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 approximate delay and Doppler orthogonalities, if the truncating windows are periodic within the support. Motivated by this, we propose a DD Nyquist pulse shaping scheme considering signals with periodicity. Finally, our conclusions are verified by using various strictly or approximately periodic pulses.

翻译：本文针对时延-多普勒（DD）通信中的脉冲整形问题展开研究。我们首先依据Zak变换的性质，在DD域中构建基函数，特别地，论证了所构建的基函数具有全局准周期性及局部扭曲移位特性，并揭示了其在时域和频域中的重要性。进一步分析了基函数的模糊函数，表明通过使用周期信号构建基函数可实现完全局域的模糊函数。更重要的是，我们证明若截断窗口在支撑集内具有周期性，则对这类基函数进行时频截断可自然逼近时延与多普勒正交性。受此启发，我们提出了一种考虑信号周期性的DD奈奎斯特脉冲整形方案。最后，通过多种严格或近似周期脉冲的仿真验证了结论。