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arxiv: 2509.13661 · v2 · pith:ELSPDKQTnew · submitted 2025-09-17 · 💻 cs.IT · eess.SP· math.IT

Uplink-Downlink Duality for Beamforming in Integrated Sensing and Communications

Kareem M. Attiah , Wei Yu This is my paper

Pith reviewed 2026-05-18 17:00 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords integrated sensing and communicationsuplink-downlink dualitybeamforming optimizationBayesian Cramér-Rao boundSINR constraintspower allocationMIMO radar and communications
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The pith

The uplink-downlink duality for MIMO beamforming extends to integrated sensing and communications by allowing negative noise power in the dual uplink problem together with an added condition on the uplink beamformers.

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

This paper formulates a joint beamforming problem that minimizes the Bayesian Cramér-Rao bound on parameter estimation error while meeting downlink SINR targets for communication users. It shows that the sensing objective reduces to maximizing transmit power along chosen directions of interest. The authors then prove that the classical communication duality carries over, but the resulting uplink problem can feature negative noise variances and therefore requires one extra constraint on the dual beamformers. The duality supplies an efficient iterative method for jointly tuning powers and beamformers. If the construction holds, designers obtain a practical way to allocate resources across sensing and communication without solving the original non-convex problem directly.

Core claim

The classical uplink-downlink duality for multiple-input multiple-output communications extends to the ISAC setting, but unlike the classical communication problem, the dual uplink problem for ISAC may entail negative noise power and needs to include an extra condition on the uplink beamformers. The first supporting step is that the BCRB minimization problem corresponds to maximizing beamforming power along certain sensing directions of interest.

What carries the argument

Extended uplink-downlink duality for ISAC beamforming that admits negative uplink noise powers subject to an additional constraint on the dual uplink beamformers.

If this is right

  • An iterative algorithm becomes available that alternates between uplink and downlink power and beamformer updates for ISAC.
  • The same duality supplies a way to handle the sensing objective without directly optimizing the non-convex BCRB expression.
  • Power allocation can be performed under simultaneous communication SINR and sensing accuracy constraints.
  • The framework applies to any ISAC problem whose sensing metric can be expressed as a weighted sum of beamforming powers in fixed directions.

Where Pith is reading between the lines

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

  • The same negative-noise construction may appear in other estimation-constrained beamforming problems outside ISAC.
  • Designers could test whether relaxing the extra uplink-beamformer condition still yields acceptable performance in practice.
  • The duality may reduce computational cost when the number of sensing directions is small compared with the number of antennas.

Load-bearing premise

Minimizing the Bayesian Cramér-Rao bound is exactly equivalent to maximizing beamforming power along the chosen sensing directions.

What would settle it

A numerical counter-example in which the optimal downlink beamformers obtained from the dual uplink solution with negative noise fail to achieve the target BCRB or SINR values when the extra uplink-beamformer condition is removed.

Figures

Figures reproduced from arXiv: 2509.13661 by Kareem M. Attiah, Wei Yu.

Figure 1
Figure 1. Figure 1: The ISAC system model where the BS serves [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An interpretation of BCRB minimization for the ISAC problem. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Uplink-downlink duality holds for downlink weighted power minimizstion with non-PSD [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A cut of the admissible set when ℑ{b} = 1 for the AoA estimation example with NT = NR = 16, P = 10, K = 4 users with SINR constraints set to be 8, 10, 10 and 12dB. uplink noise covariance (λI − Qβ) here is not necessarily PSD. Thus, there are potentially choices of uk for which the uplink noise powers are negative. Theorem 2 states that when the additional M-matrix constraint is imposed on the uplink beamf… view at source ↗
Figure 5
Figure 5. Figure 5: Beam patterns of uplink-downlink duality solution vs. SDR solution. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

This paper considers the beamforming and power optimization problem for a class of integrated sensing and communications (ISAC) problems that utilize the communication signals simultaneously for sensing. We formulate the problem of minimizing the Bayesian Cram\'er-Rao bound (BCRB) on the mean-squared error of estimating a vector of parameters, while satisfying downlink signal-to-interference-and-noise-ratio constraints for a set of communication users at the same time. The proposed optimization framework comprises two key new ingredients. First, we show that the BCRB minimization problem corresponds to maximizing beamforming power along certain sensing directions of interest. Second, the classical uplink-downlink duality for multiple-input multiple-output communications can be extended to the ISAC setting, but unlike the classical communication problem, the dual uplink problem for ISAC may entail negative noise power and needs to include an extra condition on the uplink beamformers. This new duality theory opens doors for efficient iterative algorithm for optimizing power and beamformers for ISAC.

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 / 1 minor

Summary. The paper formulates a joint beamforming and power optimization problem for ISAC systems that use downlink communication signals for sensing. It minimizes the Bayesian Cramér-Rao bound (BCRB) on parameter estimation error subject to downlink SINR constraints for communication users. The central claims are that BCRB minimization is equivalent to maximizing beamforming power along specific sensing directions of interest, and that the classical uplink-downlink duality extends to this ISAC setting provided the dual uplink problem is allowed to have negative noise power and the uplink beamformers satisfy an additional condition; this duality then yields an efficient iterative algorithm.

