PAPR of DFT-s-OTFS with Pulse Shaping

Arman Farhang; Jialiang Zhu; Sanoopkumar P. S.

REVIEW 2 major objections 5 minor 16 references

PAPR of DFT-s-OTFS with Pulse Shaping

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read DFT-s-OTFS PAPR bounds: interleaved Doppler allocation caps PAPR at the QAM constellation ceiling, while block allocation multiplies it by the spreading size.

desk verdict 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. read the letter →

arxiv 2507.12210 v1 pith:653F3TNJ submitted 2025-07-16 eess.SP

Jialiang Zhu , Sanoopkumar P. S. , Arman Farhang This is my paper · ORCID

classification eess.SP

keywords PAPROTFSDFT-spreadDopplerdivisionmultipleaccesspulseshapingrootraisedcosineresourceallocationuplink

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Bottom 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.

Referee Report

2 major / 5 minor

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).

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 (2)

[IV-A, Eqs. (20)-(23)] 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.
[V, Figs. 2-4] 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.

minor comments (5)

[Throughout] 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.
[III-B, Eq. (11)] 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.
[III-B, Eqs. (17)-(18)] 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.
[V, Fig. 4] '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.
[VI, Conclusion] 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.

Circularity Check

1 steps flagged · score 5.0 of 10

All four claimed PAPR upper bounds (Eqs. 10, 19, 23, 25) use E{|x_DT|^2} for the denominator (Eqs. 9, 15, 21, 24) instead of the per-frame average in the PAPR definition (Eq. 4), so the bounds hold for peak/expected-average power, not the stated PAPR; a valid 16-QAM frame gives PAPR ≈ 8.93 against Eq. (10)'s claimed 1.8.

other [Sec. III-A, Eqs. (4), (9), (10); same substitution at Sec. III-B Eq. (15), Sec. IV-A Eq. (21), Sec. IV-B Eq. (24); claimed bounds Eqs. (10), (19), (23), (25)]
"The average power of the transmit signal, xq(t) can be calculated as Pavg = 1/MN Σ_{i=0}^{MN−1} E{|xDT_q[i]|²} = ... = 1/Q ... Using (8) and (9), the upper bound for PAPR of the DFT-s-OTFS transmit signal with interleaved DoDMA at each user terminal is obtained as PAPR_{DFT-s-OTFS}^{interleaved} ≤ 3(√M−1)²/(M−1)."

The proof chain from (4) to (10) replaces the per-frame denominator of the PAPR ratio, (1/T)∫|xq(t)|²dt, with the ensemble expectation: Eq. (9) computes Pavg = (1/MN)Σ E{|xDT_q[i]|²} = 1/Q. For a rectangular pulse the true denominator is the random sample mean (1/MN)Σ|xDT[i]|², which on valid frames can fall below its expectation, so the derived inequality bounds peak/(E[average power]), not the PAPR defined in Eq. (4). A valid 16-QAM frame with one 1.8-energy corner symbol and 1023 minimum-energy (0.2) symbols yields PAPR ≈ 8.93, violating the claimed 1.8 bound of Eq. (10); by construction of the derivation the stated bound is just the input constellation's peak-to-expected-average ratio. The identical substitution recurs at Eqs.

full rationale

The derivation chain is mostly self-contained first-principles algebra: Eq. (6) and Eq. (20) follow from DFT geometric-sum identities, the peak-power bounds use standard QAM peak energy [13] plus triangle and Cauchy-Schwarz inequalities, and the RRC peak-sum factor g0 is taken from the external reference [15]. The block bound is re-derived independently and only afterwards compared with [10], which is external to the present authors, so the agreement is corroboration rather than a self-citation crutch. The only self-citation, [12], supports a routine note that the RF modulator raises PAPR by 3 dB and is not load-bearing. No parameter is fitted, no uniqueness theorem is imported, and no ansatz is smuggled through a self-citation. The single substantive defect is the average-power substitution flagged in the step: Eqs. (9), (15), (21), and (24) compute the ensemble expectation E{|xDT|²} and place it as the denominator, whereas Eq. (4) defines PAPR with the per-frame integral (1/T)∫|xq(t)|²dt; for rectangular pulses that integral is the sample mean of |xDT[i]|², which fluctuates and can be far below 1/Q on valid frames. Consequently the derived inequalities (10), (19), (23), and (25) control peak power divided by expected average power, a different ratio, and the deterministic upper-bound claims as stated are not established; the concrete counterexample (PAPR ≈ 8.93 against the claimed 1.8 bound) is computed entirely from the paper's own definitions and valid 16-QAM inputs. This is best read as a definitional equivocation and omitted justification in the proof rather than a fit-based or self-citation-based circularity, and the comparative conclusions (interleaved versus block differing by factor K, RRC raising the bound by g0², and the K-independence of interleaved) do not depend on the substitution and retain independent content. On the 0-10 scale, the mid-range score reflects that the absolute bound predictions reduce by construction to an input-ratio statement, while the structural derivation and external corroboration keep the claim from being wholly forced.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard DFT mathematics, a common finite-frame averaging approximation, and two external formulas (RRC peak sum, QAM peak power). No free parameters are fitted to data, and no new entities are introduced.

