{"id":"a8616182-fb63-4bad-9bc6-98e372527c87","arxiv_id":"2501.01053","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A variational and Blahut-Arimoto framework computes the MMSE-rate limit of MIMO ISAC and shows the SISO limit is unattainable except at the two SAC-optimal endpoints.","lead":"This paper derives the fundamental tradeoff between sensing accuracy and communication rate for MIMO integrated sensing and communication systems, using a modified mean-square-error metric and the Shannon rate. It supplies a Blahut-Arimoto-type algorithm and closed-form optimal waveforms, and proves that for a single-antenna channel the tradeoff curve cannot be achieved except at the sensing-only or communication-only endpoints.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence proof for Algorithm 1 does not apply to the implemented update: a sign error in the derived exponent, plus unverified compactness for the infinite-dimensional alternating maximization, leaves the central numerical claim unproven.","rationale":"The reader's weakest assumption (Section III-D channel-estimation error) is a legitimate issue, but it concerns a downstream strategy rather than the central MMSE-Rate characterization. The more load-bearing point is that Algorithm 1, which is the paper's numerical embodiment of the limit, has a convergence proof that does not align with the implemented iterations. The sign mismatch between (C.6) and Algorithm 1 step 13 is directly checkable and, if confirmed, means the proof in Appendix C does not cover the algorithm. The proof's reliance on a single concavity statement plus a citation also leaves the global convergence and epsilon-accuracy claims unsupported. I am not claiming the result is false; the proof may be repairable, and the algorithm may still work in practice. But the central claim includes 'convergence proven in Appendix C', so this is the condition that must hold. A concrete re-derivation and a small-scale convergence check would settle it. Verdict remains conditional: the paper should not be accepted as-is until the proof is corrected or the algorithm is adjusted to match the derivation.","tokens_in":34207,"tokens_out":20182,"duration_ms":192969,"concrete_test":"Re-derive the variational update for (C.2a) from scratch: with Lagrangian L(p_X) = alpha/T integral p_X D dX - (1-alpha) integral p_X Phi_tilde dX + mu1(integral p_X - 1) + mu2(integral Tr(XX^dagger)p_X - NTP0), take the functional derivative with respect to p_X and solve for p_X. Compare the sign of the Phi_tilde term with Algorithm 1 step 13. Independently, for a small finite-alphabet analogue where the global optimum can be found by exhaustive search, run Algorithm 1 both with the sign as printed in (C.6) and with the sign in step 13; if the latter matches the correct derivation and converges to the exhaustive optimum while the former does not, the published convergence proof is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 1 is the paper's route to computing the MMSE-Rate limit (11), and the claimed guarantee is 'rigorously proven' convergence to arbitrary precision. Two concrete problems undermine this. First, the update rule derived in Appendix C, Eq. (C.6), contains exp[+(1/alpha-1)T Phi_tilde(X)] in the p_X exponent, whereas Algorithm 1 step 13 updates with exp[-(1/alpha-1)T Phi_tilde(X)]. Re-solving the first-order condition of (C.2a), the sensing-penalty term enters as -T(1-alpha)/alpha Phi_tilde(X) = -T(1/alpha-1)Phi_tilde(X); the minus sign in Algorithm 1 is the correct one, so the derivation in C does not match the algorithm it is supposed to prove convergent. Second, the convergence argument in Appendix C-B only states that the functional J is concave and cites [24, Thm 9.5]; it does not verify the compactness/continuity conditions needed for alternating maximization over infinite-dimensional distribution spaces, nor does it account for the inner Newton iteration for mu or the required discretization of X. Even if concavity were enough, convergence of J to its supremum does not establish that the returned p_X achieves the MMSE-Rate pair to tolerance epsilon_J; a separate error bound is absent. Because the principal computational claim depends on this proof, the paper's central assertion should not be taken as established until the derivation is corrected or the proof strengthened.