{"id":"83487114-f4d9-44f3-b86a-445e43462ff8","arxiv_id":"2501.15652","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For open-loop joint communication and sensing with a Markov state, the rate-distortion tradeoff is characterized by a limiting mutual information formula, and the paper derives bounds for beam-switching and multi-beam strategies.","lead":"This paper analyzes a joint communication and sensing system where the state to be sensed changes over time as a Markov process, and it characterizes the tradeoff between communication rate and sensing accuracy under an open-loop transmission strategy. It applies the general result to a beam-pointing example and compares two beam control strategies: switching between sensing and communication, or using multiple beams at once.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's lower bound presumes the receiver knows the beam schedule Γ^n; the model's decoder h(Y^n,S^n) does not, so the claimed (1−λ)I(X;Y) achievability is not supported.","rationale":"The reader's weakest assumption identifies the same load-bearing concern, and I agree with that diagnosis. Theorem 1 is explicitly acknowledged in the paper as identical to prior work [26], so the paper's incremental value lies in the beam-pointing specializations. Between the two, Theorem 3's multi-beam setting uses a deterministic γ0 that is naturally known to all parties and is less exposed. Theorem 2's randomized beam schedule is where the model mismatch bites: the achievability proof treats Γ^n as a known erasure pattern, but the decoder is defined without Γ^n. The binary counterexample shows this is not a cosmetic gap; it changes whether the claimed lower bound is achievable. This is an addressable modeling/assumption issue rather than a fundamental flaw in the rate-distortion framework, so the existing CONDITIONAL verdict remains appropriate. The concrete test would settle the matter by exhibiting a parameter regime where the stated lower bound exceeds the actual channel capacity.","tokens_in":18776,"tokens_out":16216,"duration_ms":153886,"concrete_test":"Instantiate the beam-switching model of Theorem 2 with X∈{0,1}, PY|X,Γ=∞ = δ_{Y=X}, PY|X,Γ=1 = δ_{Y=0}, λ=1/2, and a state S independent of Γ. Use the decoder h(y^n,s^n)=h(y^n), since S provides no information about Γ or X from Y. Compute the capacity of the resulting memoryless channel: C* = max_{p∈[0,1]} [h_2(p(1−λ)) − p h_2(1−λ)] ≈ 0.322 bits. Compare with the claimed lower bound max_{P_X}(1−λ)I(X;Y) = 0.5 bits. Since C* < 0.5, the achievability side of (28) fails without decoder knowledge of Γ^n; if the decoder is instead given Γ^n, the erasure-channel capacity 0.5 bits is achieved, confirming the missing premise.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's distinct new contribution is the beam-switching analysis. Theorem 2 (eq. (28)) asserts that (1−λ)I(X;Y) is an achievable lower bound. The achievability proof in Appendix E models Γ^n as a symbol erasure and decodes by joint typicality of (X^n,Y^n). But the decoder in Section III is defined as h: Y^n × S^n → M; it has no access to Γ^n, and the paper never states that the receiver is told the schedule. Under γ_i=1 (prob. λ) the communication symbol is erased; under γ_i=σ it is transmitted. If Γ^n is unknown to the receiver, this is a memoryless channel with an unobserved state, whose capacity is generally strictly smaller than the erasure-channel capacity (1−λ)I(X;Y). A binary noiseless instance shows the gap: X∈{0,1}, Y=X in the communicating state, Y=0 in the sensing state, λ=1/2. The claimed lower bound is 0.5 bits/symbol, while the actual capacity of the unknown-state channel is max_{p∈[0,1]} [h_2(p(1−λ)) − p h_2(1−λ)] ≈ 0.322 bits/symbol. Thus the lower bound in (28) is not proven under the stated model; it holds only if the receiver knows Γ^n. This is an unstated premise, not an error in Theorem 1, but it affects the paper's advertised beam-switching result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a joint communication and sensing system in which the state evolves as a first-order Markov chain. A transmitter sends codewords over a memoryless state-dependent channel while also producing causal state estimates from channel measurements. Lemma 1 identifies the optimal causal estimator as the per-letter Bayesian estimator, and Theorem 1 states that the open-loop capacity-distortion trade-off is the limit of a maximized sum of conditional mutual informations under a distortion constraint. The paper then specializes the model to a