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In our first paper (Mohammed et al., IEEE BITS Inf. Theory Mag., vol. 2, no. 2, pp. 36–55, Nov. 2022.), we explained why the Zak-OTFS input–output (I/O) relation is predictable and nonfading 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. It is well known that it is possible to identify a linear time varying (LTV) channel if and only if it is underspread. The crystallization condition is more restrictive than the underspread condition, so identification is always possible. In the crystalline regime, we show that Zak-OTFS pilot sequences minimize the complexity of identifying the effective delay-Doppler (DD) domain channel filter. We demonstrate that the filter taps can simply be read off from the response to a single Zak-OTFS pilot. In general, we provide an explicit formula for reconstructing the Zak-OTFS I/O relation from a finite number of received pilot symbols in the DD domain. This reconstruction 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 estimation of the underlying physical channel) is out of reach (for example, when the channel comprises of unresolvable paths, or exhibits a continuous DD profile such as in presence of acceleration). Our study clarifies the fundamental origins of predictability by revealing how nonpredictability 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 nonpredictable components, which we refer to as the crystalline decomposition. Vanishing of the nonpredictable component of the channel is equivalent to satisfying the crystallization condition. Finally, we measure the benefits of predictability in terms of bit error rate performance. We consider two cases. In the first, we measure performance given perfect knowledge of the I/O relation. We show that performance is optimal when the crystallization condition holds, that performance approaches that of time domain modulation when the Doppler period vanishes, and approaches that of frequency domain modulation when the delay period vanishes. In the second, we measure performance given imperfect knowledge of the I/O relation, as is the case when it is not possible to learn the underlying channel. We show that model-free operation is successful when the crystallization condition holds, and that performance is only slightly worse than performance given perfect knowledge of the I/O relation. We also compare the performance of Zak-OTFS with that of a well-studied conventional multicarrier approximation to Zak-OTFS, which we refer to as MC-OTFS. We show that the I/O relation of MC-OTFS is predictable to a lesser degree than that of Zak-OTFS, and as a result the performance of MC-OTFS is inferior as the Doppler spread increases.

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