The recently proposed orthogonal delay-Doppler division multiplexing (ODDM) modulation, which is a delay-Doppler (DD) domain multi-carrier (DDMC) modulation scheme based on the DD domain orthogonal pulse (DDOP), is studied. We first revisit the linear time-varying (LTV) channel model for the wireless channel, and review the conventional multi-carrier (MC) modulation schemes and their design guidelines for both linear time-invariant (LTI) and LTV channels. We then focus on the representation of the LTV channel in an equivalent sampled DD (ESDD) domain, and propose an impulse-function-based transmission strategy for the ESDD channel. Next, we take an in-depth look into the DDOP and show that it achieves orthogonality with respect to the fine time and frequency resolutions in the ESDD domain thus behaves like an impulse function. This allows us to unveil the unique input-output relation of the resultant ODDM modulation over the ESDD channel. We point out that the conventional MC modulation design guidelines based on the Weyl-Heisenberg (WH) frame theory can be relaxed without compromising its orthogonality or violating the WH frame theory. More specifically, for a practical communication system with bandwidth and duration constraints, MC modulation signals can be designed considering so-called local or sufficient (bi)orthogonality, which refers to the (bi)orthogonality among a WH subset for the MC signal within a specific bandwidth and duration. This novel design guideline could potentially open up opportunities for developing future waveforms required by new applications such as communication systems associated with high delay and/or Doppler shifts, as well as integrated sensing and communications.

翻译：本文研究了近期提出的正交延时-多普勒域复用（ODDM）调制方案，这是一种基于延时-多普勒域正交脉冲（DDOP）的延时-多普勒（DD）域多载波（DDMC）调制方案。我们首先重新审视无线信道的线性时变（LTV）信道模型，并回顾传统多载波（MC）调制方案及其针对线性时不变（LTI）和LTV信道的设计准则。随后重点研究LTV信道在等效采样DD（ESDD）域中的表示，并提出一种基于脉冲函数的ESDD信道传输策略。接下来，我们深入分析DDOP，证明其能在ESDD域的精细时间和频率分辨率下实现正交性，从而表现出类似脉冲函数的特性。这使我们得以揭示所得ODDM调制在ESDD信道上的独特输入-输出关系。我们指出，基于Weyl-Heisenberg（WH）框架理论的传统MC调制设计准则可在不牺牲正交性或不违反WH框架理论的前提下进行放宽。具体而言，对于具有带宽和持续时间约束的实际通信系统，MC调制信号可基于所谓的局部或充分（双）正交性进行设计，即WH子集在特定带宽和持续时间内的（双）正交性。这一新颖的设计准则可能为开发适应高延时和/或多普勒偏移通信系统以及通感一体化等新型应用所需的未来波形开辟新机遇。