beam-pointing problem with a linear Gauss-Markov target: Theorem 2 gives inner and outer bounds for a beam-switching strategy, Theorem 3 gives a formula for a multi-beam strategy, and numerical comparisons are provided. The authors note in the introduction that Lemma 1 and Theorem 1 coincide with results in [26].","tokens_in":28,"tokens_out":18293,"duration_ms":236746,"significance":"The paper's main conceptual contribution is the extension of the rate-distortion JCAS framework from i.i.d. states to Markov states, and the authors correctly emphasize that single-letter formulas are not generally available. The use of the intermittent Kalman filter to convert a distortion constraint into a constraint on the sensing probability lambda, and the comparison between beam-switching and multi-beam strategies, are useful and clearly presented. The numerical section illustrates a meaningful qualitative difference between stable and unstable targets. The authors are transparent about the overlap with [26]. However, the distinctive beam-switching result is currently not proven under the stated model because the receiver is not given the beam-schedule sequence, and the proof of the general capacity formula relies on unstated finite-alphabet and information-stability restrictions.","major_comments":[{"comment":"Section III defines the decoder as h: Y^n x S^n -> M, so the receiver is not given the beam-schedule sequence Gamma^n. The achievability proof of Theorem 2 in Appendix E nevertheless models Gamma_i as a symbol erasure and decodes by joint typicality of (X^n,Y^n), which is valid only when the decoder knows which positions were erased. If Gamma^n is unknown to the receiver, the channel from X to Y is a channel with an unobserved state and its capacity is generally strictly smaller than (1-lambda)I(X;Y). For example, for a noiseless binary channel with Y=X in the communicating state and Y=0 in the sensing state and lambda=1/2, the claimed lower bound (1-lambda)I(X;Y)=1/2 bit per symbol, whereas the capacity of the averaged channel is max_p [h_2(p(1-lambda))-p h_2(1-lambda)] approximately 0.322 bits per symbol. The model should be amended to state that Gamma^n is known to the receiver (in which case a simple time-sharing argument gives the claimed lower bound), or the achievable rate should be computed for the true unknown-state channel.","section":"Section III, Appendix E, Theorem 2"},{"comment":"The converse chain contains a reversed inequality. From I(X_i,Gamma_i;Y_i|S_i) = I(Gamma_i;Y_i|S_i) + I(X_i;Y_i|S_i,Gamma_i) it follows that I(X_i,Gamma_i;Y_i|S_i) >= I(X_i;Y_i|S_i,Gamma_i), not the <= displayed in (83). Thus the displayed argument does not establish the upper bound in Theorem 2. A correct converse would need to upper-bound R directly in terms of I(X_i;Y_i|S_i,Gamma_i), typically by starting from nR <= I(W;Y^n|S^n,Gamma^n), which again requires Gamma^n to be available at the decoder. The same inequality appears in Appendix F but is harmless there because Gamma is constant.","section":"Appendix E, eqs. (82)-(86)"},{"comment":"The achievability proof for Theorem 1 fixes an i.i.d. product distribution P_{X^n} = prod p_X(x_i) and additionally assumes |Y| < infinity and an information-stability/concentration condition. The theorem statement, however, maximizes over arbitrary P_{X^n} in the cost-constrained set, and the numerical beam-pointing specialization uses Gaussian outputs (Section V.D). No argument is given that the optimum over block distributions is attained by an i.i.d. codebook, nor that the finite-alphabet information-stability argument extends to continuous alphabets and Markov states with unbounded log-likelihood ratios. Since Appendices E and F rely on the achievability analysis of Appendix B, this gap affects Theorems 2 and 3 as well.","section":"Appendix B, Theorem 1"}],"minor_comments":[{"comment":"The indicator in (75) appears to use x_i where gamma_i is intended: the expression 1{x_i in X_s} should presumably be 1{gamma_i is in the sensing mode}, and the set X_s is never defined.","section":"Appendix C, eq. (75)"},{"comment":"The notation P_Lambda_S(D) and P_Lambda_V(D) is defined as a set but the dependence on D is dropped in the surrounding text; writing the argument explicitly and denoting the sets as functions of D would improve readability.","section":"Section V.B, eq. (27)"},{"comment":"The formal encoding functions are defined as f_i: M x Z^{i-1} -> X, while Assumption 1 makes the beam-pointing channel input the pair (X_i,Gamma_i); the model should state