{"id":"aee60a64-fec1-404a-bd5d-e5a5068ba89d","arxiv_id":"2507.12210","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"DFT-s-OTFS with interleaved Doppler allocation has a PAPR upper bound equal to the QAM constellation's PAPR, independent of spreading size, while block allocation has a bound K times larger.","lead":"OTFS, a waveform for high-mobility wireless links, suffers from a high peak-to-average power ratio (PAPR). This paper derives closed-form PAPR upper bounds for a spread variant, DFT-s-OTFS, and shows that interleaved Doppler allocation keeps PAPR low while simplifying the transmitter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed PAPR upper bounds use the expected average power in the denominator, not the per-frame average of Eq. (4); an explicit 16-QAM frame gives PAPR ≈ 8.93 for interleaved allocation, exceeding the bound of 1.8 in Eq. (10).","rationale":"The paper's main deliverable is a set of closed-form PAPR upper bounds for DFT-s-OTFS. For such a claim to be an upper bound in the sense of Eq. (4), the bound must hold for every transmitted frame, because the PAPR is defined for the particular realization xq(t). The derivation instead computes the denominator as the expectation over QAM symbols. This is a real gap, not a stylistic choice: because the DFT spreading and the N-point IDFT are unitary maps, the actual frame average is the sample mean of the QAM symbol energies divided by Q. That sample mean is a random variable. For typical frames it is close to 1 for large MK, which is why the simulations appear to confirm the bounds. But for atypical but valid frames it can be much smaller, making the true PAPR larger than the claimed bound. The explicit one-corner frame yields a PAPR of about 8.93 for interleaved rectangular allocation, versus the claimed bound 1.8. This settles the concern: the bound is not an upper bound on Eq. (4) for finite frames. The same issue enters Eqs. (19), (23), and (25) through the same expectation-based average-power steps. I agree with the reader's weakest-assumption analysis; the counterexample makes it concrete and shows the error is in the central claim, not a minor typo. Nevertheless, the paper has independent value: the interleaved allocation signal structure (repetition of QAM symbols) and the qualitative comparison (interleaved lower than block, RRC higher than rectangular for typical parameters) are supported by the derivations and simulations, and the analysis could be repaired by a caveat or a concentration bound. Therefore I would not reject outright; conditional acceptance with a required revision is appropriate, matching the reader's verdict.","tokens_in":10426,"tokens_out":14017,"duration_ms":164810,"concrete_test":"Evaluate the exact PAPR of Eq. (4) for the constructed frame: set M=128, N=32, Q=4, K=8, use 16-QAM normalized to unit average energy, put one symbol at the corner of energy 1.8 (e.g., 3+3j scaled by 1/sqrt(10)) and all other Xq[m,k] at an inner point of energy 0.2 (e.g., 1+j scaled by 1/sqrt(10)); generate xDT via Eq. (2) and xq(t) with the rectangular pulse of Eq. (5), then compute the ratio max|xq|^2 / (1/T)∫|xq|^2 dt. If the result is ≈8.93 (9.5 dB), which is above the claimed bound 1.8 (2.55 dB), the per-frame upper bound in Eq. (10) is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Eqs. 10, 19, 23, 25) are presented as upper bounds on the PAPR defined in Eq. (4). The derivations replace the denominator, (1/T)∫|xq(t)|^2 dt, with E{|xDT[i]|^2}/Q (Eqs. 9, 15, 21, 24), i.e., with an expectation over the random QAM symbols rather than the actual sample average of the transmitted frame. For rectangular pulses, and more generally whenever the delay-time mapping is unitary, the true denominator is the sample mean of |Xq[m,k]|^2 divided by Q, which fluctuates from frame to frame. A deterministic 16-QAM frame for interleaved allocation with M=128, K=8, one corner symbol of energy 1.8 and the other 1023 symbols at the minimum-energy point of 0.2 gives exact PAPR = 1.8 / ((1.8 + 1023×0.2)/1024) ≈ 8.93, while Eq. (10) claims an upper bound of 1.8. Thus the stated 'upper bound' fails for a valid input frame. The block and RRC bounds share the same defect, since their average-power steps (21), (24) are also expectations. Correcting the claim requires either (i) explicitly redefining PAPR with the expected average power, or (ii) bounding the deviation of the sample mean from its expectation for finite M, K, possibly with a probability-of-exceedance statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies the peak-to-average power ratio (PAPR) of DFT-spread OTFS (DFT-s-OTFS) in the uplink with two Doppler resource allocations (interleaved and