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.

💡 Research Summary

The paper tackles a critical practical issue in the implementation of Zak‑OTFS, a modulation scheme that promises robust performance in high‑mobility, doubly‑dispersive channels. Zak‑OTFS leverages the Zak transform to express the input‑output (I/O) relationship as a twisted convolution between the transmitted delay‑Doppler (DD) symbols and the channel response. This mathematical structure implies that every DD symbol experiences the same effective channel response, enabling accurate channel estimation from a single pilot symbol in theory. However, in real systems the pulse‑shaping filter spreads this effective response, degrading the promised estimation accuracy.

The authors first derive a discrete‑time, oversampled Zak‑OTFS model that faithfully approximates the continuous‑time formulation. By inserting a cyclic prefix (CP) and using an OFDM‑based implementation, they obtain an I/O relation that remains a discrete twisted convolution, preserving the “all symbols see the same channel” property. This derivation bridges the gap between theory and practical digital signal processing, allowing realistic simulation and hardware design.

Next, the paper investigates how the ambiguity function of the time‑frequency window (the product of the transmit and receive windows) influences channel spreading. The ambiguity function’s sidelobes dictate how much the effective channel response leaks beyond its nominal support. High sidelobes, typical of conventional root‑raised‑cosine (RRC) or rectangular windows, cause significant spreading, which in turn inflates the mean‑square error (MSE) of single‑pilot channel estimation. The authors analytically link the sidelobe level to the degradation of estimation performance, confirming that low‑sidelobe windows are essential for preserving Zak‑OTFS’s predictability.

Motivated by this insight, the authors propose a novel pulse‑shaping filter constructed by synthesizing Prolate Spheroidal Wave Functions (PSWF) within the Isotropic Orthogonal Transform Algorithm (IOTA) framework. PSWFs are known to be energy‑optimal within a prescribed time‑frequency region, yielding ambiguity functions with minimal sidelobes. IOTA provides a practical way to generate orthogonal time‑frequency atoms while keeping implementation complexity manageable. By combining these two tools, the authors create a filter set that simultaneously satisfies orthogonality, spectral confinement, and low‑sidelobe ambiguity.

Extensive Monte‑Carlo simulations compare the proposed PSWF‑IOTA filter against standard RRC and rectangular windowing schemes. In the high‑SNR regime (SNR > 20 dB), the new design achieves an effective channel estimation MSE below –30 dB, a substantial improvement over the –25 dB benchmark reported for conventional Zak‑OTFS. Correspondingly, bit error rate (BER) performance shows a 2 dB SNR gain at typical target BER levels (e.g., 10⁻⁴). The gains are especially pronounced in scenarios with large Doppler spreads, such as vehicular or aerial communications, where channel spreading is most detrimental.

The paper also discusses practical constraints, including filter length, sampling rate, and spectral mask compliance. The PSWF‑IOTA filters are shown to meet these constraints while delivering superior performance, indicating feasibility for real‑world deployment and potential inclusion in future wireless standards.

In summary, the work makes three key contributions: (1) a rigorous discrete‑time I/O model for oversampled Zak‑OTFS that preserves the twisted‑convolution structure; (2) a clear analytical connection between window ambiguity‑function sidelobes and channel‑estimation degradation; and (3) a concrete pulse‑shaping design—PSWF synthesized via IOTA—that markedly improves channel estimation accuracy and BER in high‑mobility environments. This advances Zak‑OTFS from a promising theoretical construct toward a practically viable solution for next‑generation high‑speed wireless communications.