explicitly whether Gamma_i is a second encoder output, an exogenous random variable, or a known schedule, since the proofs of Theorems 2 and 3 treat it differently.","section":"Section III, Assumption 1"},{"comment":"The symbols C_mb and C_bs are used for the numerical rate expressions while C_mb(D) and C_bs(D) denote the capacity-distortion functions; using distinct names or a sentence clarifying the distinction would avoid confusion.","section":"Section V.D, eqs. (33)-(34)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the overlap with [26] is disclosed in the introduction, so I do not see a novelty issue. The paper's distinct contribution is the beam-pointing specialization, and that is exactly where the proof gaps are concentrated. The missing receiver knowledge of Gamma^n is easy to fix by an explicit assumption, and the reversed inequality in the converse of Theorem 2 plus the gap between i.i.d. codebooks and block-optimal inputs in Theorem 1 require careful rewriting. The numerical conclusions may survive after the proofs are corrected, but the current manuscript should not be accepted as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the genuinely new part is the beam-pointing analysis, and it has a load-bearing gap. Theorem 2's lower bound assumes the receiver knows the beam schedule Γ^n, which the model's decoder h(Y^n,S^n) does not get; and the converse uses I(XΓ;Y|S) ≤ I(X;Y|S,Γ), which is backwards, because conditioning on Γ can only reduce mutual information. So the proof of the paper's advertised beam-switching tradeoff is not sound as written.\n\nWhat is good: the paper is honest. It explicitly says Lemma 1 and Theorem 1 are identical to [26]. The multi-beam result in Theorem 3 looks correct, and the numerical comparison of beam-switching versus multi-beam gives useful design intuition. The Kalman/covariance bounds borrowed from Sinopoli are applied cleanly, and Lemma 3's sandwiching argument is reasonable.\n\nSoft spots, in proportion: the achievability proof of Theorem 2 treats Γ^n as a symbol erasure and decodes by joint typicality of X^n and Y^n, but an erasure pattern unknown to the decoder is not an erasure channel with side information. The capacity of the mixture can be strictly smaller than (1−λ)I(X;Y), as a simple binary example shows. The theorem may be repairable by giving the decoder Γ^n in the model, or by proving a different lower bound, but as stated the lower bound is not established. The converse step in (83) should be reversed; a valid outer bound might follow from a genie argument that reveals Γ to the decoder, but that argument is not written. Also, the finite-alphabet proof is applied to Gaussian channels without a discretization or continuous-alphabet argument; likely fixable, but absent. Minor: the estimator indexing in the code definition is off by one relative to Lemma 1, and the paper never states whether the decoder knows the beam pattern, which is exactly the issue that matters.\n\nWho this is for: information theorists working on JCAS and 6G beam management. The plots and the multi-beam specialization are worth having, but the main beam-switching theorem is not yet trustworthy.\n\nRecommendation: I would send this to review because the topic is timely and the general formula, though not new, plus the multi-beam specialization, deserve referee time. But the referee should be asked to focus on Theorem 2: either fix the decoder model, fix the converse inequality, or qualify the result. As it stands, it needs major revision before acceptance.","headline":"The beam-switching capacity bounds are the real contribution, but Theorem 2 rests on an unstated decoder-knows-Γ assumption and a backwards inequality, so the advertised bounds need major repair before they can be trusted.","tokens_in":19608,"tokens_out":6516,"would_cite":false,"duration_ms":63357,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A15","94A24","93E11","60J05","94A34"],"pacs":[],"model":"deepseek-v4-flash","headline":"Open-loop joint communication and sensing with a Markov state has a capacity-distortion limit given by a maximized sum of conditional mutual informations, with Bayesian filtering as the optimal estimator.","keywords":["joint communication and sensing","rate-distortion theory","capacity-distortion tradeoff","Markov state","Bayesian filtering","Kalman filtering","beam switching","multi-beam"],"falsifier":"A direct falsifier is to implement the beam-switching scheme on the simulated Gauss-Markov target with $A=-1.15$, set $\\lambda$ below the