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

TL;DR

This paper presents a comprehensive continuous-time framework for chirp-domain waveforms like OCDM and AFDM, analyzing their spectral properties, practical implementation issues, and channel behavior, with verification through simulations.

Contribution

It unifies chirp-domain waveforms within the Weyl-Heisenberg framework, derives their spectral density, and analyzes practical aliasing effects and input-output relations in delay-Doppler channels.

Findings

Chirp-domain waveforms can be interpreted as pulse-shaped OFDM within the WH framework.

Aliased chirps are conditionally orthogonal and can maintain orthogonality with proper pulse shaping.

The effective channel at a practical receiver deviates from ideal models, affecting waveform performance.

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 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 (I/O) relation for the implemented chirp-domain waveform over delay-Doppler (DD) channels, showing that the effective channel at a practical receiver does not, in general, conform to a superposition of pure path-wise DD components, resulting in a non-negligible deviation from the I/O relation commonly used 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.