block) and two transmit pulse shapes (rectangular and root raised cosine). The authors derive closed-form expressions that they present as PAPR upper bounds: Eq. (10) for interleaved rectangular, Eq. (19) for interleaved RRC, Eq. (23) for block rectangular, and Eq. (25) for block RRC. They further show that interleaved allocation yields a time-domain signal equal to repeated QAM symbols (Eq. (6)), claim that interleaved allocation has a PAPR bound K times smaller than block allocation, and support their results with CCDF and BER simulations. The main mathematical issue is that the 'average power' used in the derivations is the expectation over random QAM symbols, while the PAPR definition in Eq. (4) uses the per-frame sample average; consequently the derived expressions are not upper bounds on the PAPR defined in Eq. (4).","tokens_in":10696,"tokens_out":15806,"duration_ms":177054,"significance":"The paper has concrete strengths: the structural identity in Eq. (6) is cleanly derived and gives real transmitter-design insight; the peak-power upper bounds are valid for the peak numerator; the simulations are extensive and reproducible in spirit; and the comparison between allocation schemes is practically useful. If the PAPR metric is redefined to use expected average power, or if finite-frame concentration bounds are added, the analytical results would be valuable. As written, however, the central claim that Eqs. (10), (19), (23), and (25) bound the PAPR of Eq. (4) is false, and the block-allocation bound contains an additional algebraic error. The paper therefore needs a substantive revision before it can be accepted.","major_comments":[{"comment":"The derivations replace the denominator of the PAPR definition in Eq. (4), namely the per-frame integrated average power (1/T)∫|x_q(t)|^2 dt, with E{|x_q^DT[i]|^2} in Eq. (9), and similarly in Eqs. (15), (21), and (24). For the rectangular pulse the true denominator is the realized sample average (1/(MN))Σ_i |x_q^DT[i]|^2, which fluctuates from frame to frame, so Eq. (10) is not an upper bound on the PAPR defined in Eq. (4). A concrete valid frame for the paper's own simulation parameters (M=128 delay bins, Q=4, K=8, 16-QAM) gives a counterexample: with one maximum-energy symbol (1.8) and the remaining 1023 QAM symbols at minimum energy (0.2), the sample average power after the 1/Q scaling in Eq. (6) is (1.8+1023·0.2)/(128·32·4) ≈ 0.0504, the peak sample power is 1.8/4 = 0.45, and the exact PAPR is approximately 8.93, far above the claimed bound 1.8. The same expectation-versus-sample-mean issue invalidates Eqs. (19), (23), and (25) as bounds on the defined PAPR. To repair the claim, the authors must either (i) explicitly redefine the PAPR metric to use expected average power, or (ii) bound the deviation of the sample mean from its expectation for finite M and K, for example with a probability-of-exceedance statement.","section":"IV-A, Eqs. (20)-(23)"},{"comment":"The empirical CCDFs in Figs. 2-4 cannot validate the analytic expressions as 'upper bounds on PAPR' as defined by Eq. (4), because the analytic expressions use expected average power while the empirical CCDFs use per-frame average power. A simulation can only show that random frames rarely violate a given threshold; it cannot establish a deterministic upper bound of the type claimed in Eqs. (10), (19), (23), and (25). The text in Section V and the abstract should be revised to either compare against a explicitly redefined metric or to present the finite-sample results as empirical observations rather than as confirmations of deterministic bounds.","section":"V, Figs. 2-4"}],"minor_comments":[{"comment":"The symbol M is used both for the number of delay bins and for the QAM constellation order (e.g., 'M=128, ... 16 QAM'), which is confusing; please disambiguate, for example by writing M_delay and M_QAM.","section":"Throughout"},{"comment":"The root-raised-cosine impulse response in Eq. (11) is undefined at t=0 and at t=±Δτ/(4β); please state the limiting values or cite the standard definition that includes the limiting cases.","section":"III-B, Eq. (11)"},{"comment":"The definitions of A(i), B(i), C(i), and D(i) and the role of Lspan are hard to follow without consulting Ref. [15]; please state the exact expression and the value of Lspan used in computing g0 in Fig. 4, so that the analytical bounds are reproducible.","section":"III-B, Eqs. (17)-(18)"},{"comment":"'Maximum PAPR' in Fig. 4 depends on the number of random frames searched; since the paper claims to show a maximum rather than a CCDF tail, please specify the number of frames and the search procedure, and state that the plotted maximum is a sample maximum.","section":"V, Fig. 