In this article, we provide, for the first time, a comprehensive understanding of long-range (LoRa) waveform theory in order to quantify its orthogonality. We present LoRa waveform expressions in continuous- and discrete-time domains, and analyze measures of orthogonality between different LoRa spreading factors (SFs) through cross-correlation functions. The cross-correlation functions are analytically expressed in a general form and they account for diverse configuration parameters (bandwidth, SF, etc.) and different cases of signal displacements (time delay shift, frequency shift, etc.). We quantify their mean and maximum in all time domains. We highlight the impact of the temporal displacement and different bandwidths. The general result is that LoRa modulation is nonorthogonal. First, we observe that for same bandwidths, the largest maximum cross-correlation happens for same SF and is equal to 100% due to same symbols; whereas for different bandwidths, the largest maximum cross-correlation is no longer observed at the same SF. Second, the maximum cross-correlation is less than 26% between different SFs, is higher for closer SFs, and decreases as the difference between SFs increases. After downchirping, the maximum cross-correlation increases and the mean decreases compared to those before downchirping. Moreover, the maximum cross-correlation is insignificantly impacted by the temporal delay, which makes it valid to adopt for the performance analysis of both synchronous and asynchronous systems. Finally, we analyze by simulating the bit error probability statistics for different bandwidth ratios and highlighting their correlated behavior with the insights obtained from the maximum cross-correlation expressions.
Bibliographical note© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.
- Chirp modulation
- Frequency shift keying
- Internet of Things
- Time-domain analysis
- orthogonality conditions.
ASJC Scopus subject areas
- Signal Processing
- Information Systems
- Hardware and Architecture
- Computer Science Applications
- Computer Networks and Communications