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Delay-Doppler Pulse Shaping in Zak-OTFS Using Hermite Basis Functions

Fathima Jesbin, Ananthanarayanan Chockalingam

TL;DR

Zak-OTFS performance hinges on DD-domain pulse shaping under the $L^2(\mathbb{R})$ Balian–Low constraint, which motivates a Hermite-basis DD pulse design with no bandwidth/time expansion. The authors formulate a constrained SVD-based optimization to minimize $E_{\text{ISI}}$ by selecting coefficients for scaled Hermite functions, and derive closed-form expressions for $h_{\mathrm{eff}}(\tau,\nu)$ and the noise covariance. They show via Vehicular-A channel simulations that the proposed Hermite pulses achieve BER improvements over canonical sinc and Gaussian pulses and rival GS, while offering flexible control over ISI and sidelobes and reducing computation. The framework extends model-free I/O relation estimation in high-Doppler regimes by providing a tunable, efficient pulse design that balances localization with orthogonality without expanding bandwidth or time.

Abstract

The performance of Zak-OTFS modulation is critically dependent on the choice of the delay-Doppler (DD) domain pulse shaping filter. The design of pulses for $L^2(\mathbb{R})$ is constrained by the Balian-Low Theorem, which imposes an inescapable trade-off between time-frequency localization and orthogonality for spectrally efficient systems. In Zak-OTFS, this trade-off requires balancing the need for localization for input/output (I/O) relation estimation with the need for orthogonality for reliable data detection when operating without time or bandwidth expansion. The well-known sinc and Gaussian pulse shapes represent the canonical extremes of this trade-off, while composite constructions such as the Gaussian-sinc (GS) pulse shape offer a good compromise. In this work, we propose a systematic DD pulse design framework for Zak-OTFS that expresses the pulse as a linear combination of Hermite basis functions. We obtain the optimal coefficients for the Hermite basis functions that minimize the inter-symbol interference (ISI) energy at the DD sampling points by solving a constrained optimization problem via singular value decomposition. For the proposed class of Hermite pulses, we derive closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. Simulation results of Zak-OTFS with embedded pilot and model-free I/O relation estimation in Vehicular-A channels with fractional DDs demonstrate that the optimized pulse shape achieves a bit error rate performance that is significantly superior compared to those of the canonical sinc and Gaussian pulses and is on par with that of the state-of-the-art GS pulse, validating the proposed framework which provides greater design flexibility in terms of control of ISI and sidelobe energies.

Delay-Doppler Pulse Shaping in Zak-OTFS Using Hermite Basis Functions

TL;DR

Zak-OTFS performance hinges on DD-domain pulse shaping under the Balian–Low constraint, which motivates a Hermite-basis DD pulse design with no bandwidth/time expansion. The authors formulate a constrained SVD-based optimization to minimize by selecting coefficients for scaled Hermite functions, and derive closed-form expressions for and the noise covariance. They show via Vehicular-A channel simulations that the proposed Hermite pulses achieve BER improvements over canonical sinc and Gaussian pulses and rival GS, while offering flexible control over ISI and sidelobes and reducing computation. The framework extends model-free I/O relation estimation in high-Doppler regimes by providing a tunable, efficient pulse design that balances localization with orthogonality without expanding bandwidth or time.

Abstract

The performance of Zak-OTFS modulation is critically dependent on the choice of the delay-Doppler (DD) domain pulse shaping filter. The design of pulses for is constrained by the Balian-Low Theorem, which imposes an inescapable trade-off between time-frequency localization and orthogonality for spectrally efficient systems. In Zak-OTFS, this trade-off requires balancing the need for localization for input/output (I/O) relation estimation with the need for orthogonality for reliable data detection when operating without time or bandwidth expansion. The well-known sinc and Gaussian pulse shapes represent the canonical extremes of this trade-off, while composite constructions such as the Gaussian-sinc (GS) pulse shape offer a good compromise. In this work, we propose a systematic DD pulse design framework for Zak-OTFS that expresses the pulse as a linear combination of Hermite basis functions. We obtain the optimal coefficients for the Hermite basis functions that minimize the inter-symbol interference (ISI) energy at the DD sampling points by solving a constrained optimization problem via singular value decomposition. For the proposed class of Hermite pulses, we derive closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. Simulation results of Zak-OTFS with embedded pilot and model-free I/O relation estimation in Vehicular-A channels with fractional DDs demonstrate that the optimized pulse shape achieves a bit error rate performance that is significantly superior compared to those of the canonical sinc and Gaussian pulses and is on par with that of the state-of-the-art GS pulse, validating the proposed framework which provides greater design flexibility in terms of control of ISI and sidelobe energies.
Paper Structure (17 sections, 3 theorems, 47 equations, 9 figures, 1 table)

This paper contains 17 sections, 3 theorems, 47 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Zak-OTFS system model
  3. DD pulse shaping filters
  4. Model-free I/O relation estimation
  5. Proposed Hermite Pulse Shaping Filter
  6. Hermite functions as an orthonormal basis
  7. Construction of the proposed pulse shape
  8. Derivation of optimal coefficients
  9. Analysis of the characteristics of the optimized pulse
  10. I/O relation and noise covariance closed-form expressions
  11. Results and Discussions
  12. Effect of the number of basis functions ($N_c$)
  13. Performance comparison with other pulses
  14. Conclusions
  15. Proof of Theorem \ref{['thm:svd']}
  16. ...and 2 more sections

Key Result

Theorem 1

Given the SVD of the real matrix $\mathbf{\Phi}_{\tau} = \mathbf{U}_{\tau}\mathbf{\Sigma}_{\tau}\mathbf{V}_{\tau}^T$, where $\mathbf{U}_{\tau} \in \mathbb{R}^{L \times L}$ and $\mathbf{V}_{\tau} = [\mathbf{v}_{\tau,1} \ \mathbf{v}_{\tau,2} \ \dots \ \mathbf{v}_{\tau,N_c}] \in \mathbb{R}^{N_c \times Furthermore, the minimum ISI energy achieved by the resulting pulse is $E_{\text{ISI}, w_1}^{\min}

Figures (9)

  • Figure 1: Block diagram of Zak-OTFS transceiver.
  • Figure 2: Sinc, Gaussian, Gaussian-sinc, and proposed Hermite pulse shapes.
  • Figure 3: Heatmaps of the proposed Hermite pulses for different values of $N_c$. $N_c=1$ corresponds to the canonical Gaussian pulse.
  • Figure 4: ISI energy and sidelobe energy of the proposed Hermite pulses as a function of $N_c$.
  • Figure 5: Embedded pilot frame with pilot symbol, pilot region ${\mathcal{P}}$, guard region ${\mathcal{G}}$, data region ${\mathcal{D}}$, and support set of the effective channel ${\mathcal{S}}$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3