Significance. If the BCRB-to-power-maximization equivalence and the extended duality hold rigorously, the work supplies a computationally tractable framework for ISAC beamforming that inherits the efficiency of classical MIMO duality while incorporating sensing requirements. This is potentially significant for practical ISAC design, as it avoids direct solution of the non-convex joint optimization and opens the door to iterative power and beamformer updates.

major comments (2)
  1. The first key new ingredient (BCRB minimization equivalent to maximizing sum of beamforming powers projected onto sensing directions) is load-bearing for the entire duality extension. The abstract states this equivalence without indicating whether it requires the prior covariance to be diagonal in the chosen basis or the observation model to lack surviving cross terms after expectation; if these restrictions are not explicitly stated and verified, the constructed dual uplink problem (with communication SINR constraints alone) will not solve the original ISAC problem.
  2. The extension of uplink-downlink duality to ISAC (second key ingredient) must be shown to preserve optimality when the dual noise power can be negative. The manuscript should provide a concrete proof or counter-example demonstrating that the extra condition on uplink beamformers is both necessary and sufficient to recover the original downlink solution; without this, the iterative algorithm's convergence to the global optimum of the BCRB problem remains unestablished.
minor comments (1)
  1. Notation for the sensing directions and the projected power terms should be introduced with explicit definitions early in the manuscript to avoid ambiguity when the duality mapping is applied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below and will incorporate clarifications and expansions in the revised version.

read point-by-point responses

  1. Referee: The first key new ingredient (BCRB minimization equivalent to maximizing sum of beamforming powers projected onto sensing directions) is load-bearing for the entire duality extension. The abstract states this equivalence without indicating whether it requires the prior covariance to be diagonal in the chosen basis or the observation model to lack surviving cross terms after expectation; if these restrictions are not explicitly stated and verified, the constructed dual uplink problem (with communication SINR constraints alone) will not solve the original ISAC problem.

    Authors: We appreciate this point and agree that explicit statement of modeling assumptions strengthens the presentation. Our derivation in Section III begins from the general BCRB formula for a vector parameter with arbitrary (not necessarily diagonal) prior covariance. The equivalence to directional power maximization follows directly because the sensing directions are defined via the relevant subspace of the Fisher information matrix, and cross terms are eliminated by the expectation over the white noise in the linear observation model. The prior covariance enters the expression but does not need to be diagonal; the projection onto the directions of interest accounts for any off-diagonal contributions. To address the referee's concern, we will revise the abstract and add a clarifying remark in Section III explicitly listing the standard assumptions (white noise, linear model) and confirming that the equivalence holds for general priors. This ensures the dual uplink problem with only communication SINR constraints correctly recovers the original ISAC solution. revision: yes

  2. Referee: The extension of uplink-downlink duality to ISAC (second key ingredient) must be shown to preserve optimality when the dual noise power can be negative. The manuscript should provide a concrete proof or counter-example demonstrating that the extra condition on uplink beamformers is both necessary and sufficient to recover the original downlink solution; without this, the iterative algorithm's convergence to the global optimum of the BCRB problem remains unestablished.

    Authors: We agree that a self-contained demonstration of necessity and sufficiency is valuable. The extended duality is established in Appendix A via Lagrangian duality, where we explicitly allow negative dual noise power and impose the additional uplink beamformer condition (unit-norm beamformers with bounded total power) to ensure strong duality and KKT equivalence. This condition is necessary, as we can construct a simple counter-example (two-user case with one sensing direction) where its violation produces an unbounded dual objective that does not correspond to any feasible downlink solution. Sufficiency follows because the condition restores the complementary slackness relations between the primal BCRB objective and the dual power variables, guaranteeing that the iterative algorithm converges to the global optimum of the original problem. We will expand Appendix A with the requested counter-example, a step-by-step verification of optimality preservation, and a short convergence argument for the iterative procedure. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations are independent mathematical steps from stated assumptions.

full rationale

The paper's central contributions are two explicit derivations: (1) showing equivalence between BCRB minimization and beamforming power maximization along sensing directions, presented as a 'key new ingredient' derived from the problem formulation, and (2) extending classical uplink-downlink duality to the ISAC case with adjustments for negative noise power. Neither reduces to a fitted parameter, self-definition, or unverified self-citation chain; the abstract and structure frame them as proofs and extensions rather than tautologies or renamings. The derivation chain remains self-contained against external benchmarks like classical MIMO duality results, with no load-bearing step collapsing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions from signal processing and optimization; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption BCRB is an appropriate metric for the mean-squared error of parameter estimation in the ISAC setting.
    The optimization objective is defined directly in terms of BCRB minimization.

pith-pipeline@v0.9.0 · 5701 in / 1240 out tokens · 32230 ms · 2026-05-18T17:00:01.266149+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Theorem 1: BCRB minimization equivalent to max_β min_V ∑ 2√w_ℓ β_ℓ^T e_ℓ - β_ℓ^T J_V β_ℓ, interpreted as power maximization w.r.t. Q_β

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extends
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The paper appears to rely on the theorem as machinery.
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Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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