{"id":"b65f3155-8705-4502-960f-9e3af1c4367b","arxiv_id":"2507.14018","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A distortion-aware hybrid beamforming algorithm for integrated sensing and communication maximizes weighted rate under nonlinear power amplifier distortion.","lead":"A new beamforming design for base stations that both communicate and sense now accounts for distortion from nonlinear power amplifiers. In simulations the design improves combined communication and sensing rate, with the largest gains under strong amplifier nonlinearity.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The hybrid decomposition (P2) is not distortion-aware: a full-digital F is optimized under the Bussgang model, then projected to F_A F_D by Frobenius minimization, which does not preserve B, C_e, SINRs, or the power constraint, so Fig. 2 may overstate the achievable hybrid gains.","rationale":"I focused on the transition from the full-digital solution to the hybrid solution, because the abstract and title promise a hybrid beamforming contribution. The reader's Gaussian-input concern is legitimate and should be fixed by stating s ~ CN(0,I), but it affects the model's domain of validity rather than the internal comparison: if the simulations use Gaussian symbols, the Bussgang expressions are exact, and the central comparison stands within the model. The decomposition concern is more decisive for the specific numerical claim: the objective and constraints are nonlinear functions of the actual F_A F_D, while P2 minimizes a linear Euclidean distance, so there is no theoretical guarantee that the hybrid precoder inherits the optimized distortion trade-off. With only two streams and sixteen subarrays, the 4x2 blocks of F_A F_D are rank-one, so exact decomposition is impossible in general. The paper omits the decomposition details, the error, and the evaluation point for Fig. 2; without those, the reported gain cannot be attributed to the hybrid design. This does not mean the paper is wrong, but it makes the central claim conditional on a check the manuscript does not provide. I therefore keep the conditional verdict and recommend the decomposition check as the condition.","tokens_in":8275,"tokens_out":22199,"duration_ms":286365,"concrete_test":"Reproduce the P2 decomposition for the Fig. 2 channel realizations (N_t=64, N_RF=16, K=2) using the closed-form AO of [10]; compute the normalized error ||F - F_A F_D||_F^2 / ||F||_F^2 and, more importantly, re-evaluate the weighted sum G(F_A F_D) by recomputing B and C_e from Eqs. (6)-(7) at F_A F_D, renormalizing to E||phi(F_A F_D s)||^2 = P_tot, and comparing with G(F) and with the MRT/ZF/RBF curves. If the gap between the proposed and baseline curves shrinks materially or the ordering changes, the central claim is not established for the actual hybrid precoder.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is for a hybrid beamformer, but the optimization that produces the reported gains is done on a full-digital matrix F: P1 maximizes G1(F) using the Bussgang expressions (6), (7), and the SINRs (9), (11). The hybrid matrices are then obtained by solving P2, min ||F - F_A F_D||_F^2, with only the constant-modulus block-diagonal constraint (14b). Because the PA distortion model is nonlinear in the actual precoder, the projected matrix F_hyb = F_A F_D has a different B, a different C_e, different SINRs, and generally does not satisfy the nonlinear power budget (14c). Moreover, for the partially-connected array in Fig. 1, each block of F_hyb is a rank-1 outer product of a 4x1 unit-modulus vector and a 1xK row, so the decomposition of a general 64x2 optimal F is not exact. The paper gives no decomposition error, no recomputation of G at F_hyb, and no indication whether Fig. 2 uses F or F_hyb at evaluation. Thus the headline 'hybrid beamforming improves ISAC performance' is not yet supported: the numerical gain could be a property of the full-digital surrogate, not of the feasible hybrid precoder.