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fundamental estimation-information performance limits of MIMO ISAC systems with random dual-functional waveforms. It defines an MMSE-Rate region whose boundary is the MMSE-Rate limit, derives variational conditions for optimal input/output distributions (Theorem 1), proves non-achievability of the limit for fast-fading SISO channels except at the SAC-optimal endpoints (Theorem 2), proposes a Blahut-Arimoto-type algorithm for computing the limit (Algorithm 1), derives closed-form sensing- and communication-optimal waveforms (Theorems 3-6), and proposes a compound signaling strategy for coincided SAC channels (Section III-D). Numerical examples illustrate the MMSE-Rate region, the two tradeoffs (water-filling and waveform uncertainty), and gains of the compound strategy.","tokens_in":34523,"tokens_out":11093,"duration_ms":109751,"significance":"If the results are fully established, the paper would be a substantial contribution: it gives a variational characterization of a genuine ISAC performance limit with random waveforms, closed-form SAC-optimal strategies, an explicit non-achievability result for SISO channels, and a signaling strategy exploiting integrated sensing for channel acquisition. The derivations are largely self-contained, and prior work including [7] is cited with attribution. The paper makes falsifiable numerical predictions and does not disguise fitted parameters as predictions. However, the central computational claim rests on a convergence proof that, as written, does not match the implemented update and lacks the required functional-analytic hypotheses; the compound-signaling rate is evaluated under a perfect-channel-estimate assumption. These issues are load-bearing for the paper's main claims.","major_comments":[{"comment":"The update rule derived in Appendix C, Eq. (C.6), contains exp[+(1/alpha - 1)T Phi_tilde(X)] in the p_X exponent, whereas Algorithm 1 step 13 and the inner Newton update in line 7 use exp[-(1/alpha - 1)T Phi_tilde(X)]. Re-solving the first-order condition of (C.2a) gives the sensing-penalty term as -(1-alpha)Phi, i.e., an exponent of -(1/alpha - 1)T Phi_tilde(X), so the algorithm's sign is the correct one and Eq. (C.6) is not. Since Appendix C is the only proof that Algorithm 1 converges to the MMSE-Rate limit, this inconsistency must be fixed by re-deriving Eq. (C.6) consistently with the implemented iteration.","section":"Appendix C, Eq. (C.6) and Algorithm 1"},{"comment":"The convergence proof establishes only concavity of the functional J-tilde and cites [24, Theorem 9.5]. It does not verify the compactness and continuity conditions needed for alternating maximization over infinite-dimensional distribution spaces, does not account for the effect of the inner Newton iteration for the Lagrange multiplier, and provides no error bound connecting convergence of J to the returned p_X and the claimed (epsilon, R) pair to tolerance epsilon_J. The algorithm is presented for continuous random matrices, yet the numerical implementation necessarily involves discretization, and no discretization-error analysis is supplied. Therefore the statement that Algorithm 1 converges 'rigorously' to the limit 'to any desired level of precision' is not supported.","section":"Appendix C-B"},{"comment":"The compound signaling scheme evaluates the data-phase rate as R_c(X_c) with the coherent capacity expression (41) in which H is replaced by the estimate H-hat. This assumes the estimation phase yields a perfect channel estimate for the data phase. Residual estimation error from the finite-length pilot phase (T' symbols) is not propagated into the rate expression, and the water-filling precoder computed from H-hat is not necessarily optimal for the true H. Thus (53) and the rate improvements shown in Fig. 10 are optimistic upper bounds, not established achievable rates. The paper should either provide an achievable rate that accounts for channel estimation error or explicitly label the compound-signaling comparison as a genie-aided upper bound.","section":"Section III-D, Eq. (53)"},{"comment":"The non-existence proof for alpha in (0,1) rests on the Hermite expansion of log p_y and the claim that the right-hand side of (B.1) cannot be a Gaussian convolution because the Cauchy term is not a Gaussian convolution. As written, this is not fully rigorous: the left-hand side of (B.1) is analytic in x for densities with Gaussian tails, while 1/(sigma_g^{-2}+sigma_s^{-2}x^2) has poles at x = ± i sigma_s/sigma_g, and the role of the additional mu_2 x^2 term is not addressed. Since Theorem 2 is a central theoretical claim, the proof should be stated cleanly, e.g., via Fourier transforms or a rigorous analyticity argument, rather than relying on a cited example that concerns only the pure Cauchy term.","section":"Appendix B, Theorem 2"}],"minor_comments":[{"comment":"In the expression for p_y^*(y,1), the exponent is written with x^2 instead of y^2; this is a typo.","section":"Theorem 2, Eq. (17)"},{"comment":"The text contains an unresolved cross-reference 'Section ??'; the reference to the limit-achieving conditions should be fixed.","section":"Section III-C3"},{"comment":"The notation 'E {Phi [I ⊗ R*_X(alpha)]}' is ambiguous because R*_X(alpha) is a deterministic matrix; the expectation operator should be removed or clarified.","section":"Eq. (44b)"},{"comment":"The legend and axis labels in Fig. 10 are difficult to parse; please clarify which curves correspond to the compound signal components and which to the non-coherent baseline.","section":"Section