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arxiv: 2308.01802 · v2 · submitted 2023-08-03 · 💻 cs.IT · eess.SP· math.IT

Multi-Carrier Modulation: An Evolution from Time-Frequency Domain to Delay-Doppler Domain

Hai Lin , Jinhong Yuan , Wei Yu , Jingxian Wu , Lajos Hanzo This is my paper

Pith reviewed 2026-05-24 07:04 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.IT
keywords ODDMdelay-Doppler modulationmulti-carrier modulationlinear time-varying channelWeyl-Heisenberg framelocal biorthogonalityequivalent sampled DD domainintegrated sensing and communications
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The pith

Delay-Doppler orthogonal pulses behave like impulses in the equivalent sampled domain of linear time-varying channels, revealing a clean input-output relation for ODDM.

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

The paper establishes that the DD domain orthogonal pulse used in ODDM modulation achieves orthogonality with respect to the fine time and frequency resolutions of the equivalent sampled delay-Doppler representation of an LTV channel. This property makes the pulses function like impulses and produces a distinct input-output relation for the modulation scheme. The work further shows that the usual Weyl-Heisenberg frame requirements for multi-carrier signals can be relaxed to local or sufficient biorthogonality within the actual bandwidth and duration of a practical system. A reader would care because the relaxation keeps the orthogonality intact while opening waveform design options for channels with large delay or Doppler spreads.

Core claim

The DDOP achieves orthogonality with respect to the fine time and frequency resolutions in the ESDD domain thus behaves like an impulse function. This allows us to unveil the unique input-output relation of the resultant ODDM modulation over the ESDD channel. The paper points out that the conventional MC modulation design guidelines based on the WH frame theory can be relaxed without compromising its orthogonality or violating the WH frame theory, specifically by using local or sufficient (bi)orthogonality among a WH subset for the MC signal within a specific bandwidth and duration.

What carries the argument

The delay-Doppler orthogonal pulse (DDOP) that is orthogonal to the fine time-frequency grid of the equivalent sampled delay-Doppler (ESDD) representation of an LTV channel and therefore acts as an impulse.

If this is right

  • ODDM modulation over the ESDD channel has a unique and simple input-output relation based on the impulse-like DDOP.
  • Multi-carrier signals can be designed using local or sufficient (bi)orthogonality within the system's bandwidth and duration limits.
  • The relaxed design guideline preserves orthogonality and does not violate Weyl-Heisenberg frame theory.
  • The approach creates new waveform opportunities for systems with high delay or Doppler shifts and for integrated sensing and communications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-mobility links may gain robustness by moving the entire modulation design into the delay-Doppler domain where channel variations appear as simple shifts.
  • Joint communication-sensing systems could exploit the same DD representation that already matches radar observables.
  • Pulse-shaping work can now target only the local biorthogonality condition inside the occupied spectrum rather than global frame conditions.

Load-bearing premise

The linear time-varying channel admits an equivalent sampled delay-Doppler representation in which an impulse-function-based transmission strategy produces the claimed input-output relation without additional unmodeled interference or sampling artifacts.

What would settle it

A measurement or simulation over a realistic LTV channel that shows the ODDM received symbols deviate from the predicted impulse-like input-output mapping because of extra interference terms not captured by the ESDD model.

Figures

Figures reproduced from arXiv: 2308.01802 by Hai Lin, Jingxian Wu, Jinhong Yuan, Lajos Hanzo, Wei Yu.