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a partially-connected hybrid beamforming transmitter for an ISAC system in which the power amplifiers are nonlinear. Using a third-order polynomial PA model and Bussgang's theorem, the authors derive a distortion-aware weighted sum-rate objective comprising communication rates and sensing mutual information, subject to a total power constraint. The optimization problem is nonconvex, so the paper proposes a two-stage solution: first, a full-digital beamforming matrix F is optimized via a penalty method combined with manifold optimization and closed-form updates for two auxiliary variables U and V; second, F is decomposed into analog and digital matrices by Frobenius-norm minimization. Numerical results compare the proposed algorithm with MRT, ZF, and RBF precoders and show gains that grow with PA nonlinearity.","tokens_in":8557,"tokens_out":6719,"duration_ms":78929,"significance":"If the reported gains are realized by the actual hybrid beamformer, the work addresses a relevant gap in the ISAC literature, where PA distortion is usually ignored. The paper contains useful derivations, including closed-form updates for the auxiliary variables and gradient expressions, and it uses PA coefficients extracted from a measured amplifier model, which is a strength. However, the significance is conditional on two load-bearing points: the Bussgang model requires Gaussian signaling to be exact, and the hybrid decomposition is not shown to preserve the performance of the optimized full-digital solution. The manuscript does not currently establish that the headline 'distortion-aware hybrid beamforming' gains are achievable by the feasible hybrid precoder, because the numerical evaluation does not state whether the full-digital or hybrid matrices are used. These issues need to be resolved before the contribution can be fully assessed.","major_comments":[{"comment":"The Bussgang decomposition in Eq. (5), with B given by Eq. (6) and the distortion covariance C_e given by Eq. (7), is exact only when the input signal x is Gaussian. The manuscript only states E{ss^H}=I in Section II-B and does not restrict the transmitted symbols to be Gaussian. For non-Gaussian signaling such as QPSK, the expressions for C_e and for the SINRs in Eqs. (9) and (11) are not exact, and the rate expression R_k = log2(1+gamma_k) treats the distortion e as additive Gaussian noise without justification. Please state the Gaussian assumption explicitly and discuss its practical relevance, or provide numerical evidence that the design maintains its performance for constellations used in the intended system.","section":"Section II-B, Eqs. (5)-(7)"},{"comment":"The central claim concerns hybrid beamforming, but the optimization in P1 is performed on the full-digital matrix F, and the hybrid matrices are obtained by solving the Frobenius-norm decomposition problem P2. This decomposition does not preserve the distortion terms B(F) and C_e(F), the SINRs, or the nonlinear power constraint in Eq. (14c). The manuscript does not state whether the numerical results in Fig. 2 evaluate the sum rate at the full-digital F or at the feasible hybrid precoder F_A F_D. If the evaluation uses the full-digital matrix, the reported gains are for a surrogate problem and do not support the hybrid claim; if it uses the hybrid matrices, the decomposition error and the recomputed objective at F_A F_D must be reported. Because each block of F_A is a rank-1 outer product in the partially-connected structure, the decomposition is generally not exact. Please clarify this point and, ideally, report the performance of the final hybrid precoder, including the decomposition error and the constraint violation for (14c).","section":"Section III, P1/P2 and Section IV, Fig. 2"},{"comment":"The retraction formula in Eq. (23) defines c1 as the quotient [Ptot - 4Re{beta_1^* beta_3} Tr(U) - 6|beta_3|^2 Tr(V)] / |beta_1|^2 and states that c1 is the square of the desired Frobenius norm. Nothing in the algorithm ensures that c1 is nonnegative. If the auxiliary variables U and V grow large during the alternating optimization, c1 can become negative, in which case the retraction is undefined and the manifold update breaks. The paper should specify safeguards or prove that the penalty parameters and initialization guarantee c1 > 0, or the algorithm cannot be considered robust.","section":"Section III, Eq. (23)"}],"minor_comments":[{"comment":"There is a typo in the sentence 'the signal vector at the input of the the PAs stage'; 'the the' should be 'the PA'.","section":"Section II-B"},{"comment":"The derivation of the power constraint in Eq. (17b) from the original constraint in Eq. (14c) is not shown. Since the constraint involves the auxiliary variables U and V, a