{"id":"a5a57873-953a-4e84-a471-1dc36851a5fc","arxiv_id":"2506.09931","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Single-carrier faster-than-Nyquist signaling avoids spectral aliasing caused by the symbol-rate and pulse-bandwidth mismatch, yielding communication spectral-efficiency gains and fewer Doppler ambiguity peaks for integrated sensing and communications.","lead":"This paper analytically studies faster-than-Nyquist signaling for integrated sensing and communications, finding that sending symbols faster than the Nyquist rate can avoid spectral aliasing and improve both communication spectral efficiency and radar-like sensing. A reader interested in 6G waveform design may care because the same signal could carry data and sense targets more reliably without extra bandwidth.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"FTN's ISAC advantage is established only under the equal-PSD normalization Es=PξT; under peak-power or per-symbol constraints the communication-side ordering is untested and may reverse, so the headline claim needs scoping.","rationale":"The reader's weakest_assumption is the same one I would flag. The paper's analytic machinery — Szegő's theorem, the folded and twisted folded spectra, and the Dirichlet-kernel analysis of the ambiguity function — is internally consistent, and the Doppler-ambiguity result is clean: when ξ≤ξ0=1/(WT), the PRF 1/(ξT) exceeds the Doppler support W of the pulse, so the Dirichlet peaks at k/(ξT) fall outside |AFp(0,ν)| and no undesired peaks appear. The communication result also checks out: in the no-aliasing regime the bounds coincide and recover the continuous-time channel capacity with transmit filter Hp(f). The chief soft spot is the scope of the headline. The equal-PSD convention is standard for average-power-limited links, but it is a modeling choice, and the paper does not show what happens under peak-power limits, where FTN's pulse overlap can increase PAPR. This does not invalidate the paper; it means the conclusion should be stated as conditional on the average-power normalization. Since the reader's verdict is already CONDITIONAL and my concern aligns with the reader's weakest_assumption, no verdict change is needed.","tokens_in":23162,"tokens_out":32475,"duration_ms":336511,"concrete_test":"Recompute the spectral efficiency comparison of Fig. 2 (RRC β=0.3, T=1, L=3 channel) under an equal peak-power constraint: for each ξ, scale the symbol energy Es so that max_t |s(t)|^2 over a long QPSK/Gaussian symbol realization is equal for FTN and Nyquist (Monte Carlo over at least 10^4 symbols), then evaluate the FTN rate (29) and the corresponding Nyquist rate at the same allowed peak power. If FTN does not dominate Nyquist in this peak-power-limited comparison, the paper's headline must be scoped to the average-power/equal-PSD case; if FTN still dominates, the normalization concern is not practically limiting.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is the fair-comparison convention in Section II-B: Es=PξT, which equalizes the PSD and total energy of FTN and Nyquist while giving FTN 1/ξ more symbols. This is a natural average-power normalization, and the paper is transparent about it, but the abstract and title state the conclusion without this caveat. The communication Theorem 1 and the exact no-aliasing formula (35) deliver the FTN advantage as a DoF (pre-log) gain: FTN occupies the full band W in the integral, whereas Nyquist with the same RRC pulse integrates only over 1/(ξT) and pays a folding loss. Under an equal peak-power constraint, FTN's overlapping pulses can raise the PAPR, and the paper does not model this; citing the low-PAPR property of single-carrier FTN in the introduction is not an analysis. Under an equal per-symbol energy constraint, FTN's average power is larger by 1/ξ, so the comparison is no longer at the same operating point. Thus the joint 'FTN is good for ISAC' claim is proven for average-power-limited systems with the stated normalization, but not for peak-power-limited or per-symbol-energy comparisons. The sensing-side Doppler conclusion in Section IV-B does not depend on this normalization, but the communication-side claim is what the title's 'good' largely rests