Multi-Carrier Faster-Than-Nyquist Signaling for OTFS Systems

Chengwen Xing; J. Andrew Zhang; Shiqi Gong; Wenqian Shen; Xueyang Wang

REVIEW 3 major objections 8 minor 39 references

Multi-Carrier Faster-Than-Nyquist Signaling for OTFS Systems

T0 review · 3 major / 8 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that packing OTFS pulses tighter than the Nyquist grid in both time and frequency, with transmitter precoding and power allocation, raises capacity above conventional Nyquist-criterion OTFS, and that a SIC-based MIMO…

desk verdict The claimed MC-FTN-OTFS capacity gain is likely an SNR artifact from a missing sqrt(alpha*beta*E0) normalization; the paper is otherwise a competent extension with a fixable flaw. read the letter →

arxiv 2501.06750 v1 pith:ZFQZFVPP submitted 2025-01-12 eess.SP

Xueyang Wang , Shiqi Gong , Wenqian Shen , Chengwen Xing , J. Andrew Zhang This is my paper · ORCID

classification eess.SP

keywords OrthogonaltimefrequencyspacemodulationFaster-than-NyquistsignalingMulti-carrierFTNDelay-DopplerprecodingEigenvaluedecompositionOptimalpowerallocationMIMOcapacityDoubly-selectivefading

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

The paper proposes a multi-carrier faster-than-Nyquist (MC-FTN) signaling scheme for OTFS systems, where pulses are compressed in both time and frequency rather than only in time as in classical FTN. The authors argue that this dual compression, together with a carefully designed delay-Doppler precoder and optimal power allocation, significantly increases the achievable capacity of OTFS in doubly-selective (high-mobility) channels. They derive input-output models for SISO and MIMO, give an EVD-based precoding solution with water-filling-style power allocation for SISO, and a lower-complexity SIC-based precoding scheme for MIMO. Numerical results are presented as evidence that the proposed scheme beats Nyquist-criterion OTFS and classical FTN in terms of normalized capacity, with a modest BER penalty under an MMSE receiver.

What carries the argument

The load-bearing object is the MC-FTN-induced interference matrix G, whose entries are sampled cross-ambiguity functions of the transmit and receive pulses; it captures the ISI and ICI created by packing pulses at intervals αT₀ and βΔf₀. The argument proceeds by whitening the received noise with $G^{{-1/2}}$, forming the effective channel D = $G^{{-1/2}}$(F_N^H ⊗ F_M)H_DD, and then diagonalizing D^H D by EVD. The precoder P = U_D $Λ_P^{{1/2}}$ rotates and powers the data symbols along the eigenmodes, while optimal power allocation follows from a water-filling-like condition with the constraint tr(Λ_P Φ) ≤ MN, where Φ = U_D^H G U_D. For MIMO, the same machinery is applied successively to each data stream, using auxiliary matrices T_{n_t-1} that accumulate the contributions of previously solved streams.

What would settle it

Run the proposed EVD- and SIC-based precoders using channel estimates obtained from a practical DD-domain pilot scheme rather than perfect H_DD and G, then measure the mutual information or coded rate with the same parameters; if the capacity gain over Nyquist OTFS shrinks to zero or reverses at typical estimation SNRs, the central claim that MC-FTN-OTFS 'achieves significantly higher capacity' would be refuted in the realistic regime.

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Extended reading notes

Core claim

The paper's central claim is that allowing non-orthogonal pulse packing in both the time and frequency dimensions of an OTFS system, through compression factors α and β less than one, increases the normalized capacity relative to systems obeying the Nyquist criterion in either or both dimensions. The increase is achieved by precoding the delay-Doppler symbols so that the MC-FTN-induced inter-symbol and inter-carrier interference is diagonalized, after which optimal power allocation further boosts the rate. For SISO systems, an eigenvalue decomposition of the whitened channel matrix yields a precoder P = U_D $Λ_P^{{1/2}}$ with power coefficients λ_{P,k} chosen by a Lagrange multiplier method. For MIMO, the capacity maximization is split into sequential per-stream subproblems, each solved with the same EVD-plus-power-allocation machinery, giving performance close to the full water-filling solution at reduced complexity. The paper asserts that this constitutes a practical way to exploit the time-bandwidth resource more fully in high-mobility OTFS links.

