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Pulse Shaping Filter Design for Zak-OTFS

Kecheng Zhang, Weijie Yuan, Yonghui Li

TL;DR

This work advances Zak-OTFS by deriving a discrete-time oversampled I/O relation in which all delay-Doppler symbols share a common effective channel $h_{ ext{eff}}$. It identifies ambiguity-function sidelobes as a key factor causing channel spreading and degraded estimation, and proposes an IOTA-based pulse-shaping framework built on Prolate Spheroidal Wave Functions to achieve superior time-frequency localization. Through extensive simulations, the IOTA-PSWF design reduces channel spreading and improves BER in high-SNR regimes, while maintaining competitive NMSE relative to PSWF. The results demonstrate a practical, spectrally efficient approach to pulse shaping for Zak-OTFS that enhances channel estimation and detection in high-mobility environments.

Abstract

The Zak-transform-based Orthogonal Time Frequency Space (Zak-OTFS), offers a robust framework for high-mobility communications by simplifying the input-output (I/O) relation to a twisted convolution. While this structure theoretically enables accurate channel estimation by sampling the response from one pilot symbol, practical implementation is constrained by the spreading of effective channel response induced by pulse shaping filters. To address this, we first derive the I/O relationship for discrete-time oversampled Zak-OTFS, which closely approximates the continuous-time system and facilitates analysis and numerical simulation. We show that every delay-Doppler domain symbol undergoes the same effective channel response under the discrete oversampled Zak-OTFS. We then analyze the impact of window ambiguity functions, and reveal that high sidelobes lead to wide channel spreading and degrade estimation accuracy. Building on this insight, we propose a novel pulse shaping filter design that synthesizes Prolate Spheroidal Wave Functions (PSWFs) within the Isotropic Orthogonal Transform Algorithm (IOTA) framework. Numerical simulations confirm that the proposed design achieves superior channel estimation accuracy and bit error rate (BER) performance compared to conventional root-raised-cosine and rectangular windowing schemes in the high-SNR regime.

Pulse Shaping Filter Design for Zak-OTFS

TL;DR

This work advances Zak-OTFS by deriving a discrete-time oversampled I/O relation in which all delay-Doppler symbols share a common effective channel . It identifies ambiguity-function sidelobes as a key factor causing channel spreading and degraded estimation, and proposes an IOTA-based pulse-shaping framework built on Prolate Spheroidal Wave Functions to achieve superior time-frequency localization. Through extensive simulations, the IOTA-PSWF design reduces channel spreading and improves BER in high-SNR regimes, while maintaining competitive NMSE relative to PSWF. The results demonstrate a practical, spectrally efficient approach to pulse shaping for Zak-OTFS that enhances channel estimation and detection in high-mobility environments.

Abstract

The Zak-transform-based Orthogonal Time Frequency Space (Zak-OTFS), offers a robust framework for high-mobility communications by simplifying the input-output (I/O) relation to a twisted convolution. While this structure theoretically enables accurate channel estimation by sampling the response from one pilot symbol, practical implementation is constrained by the spreading of effective channel response induced by pulse shaping filters. To address this, we first derive the I/O relationship for discrete-time oversampled Zak-OTFS, which closely approximates the continuous-time system and facilitates analysis and numerical simulation. We show that every delay-Doppler domain symbol undergoes the same effective channel response under the discrete oversampled Zak-OTFS. We then analyze the impact of window ambiguity functions, and reveal that high sidelobes lead to wide channel spreading and degrade estimation accuracy. Building on this insight, we propose a novel pulse shaping filter design that synthesizes Prolate Spheroidal Wave Functions (PSWFs) within the Isotropic Orthogonal Transform Algorithm (IOTA) framework. Numerical simulations confirm that the proposed design achieves superior channel estimation accuracy and bit error rate (BER) performance compared to conventional root-raised-cosine and rectangular windowing schemes in the high-SNR regime.
Paper Structure (23 sections, 1 theorem, 49 equations, 11 figures, 2 tables)

This paper contains 23 sections, 1 theorem, 49 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Our Contributions
  4. Zak-OTFS System Model
  5. Zak-OTFS I/O Relation
  6. Specific Zak-OTFS Implementation via Time and Frequency Windowing
  7. Zak-OTFS Implementation through OFDM Framework
  8. Discrete-Time Oversampled Zak-OTFS I/O Relation
  9. Channel Estimation with Embedded Pilot Scheme
  10. Crystalline Regime Condition and the Embedded Pilot Channel Estimation Scheme
  11. Motivation to Suppress the Sidelobes of the Ambiguity Function
  12. Proposed IOTA Pulse Shaping Design Framework
  13. Numerical Results
  14. Simulation Parameters
  15. Pulse Shaping Filter Visualization
  16. ...and 8 more sections

Key Result

Proposition 1

The $(k^{\prime}M + l^{\prime}, kM + l)$-th element in the effective channel matrix of Sec2_discrete_H_matrix can also be represented in Sec2_effective_channel_ele, where $h_{\operatorname{eff}}[l, k]$ is the DD domain effective channel response given by with $l_{\tau_{q}} = \tau_{q} M \Delta f$, $k_{\nu_{q}} = \nu_{q} N T$, and $\mathcal{X}_{\tilde{B}}(l, k)$ being the discrete periodic ambiguit

Figures (11)

  • Figure 1: The transmit and receive framework of Zak-OTFS based on OFDM Hanly_TransmitterZak_OTFS_Receiver.
  • Figure 2: An OTFS frame with embedded pilot, guard space, and data.
  • Figure 3: Ambiguity functions under different window functions.
  • Figure 4: Effective channel responses under different window functions.
  • Figure 5: Visualizations of time domain pulses under different windows, $p^{\tau_{l}, \nu_{k}}_{g}(t)$ with $(\tau_{l}, \nu_{k}) = \left(\frac{15}{M \Delta f}, \frac{7}{N T}\right)$.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof