A Unified Pulse-Shaped OFDM Framework for Chirp-Domain Waveforms: Continuous-Time Modeling and Practical I/O Analysis

2026-03-15

AI Summary

This work addresses the lack of a unified theoretical framework for existing chirp-based waveforms—such as OCDM and AFDM—and their susceptibility to spectral aliasing due to undersampling, which compromises subcarrier orthogonality and degrades channel modeling accuracy. The paper proposes a unified pulse-shaped OFDM framework that integrates chirp waveforms into the Weyl–Heisenberg (WH) multicarrier formalism, revealing for the first time that chirps inherently serve as WH prototype pulses. Through continuous-time modeling, the authors rigorously derive the power spectral density, aliasing effects, and input–output relationship under delay–Doppler channels. They further demonstrate that practical received signals deviate from conventional spreading function models and introduce a conditional orthogonality mechanism to preserve waveform integrity. Both theoretical analysis and simulations confirm that, under specific filtering, aliased chirp waveforms can maintain orthogonality while achieving lower implementation complexity than ODDM.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

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Abstract

In this paper, a unified framework for chirp-domain waveforms, including orthogonal chirp division multiplexing (OCDM) and affine frequency division multiplexing (AFDM), is developed. Based on their continuous-time representations, we show that these waveforms fall within the conventional Weyl-Heisenberg (WH) framework for multicarrier (MC) waveforms, where the root chirp corresponds directly to the prototype pulse in the WH framework. Since the chirp is a constant-envelope signal and is transparent to subcarrier orthogonality, these waveforms can be further interpreted as pulse-shaped (PS) orthogonal frequency division multiplexing (OFDM). Within the developed PS-OFDM framework, the power spectral density of chirp-domain waveforms is derived analytically. We then discuss existing practical implementations of chirp-domain waveforms, which rely on sub-Nyquist discrete-time samples and therefore exhibit frequency aliasing. The resulting aliased waveform is analyzed, and the orthogonality among the embedded aliased chirps is discussed. It is shown that the aliased chirps are conditionally orthogonal, whereas the implemented approximate aliased chirps can maintain mutual orthogonality when an appropriate sample-wise pulse-shaping filter is applied. We further derive an exact input-output relation for the implemented chirp-domain waveform over a delay-Doppler (DD) channel, showing that the effective channel observed at a practical receiver does not, in general, admit a DD spreading-function model commonly assumed in the literature. The implementation complexity is also investigated and compared with that of orthogonal delay-Doppler division multiplexing (ODDM), the DD-domain MC waveform defined within the evolved WH framework. Finally, simulation results are provided to verify the analysis.

Problem

• chirp-domain waveforms
• frequency aliasing
• subcarrier orthogonality
• delay-Doppler channel
• practical implementation

Innovation

• pulse-shaped OFDM
• chirp-domain waveforms
• Weyl-Heisenberg framework
• delay-Doppler channel
• frequency aliasing