Load-bearing premise

The transmitter must know the full delay-Doppler channel matrix and the pulse shaping matrix exactly, because the precoder and power allocation are computed from them; the paper only notes this in passing and does not analyze what happens when the channel is estimated imperfectly.

Editorial extensions

If this is right

If the central claim is correct, OTFS systems can operate with sub-Nyquist time-frequency packing to raise spectral efficiency in high-mobility links without expanding bandwidth or power.
The EVD precoder with optimal power allocation provides a concrete, closed-form transmitter design for SISO MC-FTN-OTFS, replacing brute-force numerical capacity optimization.
The SIC-based MIMO precoder reduces the complexity of capacity maximization from O((MNN_T)^3) to N_T instances of O((MN)^3), making the approach scalable to large antenna arrays.
The capacity gain persists under practical MMSE equalization, though with a moderate BER penalty, suggesting the scheme is compatible with existing receiver structures.
The framework reduces to classical FTN when the frequency compression factor β = 1, and to Nyquist OTFS when α = β = 1, so it generalizes prior signaling designs.

Reading between the lines

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

A likely unstated limitation is that the normalized capacity is measured per unit of the compressed time-bandwidth product, so the apparent gain may partly reflect a different resource-counting convention rather than a strictly larger information-theoretic region; comparing total achievable bits per frame at fixed bandwidth and latency would sharpen the claim.
The strong dependence on exact knowledge of H_DD and G at the transmitter suggests that channel estimation errors, which are inevitable in practice, could erode the advantage; a natural testable extension is to evaluate the scheme with estimated channels from pilot-aided DD-domain estimators.
The same EVD-precoding rationale could be applied to other doubly-selective waveform designs, such as generalized frequency-division multiplexing or index-modulated FTN, where a Gram-type interference matrix appears in the capacity expression.
The BER results are shown for a fixed LDPC code and MMSE equalizer; iterative receivers or joint equalization-and-decoding may close the BER gap, a direction the paper does not explore.

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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

Short version: the paper's headline claim—that MC-FTN-OTFS significantly beats Nyquist OTFS capacity—is probably an artifact of a missing normalization factor. The transmitted signal in Eq. (3) includes sqrt(alpha*beta*E0), but the derived input–output relation and the capacity expression (Eqs. (8), (14), (32)) drop it. The energy constraint in (36) keeps alpha*beta*E0, so the capacity is computed at an SNR that is 1/(alpha*beta) higher than the actual one. For the simulated (0.8, 0.9), that's about 1.4 dB—roughly the size of the reported gain. This is an internal inconsistency, not a matter of convention.

What's legitimately here: the MC-FTN-OTFS signal model is new, and the EVD precoder with water-filling in the DD domain plus the SIC decomposition for MIMO is a competent extension of Ishihara and Sugiura's FTN precoding and Gao et al.'s SIC approach. The capacity derivation is standard once you accept the model, and the paper gives credit where it's due. The numerical trends (capacity grows as alpha and beta shrink) are plausible, but they need a fair comparison.

Other soft spots are minor. Perfect CSIT of the full DD channel and pulse Gram matrix is assumed, with only a one-line caveat; that's a real limitation but secondary. Appendix A's proof of Proposition 1 shows a single path instead of summing over L. The claimed SIC complexity reduction omits the cost of forming and inverting T_{nt-1} each step. The simulation description is vague about delay/Doppler statistics, making it hard to reproduce.

Bottom line: this is a fixable but load-bearing flaw. If the normalization is corrected, the capacity gain may shrink or disappear, though the scheme might still have value for packing more symbols into a given time–frequency footprint. I'd send it to review—a competent referee can catch the factor and the authors can rerun the experiments—but I wouldn't cite the current capacity numbers.

Referee Report

3 major / 8 minor

Summary. The paper proposes multi-carrier faster-than-Nyquist (MC-FTN) signaling for OTFS systems, packing data symbols with non-orthogonal pulses in both time (factor α) and frequency (factor β). It develops TF- and DD-domain input-output models for SISO and MIMO systems; for SISO it derives an EVD precoder with optimal water-filling power allocation, and for MIMO it proposes a SIC-based decomposition of the capacity maximization into per-stream subproblems. Numerical results report significantly higher normalized capacity than Nyquist-criterion-based OTFS and classical single-carrier FTN, with the SIC MIMO precoder approaching the optimal water-filling performance at reduced complexity.

