Improving waveform design to reduce unwanted signal emissions
Resolving the Discontinuity of Continuous-Time AFDM Waveforms
Information Theory
Summary
The existing method for creating certain radio signals, called continuous-time AFDM waveforms, has a hidden problem: the signal can suddenly jump in value, which causes too much signal to spill into unwanted frequencies. The authors found that these jumps happen because of how the signal is built and are responsible for poor signal quality outside the main frequency band. To fix this, they designed a new version called SFDM that keeps the signal smooth and continuous by holding the frequency steady at specific moments. This new approach reduces unwanted frequency spill and works just as well as the old method in sending information.
affine frequency division multiplexing (AFDM)continuous-time waveformcomplex envelopechirp parameterout-of-band emission (OOBE)Nyquist sampling intervalinstantaneous frequencyspectral analysisfrequency wrappingsignal discontinuity
Authors
Yewen Cao, Yulin Shao
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.