short derivation or a reference would improve readability.","section":"Section III, P3 reformulation"},{"comment":"The statement that 'the convergence of this algorithm is guaranteed' because the MO method resembles the conjugate gradient method is not a proof. The objective is nonconvex and the penalty/AO framework has no convergence guarantee; please either provide a rigorous convergence argument or soften the claim.","section":"Section IV, Fig. 3 text"},{"comment":"The beam pattern figure is only for a single user and a single target angle. It would be helpful to state how the pattern is computed, especially whether the nonlinear pattern includes the hybrid precoder or the full-digital precoder, since this affects the interpretation of the 42 dB distortion reduction.","section":"Section IV, Fig. 4"},{"comment":"The decomposition algorithm is attributed to [10], but the manuscript gives no details on initialization, number of iterations, or stopping criteria. For reproducibility, these details should be added.","section":"Section III, P2"}],"recommendation":"major_revision","confidential_remarks":"The most important issue for the revision is the ambiguity in the numerical evaluation: the paper must report the performance of the feasible hybrid precoder, not just the full-digital surrogate. If the hybrid decomposition causes a significant loss, the central claim may need to be reframed as a full-digital distortion-aware design. The Gaussian assumption is also essential to the validity of the Bussgang model and should be addressed. I recommend a major revision to resolve these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"New combination, real gap. The paper is the first in its reference set to put a Bussgang PA distortion model into a partially-connected hybrid ISAC optimization, and the alternating optimization machinery is worked out in detail. The closed-form updates for U and V are internally consistent, and the PA parameters are taken from a measured model rather than invented. That part deserves credit.\n\nThe trouble is that the actual hybrid beamformer is an afterthought. P1 optimizes a full-digital F against the distortion-aware objective and power constraint. P2 then finds FA and FD by minimizing Frobenius distance, ignoring the Bussgang distortion terms entirely. The projection does not preserve B, Ce, the SINRs, or the nonlinear power budget. For a partially-connected array the decomposition is generally inexact. The paper doesn't report the decomposition error, doesn't say whether Fig. 2 evaluates F or F_hyb, and doesn't recompute the objective at the hybrid solution. So the headline 'hybrid beamforming improves ISAC performance' is not yet supported—the gains in Fig. 2 may be coming from the full-digital matrix, not from a feasible hybrid precoder.\n\nOther soft spots are smaller. The Bussgang decomposition requires Gaussian symbols, stated nowhere. Convergence is asserted by analogy to CG, which is not a proof. No code, no error bars, and no distortion-aware baseline such as a conventional precoder with digital predistortion. These are fixable.\n\nNet: the idea is a solid direction and the derivations are mostly clean, but the central experiment doesn't validate the hybrid claim as written. A revision that either optimizes the hybrid structure directly or at least quantifies and accounts for the decomposition loss would make this a genuinely useful paper. I'd send it to review, because the combination is new and the flaws are concrete enough for a reviewer to pin down. I wouldn't cite it as-is.","headline":"New combination of Bussgang distortion with hybrid ISAC, but the reported gains may come from a full-digital surrogate, not the feasible hybrid precoder.","tokens_in":9107,"tokens_out":2123,"would_cite":false,"duration_ms":24741,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a distortion-aware hybrid beamforming algorithm for ISAC transmitters with nonlinear power amplifiers, and shows it achieves a higher weighted sum of communication rate and sensing mutual information than MRT, ZF, and…","keywords":["integrated sensing and communication","hybrid beamforming","power amplifier nonlinearity","Bussgang decomposition","manifold optimization","alternating optimization","sensing mutual information","partially-connected arrays"],"falsifier":"Measure the distortion covariance at the output of a real or