Significance. The framework is a direct and useful extension of eigendecomposition-precoded FTN (Ishihara and Sugiura, [11]) to doubly-selective channels in the OTFS setting, and the MIMO SIC decomposition addresses a genuinely high-dimensional optimization. The mathematical core is largely sound and checkable: the determinant identities in Eqs. (27) and (49)-(52) follow, the water-filling solution (39) has the expected form, and the model appears to contain no fitted parameters or circular arguments. The main qualification is that the capacity expression omits the transmit normalization sqrt(αβE0) introduced in Eq. (3); this bears directly on the headline capacity claim. If this is corrected and the related complexity and reproducibility issues are addressed, the paper would be a solid contribution to the OTFS and FTN literature.

major comments (3)

[II-A, III-A, III-B, V (Eqs. (3), (8), (14), (27), (32), (34), (40))] The capacity expression is not consistent with the transmit signal model. Eq. (3) defines s(t) with the factor sqrt(αβE0), so after the matched filter and SFFT the signal component in Eqs. (8) and (14) carries the same factor, i.e., y_DD = sqrt(αβE0) H_DD x_DD_P + z_DD. However, the covariance E[y_DD y_DD^H] in Eq. (24) and the capacity in Eqs. (27) and (32) omit this factor and use only σ_x²/N0. Since the transmit energy constraint in Eqs. (34)-(36) correctly includes αβE0 σ_x², the optimization problem (37)-(39) is internally inconsistent: the water-filling levels and the capacity in (32) correspond to a per-symbol SNR of σ_x²/N0, whereas the average transmit energy delivered by (3) corresponds to αβE0 σ_x²/N0. Consequently, in Figs. 5-9 the MC-FTN curves (αβ<1) are evaluated at an SNR that the transmitted signal does not actually attain for the reported σ_x², N0, and E0=1, so part of the shown capacity advantage over the Nyquist benchmark is an artifact of the missing factor. Please re-derive the capacity including the normalization (or, equivalently, state explicitly that σ_x² denotes the post-scaling symbol energy), re-solve the power allocation in (39), and re-run the numerical comparisons; the abstract's 'significantly higher capacity' claim should be re-verified after this correction.
[IV-A, IV-B, Algorithm 1] The claimed complexity reduction for the SIC-based MIMO precoder is not established. Section IV-A states that problem (48) with complexity O((MNN_T)³) is decomposed into N_T subproblems, each 'solved with the complexity of O((MN)³)'. However, Step 1 of Algorithm 1 requires forming Q_{nt-1} = D_MIMO^H T_{nt-1}^{-1} D_MIMO for each stream, where T_{nt-1} ∈ C^{MNN_R × MNN_R}; a direct computation of T_{nt-1}^{-1} costs O((MNN_R)³) per iteration, which dominates the O((MN)³) per-stream cost and negates the advertised saving when N_R ~ N_T. If a low-rank Woodbury update is intended, the matrix to invert has dimension (nt-1)MN, giving a total complexity on the order of (MN)³ Σ_{k=0}^{N_T-1} k³, which is not O(N_T(MN)³) and is not smaller than O((MNN_T)³). The 'low-complexity' claim in the abstract and Section IV should be supported by a clear complexity analysis of the per-iteration matrix cost.
[V (Simulation setup)] The simulation parameters in Section V are incomplete, which prevents reproduction of Figs. 5-9. The channel is described only as 'randomly generated channel coefficients according to CN(0,1/L)' with L=3; the delay and Doppler values τ_i and ν_i (their ranges, whether integer or fractional multiples of the DD grid spacings 1/(MΔf0) and 1/(NT0), and their distributions) are never specified, even though the equivalent channel H_DD in Eqs. (11)-(15) depends on them. In addition, the x-axis 'SNR(dB)' is not defined: the text never states whether SNR = σ_x²/N0, E_s/N0, or another quantity, and the noise variance N0 and symbol variance σ_x² are not given. Please specify all channel statistics and the SNR convention, and state σ_x² explicitly, so that the numerical claims can be independently verified.

minor comments (8)