Figure 1
Figure 1. Figure 1: TF grid in MC modulation has been widely adopted in wireless standards, such as the Wi-Fi [9], [10], the fourth generation mobile communication system (4G) [11], the fifth generation mobile communication system (5G) [12], and so on. In OFDM, to achieve eigenfunction-based transmission, the transmit pulses are exactly complex sinusoids termed as subcarriers and truncated by a common time-domain (TD) window … view at source ↗
Figure 2
Figure 2. Figure 2: Block diagram of a wireless communication link [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Metrics of pulse’s TF occupancy (γ, T , F) using another prototype pulse γ(t) having the same time and frequency resolutions. Meanwhile, the inverse of the frequency resolution is known as the symbol period denoted by T = 1/F. Then, the transmit waveform of MC modulation synthesized by the transmit pulses in (19) is given by x(t) = M X−1 m=0 N/ X 2−1 n=−N/2 X[m, n]g(t − mT )e j2πnF(t−mT ) , (20) where M is… view at source ↗
Figure 4
Figure 4. Figure 4: TF grid and signal localization of CP-OFDM [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Analog implementation with g(t) as transmit pulse/filter D. Implementation Methods Given T , F and g(t), how to generate the waveform in (20) for MC modulation at low complexity is of pivotal practical importance, especially when the number of subcarriers N is large. In theory, we have two direct analog approaches using N modulators associated with carrier frequencies of nF, n = − N 2 , · · · N 2 −1. As sh… view at source ↗
Figure 7
Figure 7. Figure 7: Digital implementation with g(t) as prototype pulse/window function As shown in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Oversampling based digital implementation [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Delay Doppler domain orthogonal pulse (DDOP) [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: g(t) orthogonal w.r.t F = 1 T for |n| ≤ N − 1 and fixed m to be shifted freely over the whole TF domain. As a result, g(t) is designed based only on F and T to achieve the global (bi)orthogonality in (21) and (22), and therefore bounded by the JTFR limit of R = 1. On the other hand, for MC modulations, in contrast to achieving the global (bi)orthogonality in (21) and (22), we only have to consider the suf… view at source ↗
Figure 12
Figure 12. Figure 12: Derivation of U(f) are inter-dependent. Therefore, they cannot form a real 2D domain (t, f) to let us design a 2D pulse/filter that can be denoted by g(t, f). In other words, a pulse is always a 1D function. On the other hand, we would point out that the delay variable τ and the Doppler variable ν, namely the time and frequency variables of the channel’s TF domain, are indepen￾dent. Therefore, they can fo… view at source ↗
Figure 13
Figure 13. Figure 13: Simplified TF signal localization of DDOP [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of simplified TF signal localization [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: ODDM waveform without CP and CS, when N = 4. 4 6 8 10 12 14 16 0.75 0.8 0.85 0.9 0.95 1 [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Approximate implementation of ODDM filtering/ pulse-shaping equivalent to discrete Zak transform [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Simplified approximate implementation of ODDM the low-complexity implementation of ODDM shown in [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: PSD, M = 512, N = 32, 4-QAM. 8 10 12 14 16 18 -50 -45 -40 -35 -30 NMSE (dB) [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: NMSE of approximated ODDM waveform, M = 512, N = 32, 4-QAM. 6 8 10 12 14 16 10-6 10-5 10-4 10-3 10-2 10-1 BER [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: BER comparison, M = 512, N = 32, 4-QAM [PITH_FULL_IMAGE:figures/full_fig_p022_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: BER comparison, M = 512, N = 32, 4-QAM, ρ = 0.05, Q = 16. 6 8 10 12 14 16 10-6 10-5 10-4 10-3 10-2 10-1 BER [PITH_FULL_IMAGE:figures/full_fig_p023_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: BER comparison, M = 512, 4-QAM, ρ = 0.05, Q = 16. of 80km/h, 120km/h and 500km/h. The figure demonstrates that the ODDM signals achieve almost the same BER perfor￾mance over the high-mobility channels regardless of the UE speed, which means that ODDM signals are robust against Doppler shifts. Meanwhile, [PITH_FULL_IMAGE:figures/full_fig_p023_23.png] view at source ↗
Figure 26
Figure 26. Figure 26: |Auce,u (0, ν)| for M = 32 and N = 8 [PITH_FULL_IMAGE:figures/full_fig_p023_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: uce(t) for D = 1 [PITH_FULL_IMAGE:figures/full_fig_p025_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: uce(t) for D = 2 end of the last subpulse of u(t − (M − 1)T0 M ), respectively. Recall that u(t) = N X−1 n=0 a(t − nT0), (70) we can divide P u(t) into N segments, where u(t) = N−1 n=0 un(t) and the nth segment is given by un(t) = ( u(t) nT0 ≤ t < (n + 1)T0 0 otherwise . (71) Let D = ⌈Ta/T0⌉. If D = 1, we have un(t) = a(t − nT0), (72) which implies that the periodicity within −(M − 1)T0 M ≤ t ≤ (MN − 1)T0… view at source ↗
read the original abstract

