A Unified Pulse-Shaped OFDM Framework for Chirp-Domain Waveforms: Continuous-Time Modeling and Practical I/O Analysis
A unified framework for chirp-domain waveforms, including orthogonal chirp division multiplexing (OCDM) and affine frequency division multiplexing (AFDM), is developed. Their continuous-time representations are shown to fall within the conventional Weyl-Heisenberg (WH) framework for multicarrier waveforms, with the root chirp as the prototype pulse. Since the root chirp has constant envelope and is transparent to subcarrier orthogonality, these waveforms can be further interpreted as pulse-shaped (PS) orthogonal frequency division multiplexing (OFDM) signals, whose power spectral density is derived analytically. The derived spectrum reveals that implementations based on the discrete affine Fourier transform rely on sub-Nyquist samples and exhibit frequency aliasing. We prove that the corresponding aliased chirps are only conditionally orthogonal, whereas sample-wise root-Nyquist pulse shaping of the discrete-time AFDM (DT-AFDM) sequence produces mutually orthogonal pulse-shaped chirps, resulting in the pulse-shaped AFDM (PS-AFDM) waveform. We then derive an exact waveform-level input-output (I/O) relation for PS-AFDM over delay-Doppler (DD) channels, showing that the effective channel at a practical receiver is generally not a superposition of pure path-wise DD components. Waveform simulations verify the derived relation to machine precision, while the conventional sequence-level I/O relation for DT-AFDM exhibits a substantial mismatch with waveform behavior for practical channels with continuous-valued delays.