[II-A, Proposition 1, Eq. (8)] In Eq. (8) the summation ranges are transposed: m' runs from 0 to N-1 and n' from 0 to M-1, whereas the definitions in Eqs. (2)-(3) require m'=0,...,M-1 and n'=0,...,N-1. The same transposition appears in Eq. (13) and in the intermediate display (64) of Appendix B.
[Appendix A] The proof of Proposition 1 passes from the channel response h(τ,ν)=Σ_i h_i δ(τ-τ_i)δ(ν-ν_i) to a single path with coefficient h_i at intermediate step (62), without showing the sum over L or the action of the delta functions; the sum reappears only in the final formula (63). Insert the missing steps so that the proof is valid for L>1.
[Proposition 2, Eq. (13)] In Eq. (13), the phase term 'e^{j2π(n'k'/N − m'e'/M)}' should read 'e^{j2π(n'k'/N − m'l'/M)}'; the symbol e' is undefined. The same typo occurs in Appendix B.
[III-B, Eq. (40); IV-B, Eq. (61)] The normalization in Eqs. (40) and (61) includes E0 in the denominator, which is dimensionally inconsistent with bits/s/Hz, since E0 has units of power. This does not affect the figures because E0=1 in the simulations, but the definition should be stated (e.g., capacity per unit bandwidth per unit transmit power) or E0 should be removed from the normalization.
[V, Fig. 5] In Fig. 5, the curve with (α,β)=(1,0.9) and θ=0 is labeled 'Nyquist'; with β=0.9 the frequency-domain pulse packing is non-orthogonal, so the label 'Nyquist' is misleading and should be relabeled.
[II-A] The assumption of perfect channel state information for computing the precoder and power allocation is acknowledged only by a passing sentence in Section II-A ('this aspect is beyond the scope of this paper'); since the entire capacity gain depends on exact knowledge of H_DD and G, the assumption should be stated explicitly at the start and its practical limitations discussed in the conclusions.
[IV-B, Eq. (51); Algorithm 1] The symbol P_nt is used for both the MNN_T×MN block-structured matrix and the MN×MN submatrix in Eqs. (51)-(55) and Algorithm 1; use distinct notations (for example, P̄_nt and P_nt) to avoid confusion, particularly in the definition P_nt = [0^T, P_nt^T, 0^T]^T.
[Throughout] Minor language and typographical issues include: 'vetorized' (Section II-A), 'obatin' (end of Section IV-B), and 'the (nr,nt)-th element hnr,nt(τ,ν) is modeled similarly to (5)' in Section II-B, which should reference Eq. (4). In addition, the sentence 'when the channel state information (CSI) is unknown' at the end of Section II-A appears to contradict the preceding reliance on known H_DD and G, and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the precoder and capacity derivations are self-contained and do not reduce to their inputs.

full rationale

The paper's central derivation chain is: define the MC-FTN-OTFS transmit signal in Eq. (3), derive TF and DD input-output relations in Propositions 1-3, compute mutual information, use EVD precoding with water-filling power allocation in Section III, and extend via determinant identities to a SIC-based MIMO precoder in Section IV. Each step follows from the stated Gaussian-input model, channel assumptions, and pulse shapes, rather than assuming the capacity conclusion. The EVD precoder P = U_D Lambda_P^{1/2} and the water-filling solution in Eq. (39) are obtained from Hadamard's inequality and the Lagrange conditions, not from fitting the numerical capacity curves. Self-citations such as [16]-[18], [25], and [32] are background references for existing MC-FTN and OTFS models; none is invoked as a load-bearing uniqueness or existence theorem, and the core capacity formula is re-derived in the paper itself. The apparent omission of the sqrt(alpha beta E0) normalization factor between Eq. (3) and Proposition 1/Appendix A is a numerical/consistency concern about the SNR axis used in Section V, but it is not circularity: no quantity is defined in terms of the target result, and no fitted parameter is relabeled as a prediction.