critical sensing probability predicted by the intermittent Kalman filter analysis, and check whether the sample mean-square error diverges; if distortion stays bounded, the inner bound's distortion guarantee fails.","tokens_in":18552,"feed_emoji":"📡","tokens_out":7085,"duration_ms":62224,"temperature":0.7,"pith_summary":"This paper extends the rate-distortion view of joint communication and sensing (JCAS) to settings where the estimated state is not static or i.i.d. but evolves as a Markov chain. It claims that for an open-loop encoder, the fundamental tradeoff between communication rate and sensing distortion is the limit of a maximized sum of conditional mutual informations, constrained by the expected distortion a Bayesian filter achieves. If true, this gives a capacity-distortion characterization for a general class of state-dependent channels and allows concrete comparison of beam-pointing strategies. Readers should care because prior rate-distortion JCAS results mostly assume i.i.d. states, which rules out tracking and prediction.","feed_headline":"Bayesian filtering sets the open-loop JCAS rate-distortion limit","feed_subtitle":"The formula turns the distortion constraint into a feasible input set, so beam schedules and power splits can be optimized.","key_machinery":"The load-bearing identity is Theorem 1's capacity formula, in which the rate is an average of single-letter conditional mutual informations $I(X_i;Y_i|S_i)$, and the distortion constraint enters only through the set $\\mathcal{P}_D^{(n)}$ of input distributions whose expected sensing cost $c(x^n)=\\mathbb{E}[d_{0,n}(S_0^n, g^*(X^n,Z^n))|X^n=x^n]$ stays below $D$. The mechanism that makes the result work is the causal Bayesian estimator of Lemma 1: because the state is Markov and the estimator is causal, the optimal per-symbol estimate is the minimizer of the per-letter expected distortion, so the sensing cost becomes a functional of the input distribution alone, decoupled from the communication code. For the beam examples, this machinery collapses to the error-covariance recursion of an intermittent or steady-state Kalman filter, and the feasible $\\lambda$ or $\\gamma_0$ sets are read off from algebraic Riccati equations.","core_discovery":"The central claim is Theorem 1: with the state evolving as a first-order Markov chain and the encoder operating open loop, the capacity-distortion tradeoff is $C^{\\mathrm{open}}(D) = \\lim_{n\\to\\infty} \\max_{P_{X^n} \\in \\mathcal{P}_D^{(n)}} \\frac{1}{n}\\sum_{i=1}^n I(X_i;Y_i|S_i)$, where the feasible input set $\\mathcal{P}_D^{(n)}$ is defined by the requirement that the average per-block distortion using the optimal estimator stays below $D$. Lemma 1 supplies that estimator: the optimal causal estimate at each time is the Bayes estimator $g_i^*(X^i,Z^i) = \\arg\\min_{\\hat{s}} \\mathbb{E}[d_i(S_i,\\hat{s})|X^i,Z^i]$, which in the Gauss-Markov beam example becomes a Kalman filter. The paper then specializes the general formula to two beam-pointing schemes: beam switching, where sensing and communication time-share and the capacity is bracketed by $(1-\\lambda)I(X;Y)$ evaluated at inner and outer distortion-feasible values of $\\lambda$, and multi-beam, where a constant power-sharing parameter $\\gamma_0$ yields $C_{\\mathrm{mb}}(D) = \\max I(X;Y|\\Gamma=\\gamma_0)$ over $\\gamma_0$ satisfying the distortion constraint. The numerical comparison shows multi-beam dominating beam switching, especially for unstable target dynamics.","pith_inferences":["Editorial extension: if Theorem 1 is right, the same limit formula should extend to closed-loop encoders only when feedback does not change the state posterior faster than the Bayesian filter; otherwise new terms coupling past estimates to future inputs must appear.","Editorial extension: the beam-switching bounds suggest a testable design rule: allocate sensing time only up to the point where the Kalman error covariance saturates, since additional measurements beyond that point buy no communication-rate reduction.","Editorial extension: the exact multi-beam characterization could serve as a benchmark for adaptive beamforming; if an adaptive strategy outperforms the constant-$\\gamma_0$ optimum, it must be exploiting state prediction in a way the open-loop formula does not capture.","Editorial extension: the paper's Remark 2 points toward a stronger per-step distortion constraint that would guarantee target tracking is never lost; combining that constraint with Theorem 1 would yield a rate-loss tradeoff with