on. This is a scoping concern, not an internal inconsistency: the mathematics under the stated assumptions appears sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper provides an analytical study of single-carrier faster-than-Nyquist (FTN) signaling for integrated sensing and communications (ISAC). On the communications side, it models the effective channel matrix as asymptotically Hermitian Toeplitz, applies Szegő's theorem, and derives spectral-efficiency expressions with upper and lower bounds for time-invariant multipath channels; in the no-aliasing regime where the symbol rate is at least the pulse bandwidth, the bounds coincide and an exact formula is given. On the sensing side, the paper derives the expected squared ambiguity function for random symbols, defines an accumulated ISI function for ranging and a periodic Doppler variation function, and argues that FTN avoids the Doppler-dimension peaks caused by spectral aliasing in Nyquist signaling with excess bandwidth. Numerical simulations of spectral efficiency, delay slices, Doppler slices, and two-target Doppler estimation support the analytical claims.","tokens_in":23410,"tokens_out":22090,"duration_ms":239920,"significance":"If the claims are taken in their stated scope, the paper is a valuable analytical contribution: it gives parameter-free spectral-efficiency bounds, an exact no-aliasing formula, and a crisp explanation of FTN's sensing benefit through avoidance of spectral aliasing. The Doppler peak-avoidance prediction is falsifiable and is verified in simulation. The derivation is mostly self-contained, with the main tools being Toeplitz asymptotics, Szegő's theorem, and the fourth-order moment expansion of the ambiguity function. The equal-PSD comparison convention is stated explicitly and is standard in the FTN literature. The main weakness is that the title and abstract claim a general 'FTN is good for ISAC' advantage, whereas the communication-side result is proven only under an equal-average-power/equal-PSD normalization, and the ranging advantage is supported more by numerical evidence than by a formal theorem.","major_comments":[{"comment":"The communication-side advantage is established only under the equal-PSD/average-power convention Es=PξT. Under a per-symbol-energy or peak-power constraint, FTN would consume more average power or face an unmodeled PAPR increase, so the ordering in Theorem 1 is not proven for those operating points. The title and abstract state 'FTN is advantageous for ISAC' without this qualification. Because the 'good' in the title rests on this comparison, I ask the authors to scope the claims explicitly, or to add an analysis (or at least a discussion) of the alternative normalizations.","section":"II-B and Theorem 1"},{"comment":"The claim that FTN signals 'generally enjoy a more robust ranging performance' is not proven. The analysis of the accumulated ISI function X(τ) shows that its fluctuation is reduced when ξ≤ξ0, but the normalized delay slice of the expected squared ambiguity function also contains the term N²|AF_p(-τ,0)|² and a kurtosis-dependent term, and no analytical ordering of the normalized slice is derived. The numerical comparison in Fig. 6 shows similar behavior rather than a general advantage. Please either prove a formal statement about the normalized ambiguity function or weaken the claim to the observed fluctuation property of X(τ).","section":"IV-A"},{"comment":"The contribution bullet states that 'spectral aliasing will introduce undesired peaks' along the Doppler dimension, but the analysis of Y(ν) in (44) only shows that the Dirichlet kernel has peaks at multiples of 1/(ξT). An actual peak in E|AF_s(0,ν)|² requires AF_p(0,ν) to be nonzero at those Doppler offsets; this is not proven for general p(t) and is only demonstrated numerically for the RRC pulse. A sufficient condition (e.g., RRC with β>0 and 1/(ξT)<W) should be stated and proved, or the claim should be restricted to the pulse family for which the condition holds.","section":"IV-B"}],"minor_comments":[{"comment":"Equation (35) appears to contain a typo: the expression should be |E[AF_s(τ,ν)]|² = E_s² |Σ_{n=1}^N e^{-j2πnνξT}|² |AF_p(-τ,ν)|². As printed it lacks the absolute