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

The capacity result rests on the Gaussian input assumption, the discrete-path channel model, the invertibility of G under α ≥ 1/(1+θ), and the matched-filter receiver. No free parameters are fitted; α, β, and θ are system design parameters, and the power allocation is water-filling.

assumptions (4)

domain assumption The DD channel consists of L discrete paths with complex Gaussian gains h_i ~ CN(0,1/L) and known delays τ_i and Dopplers ν_i.
Stated in Section II-A, Eq. (4); the precoder design requires exact H_DD, which depends on these parameters.
domain assumption Input symbols are Gaussian with average power σ_x^2, making the differential entropy bound tight and the capacity formula exact for Gaussian codebooks.
Stated in Section II-A; used in (19)-(26).
domain assumption The pulse shape is a root-raised-cosine with roll-off θ and the time compression factor satisfies α ≥ 1/(1+θ), guaranteeing the FTN Gram matrix G is invertible.
Stated in Section III-A: 'we assume α ≥ 1/(1+θ) to guarantee that G is an invertible matrix'; G invertibility is required for the whitening step.
domain assumption The receiver uses a matched filter g_rx(t)=g_tx*(-t) and samples on the compressed grid (mβΔf0, nαT0).
Stated in Section II-A, Eqs. (6)-(7); the input-output relationship and noise correlation (22) depend on this structure.

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Pith. "Pith review of Multi-Carrier Faster-Than-Nyquist Signaling for OTFS Systems." pith.science (2026). https://pith.science/paper/ZFQZFVPP

@misc{pith2026250106750,
  author       = {Pith},
  title        = {Pith review of: Multi-Carrier Faster-Than-Nyquist Signaling for OTFS Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFQZFVPP}},
  note         = {Machine review of arXiv:2501.06750}
}

read the original abstract

Orthogonal time frequency space (OTFS) modulation technique is promising for high-mobility applications to achieve reliable communications. However, the capacity of OTFS systems is generally limited by the Nyquist criterion, requiring orthogonal pulses in both time and frequency domains. In this paper, we propose a novel multi-carrier faster-than-Nyquist (MC-FTN) signaling scheme for OTFS systems. By adopting non-orthogonal pulses in both time and frequency domains, our scheme significantly improves the capacity of OTFS systems. Specifically, we firstly develop the signal models for both single-input single-output (SISO) and multiple-input multiple-output (MIMO) OTFS systems. Then, we optimize the delay-Doppler (DD) domain precoding matrix at the transmitter to suppress both the inter-symbol interference (ISI) and inter-carrier interference (ICI) introduced by the MC-FTN signaling. For SISO systems, we develop an eigenvalue decomposition (EVD) precoding scheme with optimal power allocation (PA) for achieving the maximum capacity. For MIMO systems, we develop a successive interference cancellation (SIC)-based precoding scheme via decomposing the capacity maximization problem into multiple sub-capacity maximization problems with largely reduced dimensions of optimization variables. Numerical results demonstrate that our proposed MC-FTN-OTFS signaling scheme achieves significantly higher capacity than traditional Nyquist-criterion-based OTFS systems. Moreover, the SIC-based precoding scheme can effectively reduce the complexity of MIMO capacity maximization, while attaining performance close to the optimal EVD-based precoding scheme.

Figures

Figures reproduced from arXiv: 2501.06750 by the authors.

**Figure 2.** The diagram of the MIMO MC-FTN-OTFS system. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The diagram of the proposed SIC-based precoding sche [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The signal processing flowchart of the proposed MC-FTN-OTFS transmitter and receiver. 0 4 8 12 16 20 SNR(dB) 1 2 3 4 5 6 7 8 9 Normalized Capacity ( , ) = (1,0.9), Nyquist with PA, = 0 ( , ) = (1,0.9), Nyquist without PA, = 0 ( , ) = (0.8,0.9), MC-FTN with PA ( , ) = (…

**Figure 5.** Figure 5: Normalized capacity CSISO nor of our proposed MC-FTN-OTFS signaling scheme with and without PA. The compression factors are set as (α, β) = (1, 1), (α, β) = (0.9, 1), (α, β) = (0.9, 0.9), (α, β) = (0.8, 0.9) and (α, β) = (1, 0.9). A. MC-FTN-OTFS Signaling Scheme for th…

**Figure 7.** Figure 7: Normalized capacity CMIMO nor of our proposed MC-FTN-OTFS signaling with SIC-based and WF-based precoding schemes with different numbers of antennas, which are set as (NT , NR) = (1, 1), (NT , NR) = (2, 2) and (NT , NR) = (4, 4). The compression factors are set as (α, …

**Figure 8.** Figure 8: Normalized capacity CMIMO nor of our proposed MC-FTN-OTFS signaling with SIC-based and WF-based precoding schemes under different compression factors, i.e., (α, β) = (1, 1), (α, β) = (0.9, 0.9) and (α, β) = (0.8, 0.8). The number of transmitted and received antennas a…

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