Stepped Frequency Division Multiplexing: A Jump-Free Continuous-Time AFDM Waveform

2026-05-12

AI Summary

This work addresses the high out-of-band emission (OOBE) and receiver sensitivity to fractional delays in conventional continuous-time AFDM waveforms, which arise from envelope discontinuities caused by frequency wrapping. To resolve this, the authors propose Staircase Frequency Division Multiplexing (SFDM), which eliminates waveform discontinuities by fixing the instantaneous frequency at the midpoint of the wrapped chirp within each sampling interval while maintaining continuous phase accumulation—all without altering the IDAFT output or overall system architecture. The study provides the first analytical characterization of the frequency-wrapping mechanism underlying OOBE in AFDM and introduces a unique sample-preserving midpoint frequency selection strategy. Theoretical analysis demonstrates that SFDM substantially suppresses spectral regrowth, significantly reduces OOBE, and enhances receiver robustness under both high-percentile and worst-case channel conditions, thereby enabling more reliable and spectrally cleaner physical-layer transmission.

Abstract

Affine frequency division multiplexing (AFDM) has emerged as a promising modulation scheme for doubly selective channels, but its canonical continuous-time realization, referred to herein as piecewise continuous AFDM (PC-AFDM), has been observed to exhibit high out-of-band emission (OOBE) whose mechanism has not been analytically characterized. This paper shows that the underlying cause is frequency wrapping, which introduces internal envelope jumps between AFDM sampling instants and generates a high-frequency spectral tail distinct from ordinary block truncation. To eliminate these discontinuities without altering the inverse discrete affine Fourier transform (IDAFT) output sequence, we propose stepped frequency division multiplexing (SFDM). In SFDM, the instantaneous frequency is kept constant at the midpoint of the wrapped chirp within each sampling interval, while the phase is continuously accumulated across interval boundaries. We prove that, under continuous phase accumulation and without additional phase correction, the midpoint choice is the unique sample-preserving choice for arbitrary chirp-rate parameter. The resulting waveform is continuous within each AFDM block, reduces OOBE, and preserves the standard AFDM modulation matrix, guard-interval structure, and receiver processing. Moreover, under fractional-delay propagation, SFDM mitigates the receiver sensitivity that arises when delayed sampling points fall near wrapping-induced discontinuities in PC-AFDM. Numerical results verify the theoretical tail coefficients, demonstrate OOBE reduction, and show improved receiver robustness in the high-percentile and worst-case regimes. These findings establish SFDM as a spectrally cleaner and more reliable physical layer for AFDM systems.

Problem

• out-of-band emission
• AFDM
• frequency wrapping
• envelope discontinuity
• spectral leakage

Innovation

• Stepped Frequency Division Multiplexing
• Affine Frequency Division Multiplexing
• Out-of-Band Emission
• Continuous-Time Waveform
• Frequency Wrapping