The recently proposed orthogonal delay-Doppler division multiplexing (ODDM) modulation, which is a delay-Doppler (DD) domain multi-carrier (DDMC) modulation scheme based on the DD domain orthogonal pulse (DDOP), is studied. We first revisit the linear time-varying (LTV) channel model for the wireless channel, and review the conventional multi-carrier (MC) modulation schemes and their design guidelines for both linear time-invariant (LTI) and LTV channels. We then focus on the representation of the LTV channel in an equivalent sampled DD (ESDD) domain, and propose an impulse-function-based transmission strategy for the ESDD channel. Next, we take an in-depth look into the DDOP and show that it achieves orthogonality with respect to the fine time and frequency resolutions in the ESDD domain thus behaves like an impulse function. This allows us to unveil the unique input-output relation of the resultant ODDM modulation over the ESDD channel. We point out that the conventional MC modulation design guidelines based on the Weyl-Heisenberg (WH) frame theory can be relaxed without compromising its orthogonality or violating the WH frame theory. More specifically, for a practical communication system with bandwidth and duration constraints, MC modulation signals can be designed considering so-called local or sufficient (bi)orthogonality, which refers to the (bi)orthogonality among a WH subset for the MC signal within a specific bandwidth and duration. This novel design guideline could potentially open up opportunities for developing future waveforms required by new applications such as communication systems associated with high delay and/or Doppler shifts, as well as integrated sensing and communications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies orthogonal delay-Doppler division multiplexing (ODDM) based on the delay-Doppler orthogonal pulse (DDOP). It revisits LTV channel models and conventional MC schemes, introduces an equivalent sampled DD (ESDD) representation of the LTV channel, proposes an impulse-function-based transmission strategy, shows that DDOP achieves orthogonality w.r.t. fine time/frequency resolutions in the ESDD domain (thus behaving like an impulse), derives the resulting input-output relation, and argues that Weyl-Heisenberg (WH) frame theory design guidelines can be relaxed to 'local or sufficient (bi)orthogonality' within bandwidth/duration constraints without compromising orthogonality or violating the theory. This is positioned as enabling new waveforms for high-mobility and ISAC applications.

Significance. If the ESDD equivalence and impulse-like property hold rigorously, the work offers a conceptual bridge from TF-domain to DD-domain MC modulation with relaxed design rules that could impact waveform design for high Doppler/delay channels. The explicit credit for the manuscript is its attempt to connect standard channel models and frame theory to a practical relaxation; however, the absence of visible derivations, error bounds, or numerical verification in the abstract limits immediate impact.

major comments (2)
  1. [ESDD channel and impulse-function-based strategy paragraphs] The ESDD representation of the LTV channel and the impulse-function-based transmission strategy (paragraphs following the review of conventional MC schemes): the central claim that DDOP behaves like an impulse yielding a unique input-output relation rests on this equivalence being exact. Discretization from continuous LTV to sampled DD can introduce aliasing or cross-bin interference terms not canceled by the claimed local biorthogonality; the manuscript must explicitly derive or bound these terms to confirm the impulse property.
  2. [paragraph on conventional MC modulation design guidelines] The relaxation of WH frame theory to local/sufficient (bi)orthogonality for a WH subset within bandwidth/duration constraints (final paragraph on design guidelines): while the paper asserts this does not violate WH theory, it is unclear whether subset biorthogonality within the ESDD grid guarantees the required frame bounds or impulse behavior for the full ODDM signal when the ESDD mapping is inexact. A concrete counter-example or proof that the relaxation preserves the input-output relation is needed.
minor comments (1)
  1. The abstract states claims conceptually without referencing specific equations or sections for the orthogonality proof or input-output relation; adding forward references would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify the technical details of our work. We respond to each major comment below.

