Pulse Shaping Filter Design for Zak-OTFS

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.