IV-B3"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely topic and the variational characterization is a promising direction, but the manuscript currently overstates the rigor of its algorithmic convergence proof and the achievability of the compound-signaling rate. These are fixable but require substantial revision of the proofs and of the claims in Sections III-D and IV-B3. The use of [7] is appropriate and the authors are transparent about relying on it for Theorem 5. I would not recommend rejection if the authors can provide the missing convergence analysis or carefully restate the guarantees, and either prove or explicitly upper-bound the compound-signaling rate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere is my read. The paper gives a variational characterization of the MMSE-rate Pareto limit for MIMO ISAC, a Blahut-Arimoto-type algorithm for computing it, a SISO unattainability result, closed-form sensing-optimal waveforms under wide-separation sensing, and a compound signaling idea for G=H. The closed-form sensing-optimal waveform (Theorem 4) and the water-filling / waveform-uncertainty tradeoffs are genuinely useful, and the paper is honest about which parts come from Xiong et al. The SISO negative result is interesting and I largely buy it.\n\nThe soft spots, in order of severity. The biggest is the convergence claim for Algorithm 1. Appendix C's derivation, Eq. (C.6), has exp[+(1/alpha-1)T Phi(X)] in the exponent, while Algorithm 1 updates with exp[-(1/alpha-1)T Phi(X)]. Re-doing the first-order condition from (C.2a) gives the negative sign, so the algorithm implements the correct update but the proof does not prove the algorithm as written. On top of that, the convergence argument stops at concavity plus a citation; it does not verify the compactness/continuity needed for alternating maximization over infinite-dimensional distributions, and it gives no error bound between the discretized p_X and the true optimizer. So \"converges to arbitrary precision\" is an overclaim, even if the method may work numerically.\n\nSecond, the compound signaling section treats the estimated channel as perfect when computing the coherent rate (41) after estimation. Residual estimation error is ignored, so the rate advantage over non-coherent capacity in Fig. 10 is optimistic. It is a modeling assumption, but it should be stated as an upper-bound-style analysis, not as an achieved rate.\n\nThird, the Theorem 2 proof's Hermite/Cauchy step is compressed. It is probably fixable, but the non-existence claim for intermediate alpha deserves a careful rewrite rather than a power-series truncation plus a quote.\n\nI do not see circularity: Theorem 5 is taken from [7] with attribution, and the central variational results are self-contained. The citation pattern is fine.\n\nWho should read this: ISAC theorists who want the closed-form waveforms and the SISO negative result. The numerical algorithm needs testing and repair before I would rely on it. I would accept for peer review, with the convergence proof and compound-signaling assumptions as the main revision targets.","headline":"Serious ISAC limits paper with real new results, but the central convergence proof has a sign mismatch and the compound-signaling gain is optimistic; worth refereeing, not taking at face value.","tokens_in":96,"tokens_out":3127,"would_cite":true,"duration_ms":68921,"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 characterizes the MIMO ISAC performance limit as a Pareto boundary and proves that the SISO boundary is unreachable except at its endpoints.","keywords":["integrated sensing and communication","MIMO","MMSE-rate tradeoff","Pareto optimality","Blahut-Arimoto algorithm","water-filling","waveform uncertainty","channel estimation"],"falsifier":"Run a Monte Carlo simulation of the compound signaling scheme of Section III-D in which the data precoder is built from the actual MMSE estimate $\\hat{\\mathbf{H}}$ obtained from the sensing phase, rather than from the true channel, and compare the achieved rate with $\\breve{R}$ in (53). If the achieved rate lies persistently below $\\breve{R}$ by a gap that does not vanish as $T'$ grows, the perfect-estimate premise is false. Alternatively, for the SISO case, a square-integrable $\\log p_y^\\star(y,\\alpha)$ satisfying (13) for any $\\alpha\\in(0,1)$ would contradict Theorem 2.","tokens_in":34017,"feed_emoji":"📡","tokens_out":11360,"duration_ms":104352,"temperature":0.7,"pith_summary":"The paper aims to pin down the fundamental sensing-communication limit of a MIMO integrated sensing and communication (ISAC) system, where one random waveform has to estimate a sensing channel and transmit data at the same time. It defines the limit as the Pareto boundary of all achievable pairs of a modified MMSE sensing metric and an ergodic coherent rate, and derives variational conditions that any boundary-achieving input and output distribution must satisfy. For a fast-fading SISO channel, it proves that every interior point of this boundary is unattainable, so the limit is a supremum that practical signals can approach but cannot reach, except at the sensing-only and communication-only endpoints. It then contributes a provably convergent alternating-maximization algorithm for computing boundary points, closed-form endpoint waveforms that exhibit a water-filling tradeoff and a waveform-uncertainty tradeoff, and a compound signaling scheme for coincided sensing and communication channels. If these arguments hold, the MMSE-Rate limit becomes a computable benchmark rather than an abstract multi-objective optimization, and system designers know both what is achievable and which purported bounds are loose.","feed_headline":"For SISO, every interior point of ISAC's frontier is unreachable","feed_subtitle":"Variational conditions and a convergent alternating-maximization algorithm pin down the optimal sensing-communication tradeoff.","key_machinery":"The object that carries the argument is the MMSE-Rate limit set (11) together with the variational equation (13) that characterizes its boundary. Equation (13) is the statement that the Gaussian-blurred logarithm of the optimal output density must equal a sensing-dependent term plus Lagrange-multiplier terms; it converts the functional Pareto optimization into a solvability condition for $p_Y^\\star$. The computational machinery is an alternating-maximization algorithm over the input density $p_X$ and an auxiliary backward channel $\\phi$, with a Newton-Raphson update for the Lagrange multiplier that enforces the average power constraint and a proof that the objective is concave, so the iterates converge to the supremum. At the two endpoints the machinery reduces to standard water-filling: sensing optimality forces a deterministic sample correlation matrix and an isometry waveform, while communication optimality forces a Gaussian waveform whose covariance water-fills the channel.","core_discovery":"On its own terms, the paper's central claim is that the MMSE-Rate limit, i.e. the boundary of the set in (11), is characterized by Theorem 1: for each weight $\\alpha\\in(0,1]$, any optimal output distribution $p_Y^\\star(\\mathbf{Y},\\alpha)$ must solve the convolutional equation $\\int p_{\\mathbf{Z}_c}(\\mathbf{Y}-\\mathbf{H}\\mathbf{X})\\log p_Y^\\star(\\mathbf{Y},\\alpha)\\,d\\mathbf{Y} = T(1-1/\\alpha)\\operatorname{Tr}\\!\\big[(\\bar{\\Sigma}_g^{-1}+\\sigma_s^{-2} \\mathbf{I}_{N_s}\\otimes\\mathbf{X}\\mathbf{X}^\\dagger)^{-1}\\big]+\\mu_1(\\alpha)+\\mu_2(\\alpha)\\operatorname{Tr}(\\mathbf{X}\\mathbf{X}^\\dagger)$. For $\\alpha=0$, a deterministic sample correlation matrix is sufficient. Theorem 2 then shows that in the fast-fading SISO case no square-integrable output density solves the equation for $\\alpha\\in(0,1)$, so interior points of the limit curve are not achievable; only the Gaussian communication-optimal endpoint and the binary sensing-optimal endpoint are. Algorithm 1, a constrained alternating-maximization algorithm of Blahut-Arimoto type, is proven to converge to the supremum and is used to plot the boundary numerically. The paper also derives closed-form endpoint waveforms: an isometry waveform with water-filling over the sensing channel statistics at the sensing-optimal point, and a Gaussian waveform with water-filling over the communication channel realization at the communication-optimal point. For coincided channels it proposes a compound signal that first sends the sensing-optimal waveform for channel estimation and then the communication-optimal waveform for data, claiming a rate improvement over non-coherent capacity.","pith_inferences":["Inference: The SISO unattainability result hints that for general MIMO the boundary may also be an unattainable supremum; if that is proved, the design target shifts from operating on the frontier to approaching the frontier from inside the region.","Inference: The perfect-estimate simplification could be relaxed by modelling $\\hat{\\mathbf{H}}$ as a noisy version of $\\mathbf{H}$; re-deriving the data-phase rate with the induced estimation error would likely reduce the claimed gain over non-coherent capacity at small $T'$.","Inference: The same variational-plus-alternating-maximization template applies to other sensing metrics that the paper mentions, such as the modified Cramér-Rao bound or the estimation rate, yielding analogous Pareto frontiers.","Inference: The numerical finding that $\\breve{R}_c$ saturates once $T'=N$ suggests a simple design rule for pilot length in this scheme: use $N$ symbols to estimate an $N$-column channel, because additional pilots cost rate without improving the data-phase rate."],"forward_implications":["Any candidate ISAC waveform can be checked against (13): if its output density fails the variational condition, that waveform cannot lie on the MMSE-Rate limit for that weight $\\alpha$.","For fast-fading SISO channels, interior