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

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.