square and mislabels the expectation, which is inconsistent with the decomposition in (33)-(36).","section":"Eq. (35)"},{"comment":"The legend in Fig. 2 appears to list 'ξ=0.75, achievable rate' three times; please check and correct the legend entries.","section":"Fig. 2"},{"comment":"The wording of the Nyquist no-ISI theorem in footnote 2 is confusing; for the two-sided bandwidth W used in the paper, the no-ISI condition is 1/T ≤ W, and the text should state this explicitly.","section":"Footnote 2"},{"comment":"The approximation in (15) is stated without quantifying its accuracy; a brief comment on when the ratio of expectations is a good approximation to the expectation of the ratio would be helpful.","section":"Section II-B, Eq. (15)"},{"comment":"The phrase 'oscillated values' should read 'oscillatory values'.","section":"Section IV-A"}],"recommendation":"major_revision","confidential_remarks":"The technical core of the paper appears sound under the stated equal-PSD normalization. My main concern is that the headline claims are broader than what is proven: the communication advantage is normalization-dependent, the ranging advantage is not formally established, and the Doppler-peak statement needs a precise sufficient condition. These are fixable with scoping and additional statements rather than new fundamental derivation. I would not reject the paper; a major revision that tightens the claims (or adds the missing analysis) would make it suitable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you read it. First, the sensing analysis is the real contribution: it is the first analytical treatment of the expected squared ambiguity function for non-Nyquist FTN random signaling, and it proves the Doppler peak-avoidance mechanism. That part is self-contained, starts from the definition, and does not fit any parameters. Second, the headline claim \"FTN is good for ISAC\" is only established under the equal-PSD normalization Es = PξT. The authors state this clearly in Section II-B, but the abstract and title do not carry the caveat, and the communication-side ordering is untested under peak-power or per-symbol-energy constraints. The stress-test note is right that this is a scoping concern, not an internal inconsistency.\n\nWhat the paper does well: the Toeplitz/Szegő derivation is standard and internally consistent, and the bounds line up with the Monte Carlo results. The connection between spectral aliasing and the appearance of undesired Doppler peaks is clearly demonstrated, and the numerical examples support the qualitative picture. I also give credit for the honest acknowledgment of practical drawbacks (complexity, lack of good codes) in the conclusion.\n\nSoft spots, in proportion: the lower bound in (34) uses a max{·, 0} that deserves scrutiny; it may be trivial or loose in some regimes, and the paper does not discuss when it is tight. The MSE experiment in Fig. 9 does not specify the estimator, which makes the \"Nyquist fails\" claim hard to reproduce. No code or detailed simulation parameters are provided, so the figures are not fully verifiable. The communication-side bounds extend the authors' own folded-spectrum framework [42] to SISO multipath with delay-dependent bounds; that is a legitimate extension, and the self-citation is not a problem.\n\nThese are presentation and scoping issues, not load-bearing flaws. The mathematics under the stated assumptions appears sound, and the sensing-side conclusion does not depend on the contested normalization. The paper would benefit from a revised abstract that explicitly says the communication advantage is for average-power-limited systems with the equal-PSD convention, and from a footnote or appendix that checks the lower-bound tightness.