On the Time-Frequency Localization Characteristics of the Delay-Doppler Plane Orthogonal Pulse

TL;DR

This paper analyzes the time-frequency localization of the delay-Doppler orthogonal pulse, revealing its high energy spread and potential benefits for diversity and sensing, supported by theoretical derivations and numerical validation.

Contribution

It derives TF localization metrics for the DD orthogonal pulse and explores its energy spread characteristics and implications for DD modulation schemes.

Findings

High energy spread in time, frequency, and joint TF domains.

Adherence to the Heisenberg uncertainty principle.

Insights into pulse design for diversity and sensing applications.

Abstract

In this work, we study the time-frequency (TF) localization characteristics of the prototype pulse of orthogonal delay-Doppler (DD) division multiplexing modulation, namely, the DD plane orthogonal pulse (DDOP). The TF localization characteristics examine how concentrated or spread out the energy of a pulse is in the joint TF domain, the time domain (TD), and the frequency domain (FD). We first derive the TF localization metrics of the DDOP, including its TF area, its time and frequency dispersions, and its direction parameter. Based on these results, we demonstrate that the DDOP exhibits a high energy spread in the TD, FD, and the joint TF domain, while adhering to the Heisenberg uncertainty principle. Thereafter, we discuss the potential advantages brought by the energy spread of the DDOP, especially with regard to harnessing both time and frequency diversities and enabling fine-resolution sensing. Subsequently, we examine the relationships between the time and frequency dispersions of the DDOP and those of the envelope functions of DDOP's TD and FD representations, paving the way for simplified determination of the TF localization metrics for more generalized variants of the DDOP and the pulses used in other DD domain modulation schemes. Finally, using numerical results, we validate our analysis and find further insights.