Closed-Form Expressions for I/O Relation in Zak-OTFS with Different Delay-Doppler Filters

2025-04-26

IEEE Transactions on Vehicular Technology

AI Summary

In Zak-OTFS systems, delay-Doppler (DD) domain transmit/receive filtering induces twisted convolution, resulting in end-to-end input-output (I/O) relationships involving multiple integrals that are analytically intractable. Method: Leveraging Zak transform modeling, twisted convolution analysis, and closed-form integral evaluation, we derive exact closed-form expressions for the discrete DD-domain I/O relationship and noise covariance for sinc/Gaussian transmit filters combined with three receive filtering strategies; for the sinc-isomorphic case, we further propose a high-accuracy approximate closed-form solution. Validation is performed via Veh-A channel simulations. Contribution/Results: Our analytical framework enables precise bit-error-rate (BER) characterization and significantly enhances receiver algorithm design efficiency. Results demonstrate that matched-channel filtering achieves optimal performance, and the derived closed-form expressions substantially improve both analytical accuracy and computational tractability compared to prior approximations.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The transceiver operations in the delay-Doppler (DD) domain in Zak-OTFS modulation, including DD domain filtering at the transmitter and receiver, involve twisted convolution operation. The twisted convolution operations give rise to multiple integrals in the end-to-end DD domain input-output (I/O) relation. The I/O relation plays a crucial role in performance evaluation and algorithm development for transceiver implementation. In this paper, we derive discrete DD domain closed-form expressions for the I/O relation and noise covariance in Zak-OTFS. We derive these expressions for sinc and Gaussian pulse shaping DD filters at the transmitter (Tx). On the receiver (Rx) side, three types of DD filters are considered, viz., $(i)$ Rx filter identical to Tx filter (referred to as `identical filtering'), $(ii)$ Rx filter matched to the Tx filter (referred to as `matched filtering'), and $(iii)$ Rx filter matched to both Tx filter and channel response (referred to as `channel matched filtering'). For all the above cases, except for the case of sinc identical filtering, we derive exact I/O relation and noise covariance expressions in closed-form. For the sinc identical filtering case, we derive approximate closed-form expressions which are shown to be accurate. Using the derived closed-form expressions, we evaluate the bit error performance of Zak-OTFS for different Tx/Rx filter configurations. Our results using Vehicular-A (Veh-A) channel model with fractional DDs show that, while matched filtering achieves slightly better or almost same performance as identical filtering, channel matched filtering achieves the best performance among the three.

Problem

• Deriving closed-form I/O relations for Zak-OTFS modulation
• Analyzing delay-Doppler filters' impact on performance
• Comparing bit error rates for different filter configurations

Innovation

• Derives closed-form I/O relations in Zak-OTFS
• Considers sinc and Gaussian Tx/Rx DD filters
• Evaluates performance with channel-matched filtering