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.

翻译：本文发展了一个统一的啁啾域波形框架，涵盖正交啁啾分割复用（OCDM）与仿射频率分割复用（AFDM）。基于这些波形的连续时间表示，我们证明它们隶属于传统多载波（MC）波形框架中的Weyl-Heisenberg（WH）框架，其中根啁啾对应于WH框架中的原型脉冲。由于啁啾是恒包络信号且不影响子载波正交性，这些波形可进一步理解为脉冲成形（PS）正交频分复用（OFDM）。在该PS-OFDM框架内，解析推导了啁啾域波形的功率谱密度。随后讨论现有依赖亚Nyquist离散时间采样、因而存在频率混叠的啁啾域波形实际实现方案。分析了所得混叠波形，并探讨了嵌入混叠啁啾间的正交性。研究表明，混叠啁啾具有条件正交性，而实际近似的混叠啁啾在施加适当逐采样脉冲成形滤波器后可保持相互正交。我们进一步推导了实现后的啁啾域波形在时延-多普勒（DD）信道上的精确输入输出（I/O）关系，表明实际接收机处的等效信道通常不服从纯路径DD分量的叠加形式，导致与文献中常用I/O关系存在不可忽略的偏差。同时研究了实现复杂度，并与演进WH框架内定义的DD域多载波波形——正交时延-多普勒分割复用（ODDM）进行了对比。最后通过仿真结果验证了理论分析。