In this paper, we study the pulse shaping for delay-Doppler (DD) communications. We start with constructing a basis function in the DD domain following the properties of the Zak transform. Particularly, we show that the constructed basis functions are globally quasi-periodic while locally twisted-shifted, and their significance in time and frequency domains are then revealed. We further analyze the ambiguity function of the basis function, and show that fully localized ambiguity function can be achieved by constructing the basis function using periodic signals. More importantly, we prove that time and frequency truncating such basis functions naturally leads to delay and Doppler orthogonalities, if the truncating windows are orthogonal or periodic. Motivated by this, we propose a DD Nyquist pulse shaping scheme considering signals with periodicity. Finally, our conclusions are verified by using various orthogonal and periodic pulses.

翻译：本文研究时延-多普勒（DD）通信中的脉冲成形问题。首先根据Zak变换的性质构建DD域基函数，特别地，我们证明所构建的基函数具有全局准周期性和局部扭曲平移特性，并揭示其在时域和频域中的重要意义。进一步分析基函数的模糊函数，表明利用周期信号构建基函数可实现完全局域化的模糊函数。更重要的是，我们证明若截断窗口具有正交性或周期性，对这类基函数进行时频截断将自然产生时延和多普勒正交性。受此启发，我们提出一种考虑信号周期性的DD奈奎斯特脉冲成形方案。最后，通过多种正交与周期脉冲验证了我们的结论。