How Many Simultaneous Beamformers are Needed for Integrated Sensing and Communications?

Kareem M. Attiah; Wei Yu

REVIEW 3 major objections 4 minor 37 references

How Many Simultaneous Beamformers are Needed for Integrated Sensing and Communications?

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read At most $K+\lfloor\sqrt{L(L+1)/2}\rfloor$ beamformers suffice for simultaneous sensing and communication; without interference cancellation the cap is $\lfloor\sqrt{K^2+L(L+1)/2}\rfloor$.

desk verdict Solid answer to a basic ISAC counting question; the main theorems hold up after a minor fix in one lemma's scaling step. read the letter →

arxiv 2507.14982 v3 pith:SRN27DWB submitted 2025-07-20 eess.SP cs.ITmath.IT

Kareem M. Attiah , Wei Yu This is my paper · ORCID

classification eess.SPcs.ITmath.IT MSC 90C2294A12

keywords integratedsensingandcommunicationsbeamformingminimumnumberofbeamformersCramér-RaoboundBayesianFisherinformationmatrixrankreductionsemidefiniterelaxationMIMOradar

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 asks how many transmit beamformers (spatial weight vectors) a base station needs to simultaneously serve $K$ communication users and estimate $L$ environmental parameters. It proves a pair of universal upper bounds: with interference cancellation at the users, $K+\lfloor\sqrt{L(L+1)/2}\rfloor$ beamformers always suffice; without cancellation, $\lfloor\sqrt{K^2+L(L+1)/2}\rfloor$ suffice. The bounds hold for every channel realization, every number of antennas, and every feasible combination of sensing accuracy and user SINR. If the bounds are right, massive-antenna ISAC transmitters can be designed and optimized with far fewer beamforming chains than antennas, and the extra sensing beams are often unnecessary.

What carries the argument

The engine is a rank-reduction argument that turns performance preservation into a counting problem. Starting from an optimal full beamformer set, the paper writes the requirement that a new set keep the same BFIM, SINRs, and power as quadratic equations, then substitutes $a_k = 1-|d_k|^2$ and $M_s = I_{N_s}-U_s U_s^H$ to obtain a homogeneous linear system. This system has $K+L(L+1)/2$ equations and $K+N_s^2$ unknowns, so whenever $N_s^2 > L(L+1)/2$ a nonzero solution exists; scaling it by its largest component makes $I_{N_s}-M_s$ positive semidefinite and singular, yielding a shorter beamformer matrix with identical performance. In the no-cancellation case a separate lemma forces the optimal sensing beams to be orthogonal to every user channel, which lets the communication beams absorb sensing beams and shifts the counting threshold to $N^2 > K^2+L(L+1)/2$. Iterating until the counting condition fails leaves at most the advertised number of beamformers.

What would settle it

A counterexample would be a feasible ISAC instance with $N_s^2 > L(L+1)/2$ in which every nonzero solution of the homogeneous system (109)-(110) fails, after scaling, to make $I_{N_s}-M_s$ positive semidefinite and singular with $1-a_k \ge 0$ for all $k$; searching random channel and target realizations for such a case would settle whether the sum bound is universal.

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Extended reading notes

Core claim

The paper establishes that the minimum number of downlink beamformers for ISAC is governed by the number of distinct quadratic terms in the performance metrics rather than by the number of antennas. For Bayesian Cramér-Rao bound (BCRB) estimation of $L$ real parameters, any pair consisting of a Bayesian Fisher information matrix (BFIM) and user SINRs that is achievable with the full $N_T+K$ beamformers is also achievable with at most $K+\lfloor\sqrt{L(L+1)/2}\rfloor$ beamformers when users cancel sensing interference, and with at most $\lfloor\sqrt{K^2+L(L+1)/2}\rfloor$ when they cannot. Because the second bound is less than the sum $K+\sqrt{L(L+1)/2}$, joint operation can require strictly fewer beamformers than the two tasks taken separately. For any sensing metric depending on $d$ quadratic trace terms, the same argument gives $K+\lfloor\sqrt{d}\rfloor$ and $\lfloor\sqrt{K^2+d}\rfloor$; for $N_{\mathrm{tr}}$ line-of-sight targets this scales roughly as $K+1.871N_{\mathrm{tr}}$ with cancellation, and without cancellation the single-target worst case is exactly 2 beamformers for $K=0$ or 1 and $K$ beamformers for $K \ge 2$.

Load-bearing premise

The whole result rests on the scaling step: whenever the homogeneous linear system has more unknowns than equations, a nonzero solution can always be scaled so that deleting one beamformer does not change the BFIM, SINRs, or power, and this must hold for every feasible configuration.

Editorial extensions

If this is right

Any feasible ISAC performance pair can be realized with at most the stated number of beamformers, so optimizing directly in beam space loses nothing compared with a full-antenna semidefinite relaxation.
Without interference cancellation, if $K \ge L(L+1)/4$ then communication beamformers alone achieve any feasible BFIM and SINR targets.
For single-target line-of-sight sensing without cancellation, the worst-case minimum is exactly 2 beamformers for $K=0$ or 1 and exactly $K$ beamformers for $K \ge 2$.
For radar SNR or SCNR, which are $d$-quadratic metrics with $d=1$ or 2, no extra sensing beamformers are needed when at least one user is served without cancellation.
For beam-pattern probing over $N_g$ grid points, the bounds give at most $K+\lfloor\sqrt{N_g+1}\rfloor$ beamformers with cancellation and $\lfloor\sqrt{K^2+N_g+1}\rfloor$ without.

Reading between the lines

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

The same counting logic suggests that any sensing metric whose Fisher information or objective is a function of $m$ independent quadratic forms will inherit bounds of order $\sqrt{m}$ or $\sqrt{K^2+m}$, even when the underlying unknowns are not angles or path losses.
Because the bounds are worst-case over channels, typical instances need far fewer beams (the paper's own simulations show 2-3 beams in many cases); a practical design rule is to start from the upper bound and then prune using the same reduction algorithm.
The paper flags that its tighter AoA-only bound relies on zero-mean path-loss priors; if that caveat bites, active or multi-stage sensing that updates priors to nonzero means should revert to the larger $O(N_{\mathrm{tr}})$ scaling, which is a testable prediction for multi-stage ISAC.
The result gives a direct argument for reduced-RF or hybrid architectures: since the needed number of beamformer ports is bounded independently of $N_T$, the beamformer count can be fixed before the antenna-array size is chosen.

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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

Colleague—

This paper gives clean worst-case upper bounds on the number of beamformers needed for integrated sensing and communication: K + sqrt(L(L+1)/2) with interference cancellation, and sqrt(K^2 + L(L+1)/2) without. Those are genuinely new, especially the hypotenuse bound, which is tighter than the sum bound and leads to the nice corollary that communication beamformers alone suffice once K >= L(L+1)/4. The d-quadratic extension covers radar SNR/SCNR and beam-pattern metrics and is a useful unification.

The proof strategy is sound. The key is a rank-reduction procedure: for the no-cancellation case, Lemma 2 first shows the sensing beams can be taken orthogonal to all communication channels, which makes the SINR constraints invariant to the transformation, and then a counting argument gives the bound. The appendices are detailed and the strong-duality groundwork is handled properly. I also appreciate that the numerical section says plainly that these are worst-case bounds and that typical numbers are often much smaller.

The soft spot is in Appendix B, Lemma 1. The paper chooses a scaling delta whose magnitude is the largest among the solution components, and then asserts I - M_s is PSD and singular. To get singularity, it needs delta to be one of the eigenvalues of M'_s, not one of the a'_k. The proof sketches a contradiction for the a'_k case, but it doesn't fully spell out the sign choice when the largest magnitude is attained by a negative component. The stress-test note gives a correct fix: if scaling makes d_k = 0 for some user, that beamformer vanishes and the positive SINR target is violated, so the maximum must be an eigenvalue. With that repair, the lemma goes through. This is a minor presentation gap, not a structural flaw.

A smaller issue: the d-quadratic generalization is largely mechanical once the BCRB case is done. That's fine, but the novelty there is modest.

Overall, the counting question is basic and the answer is clean. I'd bring it to a reading group focused on ISAC or MIMO beamforming, and I'd cite it. The paper deserves a serious referee; the math is self-contained and the claims are explicit about what is tight and what is not.

Referee Report

3 major / 4 minor

Summary. The paper studies a downlink ISAC system in which a base station uses linear beamforming to serve K single-antenna communication users while estimating L real-valued sensing parameters, and asks how many beamformers are needed to attain the same joint performance as a full set of N_T + K beamformers. The central results are universal upper bounds on this minimum number: with interference cancellation at the users, at most K + floor(sqrt(L(L+1)/2)) beamformers suffice; without interference cancellation, at most floor(sqrt(K^2 + L(L+1)/2)) suffice. These bounds are extended to a class of d-quadratic sensing metrics, yielding K + floor(sqrt(d)) and floor(sqrt(K^2 + d)), respectively. The paper also derives applications to channel-matrix estimation, estimation of LoS target parameters, radar SNR/SCNR detection, and beam-pattern matching, with several tightness results, most notably the exact characterization for one LoS target without interference cancellation.

