Zak-OTFS: A Predictable Physical Layer for Communications and Sensing

TL;DR

This paper explores Zak-OTFS waveforms, demonstrating their predictability and non-selectivity in delay-Doppler channels, and relates their structure to other 6G waveforms like AFDM and ODDM.

Contribution

It provides a mathematical foundation for Zak-OTFS as geometric HW group modes, establishing conditions for predictability and linking it to other 6G waveforms.

Findings

Zak-OTFS waveforms are quasi-periodic localized functions in the DD domain.

Under certain channel conditions, Zak-OTFS I/O response is predictable across the DD frame.

The pulse train modulated by a Hadamard matrix is common to Zak-OTFS, AFDM, OTSM, and ODDM.

Abstract

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.