\n\nWho is this for: people working on 6G physical layer, ISAC waveform design, and FTN theory. A serious referee should engage with it. My recommendation: send it to peer review. Ask the reviewers to verify the lower bound and the MSE estimator, but do not desk-reject.","headline":"The sensing-side analysis is genuinely new and holds up; the communication-side 'FTN is good' claim needs to be scoped to the equal-PSD normalization, but this is a fixable scoping issue, not a fatal flaw.","tokens_in":23983,"tokens_out":1713,"would_cite":true,"duration_ms":19268,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A12","94A17","15B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Faster-than-Nyquist signaling—packing symbols tighter than the classical rate—improves both spectral efficiency over multipath channels and sensing reliability, because it avoids the spectral aliasing that practical Nyquist pulses suffer.","keywords":["faster-than-Nyquist signaling","integrated sensing and communications","spectral aliasing","folded spectrum","spectral efficiency","ambiguity function","Doppler estimation","single-carrier transmission"],"falsifier":"Repeat the paper's spectral-efficiency comparison of $\\xi = 1$ versus $\\xi = 0.75$ with a root-raised-cosine pulse ($\\beta = 0.3$, $T = 1$) under a fixed peak-power or per-symbol-energy budget instead of the equal-PSD rule $E_s = P \\xi T$: if the Nyquist signal then matches or beats FTN, the communication-side advantage holds only under the paper's normalization. As a separate check of the mechanism, sharply filter the transmitted pulse so that the Nyquist-rate signal has no spectral aliasing and observe whether the predicted Doppler peaks at multiples of $1/(\\xi T)$ disappear at $\\xi = 1$.","tokens_in":22943,"feed_emoji":"📡","tokens_out":13682,"duration_ms":123656,"temperature":0.7,"pith_summary":"This paper argues that faster-than-Nyquist (FTN) signaling—deliberately packing symbols at a rate above the classical Nyquist limit—is a benefit rather than a liability for integrated sensing and communications (ISAC). The claimed mechanism is that FTN avoids spectral aliasing: practical shaping pulses have excess bandwidth, and at the Nyquist rate that excess spectrum folds back onto itself, corrupting both the multipath communication channel and the sensing ambiguity function, whereas a compression factor $\\xi \\leq \\xi_0 = 1/(WT)$ eliminates the overlap. The authors derive upper and lower bounds on the spectral efficiency of single-carrier FTN over time-invariant multipath channels, prove that the bounds coincide once the symbol rate reaches the pulse bandwidth, and show that the Doppler slice of the expected squared ambiguity function then has no spurious peaks, so velocity estimation improves. If the analysis is right, one FTN waveform outperforms an otherwise identical Nyquist waveform for both tasks under a fair equal-power, equal-bandwidth comparison.","feed_headline":"Transmitting faster than Nyquist sharpens both comms and radar","feed_subtitle":"The same waveform avoids the spectral aliasing that blurs multipath data and creates false Doppler peaks in sensing.","key_machinery":"The load-bearing objects are the folded spectrum $|H_{\\mathrm{fo}}(f)|^2$ and the twisted folded spectrum $|H_{\\mathrm{tfo}}(f)|^2$, two extreme cases of what happens when the periodic aliasing of the pulse spectrum is constructive or destructive; they coincide with the pulse spectrum $|H_p(f)|^2$ exactly when $\\xi \\leq \\xi_0 = 1/(WT)$, which is the no-aliasing regime the whole argument exploits. On the communication side, the machinery is the observation that the effective channel matrices $G_l$ and their cross terms are asymptotically Hermitian Toeplitz, so Szegő's theorem converts the mutual-information determinant into a one-dimensional frequency integral whose integrand is an SNR per frequency component, with the folded spectra providing the bounds on that SNR. On the sensing side, the machinery is the decomposition of the expected squared ambiguity function into a squared-mean (iceberg) term and a variance term, together with the squared Dirichlet kernel $A(\\Delta f, N, \\xi) = \\sin^2(\\pi N \\Delta f \\xi T)/\\sin^2(\\pi \\Delta f \\xi T)$ that appears in the accumulated ISI function $X(\\tau)$ and in the periodic Doppler variation function $Y(\\nu)$; whether the kernel's peaks at multiples of $1/(\\xi T)$ fall inside the pulse's frequency support is exactly what spectral aliasing decides.","core_discovery":"The paper's central claim is that a single-carrier ISAC waveform that transmits faster than the Nyquist rate is strictly better than its Nyquist counterpart in both functionalities, and that the cause