read point-by-point responses

  1. Referee: [ESDD channel and impulse-function-based strategy paragraphs] The ESDD representation of the LTV channel and the impulse-function-based transmission strategy (paragraphs following the review of conventional MC schemes): the central claim that DDOP behaves like an impulse yielding a unique input-output relation rests on this equivalence being exact. Discretization from continuous LTV to sampled DD can introduce aliasing or cross-bin interference terms not canceled by the claimed local biorthogonality; the manuscript must explicitly derive or bound these terms to confirm the impulse property.

    Authors: The ESDD representation is constructed precisely as the sampled equivalent of the continuous LTV channel at the fine delay-Doppler resolutions matching the time-frequency grid. In the manuscript we derive the input-output relation by establishing that the DDOP is orthogonal exactly at these sampled points in the ESDD domain, which by construction precludes aliasing and cross-bin terms within the sampled grid. The local biorthogonality is defined with respect to those same points. To make the absence of residual interference fully explicit, we will add a step-by-step derivation of the input-output relation together with a bound on any discretization error in the revised manuscript. revision: yes

  2. Referee: [paragraph on conventional MC modulation design guidelines] The relaxation of WH frame theory to local/sufficient (bi)orthogonality for a WH subset within bandwidth/duration constraints (final paragraph on design guidelines): while the paper asserts this does not violate WH theory, it is unclear whether subset biorthogonality within the ESDD grid guarantees the required frame bounds or impulse behavior for the full ODDM signal when the ESDD mapping is inexact. A concrete counter-example or proof that the relaxation preserves the input-output relation is needed.

    Authors: WH frame theory supplies sufficient conditions on the infinite lattice; for any practical finite-bandwidth, finite-duration signal only a finite subset is transmitted. The manuscript shows that the input-output relation follows directly from the ESDD-sampled orthogonality of the DDOP, which depends only on the utilized subset. Because the ESDD mapping is defined within the same bandwidth-duration support, the local (bi)orthogonality is sufficient to preserve the relation. We will insert a short proof sketch in the revision demonstrating that the finite-support case inherits the impulse property without requiring the full-frame bounds. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard models

full rationale

The paper reviews established LTV channel models and WH frame theory, then introduces an ESDD representation and DDOP properties to derive the ODDM input-output relation and a relaxed local biorthogonality guideline. No equations reduce the claimed relation to a definition by construction, no parameters are fitted to data and renamed as predictions, and no load-bearing step relies on a self-citation chain whose validity is internal to the paper. The argument is self-contained against external benchmarks of channel sampling and frame theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited visibility into parameters or assumptions; the central claims rest on standard LTV channel models and WH frame theory without visible new fitted constants or invented entities.

axioms (2)
  • domain assumption The linear time-varying wireless channel admits an equivalent sampled delay-Doppler representation in which impulse-like pulses produce a direct input-output mapping.
    Invoked in the abstract paragraphs on ESDD channel and impulse-function-based transmission strategy.
  • domain assumption Weyl-Heisenberg frame theory permits local or sufficient biorthogonality within bandwidth and duration constraints without violating overall frame properties.
    Stated in the final paragraph on relaxed design guidelines.

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Forward citations

Cited by 2 Pith papers

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  2. Low-complexity Frequency Domain Equalization for filtered-AFDM over General Physical Channels

    eess.SP 2026-04 unverdicted novelty 5.0

    A two-stage frequency-domain equalizer for filtered-AFDM delivers near full-block LMMSE performance at far lower complexity and beats time-domain methods in wideband high-mobility settings.

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