The recently proposed orthogonal time frequency space (OTFS) modulation, which is a typical Delay-Doppler (DD) communication scheme, has attracted significant attention thanks to its appealing performance over doubly-selective channels. In this paper, we present the fundamentals of general DD communications from the viewpoint of the Zak transform. We start our study by constructing DD domain basis functions aligning with the time-frequency (TF)-consistency condition, which are globally quasi-periodic and locally twisted-shifted. We unveil that these features are translated to unique signal structures in both time and frequency, which are beneficial for communication purposes. Then, we focus on the practical implementations of DD Nyquist communications, where we show that rectangular windows achieve perfect DD orthogonality, while truncated periodic signals can obtain sufficient DD orthogonality. Particularly, smoothed rectangular window with excess bandwidth can result in a slightly worse orthogonality but better pulse localization in the DD domain. Furthermore, we present a practical pulse shaping framework for general DD communications and derive the corresponding input-output relation under various shaping pulses. Our numerical results agree with our derivations and also demonstrate advantages of DD communications over conventional orthogonal frequency-division multiplexing (OFDM).

翻译：近期提出的正交时频空间（OTFS）调制作为一种典型的延迟-多普勒（DD）通信方案，因其在双选择性信道上的卓越性能而备受关注。本文从Zak变换视角阐述广义DD通信的基本原理。我们首先通过构建与时频（TF）一致性条件匹配的DD域基函数展开研究，这些基函数具有全局准周期性和局部扭曲平移特性。研究表明，这些特性会转化为时域与频域的独特信号结构，对通信系统设计具有重要价值。随后重点研究DD奈奎斯特通信的实用实现，证明矩形窗能够实现完美DD正交性，而截断周期信号可获得足够的DD正交性。特别地，采用具有过剩带宽的平滑矩形窗虽会略微降低正交性，但能显著提升DD域的脉冲定位精度。此外，本文提出面向广义DD通信的实用脉冲成形框架，推导了多种成形脉冲下的输入-输出关系。仿真结果验证了理论推导的正确性，并展示了DD通信相对于传统正交频分复用（OFDM）的优势。