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arxiv: 2604.22392 · v1 · submitted 2026-04-24 · 💻 cs.IT · eess.SP· math.IT

Multi-User ISAC with Heterogeneous Unknown Parameters: Optimal Beamforming based on Distribution Information

Chan Xu , Shuowen Zhang This is my paper

Pith reviewed 2026-05-08 09:50 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords ISACbeamformingsensingcommunicationPCRBsemi-definite relaxationLagrange dualitymulti-user
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The pith

Optimal beamforming in multi-user ISAC with unknown target reflection needs at most one dedicated sensing beam.

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

This paper examines beamforming in an integrated sensing and communication system where a multi-antenna base station serves multiple users while estimating a target's unknown random angle from its prior distribution and echo signals. The target's reflection coefficient is treated as an additional unknown without prior information, creating heterogeneous parameters. The design minimizes the periodic posterior Cramer-Rao bound on angle estimation error subject to per-user rate constraints. Semi-definite relaxation and Lagrange duality yield the optimal solution and establish that one sensing beam suffices.

Core claim

For the downlink multi-user ISAC setup with heterogeneous unknowns, the transmit beamforming that minimizes the periodic PCRB on the angle parameter while satisfying individual communication rate constraints admits an optimal solution obtained through semi-definite relaxation and Lagrange duality theory; this solution requires at most one dedicated sensing beam.

What carries the argument

Semi-definite relaxation of the non-convex beamforming optimization combined with Lagrange duality to recover the optimal communication and sensing beams.

If this is right

  • The sensing error bound is minimized without allocating extra beams beyond one dedicated sensing beam.
  • Each communication user meets its rate target after canceling the known sensing signals.
  • The optimization incorporates prior distribution information only on the angle, not on the reflection coefficient.
  • The resulting beamformer structure separates communication beams from at most one sensing beam.
  • Numerical validation confirms both the closed-form optimality and the single-beam sufficiency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hardware implementations could simplify sensing hardware by limiting dedicated beams to one without performance loss.
  • Relaxing perfect interference cancellation would likely increase the number of required beams or degrade the PCRB.
  • The same duality approach could be tested on other periodic estimation tasks such as velocity sensing under unknown amplitudes.
  • Real deployments might replace the periodic PCRB with empirical mean-cyclic error measurements to check bound tightness.

Load-bearing premise

Users can perfectly cancel interference from the pre-determined sensing signals, and the periodic PCRB is a tight lower bound on the mean-cyclic error for the periodic angle.

What would settle it

An experiment or simulation in which the minimal achievable periodic PCRB requires two or more dedicated sensing beams under the stated rate constraints would disprove the optimality claim.

Figures

Figures reproduced from arXiv: 2604.22392 by Chan Xu, Shuowen Zhang.

Figure 1
Figure 1. Figure 1: Illustration of multi-user ISAC with sensing interference cancellation view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of radiation power pattern and target angle PDF view at source ↗
read the original abstract

This paper studies an integrated sensing and communication (ISAC) system where a multi-antenna base station (BS) communicates with multiple single-antenna users in the downlink and senses the unknown and random angle information of a target based on its prior distribution information and the received echo signals. We focus on a challenging scenario with heterogeneous unknown parameters where the target's reflection coefficient is also unknown with no prior information. We consider a general transmit beamforming structure with both communication beams and dedicated sensing beams, where the communication users can cancel the interference caused by the pre-determined sensing signals. By adopting the periodic posterior Cramer-Rao bound (PCRB) to quantify a lower bound of the mean-cyclic error (MCE) for sensing the periodic angle parameter, we optimize the transmit beamforming to minimize the periodic PCRB, subject to individual communication user rate constraints, which is a non-convex problem. By leveraging the semi-definite relaxation (SDR) technique and Lagrange duality theory, we derive the optimal solution and prove that at most one dedicated sensing beam is needed. Numerical results validate our analysis and effectiveness of the proposed beamforming design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a multi-user downlink ISAC system in which a multi-antenna BS transmits both communication beams and dedicated sensing beams to serve single-antenna users while estimating a target's random angle (with known prior) from echo signals. The reflection coefficient is treated as a deterministic unknown nuisance parameter with no prior. Communication users are assumed able to perfectly cancel interference from the pre-determined sensing signals. The design objective is to minimize the periodic PCRB (as a lower bound on mean-cyclic error for the periodic angle) subject to per-user rate constraints. The resulting non-convex problem is solved via SDR and Lagrange duality, yielding an optimal beamforming solution together with a proof that at most one dedicated sensing beam is required.