4"},{"comment":"The conclusion says that the RRC-induced PAPR increase 'can be mitigated by increasing the roll-off factor,' but Fig. 4 shows that the maximum PAPR is roughly flat for β>0.4; please qualify this statement to reflect the actual dependence on β.","section":"VI, Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The paper has a useful structural result (Eq. (6)) and a clear practical message, but the headline analytical claim is not valid for the PAPR definition in Eq. (4). I would be willing to see a revised version that either redefines the PAPR metric to use expected average power or supplies a finite-frame probabilistic bound, and that fixes the block-allocation algebra. If the authors choose to keep the current deterministic upper-bound claim unchanged, I would recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid extension of DFT-s-OTFS, with clean derivations for interleaved versus block allocation and rectangular versus RRC pulse shaping. The new pieces—the interleaved-allocation bound, the periodic-repetition observation, and the RRC peak-sum analysis—are legitimate and useful. But the headline claim that these are PAPR upper bounds does not hold up when you read carefully. The derivations replace the actual per-frame average power in the PAPR denominator with the expectation E{|x^DT|^2}. In a finite frame, the sample mean fluctuates, so the true PAPR can exceed the stated bound. The stress-test example makes this concrete: a 16-QAM frame with one corner symbol and 1023 low-energy symbols gives PAPR about 8.9 for interleaved allocation, against the claimed bound of 1.8. That is a real gap in the central claim, though it does not break the comparative conclusions. The interleaved-versus-block ordering and the RRC-versus-rectangular ordering are still credible and the simulations support them.\n\nWhat is genuinely new: Eq. (6) shows interleaved allocation produces Q repetitions of the QAM symbols, a clean structural result that simplifies the transmitter. The block bound is re-derived independently and matches [10], which is good practice. The RRC analysis gives a usable bound for different roll-off factors, even though it leans on the peak-sum formulas from [15] without derivation.\n\nSoft spots beyond the main one: the RRC formulas (17) and (18) are hard to parse, and the choice of Lspan is not explained. The paper would be improved by stating the finite-frame deviation explicitly, either by redefining PAPR with the expected average power or by adding a probabilistic bound.\n\nBottom line: this paper deserves peer review. The fix is straightforward—make the normalization in the bound statements precise—and the underlying analysis is worth publishing for the periodic-repetition result and the allocation comparison. For people working on OTFS PAPR and waveform design, this is a useful reference.","headline":"Useful closed-form PAPR bounds for DFT-s-OTFS, but the 'upper bound' claim is not strictly true because the denominator is an expectation, not the per-frame average; a concrete 16-QAM frame exceeds the stated limit.","tokens_in":11284,"tokens_out":3536,"would_cite":true,"duration_ms":38961,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"DFT-s-OTFS PAPR bounds: interleaved Doppler allocation caps PAPR at the QAM constellation ceiling, while block allocation multiplies it by the spreading size.","keywords":["PAPR","OTFS","DFT-spread OTFS","Doppler division multiple access","pulse shaping","root raised cosine","resource allocation","uplink"],"falsifier":"For the rectangular-pulse interleaved case with, say, $M=128$, $N=32$, $Q=4$, $K=8$, and 16-QAM, generate many independent frames and compute the PAPR of each frame exactly as defined in equation (4), using that frame's realized average power in the denominator; if any frame exceeds $3(\\sqrt{M}-1)^2/(M-1)$ (about 2.55 dB), the universal upper bound as stated fails for that definition.","tokens_in":10169,"feed_emoji":"📡","tokens_out":11051,"duration_ms":113298,"temperature":0.7,"pith_summary":"This paper asks how much the peak-to-average power ratio (PAPR) of OTFS can be reduced by DFT-spreading data across the Doppler dimension before assigning subcarriers to uplink users. It derives closed-form upper bounds on PAPR for two Doppler resource allocation schemes: with interleaved allocation and a rectangular transmit pulse, the ceiling is set only by the QAM constellation, $3(\\sqrt{M}-1)^2/(M-1)$, independent of the number of users $Q$ and the spreading size $K$; with block allocation the ceiling is $K$ times larger. Interleaving also turns the time-domain waveform into a periodic repetition of the original QAM symbols, which removes the need