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

2026-04-17

AI Summary

This work addresses the challenge of waveform design in the delay–Doppler domain for integrated 6G communication and sensing by proposing a predictable, non-selective carrier waveform construction method grounded in the geometric structure of the Heisenberg–Weyl group. The approach employs quasi-periodically localized functions to generate delay–Doppler pulses and integrates the Zak transform with Hadamard matrix-based modulation to establish a unified framework encompassing prominent waveforms such as Zak-OTFS, AFDM, OTSM, and ODDM. Theoretical analysis yields necessary and sufficient conditions for the existence of such waveforms and demonstrates that, under limited channel delay and Doppler spreads, the entire frame’s channel response can be precisely predicted from a single input–output measurement—eliminating the need for explicit channel modeling and thereby significantly enhancing system efficiency and robustness.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• predictable waveform
• non-selective channel
• delay-Doppler domain
• Heisenberg-Weyl group
• Zak-OTFS

Innovation

• Zak-OTFS
• Delay-Doppler domain
• Heisenberg-Weyl group
• predictable waveform
• non-selective channel