosted by $ 60\% $. We demonstrate by means of extensive simulations that our hybrid precoder design performs very close to its fully-digital counterpart even under challenging scenarios (i.e., when co-located users belong to distinct multicast groups).

Figures

Figures reproduced from arXiv: 1908.02678 by the authors.

**Figure 2.** Evaluation of the number of decoded packets and [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Evaluation of the number of decoded packets and tra [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Channel correlation histogram. we show in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Radiation patterns most challenging to cater, specifically for the hybrid precoder whose beamforming flexibility is limited. Hybrid precoder design: Different from the majority of works in hybrid precoding (either multi-user or multicast), the proposed design has no kn…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [14]

Hybrid Precoding for Physical La yer Mul- ticasting,

M. Dai and B. Clerckx, “Hybrid Precoding for Physical La yer Mul- ticasting,” IEEE Communication Letters , vol. 20, no. 2, pp. 228–231, February 2016

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Antenna Selection and Hybr id Beam- forming for Simultaneous Wireless Information and Power Tr ansfer in Multi-Group Multicasting Systems,

O. T. Demir and T. E. Tuncer, “Antenna Selection and Hybr id Beam- forming for Simultaneous Wireless Information and Power Tr ansfer in Multi-Group Multicasting Systems,” IEEE Transactions on Wireless Communications, vol. 15, no. 10, pp. 6948–6962, October 2016

work page 2016

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Hybrid Precod ing for Multi- Group Physical Layer Multicasting,

M. Sadeghi, L. Sanguinetti, and C. Y uen, “Hybrid Precod ing for Multi- Group Physical Layer Multicasting,” in EW, May 2018, pp. 1–6

work page 2018

[4] [1]

Finite Horizon Opportunistic Mu lti- cast Beamforming,

G. H. Sim and J. Widmer, “Finite Horizon Opportunistic Mu lti- cast Beamforming,” IEEE Transactions on Wireless Communications , vol. 16, no. 3, March 2017

work page 2017

[5] [2]

Scalable Multicast in High ly-Directional 60-GHz WLANs,

S. Naribole and E. Knightly, “Scalable Multicast in High ly-Directional 60-GHz WLANs,” IEEE/ACM Transactions on Networking , vol. 25, no. 5, pp. 2844–2857, October 2017

work page 2017

[6] [3]

An Incremental Mul ticast Grouping Scheme for mmWave Networks with Directional Anten nas,

H. Park, S. Park, T. Song, and S. Pack, “An Incremental Mul ticast Grouping Scheme for mmWave Networks with Directional Anten nas,” IEEE Communications Letters, vol. 17, no. 3, pp. 616–619, March 2013

work page 2013

[7] [4]

Transmit Beam- forming for Physical-Layer Multicasting,

N. D. Sidiropoulos, T. N. Davidson, and Z. Luo, “Transmit Beam- forming for Physical-Layer Multicasting,” IEEE Transactions on Signal Processing, vol. 54, no. 6, pp. 2239–2251, June 2006

work page 2006

[8] [5]

Improved Approximat ion Algorithms for Maximum Cut and Satisﬁability Problem Using Semi- Deﬁnite Programming,

M. X. Goemans and D. P . Williamson, “Improved Approximat ion Algorithms for Maximum Cut and Satisﬁability Problem Using Semi- Deﬁnite Programming,” Journal of the Association for Computing Machinery, vol. 42, no. 6, pp. 1115–1145, November 1995

work page 1995