for explicit DFT-IDFT processing at the transmitter. A root-raised-cosine pulse raises both ceilings by a factor $g_0^2$ that depends on the roll-off factor, and the paper gives explicit formulas for $g_0$. These bounds give a system designer a direct way to compare allocation schemes and pulse shapes before implementation, and BER simulations show the PAPR reduction does not cost error-rate performance.","feed_headline":"Interleaved Doppler bins cap OTFS PAPR at a QAM-only bound","feed_subtitle":"A closed-form ceiling, independent of user count and spreading size, tells uplink designers which allocation and pulse to pick.","key_machinery":"The load-bearing mechanism is a DFT precoder applied along the Doppler dimension before Doppler-bin allocation, so each user spreads its QAM symbols over $K$ Doppler bins. In interleaved allocation the mapping $n=Qk+q$ makes the subsequent Doppler IDFT collapse into the identity $X_q^{DT}[m,l]=\\frac{1}{\\sqrt{Q}}e^{j2\\pi ql/N}X_q[m,((l))_K]$, meaning the time samples are $Q$ repetitions of the original QAM symbols scaled by $1/\\sqrt{Q}$; this single identity supplies both the peak-power bound and the transmitter-simplification result. In block allocation the same substitution leaves a superposition over all $K$ symbols, and a Cauchy-Schwarz bound over the resulting $K\\times K$ phase matrix is what injects the factor $K$. For root-raised-cosine pulses the work is done by the inequality $\\max_t\\left|\\sum_i x_i g(t-i\\Delta\\tau)\\right|\\le g_0\\max_i|x_i|$, with $g_0=\\max_t\\sum_i|g(t-i\\Delta\\tau)|$ computed from the roll-off formulas.","core_discovery":"The central claim is that the PAPR of DFT-s-OTFS in the uplink obeys deterministic ceilings given by equations (10), (19), (23), and (25) of the paper. For interleaved Doppler division multiple access with rectangular pulse shaping, the peak power is at most the largest QAM symbol power divided by $Q$, while the average power is $1/Q$; the quotient leaves the constellation-only bound $\\mathrm{PAPR}\\le 3(\\sqrt{M}-1)^2/(M-1)$. For block allocation, coherent superposition of the $K$ spread symbols at the Doppler IDFT output introduces a factor $K$, so the ceiling becomes $3K(\\sqrt{M}-1)^2/(M-1)$. With a root-raised-cosine transmit filter, both ceilings are multiplied by $g_0^2$, where $g_0$ is the maximum over time of the sum of absolute pulse replicas; the paper derives $g_0$ separately for roll-off factors below and above $0.4$. It further claims that interleaved allocation produces a time-domain signal that is just $Q$ scaled repetitions of the original QAM symbols, simplifying the transmitter, and that bit error rate is essentially unchanged relative to OTFS without DFT spreading.","pith_inferences":["Editorial extension: because the bounds divide by the ensemble average power, a system that measures PAPR per frame with the realized average can see values above these ceilings on short frames; a practical rule would normalize by frame energy or add a small back-off.","Editorial extension: the periodic repetition property of interleaved allocation is an unused degree of freedom; permuting, rotating, or scrambling the $Q$ repeated copies of each QAM symbol could shape the peak further without changing the average power.","Editorial extension: the same $g_0$ triangle-inequality machinery applies to any square-root Nyquist pulse, so a designer could precompute $g_0$ for other filters and immediately obtain analogous PAPR ceilings.","Editorial extension: the block-allocation bound appears loose in the paper's own simulations; a sharper bound might follow from exploiting the phase structure of the $K\\times K$ superposition matrix instead of Cauchy-Schwarz, likely replacing the worst-case factor $K$ with something closer to the average-case peak."],"forward_implications":["For interleaved Doppler allocation, the PAPR ceiling does not grow with the number of users $Q$ or the spreading size $K$; the paper's simulations show changing $K$ moves the PAPR distribution by only about 0.5 dB.","Block allocation carries a linear penalty: its ceiling is $K$ times the interleaved ceiling, so for a fixed frame, interleaved allocation is the better uplink choice for PAPR.","The interleaved time-domain signal is $Q$ scaled repetitions of the original QAM symbols, so the transmitter can skip explicit DFT-IDFT processing and simply repeat and scale symbols.","A root-raised-cosine pulse multiplies the ceiling by $g_0^2$, and the roll-off value near $\\beta=0.4$ gives the lowest maximum PAPR, with a slight rise toward $\\beta=1$.","DFT spreading does not change bit error rate relative to plain OTFS under the simulated EVA channel, so the PAPR gain is not