Significance. If the proofs are completed as claimed, this is a substantial contribution. The bounds are parameter-free, channel-independent, and expressed directly in terms of the number of users K, the number of sensing parameters L, or the number of quadratic terms d, which makes them directly useful for system design and for avoiding over-parameterized SDR-based optimization. The paper improves on prior results such as the 2N_tr bound of [7] and generalizes earlier special-case results of [8], [9], [14]. The derivation is self-contained, builds on standard rank-reduction ideas of Pataki and Huang-Palomar, and the authors explicitly disclose the relation to their conference version [1] and to the concurrent work [18]. The numerical examples are honest about the gap between worst-case bounds and typical behavior, and several tightness cases are identified. The main reservations concern proof completeness in three load-bearing places rather than the validity of the overall approach.

major comments (3)

[Appendix B, proof of Lemma 1, scaling step after Eqs. (112)-(113)] The proof that I_Ns - M_s is singular is under-specified. The text chooses delta with |delta| equal to the largest magnitude among the a'_k and the eigenvalues of M'_s, then asserts that if I_Ns - M_s were nonsingular, delta would equal some a'_i and the corresponding v'_i would be zero, violating the SINR constraint. As written, this does not explain why the case of a largest-magnitude eigenvalue is handled, nor why delta = a'_i is impossible. The missing argument is: after scaling so that a_i = 1, the i-th equation of (110) gives |h_i^H vhat_i|^2 / gamma'_i = sum_{n != i} a_n |h_i^H vhat_n|^2, while the definition of gamma'_i gives |h_i^H vhat_i|^2 / gamma'_i = sum_{n != i} |h_i^H vhat_n|^2 + sigma^2, a contradiction because |a_n| <= 1 and sigma^2 > 0. Please add this inequality (or an equivalent argument) and specify the sign convention for delta. This is load-bearing because Lemma 1 is the engine of Theorem 1.
[Section IV-B, proof of Theorem 3] The proof invokes Lemma 1 for d-quadratic constraints, but Lemma 1 is stated and proved only for the BFIM constraint J_V = J, and its proof uses strong duality of P_IC_N together with the dual stationarity conditions (115)-(117) to establish power preservation via (118). For the d-quadratic problem (57a)-(57c), the analogous strong-duality result, the analogous dual problem, and the power-preservation identity are not stated or proved. The extension is plausible and likely follows by repeating the SDR-tightness argument of Lemma 4, since the constraints tr(Q_i V V^H) = c_i are linear in R = V V^H, but as the text stands Theorem 3 rests on an unproved generalization of the central rank-reduction lemma. Please add a formal lemma covering d linear quadratic constraints, or expand the proof of Theorem 3 to include the strong-duality and scaling steps for both the IC and NIC cases.
[Appendix D, proof of Lemma 2, limiting argument near Eqs. (139)-(143)] The proof of Lemma 2 uses a perturbation argument in which the dual problem D_epsilon is solved for every epsilon > 0, and then a limit point V'_c of the sequence {V'_{c,epsilon}} is taken as epsilon -> 0. The text asserts that 'since epsilon -> 0, V'_c must achieve the same optimal value' as the unperturbed problem (132), but no continuity or compactness argument is supplied to justify convergence of the optimal values or feasibility and optimality of the limit point. Since Lemma 2 provides the orthogonality condition (33) on which Theorem 2 depends, this is a load-bearing step. Please add a short argument showing that the optimal value of R_epsilon converges to that of R_0 and that the limit point is optimal for (127), or replace the perturbation step with a direct KKT-based proof.

minor comments (4)

[Appendix B, dual problem (114)] The identity matrices in (114b) and (114c) should be I_{N_T}, not I_N, since the matrices tilde{G}_{i,j} and the channel vectors h_k act on the N_T-dimensional transmit space; the dual problem is independent of the number of beamformers N.
[Theorem 2, Eq. (39)] The displayed constraint uses vSINR_IC, but the theorem is for the no-interference-cancellation scenario and should use vSINR_NIC.
[Appendix F] There is a factor-of-two inconsistency with Eq. (66): (66) defines J_V with 2Upsilon/sigma^2 in the off-diagonal blocks, but Appendix F defines B_1, B_2, and bar{J} with Upsilon/sigma^2, and the final trace expression misses the factor 2 inside the inverse. Please align Appendix F with Eq. (67a), which appears to have the correct factor.
[Figure 5] The label 'Hypoenuse Bound' in the figure legend should read 'Hypotenuse Bound'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the beamformer-count upper bounds are proved by rank-reduction counting arguments, with self-citations used only for comparison, not as load-bearing assumptions.