points of the limit curve are a supremum, not achievable operating points; a practical SISO ISAC system must either time-share between the sensing-optimal and communication-optimal endpoints or operate strictly inside the region.","Algorithm 1 supplies a numerical benchmark that converges with a proven guarantee, so comparisons against the true limit no longer have to rely on loose Gaussian/isometry outer and inner bounds.","The closed-form endpoint waveforms make the two tradeoffs explicit: power is divided between sensing water-filling and communication water-filling, and randomness is divided between a predictable isometry signal and an entropy-maximizing Gaussian signal.","For coincided channels, the compound scheme implies that investing a short sensing-optimal pilot phase can raise the achievable rate above the non-coherent capacity, quantifying the integration gain of ISAC at finite coherence time."],"supporting_citations":[{"why":"Supplies the baseline random-signal ISAC framework and the SAC-optimal approximations that the paper tightens and partly rectifies.","marker":"[7]"},{"why":"Provides the alternating-maximization capacity computation that Algorithm 1 extends to the joint MMSE-Rate objective.","marker":"[12, 13]"},{"why":"Gives the MIMO capacity water-filling result on which the communication-optimal waveform of Theorem 6 is built.","marker":"[18, 38]"},{"why":"Supplies the non-coherent capacity and Grassmann-manifold rate analysis used for the sensing-limited rate and the compound-signal comparison.","marker":"[19]"},{"why":"Provides the Bayesian MMSE estimator and conditional-MSE formulas used to define the modified MMSE sensing metric.","marker":"[21]"},{"why":"Earlier CRB-rate tradeoff analyses that neglect waveform randomness, against which the random-waveform limit is contrasted.","marker":"[5, 6]"},{"why":"Justifies the sample average approximation used to average the per-realization optimizations over the communication channel distribution.","marker":"[23]"},{"why":"Supplies the alternating-optimization convergence result used in the proof that Algorithm 1 converges to the MMSE-Rate limit.","marker":"[24]"}],"fun_headline_variants":["SISO ISAC: interior points of MMSE-Rate frontier unattainable","Only endpoints achievable in fast-fading SISO ISAC tradeoff","Alternating-max algorithm pinpoints ISAC sensing-communication limit","MIMO ISAC: variational conditions nail estimation-information frontier","Theorem: SISO ISAC interior points unreachable, only extremes hold"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The compound-signaling rate calculation in Section III-D uses the channel estimate $\\hat{\\mathbf{H}}$ as if it were the true channel when evaluating the coherent data rate, so residual estimation error and its effect on the water-filling precoder are not modelled.","fun_headline_variants_meta":{"raw":{"variants":["SISO ISAC: interior points of MMSE-Rate frontier unattainable","Only endpoints achievable in fast-fading SISO ISAC tradeoff","Alternating-max algorithm pinpoints ISAC sensing-communication limit","MIMO ISAC: variational conditions nail estimation-information frontier","Theorem: SISO ISAC interior points unreachable, only extremes hold"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1325,"prompt_tokens":919,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":312}},"tokens_in":535,"tokens_out":406,"duration_ms":4126,"temperature":1.0,"reasoning_tokens":312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:36:49.737934+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a Monte Carlo simulation of the compound signaling scheme of Section III-D in which the data precoder is built from the actual MMSE estimate $\\hat{\\mathbf{H}}$ obtained from the sensing phase, rather than from the true channel, and compare the achieved rate with $\\breve{R}$ in (53). If the achieved rate lies persistently below $\\breve{R}$ by a gap that does not vanish as $T'$ grows, the perfect-estimate premise is false. Alternatively, for the SISO case, a square-integrable $\\log p_y^\\star(y,\\alpha)$ satisfying (13) for any $\\alpha\\in(0,1)$ would contradict Theorem 2.","supporting_citations":[{"cited_title":"On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline random-signal ISAC framework and the SAC-optimal approximations that the paper tightens and partly rectifies."},{"cited_title":"Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel,","cited_arxiv_id":null,"evidence_quote":"Supplies the non-coherent capacity and Grassmann-manifold rate analysis used for the sensing-limited rate and the compound-signal comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Bayesian MMSE estimator and conditional-MSE formulas used to define the modified MMSE sensing metric."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the alternating-optimization convergence result used in the proof that Algorithm 1 converges to the MMSE-Rate limit."}],"review_version":1}