Resolving the Discontinuity of Continuous-Time AFDM Waveforms
Abstract
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.
Related
- Fast, cheap and sustainable: armament portfolio arbitration under attrition, lead time and political discontinuity
- Model-Order-Adaptive Channel Estimation for AFDM Systems with Fractional Delay and Doppler
- Dual-Branch Vector-Quantization-Aided Satellite Digital Semantic Communication with Index Compression for High-Resolution RSI Over AFDM
- Efficient estimation methods for non-Gaussian regression models in continuous time
- Continuous Delayed-Memory Stochastic Gradient Descent and Continuous-Time Reinforcement Learning from History of Astrophysical Time Series Studies
BibTeX
@misc{cao2026resolving,
title = {{Resolving the Discontinuity of Continuous-Time AFDM Waveforms}},
author = {Yewen Cao and Yulin Shao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07480},
eprint = {2609.07480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07480},
}
RIS
TY - UNPB
TI - Resolving the Discontinuity of Continuous-Time AFDM Waveforms
AU - Yewen Cao
AU - Yulin Shao
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07480
UR - https://arxiv.org/abs/2609.07480
AB - 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.
ER -