Fundamentals of Delay-Doppler Communications: Practical Implementation and Extensions to OTFS

TL;DR

This paper explores the fundamentals and practical implementation of Delay-Doppler (DD) communications, especially OTFS, using the Zak transform, and demonstrates their advantages over traditional OFDM in doubly-selective channels.

Contribution

It introduces a Zak transform-based framework for DD communications, constructs DD basis functions, and proposes practical pulse shaping methods with improved orthogonality and localization.

Findings

Rectangular windows achieve perfect DD orthogonality.

Truncated periodic signals provide sufficient DD orthogonality.

DD communications outperform OFDM in doubly-selective channels.

Abstract

The recently proposed orthogonal time frequency space (OTFS) modulation, which is a typical Delay-Doppler (DD) communication scheme, has attracted significant attention thanks to its appealing performance over doubly-selective channels. In this paper, we present the fundamentals of general DD communications from the viewpoint of the Zak transform. We start our study by constructing DD domain basis functions aligning with the time-frequency (TF)-consistency condition, which are globally quasi-periodic and locally twisted-shifted. We unveil that these features are translated to unique signal structures in both time and frequency, which are beneficial for communication purposes. Then, we focus on the practical implementations of DD Nyquist communications, where we show that rectangular windows achieve perfect DD orthogonality, while truncated periodic signals can obtain sufficient DD orthogonality. Particularly, smoothed rectangular window with excess bandwidth can result in a slightly worse orthogonality but better pulse localization in the DD domain. Furthermore, we present a practical pulse shaping framework for general DD communications and derive the corresponding input-output relation under various shaping pulses. Our numerical results agree with our derivations and also demonstrate advantages of DD communications over conventional orthogonal frequency-division multiplexing (OFDM).