In our first paper [2] we explained why the Zak-OTFS input-output (I/O) relation is predictable and non-fading when the delay and Doppler periods are greater than the effective channel delay and Doppler spreads, a condition which we refer to as the crystallization condition. We argued that a communication system should operate within the crystalline regime. In this paper, we provide an explicit formula for reconstructing the Zak-OTFS I/O relation from a finite number of received pilot symbols in the delay-Doppler (DD) domain. This formula makes it possible to study predictability of the Zak-OTFS I/O relation for a sampled system that operates under finite duration and bandwidth constraints. We analyze reconstruction accuracy for different choices of the delay and Doppler periods, and of the pulse shaping filter. Reconstruction accuracy is high when the crystallization condition is satisfied, implying that it is possible to learn directly the I/O relation without needing to estimate the underlying channel. This opens up the possibility of a model-free mode of operation, which is especially useful when a traditional model-dependent mode of operation (reliant on channel estimation) is out of reach (for example, when the channel comprises of unresolvable paths, or exhibits a continuous delay-Doppler profile such as in presence of acceleration). Our study clarifies the fundamental origins of predictability by revealing how non-predictability appears as a consequence of aliasing in the DD domain. This perspective leads to a canonical decomposition of the effective DD channel as a sum of predictable and non-predictable components, which we refer to as the crystalline decomposition.

翻译：在上一篇论文[2]中，我们解释了当延迟周期和多普勒周期大于有效信道延迟扩展和多普勒扩展时（这一条件称为结晶条件），Zak-OTFS输入输出（I/O）关系具有可预测性且无衰落。我们提出通信系统应在结晶状态中运行。本文给出了从延迟-多普勒（DD）域中有限数量的接收导频符号重构Zak-OTFS I/O关系的显式公式。该公式使我们能够研究在有限持续时间和带宽约束下运行的采样系统的Zak-OTFS I/O关系的可预测性。我们分析了不同延迟周期、多普勒周期以及脉冲成形滤波器选择下的重构精度。当结晶条件满足时，重构精度较高，这意味着可直接学习I/O关系而无需估计底层信道。这开启了无模型运行模式的可能性，尤其适用于传统依赖信道估计的模型驱动模式无法实现的情况（例如，信道包含不可分辨路径，或存在连续延迟-多普勒轮廓如加速度场景）。本研究通过揭示非可预测性如何作为DD域混叠的结果而出现，阐明了可预测性的基本起源。该视角导出了有效DD信道作为可预测分量与不可预测分量之和的规范分解，我们称之为结晶分解。