is the removal of spectral aliasing. Concretely, the paper proves that the effective channel matrices of FTN are asymptotically Hermitian Toeplitz, which lets Szegő's theorem turn the constrained capacity into a frequency integral of a per-frequency SNR; the upper and lower bounds on that integral are expressed through the folded spectrum and the twisted folded spectrum, which capture constructive and destructive superposition of aliased spectrum copies, and the bounds coincide exactly when $1/(\\xi T) \\geq W$, the regime where the SNR variation induced by multipath delay vanishes and the system degrees of freedom are maximal. On the sensing side, the paper derives the expected squared ambiguity function of FTN and shows that its delay slice fluctuates less than the Nyquist one, while the Doppler slice of Nyquist signaling carries undesired peaks at multiples of $1/(\\xi T)$ that appear precisely because of spectral aliasing and disappear for FTN with $\\xi \\leq \\xi_0$. The conclusion the authors draw is that a single FTN signal, with the same power spectral density and the same time-frequency footprint as a Nyquist baseline, offers simultaneously higher spectral efficiency and more reliable range and Doppler estimation.","pith_inferences":["A design choice the paper stops short of recommending: operate at the saturation threshold $\\xi = \\xi_0$ itself, rather than below it, since smaller $\\xi$ buys no additional aliasing avoidance and only raises equalization complexity.","Because the spurious Doppler peaks sit on a squared Dirichlet kernel, their width scales as $1/(N\\xi T)$: longer coherent integration does not remove the Nyquist ambiguity but sharpens it into narrow spikes, an effect the paper's formulas imply but do not quantify.","Under the paper's equal-PSD normalization the FTN advantage should widen with the roll-off factor $\\beta$, since $W = (1+\\beta)/T$ makes the aliasing-prone excess bandwidth grow; this scaling is visible in $\\xi_0 = 1/(1+\\beta)$ but is not drawn out.","The ranging gap between FTN and Nyquist is likely to shrink when the pulse is optimized for sensing, but the Doppler-side advantage should persist, because the peak mechanism is the symbol-rate/bandwidth mismatch rather than the pulse's sidelobe structure."],"forward_implications":["In the regime $\\xi \\leq 1/(WT)$ the derived spectral-efficiency bounds coincide, giving an exact rate expression: the multipath-induced SNR variation vanishes and the system attains its maximum degrees of freedom (pre-log factor).","The gap between the upper and lower spectral-efficiency bounds at the Nyquist rate is a direct quantitative measure of the SNR uncertainty that channel delay injects through spectral aliasing, and it shrinks as the symbol rate grows.","FTN ranging benefits from a less fluctuated delay slice of the expected squared ambiguity function, which the paper identifies with the accumulated ISI function's weaker oscillation at high symbol rates.","At or below the saturation threshold there are no undesired peaks in the Doppler slice, so a weak target at a nearby Doppler is not masked by periodic ambiguity; the numerical Doppler estimation confirms FTN's mean squared error decreases with SNR while Nyquist's does not.","The analysis holds for any band-limited pulse for which the folded and twisted folded spectra are defined, and becomes exact for systems with a cyclic prefix longer than the delay spread, where Toeplitz matrices become circulant."],"supporting_citations":[{"why":"Supplies the folded-spectrum definition, the saturation threshold $\\xi_0 = 1/(WT)$, and the degree-of-freedom arguments for FTN spectral efficiency that the communication analysis builds on.","marker":"[28]"},{"why":"Supplies the twisted folded-spectrum definition and the DTFT derivations of Toeplitz coefficients that Lemma 2 and Corollary 1 rest on.","marker":"[42]"},{"why":"Establishes the asymptotic Hermitian-Toeplitz structure and invertibility properties used to apply Szegő's theorem to the FTN mutual information.","marker":"[29]"},{"why":"Gives the iceberg decomposition of the expected squared ambiguity function into squared mean plus variance, which the sensing analysis extends to