guaranteed tracking, which the average-distortion result does not address."],"forward_implications":["Any open-loop JCAS system with a Markov state has an asymptotically optimal rate-versus-distortion tradeoff given by Theorem 1, provided the optimal estimator can be evaluated.","For beam switching, the capacity lies between $(1-\\lambda)I(X;Y)$ evaluated at the two distortion-derived thresholds $\\lambda_S$ and $\\lambda_V$, so the sensing schedule can be optimized at the level of the switching probability.","For multi-beam, the tradeoff is fully determined by the power-sharing parameter $\\gamma_0$ through a steady-state Riccati equation, so no codebook optimization beyond the usual channel capacity is needed.","When the target dynamics are unstable, beam switching has a critical sensing probability below which distortion diverges, while multi-beam degrades more gracefully at high communication rates.","The results extend the i.i.d.-state rate-distortion JCAS framework to predictive tracking, so the same analysis can be reused when the state evolves by any known dynamical model."],"supporting_citations":[{"why":"supplies the rate-distortion reinterpretation of state-estimation cost as a distortion constraint, which this model adapts.","marker":"[17]"},{"why":"gives the i.i.d.-state JCAS capacity-distortion framework that Theorem 1 generalizes to Markov states.","marker":"[18]"},{"why":"contains the same optimal-estimator and capacity results, which the authors cite as independently discovered.","marker":"[26]"},{"why":"provides the intermittent Kalman filter analysis and the bounds $S_n \\leq \\mathbb{E}[P_n] \\leq V_n$ used in Lemma 3.","marker":"[28]"},{"why":"supplies the Hoeffding-type concentration inequality for Markov chains that makes the achievability proof rigorous.","marker":"[29]"},{"why":"is the Bayesian filtering reference behind the predict/update equations used for the Kalman filter and Lemma 1.","marker":"[27]"}],"fun_headline_variants":["Open-loop JCAS: Bayesian filter is optimal sensing","Rate-distortion bound for joint comms and sensing","Multi-beam beats beam-switching in JCAS tracking","Markov states: how to split rate and sensing distortion","JCAS open loop: Kalman filtering sets the tradeoff"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The beam-switching theorems rest on modeling sensing intervals as erasures the receiver can detect without being told the beam schedule; if the receiver must know $\\Gamma^n$ to interpret $Y^n$, the stated rates need rework.","fun_headline_variants_meta":{"raw":{"variants":["Open-loop JCAS: Bayesian filter is optimal sensing","Rate-distortion bound for joint comms and sensing","Multi-beam beats beam-switching in JCAS tracking","Markov states: how to split rate and sensing distortion","JCAS open loop: Kalman filtering sets the tradeoff"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":1173,"prompt_tokens":976,"completion_tokens":197,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":118}},"tokens_in":592,"tokens_out":197,"duration_ms":2828,"temperature":1.0,"reasoning_tokens":118,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:04:44.654871+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct falsifier is to implement the beam-switching scheme on the simulated Gauss-Markov target with $A=-1.15$, set $\\lambda$ below the critical sensing probability predicted by the intermittent Kalman filter analysis, and check whether the sample mean-square error diverges; if distortion stays bounded, the inner bound's distortion guarantee fails.","supporting_citations":[{"cited_title":"A memory-based reinforcement learning approach to integrated sensing and communication,","cited_arxiv_id":null,"evidence_quote":"contains the same optimal-estimator and capacity results, which the authors cite as independently discovered."},{"cited_title":"Kalman Filtering With Intermittent Observations,","cited_arxiv_id":null,"evidence_quote":"provides the intermittent Kalman filter analysis and the bounds $S_n \\leq \\mathbb{E}[P_n] \\leq V_n$ used in Lemma 3."},{"cited_title":"Hoeffding’s inequality for general markov chains and its applications to statistical learning,","cited_arxiv_id":null,"evidence_quote":"supplies the Hoeffding-type concentration inequality for Markov chains that makes the achievability proof rigorous."},{"cited_title":"Särkkä and L","cited_arxiv_id":null,"evidence_quote":"is the Bayesian filtering reference behind the predict/update equations used for the Kalman filter and Lemma 1."}],"review_version":1}