full rationale

The paper's central claims are upper bounds on the minimum number of beamformers, derived constructively by fixing the BFIM and SINR values and applying a rank-reduction argument that counts equations versus unknowns. The engine of the proof is Lemmas 1 and 3, which set up homogeneous linear systems: the BFIM contributes L(L+1)/2 equations, the SINR constraints contribute K equations, and the transformation variables contribute K + N_s^2 unknowns; whenever the number of unknowns exceeds the number of equations, a nonzero solution exists and can be scaled to remove a beamformer while preserving the performance pair. This counting argument is self-contained in Appendices B and E, and it does not assume the theorem it is proving. The external results [19], [20] are cited as prior rank-reduction techniques, but the paper reproduces the needed proof steps rather than invoking the conclusion as a black box. Self-citations, notably the conference version [1], are used only to state what was previously established and to contrast the improved bounds; they are not load-bearing for the new theorems. The d-quadratic extension in Theorem 3 is the same counting argument with d replacing L(L+1)/2, not a renaming of a known result. No parameters are fitted to data, and no simulated quantity is promoted to a prediction; the numerical sections merely illustrate tightness of the proved bounds. One subtle point in Appendix B is that the scaling constant δ must be chosen with sign so that the largest-magnitude component is exactly δ, which is only implicit in the text; this is a proof-presentation issue, not circularity, and it does not make any theorem equivalent to its inputs. Overall, the derivation chain is independent and self-contained, so no circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted. The bounds are proven, not optimized against data. The only inputs are the problem sizes K, L, d, the prior and channel statistics, and standard background results from SDP and estimation theory.

assumptions (7)

standard math Standard SDP rank-reduction theorems of Pataki [20] and Huang-Palomar [19]
Used in Lemma 1 and Theorem 1 to reduce the number of beamformers; accepted background results.
standard math Bayesian Cramer-Rao bound for Gaussian observation model
The BFIM expression (13)-(15) is derived in Appendix A from standard Gaussian Fisher information [37].
domain assumption Narrowband block-fading channel with small delay and Doppler spread, full-duplex with self-interference cancellation
Section II-B; needed to reduce the channel model to (8).
domain assumption Sensing channel G(eta) is a deterministic known function of the parameters, with known prior f(eta)
Section II-B; defines the parameter estimation problem and enables the BFIM decomposition.
domain assumption Communication and sensing channels are generated from non-degenerate random distributions
Section II-C; used to avoid degenerate cases and allow spatial separation, though the bounds are claimed uniformly.
domain assumption Users either fully cancel sensing interference or treat it as noise
Defines the two SINR models (17)-(18); the two bounds correspond to these two operational regimes.
domain assumption The scalar sensing objective h is nondecreasing in the PSD cone
Used in problem (20) to define the minimum number of beamformers.

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Cite this review

Pith. "Pith review of How Many Simultaneous Beamformers are Needed for Integrated Sensing and Communications?." pith.science (2026). https://pith.science/paper/SRN27DWB

@misc{pith2026250714982,
  author       = {Pith},
  title        = {Pith review of: How Many Simultaneous Beamformers are Needed for Integrated Sensing and Communications?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRN27DWB}},
  note         = {Machine review of arXiv:2507.14982}
}

abstract

Consider a downlink integrated sensing and communications (ISAC) system in which a base station employs linear beamforming to communicate to $K$ users, while simultaneously uses sensing beams to perform a sensing task of estimating $L$ real parameters. How many beamformers are needed to achieve the best performance for both sensing and communications? This paper establishes bounds on the minimum number of downlink beamformers, in which sensing performance is measured in terms of the Cram\'{e}r-Rao bound for parameter estimation and communications performance is measured in terms of the signal-to-interference-and-noise ratios. We show that an ISAC system requires at most $K + \sqrt{\frac{L(L+1)}{2}}$ beamformers if the remote users have the ability to cancel the interference caused by the sensing beams. If cancelling interference due to the sensing beams is not possible, the bound becomes $\sqrt{K^2 + \frac{L(L+1)}{2}}$. Interestingly, in the latter case, the bound on the number of beamformers is less than the sum of the bounds for each task individually. These results can be extended to sensing tasks for which the performance is measured as a function of $d$ quadratic terms in the beamformers. In this case, the bound becomes $K + \sqrt{d}$ and $\sqrt{K^2 + d}$, respectively. Specifically, for estimating complex path losses and angles-of-arrival of $N_\text{tr}$ targets while communicating to $K$ users, the bound on the minimum number of beamformers scales linearly in $K$ and in $N_\text{tr}$, assuming interference from sensing can be cancelled. When interference cancellation is not possible, the following exact characterization for the case of $N_\text{tr} = 1$ can be obtained: when $K=0$ or $1$, two beamformers should be used; when $K \ge 2$, exactly $K$ beamformers should be used, i.e., communication beamformers alone are already sufficient.

Figures

Figures reproduced from arXiv: 2507.14982 by the authors.

**Figure 2.** The number of sensing beamformers obtained by simulations with [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Histograms of the number of sensing beamformers obtained by simulation for sensing the parameters of a channel with [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: The number of beamformers obtained from simulations versus [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: The number of sensing beamformers obtained from simulations [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

Discussion (0). Continue with ORCID to comment.

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