bought with error-rate loss."],"supporting_citations":[{"why":"It introduces DFT-s-OTFS with block Doppler allocation, the baseline scheme whose block PAPR bound the paper re-derives and extends.","marker":"[10]"},{"why":"It establishes the high PAPR of OTFS and serves as the comparison waveform for the PAPR simulations.","marker":"[5]"},{"why":"It supplies the DFT-spreading technique for PAPR reduction in single-carrier FDMA that the paper transfers to the Doppler dimension.","marker":"[9]"},{"why":"It supports the claim that pulse shaping materially changes PAPR, motivating the rectangular and RRC pulse analysis.","marker":"[11]"},{"why":"It provides the RRC pulse formulas, the pulse-peak sums for roll-off below and above 0.4, and the average-power relation used in the RRC bounds.","marker":"[15]"},{"why":"It gives the maximum power of an M-QAM constellation that enters the peak-power bounds.","marker":"[13]"},{"why":"It defines the EVA channel model used in the BER simulations that show no error-rate loss versus plain OTFS.","marker":"[16]"}],"fun_headline_variants":["Interleaved Doppler bins cap OTFS PAPR at QAM-only bound","DFT-s-OTFS PAPR bound: interleaved beats block by factor K","Closed-form PAPR ceilings for DFT-s-OTFS with pulse shaping","Interleaved Doppler allocation: lower PAPR and simpler transmitter","Pulse shaping impacts PAPR bound: rectangular beats RRC in DFT-s-OTFS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's PAPR upper bounds replace a finite frame's actual average power in the denominator with its statistical expectation over random QAM symbols; if one particular frame has an average power below the ensemble mean, that frame's true PAPR can exceed the claimed ceiling.","fun_headline_variants_meta":{"raw":{"variants":["Interleaved Doppler bins cap OTFS PAPR at QAM-only bound","DFT-s-OTFS PAPR bound: interleaved beats block by factor K","Closed-form PAPR ceilings for DFT-s-OTFS with pulse shaping","Interleaved Doppler allocation: lower PAPR and simpler transmitter","Pulse shaping impacts PAPR bound: rectangular beats RRC in DFT-s-OTFS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001093,"raw_usage":{"total_tokens":4607,"prompt_tokens":1027,"completion_tokens":3580,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":3477}},"tokens_in":643,"tokens_out":3580,"duration_ms":32521,"temperature":1.0,"reasoning_tokens":3477,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:52:24.674606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the rectangular-pulse interleaved case with, say, $M=128$, $N=32$, $Q=4$, $K=8$, and 16-QAM, generate many independent frames and compute the PAPR of each frame exactly as defined in equation (4), using that frame's realized average power in the denominator; if any frame exceeds $3(\\sqrt{M}-1)^2/(M-1)$ (about 2.55 dB), the universal upper bound as stated fails for that definition.","supporting_citations":[{"cited_title":"DFT-Spread orthogonal time f requency space modulation design for terahertz communications,","cited_arxiv_id":null,"evidence_quote":"It introduces DFT-s-OTFS with block Doppler allocation, the baseline scheme whose block PAPR bound the paper re-derives and extends."},{"cited_title":"Pea k-to- average power ratio of OTFS modulation,","cited_arxiv_id":null,"evidence_quote":"It establishes the high PAPR of OTFS and serves as the comparison waveform for the PAPR simulations."},{"cited_title":"Peak-to-average p ower ratio of single carrier FDMA signals with pulse shaping,","cited_arxiv_id":null,"evidence_quote":"It supplies the DFT-spreading technique for PAPR reduction in single-carrier FDMA that the paper transfers to the Doppler dimension."},{"cited_title":"Peak-to-average power rat io and inter- symbol interference reduction by nyquist pulse optimizati on,","cited_arxiv_id":null,"evidence_quote":"It supports the claim that pulse shaping materially changes PAPR, motivating the rectangular and RRC pulse analysis."},{"cited_title":"Root-raised cosine filter influences on papr distribution of single carrier signals,","cited_arxiv_id":null,"evidence_quote":"It provides the RRC pulse formulas, the pulse-peak sums for roll-off below and above 0.4, and the average-power relation used in the RRC bounds."},{"cited_title":"V asudevan, Digital communications and signal processing","cited_arxiv_id":null,"evidence_quote":"It gives the maximum power of an M-QAM constellation that enters the peak-power bounds."},{"cited_title":"5G; study on channel model for frequencies from 0 .5 to 100 GHz,","cited_arxiv_id":null,"evidence_quote":"It defines the EVA channel model used in the BER simulations that show no error-rate loss versus plain OTFS."}],"review_version":1}