OTFS -- Predictability in the Delay-Doppler Domain and its Value to Communication and Radar Sensing

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.