assumptions (4)

standard math DFT spreading and IDFT operations are unitary with orthogonality relation sum_k e^{j2π k(l−κ)/K} = K δ_{((l))_K,κ}.
Used to derive Eq. (6) for interleaved allocation.
domain assumption The average power in the PAPR definition can be replaced by the expectation E{|x^DT[i]|^2} over random QAM symbols.
In Eqs. (9), (15), (21), and (24), the frame average power is computed as the ensemble average. For finite MN, the sample average fluctuates, so the derived bounds are on peak-to-expected-average ratio, not strictly on per-frame PAPR.
domain assumption The maximum over t of the sum of absolute RRC pulse values, g0, is given by the expressions in (17) and (18) from reference [15].
Used to bound the peak power for RRC pulse shaping in (16), (19), and (25). Correctness depends on the external reference.
standard math The maximum instantaneous power of a unit-average-power M-QAM symbol is 3(√M−1)^2/(M−1).
Used in Eqs. (8) and (22) to bound the peak power. This is a standard result from [13].

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Pith. "Pith review of PAPR of DFT-s-OTFS with Pulse Shaping." pith.science (2026). https://pith.science/paper/653F3TNJ

@misc{pith2026250712210,
  author       = {Pith},
  title        = {Pith review of: PAPR of DFT-s-OTFS with Pulse Shaping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/653F3TNJ}},
  note         = {Machine review of arXiv:2507.12210}
}

read the original abstract

Orthogonal Time Frequency Space (OTFS) suffers from high peak-to-average power ratio (PAPR) when the number of Doppler bins is large. To address this issue, a discrete Fourier transform spread OTFS (DFT-s-OTFS) scheme is employed by applying DFT spreading across the Doppler dimension. This paper presents a thorough PAPR analysis of DFT-s-OTFS in the uplink scenario using different pulse shaping filters and resource allocation strategies. Specifically, we derive a PAPR upper bound of DFT-s-OTFS with interleaved and block Doppler resource allocation schemes. Our analysis reveals that DFT-s-OTFS with interleaved allocation yields a lower PAPR than that of block allocation. Furthermore, we show that interleaved allocation produces a periodic time-domain signal composed of repeated quadrature amplitude modulated (QAM) symbols which simplifies the transmitter design. Based on our analytical results, the root raised cosine (RRC) pulse generally results in a higher maximum PAPR compared to the rectangular pulse. Simulation results confirm the validity of the derived PAPR upper bounds. Furthermore, we also demonstrate through BER simulation analysis that the DFT-s-OTFS gives the same performance as OTFS without DFT spreading.

Figures

Figures reproduced from arXiv: 2507.12210 by the authors.

**Figure 2.** CCDF of PAPR of DFT-spread OTFS and OTFS for different [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** CCDF of PAPR of DFT-spread OTFS with different alloca [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: BER performance of DFT-sOTFS and OTFS versus SNR. From [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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DFT-Spread orthogonal time f requency space modulation design for terahertz communications,

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work page 2008

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OTFS-based ISAC: How delay doppler channel estimation assists environment sens ing?

B. Li, W. Y uan, F. Liu, N. Wu, and S. Jin, “OTFS-based ISAC: How delay doppler channel estimation assists environment sens ing?” IEEE Wireless Communications Letters, vol. 13, no. 12, pp. 3563–3567, 2024

work page 2024

[4] [2]

Orthogonal time frequency spa ce modu- lation,

R. Hadani, S. Rakib, M. Tsatsanis, A. Monk, A. J. Goldsmit h, A. F. Molisch, and R. Calderbank, “Orthogonal time frequency spa ce modu- lation,” in IEEE Wireless Communications and Networking Conference (WCNC), 2017, pp. 1–6

work page 2017

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A look at OTFS from a h ybrid carrier perspective,

Y . Dong, J. Lei, Y . Huang, and K. Lai, “A look at OTFS from a h ybrid carrier perspective,” in 10th International W orkshop on Signal Design and Its Applications in Communications (IWSDA) , 2022, pp. 1–5

work page 2022

[6] [4]

Peak to average power ratio anal ysis of multi-carrier and multi-standard signals in software radi o context,

S. Hussain and Y . Louet, “Peak to average power ratio anal ysis of multi-carrier and multi-standard signals in software radi o context,” in 3rd International Conference on Information and Communica tion Technologies: From Theory to Applications , 2008, pp. 1–5

work page 2008

[7] [5]

Pea k-to- average power ratio of OTFS modulation,

G. D. Surabhi, R. M. Augustine, and A. Chockalingam, “Pea k-to- average power ratio of OTFS modulation,” IEEE Communications Let- ters, vol. 23, no. 6, pp. 999–1002, 2019

work page 2019

[8] [6]

An expa nded precoding scheme for PAPR reduction in OTFS modulation,

J. Xiong, J. Zhu, J. Du, Y . Zhou, C. Yi, and Y . Tang, “An expa nded precoding scheme for PAPR reduction in OTFS modulation,” in IEEE Wireless Communications and Networking Conference , 2024, pp. 1–5

work page 2024