simulated nonlinear PA with non-Gaussian symbols, such as 16-QAM, and compare it with Eq. (7); if the observed covariance deviates materially from $2|\\beta_3|^2 \\mathbf{F}\\mathbf{F}^H \\odot |\\mathbf{F}\\mathbf{F}^H|^2$, the optimized weights are built on a model that does not describe the actual distortion, and the reported gains may not survive outside the Gaussian-symbol assumption.","tokens_in":8070,"feed_emoji":"📡","tokens_out":5981,"duration_ms":62087,"temperature":0.7,"pith_summary":"Integrated sensing and communication (ISAC) base stations normally design hybrid beamforming as if the power amplifiers were perfectly linear. This paper argues that the nonlinear distortion from real PAs should be part of the design objective, and proposes an algorithm that maximizes a weighted sum of per-user communication rate and sensing mutual information under a total-power constraint. The transmitter uses a partially-connected hybrid array, with one RF chain per subarray, so the sensing target is illuminated by the same communication waveforms. Numerical benchmarks show the resulting distortion-aware precoder achieves a higher ergodic sum rate than MRT, ZF, and RBF precoders, and the advantage widens as the amplifier nonlinearity strengthens and as SNR rises. If true, this gives a practical route to recovering ISAC performance that linear-only designs throw away when amplifiers are driven near saturation.","feed_headline":"Distortion-aware ISAC beamforming tops MRT, ZF, RBF","feed_subtitle":"Accounting for PA nonlinearity in beamforming lifts the combined communication and sensing rate.","key_machinery":"The engine of the method is the Bussgang decomposition of the nonlinear PA output, $\\varphi(\\mathbf{x}) = \\mathbf{B}\\mathbf{x} + \\mathbf{e}$, which turns the distorted channel into an effective linear channel $\\mathbf{B}\\mathbf{F}$ plus a distortion covariance $\\mathbf{C}_e = 2|\\beta_3|^2 \\mathbf{F}\\mathbf{F}^H \\odot |\\mathbf{F}\\mathbf{F}^H|^2$ that enters the SINR denominators. On top of that, the constrained optimization is recast with a penalty method into a problem over three coupled variables, and the full-digital update is run on a Riemannian manifold of matrices with fixed Frobenius norm, using the retraction $\\operatorname{retr}_F(tL) = \\sqrt{c_1}(F + tL)/\\|F + tL\\|_F$ to stay inside the power budget. The closed-form updates for the auxiliary variables are what make the alternating loop cheap, and the final hybrid decomposition converts the optimized full-digital precoder back into the partially-connected analog/digital structure.","core_discovery":"The central claim is that a hybrid beamformer designed with explicit knowledge of PA nonlinearity can substantially outperform conventional linear precoders in a monostatic ISAC system. The paper models nonlinear amplification with a third-order polynomial and applies Bussgang's theorem to split the output into a linear signal term plus an uncorrelated distortion term, yielding closed-form signal-to-interference-noise-and-distortion ratios for each user and for the sensing receiver. It then solves the non-convex weighted-sum problem by first finding a full digital matrix $\\mathbf{F}$ through alternating optimization: a manifold-optimization step over the constant-Frobenius-norm manifold updates $\\mathbf{F}$, and closed-form KKT solutions update two penalty auxiliary variables $\\mathbf{U}$ and $\\mathbf{V}$. The full digital solution is decomposed into constant-modulus analog weights and low-dimensional digital weights by a second alternating minimization. Numerical results show the known-PA-model version of the algorithm beats MRT, ZF, and RBF over the simulated range, with the gain growing with distortion strength and SNR.","pith_inferences":["A natural extension the paper does not run is a symbol-level simulation with QAM or PSK inputs: the Bussgang-based SINR expressions are exact for Gaussian symbols, and the achievable-rate gains could differ for finite constellations. One could test whether the distortion-aware precoder still dominates MRT and ZF under such inputs.","The same Bussgang-plus-manifold recipe could be adapted to other PA models, such as memory polynomials or soft-limiter models, since only the expressions for $\\mathbf{B}$ and $\\mathbf{C}_e$ change; the optimization machinery would survive as long as the distortion covariance has a tractable