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Resolving the Discontinuity of Continuous-Time AFDM Waveforms

Yewen Cao, Yulin Shao

arXiv:2609.07480v1eess.SPcs.ITmath.IT

TL;DR

Discrete AFDM specifies samples but not the continuous-time waveform transmitted between them, and the conventional frequency-wrapped realization has envelope jumps for generic chirp parameters. The paper proposes SFDM, which steps frequency at each Nyquist interval's chirp midpoint while preserving the discrete samples. The paper reports that SFDM achieves lower OOBE across a wide range of chirp rates.

  • Problem

    Discrete AFDM specifies values at Nyquist sampling instants, but the continuous-time waveform between samples is not uniquely determined, despite its effect on spectral properties and implementation.

    Source: §I · p. 1

  • Method

    SFDM uses piecewise-constant instantaneous frequency set to the underlying chirp midpoint in each Nyquist interval, preserving the IDAFT samples while producing a continuous complex envelope.

    Source: §I · pp. 1–2

  • Results

    SFDM consistently achieves lower OOBE across a wide range of chirp rates, while the conventional construction has envelope jumps for generic chirp parameters.

    Source: §I · pp. 1–2

  • Takeaways & Limitations

    The continuous-time realization is a fundamental design choice for chirp-based multicarrier systems because it directly governs spectral containment.

    Source: §V · p. 6

Abstract

from arXiv · show

Continuous-time affine frequency division multiplexing (AFDM) waveforms, constructed via frequency wrapping and phase correction, are known to be sample-wise equivalent to the widely adopted discrete AFDM framework. In this paper, we uncover a fundamental and previously overlooked flaw in this construction: its complex envelope is inherently discontinuous for generic chirp parameters. We show that these discontinuities are the direct cause of the high out-of-band emission (OOBE). To resolve this issue, we propose a fundamentally different continuous-time waveform, termed stepped frequency division multiplexing (SFDM). Unlike conventional approaches that allow continuous frequency variation, SFDM freezes the instantaneous frequency at the midpoint of the underlying chirp trajectory within each Nyquist sampling interval. This design yields a complex envelope that is strictly continuous over the entire symbol duration while preserving exact sample-wise equivalence with discrete AFDM. A unified spectral analysis reveals that the superior OOBE performance of SFDM stems from the absence of internal jump discontinuities, which otherwise dominate the far-out spectral roll-off. Numerical results confirm that SFDM consistently achieves significantly lower OOBE across a wide range of chirp rates.

I. INTRODUCTION

Discrete AFDM specifies samples but leaves the transmitted waveform between them unspecified, even though that continuous-time realization affects spectral properties and implementation. The existing frequency-wrapped construction preserves the samples but has envelope jumps for generic chirp parameters; SFDM instead uses midpoint-frequency steps to retain sample equivalence with a continuous envelope.

I. INTRODUCTION

  • The discrete AFDM definition leaves the between-sample waveform unspecified, although its continuous-time realization affects spectral properties, hardware implementation, and regulatory compliance.Source: §I · p. 1
  • Frequency wrapping and phase correction make the conventional continuous-time AFDM waveform sample-wise equivalent to discrete AFDM, but introduce envelope jumps at wrapping boundaries for generic chirp parameters.Source: §I · p. 1
  • Those discontinuities cause spectral splatter and elevated OOBE, potentially degrading spectral containment and violating spectrum-mask requirements.The paper also identifies them as a theoretical explanation for previously observed high OOBE.Source: §I · p. 1
  • SFDM uses piecewise-constant instantaneous frequency set to the underlying chirp's midpoint value in each Nyquist interval, yielding a continuous envelope while exactly reproducing the IDAFT samples for arbitrary chirp parameters.Source: §I · pp. 1–2
  • SFDM changes only the inter-sample trajectory and is an intrinsic multicarrier waveform, unlike radar stepped-frequency designs that vary frequency pulse by pulse or symbol by symbol.The paper also distinguishes SFDM from pulse-shaping and hardware-impairment studies, which it describes as complementary work that does not directly address the discontinuity problem.Source: §I · p. 1
  • The paper develops unified spectral analysis for both waveforms and reports that SFDM consistently achieves lower OOBE across a wide range of chirp rates.Source: §I · p. 2

II. SYSTEM MODEL

The section introduces a review of conventional piecewise-continuous AFDM, its discontinuity, and the subsequent SFDM waveform.

II. SYSTEM MODEL

  • The section reviews conventional piecewise-continuous AFDM and analyzes its discontinuity before introducing SFDM.Source: §II · p. 2

A. Conventional PC-AFDM Waveform

The conventional PC-AFDM waveform uses frequency wrapping and phase correction to constrain its chirp subcarriers to the baseband while reproducing discrete AFDM samples. Its complex envelope is continuous only for restrictive chirp-parameter cases; generic parameters produce wrapping-boundary jumps.