Significance. If the periodic PCRB is shown to be a valid and tight lower bound on the intended mean-cyclic error under the hybrid (random + deterministic) parameter model, the work supplies a concrete, optimally solved beamforming design for a practically relevant ISAC setting together with a useful structural result on the number of sensing beams. The explicit use of the angle prior, the interference-cancellation assumption, and the SDR-duality approach constitute technical strengths that would be of interest to the ISAC community.

major comments (2)
  1. [§III] §III (Problem Formulation and PCRB): The periodic PCRB is adopted directly as the sensing objective without an explicit hybrid-CRB adjustment or marginalization over the deterministic unknown reflection coefficient β. Standard estimation theory indicates that when parameters are heterogeneous (random angle with prior, deterministic nuisance β), the appropriate bound is the hybrid CRB or a nuisance-parameter-conditioned periodic bound; the expression used here appears to follow the all-random PCRB template. Because the entire optimization and the subsequent optimality proof rest on this objective, the mismatch is load-bearing for the central claim that the derived beamformer minimizes a valid lower bound on MCE.
  2. [§IV] §IV (SDR and Duality Solution): The proof that at most one dedicated sensing beam suffices is obtained from the structure of the dual solution under the chosen PCRB objective. If the PCRB expression does not correctly lower-bound the mean-cyclic error once the unknown deterministic reflection coefficient is properly accounted for, the structural result does not necessarily transfer to the intended performance metric. A concrete verification (e.g., comparison of the derived PCRB against the hybrid CRB or Monte-Carlo MCE) is needed to confirm the claim.
minor comments (2)
  1. [§II] The assumption that users perfectly cancel sensing-signal interference is stated without a detailed protocol or channel-knowledge requirement; a brief remark on how the sensing beams are made known to the users would improve clarity.
  2. [§II] Notation for the periodic angle parameter and its prior distribution should be introduced consistently in the system model before the PCRB derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised about the treatment of heterogeneous parameters in the PCRB derivation and the transferability of the structural result are important. We address each major comment below and describe the revisions we will undertake.

read point-by-point responses

  1. Referee: [§III] §III (Problem Formulation and PCRB): The periodic PCRB is adopted directly as the sensing objective without an explicit hybrid-CRB adjustment or marginalization over the deterministic unknown reflection coefficient β. Standard estimation theory indicates that when parameters are heterogeneous (random angle with prior, deterministic nuisance β), the appropriate bound is the hybrid CRB or a nuisance-parameter-conditioned periodic bound; the expression used here appears to follow the all-random PCRB template. Because the entire optimization and the subsequent optimality proof rest on this objective, the mismatch is load-bearing for the central claim that the derived beamformer minimizes a valid lower bound on MCE.

    Authors: We appreciate this observation on the hybrid nature of the parameters. In Section III, the periodic PCRB is derived from the posterior information matrix of the random angle θ (using its known prior), with the observation model conditioned on the deterministic unknown β. This yields a bound that holds for any fixed β without explicit marginalization. While the manuscript does not invoke the hybrid CRB terminology, the resulting expression is the standard periodic PCRB for the parameter of interest in the presence of a nuisance parameter. To eliminate any ambiguity, we will revise §III to explicitly state that the bound is conditioned on β, reference the relevant hybrid estimation literature, and clarify why the chosen objective remains a valid (if conservative) lower bound on MCE for the periodic angle. These clarifications will not change the optimization problem or its solution. revision: partial

  2. Referee: [§IV] §IV (SDR and Duality Solution): The proof that at most one dedicated sensing beam suffices is obtained from the structure of the dual solution under the chosen PCRB objective. If the PCRB expression does not correctly lower-bound the mean-cyclic error once the unknown deterministic reflection coefficient is properly accounted for, the structural result does not necessarily transfer to the intended performance metric. A concrete verification (e.g., comparison of the derived PCRB against the hybrid CRB or Monte-Carlo MCE) is needed to confirm the claim.

    Authors: We agree that empirical verification strengthens the claim. The analytical proof in §IV follows directly from the KKT conditions and dual solution structure under the adopted PCRB objective. In the revision we will add numerical results that (i) compare the periodic PCRB used in the paper with the corresponding hybrid-CRB expression and (ii) report Monte-Carlo estimates of the mean-cyclic error achieved by the optimized beamformer. These additions will confirm that the derived solution remains effective with respect to the MCE and that the “at most one sensing beam” property is observed in practice, thereby supporting the transferability of the structural result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard convex optimization applied to externally defined PCRB

full rationale

The paper adopts the periodic PCRB as a lower bound on MCE from prior literature on periodic parameters and applies SDR plus Lagrange duality to minimize it under rate constraints. The structural result (at most one dedicated sensing beam) follows directly from the KKT conditions of the relaxed problem without reducing to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The objective function is specified independently of the beamforming solution itself, and no step equates a derived quantity to its own input by construction. This is a standard application of convex relaxation techniques to a well-posed (if potentially debatable) estimation bound.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard optimization theory and domain assumptions from ISAC literature; no free parameters are explicitly fitted in the abstract, and no new entities are postulated.

axioms (2)
  • domain assumption Periodic PCRB provides a valid lower bound on mean-cyclic error for the periodic angle parameter.
    Used to quantify sensing performance in the optimization objective.
  • domain assumption Users can perfectly cancel interference from pre-determined sensing signals.
    Enables the general beamforming structure without residual interference.

pith-pipeline@v0.9.0 · 5497 in / 1282 out tokens · 27862 ms · 2026-05-08T09:50:54.715841+00:00 · methodology

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