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Zak-OTFS with Interleaved Pilots to Extend the Region of Predictable Operation

Jinu Jayachandran, Imran Ali Khan, Saif Khan Mohammed, Ronny Hadani, Ananthanarayanan Chockalingam, Robert Calderbank

TL;DR

This work introduces a DD-domain pilot design framework for Zak-OTFS that uses interleaved pilots to extend the region of predictable operation across users with different delay-Doppler characteristics without altering Zak-OTFS periods. By translating I/O reconstruction into solving small linear systems (LS) built from interleaved pilot responses, the method provides a scalable, model-free alternative to full channel estimation. The paper analyzes both linear LS and ML cross-ambiguity approaches, derives effective crystallization conditions, and shows via simulations that increasing the number of interleaved pilots expands the Doppler range over which reliable communication is possible, while reducing PAPR. Practical trade-offs include increased pilot/guard overhead and computational complexity, but the approach offers flexible, DD-domain-adaptive operation suitable for high-Doppler and high-delay scenarios.

Abstract

When the delay period of the Zak-OTFS carrier is greater than the delay spread of the channel, and the Doppler period of the carrier is greater than the Doppler spread of the channel, the effective channel filter taps can simply be read off from the response to a single pilot carrier waveform. The input-output (I/O) relation can then be reconstructed for a sampled system that operates under finite duration and bandwidth constraints. We introduce a framework for pilot design in the delay-Doppler (DD) domain which makes it possible to support users with very different delay-Doppler characteristics when it is not possible to choose a single delay and Doppler period to support all users. The method is to interleave single pilots in the DD domain, and to choose the pilot spacing so that the I/O relation can be reconstructed by solving a small linear system of equations.

Zak-OTFS with Interleaved Pilots to Extend the Region of Predictable Operation

TL;DR

This work introduces a DD-domain pilot design framework for Zak-OTFS that uses interleaved pilots to extend the region of predictable operation across users with different delay-Doppler characteristics without altering Zak-OTFS periods. By translating I/O reconstruction into solving small linear systems (LS) built from interleaved pilot responses, the method provides a scalable, model-free alternative to full channel estimation. The paper analyzes both linear LS and ML cross-ambiguity approaches, derives effective crystallization conditions, and shows via simulations that increasing the number of interleaved pilots expands the Doppler range over which reliable communication is possible, while reducing PAPR. Practical trade-offs include increased pilot/guard overhead and computational complexity, but the approach offers flexible, DD-domain-adaptive operation suitable for high-Doppler and high-delay scenarios.

Abstract

When the delay period of the Zak-OTFS carrier is greater than the delay spread of the channel, and the Doppler period of the carrier is greater than the Doppler spread of the channel, the effective channel filter taps can simply be read off from the response to a single pilot carrier waveform. The input-output (I/O) relation can then be reconstructed for a sampled system that operates under finite duration and bandwidth constraints. We introduce a framework for pilot design in the delay-Doppler (DD) domain which makes it possible to support users with very different delay-Doppler characteristics when it is not possible to choose a single delay and Doppler period to support all users. The method is to interleave single pilots in the DD domain, and to choose the pilot spacing so that the I/O relation can be reconstructed by solving a small linear system of equations.
Paper Structure (8 sections, 1 theorem, 48 equations, 18 figures, 1 table)

This paper contains 8 sections, 1 theorem, 48 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. System model
  3. Two Interleaved Pilots
  4. Multiple Interleaved Pilots
  5. Auto-ambiguity of Interleaved Pilots
  6. Numerical simulations
  7. Conclusions
  8. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

With $Q$ interleaved pilots and $k_{p_i} = (i-1)M/Q$, $i=1,2,\cdots, Q$, the auto-ambiguity function $A_{x_p,x_p}[k,l]$ in (eqn83652) is supported on the DD domain lattice $\Lambda_Q$ given by

Figures (18)

  • Figure 1: Orthogonal/non-overlapping allocation of time-frequency (TF) resource to different users having distinct delay-Doppler profile.
  • Figure 2: Zak-OTFS transceiver processing.
  • Figure 3: Zak-OTFS DD frame with single pilot (depicted by a red dot) at $(k_p, l_p)$. The pink shaded ellipse depicts the support set of the channel response to the pilot (i.e., ${\mathcal{S}} \, + \, (k_p, l_p)$).
  • Figure 4: Doppler domain aliasing. Assuming that the channel path Doppler shift lies in $[-\nu_{max} \,,\, \nu_{max}]$, the maximum Doppler spread is $2 \nu_{max}$. The Doppler resolution is $\nu_p/N$ (size of each Doppler bin) and the spread of the channel response is roughly $2 \nu_{max}N/\nu_p$ bins along the Doppler axis. (a) Doppler spread is less than $N$ Doppler axis bins, and the ellipses representing the channel response to constituent pilot impulses do not overlap. (b) Doppler spread is greater than $N$ bins, and the ellipses overlap, preventing accurate estimation of the effective channel $h_{\hbox{\scriptsize{eff}}}[k,l]$.
  • Figure 5: Zak-OTFS subframe with two interleaved pilots (indicated by red dots) at $(k_{p_1}, 0)$ and $(k_{p_2}, 0)$.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Theorem 1