A. Conventional PC-AFDM Waveform

  • For N subcarriers over bandwidth B, the block duration is T = N/B and sampling instants are t_n = n/B; the raw subcarrier frequency is wrapped to remain in [0, B).Source: A. Conventional PC-AFDM Waveform · p. 2
  • The conventional construction uses an integer wrap-count and phase correction, which reproduces the standard discrete-time IDAFT basis at sampling instants but introduces jumps between samples.Source: A. Conventional PC-AFDM Waveform · p. 2
  • PC-AFDM is continuous across the block only if there is no internal wrapping or if the chirp parameter satisfies the stated discrete set; generic parameters yield jump discontinuities.The stated continuity cases are c1 ≤ 1/(2N^2) for no wrapping, or c1 = 1/(2kN) for positive integer k when wrapping occurs.Source: A. Conventional PC-AFDM Waveform · p. 2
  • The resulting jump discontinuities are undesirable because they significantly increase OOBE.Source: A. Conventional PC-AFDM Waveform · p. 2

B. The SFDM Waveform

SFDM uses a stepped-frequency construction that eliminates phase discontinuities while preserving full parameter flexibility and exact discrete-time equivalence.

B. The SFDM Waveform

  • Within each Nyquist interval, SFDM freezes instantaneous frequency at the underlying chirp midpoint and accumulates continuous phase from that step-frequency trajectory.Source: B. The SFDM Waveform · p. 2
  • For arbitrary c1, SFDM’s sampled subcarrier basis exactly matches the standard discrete-time AFDM/IDAFT basis.Source: B. The SFDM Waveform · p. 3
  • Exact sample matching preserves perfect discrete-time subcarrier orthogonality and compatibility with the standard AFDM transceiver without modifications.Source: B. The SFDM Waveform · p. 3
  • The midpoint choice gives a centered piecewise-constant representative, unlike locally one-sided left- or right-endpoint choices.Source: B. The SFDM Waveform · p. 3

III. CONTINUOUS-TIME SPECTRAL CHARACTERIZATION

The section gives exact spectral forms for SFDM and PC-AFDM, then explains how their inter-sample trajectories and continuity determine their spectral behavior. At small chirp rates their phase difference can be locally small, but internal jumps in generic PC-AFDM contribute to its far-out spectrum, unlike continuous SFDM.

Exact spectral representations

  • SFDM’s spectrum is a coherent sum of sinc-shaped terms from constant-frequency intervals, while PC-AFDM’s is a coherent sum of segment-dependent Fresnel-type terms induced by wrapping boundaries.Both representations are exact, but only SFDM admits a compact representation directly tied to intervalwise constant-frequency segments.Source: §III · pp. 3–4

Small-α comparison

  • On an interval without wrapping, SFDM uses the midpoint frequency while PC-AFDM’s frequency varies affinely, yielding a locally bounded quadratic phase difference.This explains why the spectra are close for small α, but local closeness alone does not establish a global OOBE ordering.Source: §III · p. 4

Far-out spectral behavior

  • At high frequencies, generic PC-AFDM has an f^-1-order leading contribution from internal jumps, whereas SFDM’s leading term has no internal-jump contribution.The difference follows from PC-AFDM’s internal jump discontinuities versus SFDM’s continuity.Source: §III · p. 4
  • Internal wrapping adds a generically nonzero far-out penalty to PC-AFDM unless the discrete continuity condition makes every internal jump zero.Wrapping may still occur under that continuity condition, but then it contributes no internal-jump term to the 1/f leading-order expansion.Source: §III · pp. 4–5

IV. NUMERICAL RESULTS

The numerical evaluation compares continuous-time SFDM with PC-AFDM using trajectory, ESD, and OOBE analyses. The passage specifies the simulation setup and computational procedures.

Simulation setup

  • The evaluation compares the continuous-time behavior and OOBE of SFDM against the conventional PC-AFDM baseline.It uses normalized bandwidth B = 1; trajectory plots use N = 10, while ESD and OOBE evaluations use N = 64.Source: §IV · p. 5
  • Continuous-time waveforms are numerically represented with oversampling factor L_os.The simulation settings are summarized in Table I.Source: §IV · p. 5

Evaluation procedure

  • Experiment 2 computes average ESD deterministically from (7) via a zero-padded FFT and evaluates the OOBE integral numerically over the Table I truncation range.Source: §IV · p. 5

A. Time-Frequency Trajectories and Phase Continuity

SFDM freezes instantaneous frequency at each Nyquist interval’s midpoint and remains continuous, while PC-AFDM has jumps at generic chirp rates. Both waveforms match the discrete IDAFT samples at sampling instants; at α = 0.5, both preserve continuity.

A. Time-Frequency Trajectories and Phase Continuity

B. Out-of-Band Emission Performance

SFDM has lower average OOBE than PC-AFDM over most evaluated chirp rates, with near-overlap at small α and at specific continuity points. The ESD comparison shows stronger far-out leakage for PC-AFDM at α = 0.8, where internal jumps contribute to the far-out spectral tail.

B. Out-of-Band Emission Performance

V. CONCLUSION

The paper identifies jump discontinuities in continuous-time AFDM as a source of high OOBE and proposes SFDM to eliminate them while preserving sample-wise equivalence. It presents continuous-time realization as a design choice governing spectral containment, with implications for spectrum-constrained AFDM applications.

V. CONCLUSION

  • SFDM eliminates the complex-envelope discontinuities identified in continuous-time AFDM while preserving exact sample-wise equivalence.Source: §V · p. 6
  • Figure 5 plots average OOBE against normalized chirp rate α and marks theoretical continuity points α = 1/(2k).Source: Figure 5 · p. 6
  • For chirp-based multicarrier systems such as AFDM, continuous-time realization directly governs spectral containment.Source: §V · p. 6
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