FTN.","marker":"[16]"},{"why":"Provides the kurtosis framework, the expected normalized squared ambiguity function, and the approximation in (15) used for the ranging and Doppler slice evaluations.","marker":"[15]"},{"why":"Proves the full-rank and positive-definiteness of the folded-spectrum matrix $G_0$ in the asymptotic regime, which the spectral efficiency formula requires.","marker":"[41]"},{"why":"States Szegő's theorem on Toeplitz determinants, the tool that converts the rate expression into a frequency-domain integral.","marker":"[44]"}],"fun_headline_variants":["FTN signaling avoids aliasing, sharpening both comms and radar","Faster-than-Nyquist improves single-carrier ISAC in both domains","FTN beats Nyquist for spectral efficiency and Doppler clarity","No spectral aliasing: FTN gives better comms and sensing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the fairness convention that FTN and Nyquist signals share the same power spectral density and the same time-frequency footprint, enforced by $E_s = P \\xi T$; if one instead compares at equal per-symbol energy or equal peak power, the claimed spectral-efficiency ordering is not proven and could reverse, though the sensing conclusions do not depend on this normalization.","fun_headline_variants_meta":{"raw":{"variants":["FTN signaling avoids aliasing, sharpening both comms and radar","Faster-than-Nyquist improves single-carrier ISAC in both domains","FTN beats Nyquist for spectral efficiency and Doppler clarity","No spectral aliasing: FTN gives better comms and sensing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1615,"prompt_tokens":1072,"completion_tokens":543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":467}},"tokens_in":688,"tokens_out":543,"duration_ms":5462,"temperature":1.0,"reasoning_tokens":467,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:40:50.663803+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the paper's spectral-efficiency comparison of $\\xi = 1$ versus $\\xi = 0.75$ with a root-raised-cosine pulse ($\\beta = 0.3$, $T = 1$) under a fixed peak-power or per-symbol-energy budget instead of the equal-PSD rule $E_s = P \\xi T$: if the Nyquist signal then matches or beats FTN, the communication-side advantage holds only under the paper's normalization. As a separate check of the mechanism, sharply filter the transmitted pulse so that the Nyquist-rate signal has no spectral aliasing and observe whether the predicted Doppler peaks at multiples of $1/(\\xi T)$ disappear at $\\xi = 1$.","supporting_citations":[{"cited_title":"Constrained capacities for faster-than- Nyquist signaling,","cited_arxiv_id":null,"evidence_quote":"Supplies the folded-spectrum definition, the saturation threshold $\\xi_0 = 1/(WT)$, and the degree-of-freedom arguments for FTN spectral efficiency that the communication analysis builds on."},{"cited_title":"Faster-than-Nyquist asynchronous NOMA outperforms synchronous NOMA,","cited_arxiv_id":null,"evidence_quote":"Supplies the twisted folded-spectrum definition and the DTFT derivations of Toeplitz coefficients that Lemma 2 and Corollary 1 rest on."},{"cited_title":"Faster-than-Nyquist broadcast- ing in Gaussian channels: Achievable rate regions and coding,","cited_arxiv_id":null,"evidence_quote":"Establishes the asymptotic Hermitian-Toeplitz structure and invertibility properties used to apply Szegő's theorem to the FTN mutual information."},{"cited_title":"Uncovering the iceberg in the sea: Fundamentals of pulse shaping and modulation design for random ISAC signals,","cited_arxiv_id":null,"evidence_quote":"Gives the iceberg decomposition of the expected squared ambiguity function into squared mean plus variance, which the sensing analysis extends to FTN."},{"cited_title":"Properties of faster-than-Nyquist channel matrices and folded-spectrum, and their applications,","cited_arxiv_id":null,"evidence_quote":"Proves the full-rank and positive-definiteness of the folded-spectrum matrix $G_0$ in the asymptotic regime, which the spectral efficiency formula requires."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States Szegő's theorem on Toeplitz determinants, the tool that converts the rate expression into a frequency-domain integral."}],"review_version":1}