form.","The beam-pattern notch at the target direction suggests a distortion-aware design could deliberately steer nonlinear distortion away from sensitive directions, which the paper does not formulate as an explicit constraint but which follows from the objective."],"forward_implications":["If the claim holds, practical ISAC transmitters can reclaim a large part of the rate loss caused by driving power amplifiers near saturation, without changing the array hardware.","Because the gain widens with SNR, distortion-aware design matters most exactly in the high-SNR regime where ISAC links are expected to operate.","The 42 dB suppression of nonlinear power at the user direction shown in the beam-pattern experiment means distortion-aware weights can shape not only the signal but also the distortion field, which is useful for interference-limited operation.","The algorithm's decomposition stage means the improvement is available to partially-connected hybrid arrays with only a modest number of RF chains, not just to fully digital transmitters."],"supporting_citations":[{"why":"Supplies the Bussgang-based nonlinear PA distortion model, including the distortion covariance $\\mathbf{C}_e$ and the distortion-aware SINR expressions that the optimization builds on.","marker":"[7]"},{"why":"Gives the measured third-order polynomial PA parameters $\\beta_1$ and $\\beta_3$ used in the numerical evaluation.","marker":"[11]"},{"why":"Provides the alternating-minimization hybrid decomposition used to split the full-digital solution into analog and digital beamforming matrices.","marker":"[10]"},{"why":"Defines the partially-connected hybrid-beamforming ISAC setting that this paper extends to nonlinear amplification.","marker":"[5]"},{"why":"Supplies the conditional sensing mutual-information metric adopted as the sensing performance measure.","marker":"[4]"}],"fun_headline_variants":["PA-aware hybrid beamforming outperforms linear precoders in ISAC","Accounting for PA distortion in beamforming improves ISAC performance","Distortion-aware hybrid precoding beats MRT, ZF, and RBF for ISAC","Modeling amplifier nonlinearity boosts hybrid beamforming for ISAC"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole distortion model is exact only for Gaussian transmit symbols; if the actual constellation is non-Gaussian, the distortion covariance and the SINR used for optimization no longer match the true transmitter behavior.","fun_headline_variants_meta":{"raw":{"variants":["PA-aware hybrid beamforming outperforms linear precoders in ISAC","Accounting for PA distortion in beamforming improves ISAC performance","Distortion-aware hybrid precoding beats MRT, ZF, and RBF for ISAC","Modeling amplifier nonlinearity boosts hybrid beamforming for ISAC"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000365,"raw_usage":{"total_tokens":1916,"prompt_tokens":849,"completion_tokens":1067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":989}},"tokens_in":465,"tokens_out":1067,"duration_ms":11863,"temperature":1.0,"reasoning_tokens":989,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:12:19.385979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the distortion covariance at the output of a real or simulated nonlinear PA with non-Gaussian symbols, such as 16-QAM, and compare it with Eq. (7); if the observed covariance deviates materially from $2|\\beta_3|^2 \\mathbf{F}\\mathbf{F}^H \\odot |\\mathbf{F}\\mathbf{F}^H|^2$, the optimized weights are built on a model that does not describe the actual distortion, and the reported gains may not survive outside the Gaussian-symbol assumption.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Bussgang-based nonlinear PA distortion model, including the distortion covariance $\\mathbf{C}_e$ and the distortion-aware SINR expressions that the optimization builds on."},{"cited_title":"Brihuega, L","cited_arxiv_id":null,"evidence_quote":"Gives the measured third-order polynomial PA parameters $\\beta_1$ and $\\beta_3$ used in the numerical evaluation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the alternating-minimization hybrid decomposition used to split the full-digital solution into analog and digital beamforming matrices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the partially-connected hybrid-beamforming ISAC setting that this paper extends to nonlinear amplification."}],"review_version":1}