This tutorial derives the mathematical foundations of what it means for a carrier waveform to be predictable and non-selective. We focus on Zak-OTFS, where each carrier waveform is a pulse in the delay-Doppler (DD) domain, formally a quasi-periodic localized function with specific periods along delay and Doppler. Viewed in the time domain, the Zak-OTFS carrier is realized as a pulse train modulated by a tone (termed a pulsone). We start by providing physical intuition, describing what it means for the Zak-OTFS carrier waveforms to be geometric modes of the Heisenberg-Weyl (HW) group of discrete delay and Doppler shifts that define the discrete-time communication model. In fact, we show that these geometric modes are common eigenvectors of a maximal commutative subgroup of our discrete HW group. When the channel delay spread is less than the delay period, and the channel Doppler spread is less than the Doppler period, we show that the Zak-OTFS input-output (I/O) relation is predictable and non-selective. Given the I/O response at one DD point in a frame, it is possible to predict the I/O response at all other points, without recourse to some mathematical model of the channel. While it may be intuitive that geometric modes of the HW group are predictable and non-selective wireless carriers, this is not a requirement. We provide a necessary and sufficient condition that depends on the ambiguity properties of the basis of carrier waveforms. In fact, we show that the structure of a pulse train modulated by a Hadamard matrix is common to several families of waveforms proposed for 6G, including Zak-OTFS, AFDM, OTSM and ODDM.

翻译：本教程推导了载波波形具有可预测性与非选择性的数学基础。我们重点研究Zak-OTFS，其每个载波波形均为延迟-多普勒（DD）域中的脉冲，形式上是一种沿延迟轴和多普勒轴具有特定周期的准周期局部化函数。从时域角度看，Zak-OTFS载波可表示为受单音调制的脉冲串（称为pulsone）。我们首先提供物理直观理解，阐述Zak-OTFS载波波形作为定义离散时间通信模型的离散延迟和多普勒平移的海森堡-魏尔（HW）群的几何模态的含义。事实上，我们证明这些几何模态是我们离散HW群中最大交换子群的公共特征向量。当信道延迟扩展小于延迟周期，且信道多普勒扩展小于多普勒周期时，我们证明Zak-OTFS的输入-输出（I/O）关系具有可预测性与非选择性。给定帧中一个DD点的I/O响应，即可预测所有其他点的I/O响应，而无需依赖信道的某种数学模型。尽管HW群的几何模态直观上应作为可预测且非选择性的无线载波，但这并非必要条件。我们提出了一个依赖于载波波形基的模糊性特性的充要条件。实际上，我们证明受阿达玛矩阵调制的脉冲串结构是包括Zak-OTFS、AFDM、OTSM和ODDM在内的多种面向6G提出的波形族的共同特征。