Channel Characterization and Modeling for Vehicular Communications (Wireless Networks) 3031474392, 9783031474392

Table of contents :
Preface
Contents
Acronyms
1 Introduction of Vehicular Communications
1.1 Overview of Vehicular Communications
1.2 Propagation Characteristics of VehicularCommunication Channels
1.3 Classification of Vehicular Channel Models
1.4 Organization of the Monograph
References
2 A NGSM for SISO V2V Channels
2.1 Framework of SISO V2V NGSM
2.1.1 Introduction and Contributions of Proposed SISO V2V NGSM
2.1.2 Capturing Severe Fading Characteristic and Including LoS Component
2.2 Modeling of Time-Frequency Non-stationarity
2.2.1 Modeling of Time Non-stationarity by Markov Chains
2.2.2 Modeling of Frequency Non-stationarity by Generating Correlated Taps
2.3 Simulations and Discussions
2.3.1 Parameter Setting
2.3.2 Simulation Results
2.3.2.1 Power Delay Profiles
2.3.2.2 Tap Correlation Coefficient Matrix
2.3.2.3 Doppler Power Spectral Density
2.3.3 Comparisons of Conventional NGSMs and Proposed Model via Simulation
2.4 Summary
References
3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V Channels
3.1 Framework of Massive MIMO Vehicular RS-GBSM
3.1.1 Introduction and Contributions of Proposed RS-GBSM
3.1.2 Geometrical Representation and Channel Impulse Response of Proposed RS-GBSM
3.1.2.1 For LoS Component
3.1.2.2 For Ground Reflection Component
3.1.2.3 For the Double-Bounced Component Through Two Dynamic Clusters
3.1.2.4 For Double-Bounced Component Through Dynamic Clusters and Static Clusters
3.2 Space-Time Non-stationary Modeling with UniformPlanar Antenna
3.3 Simulations and Discussions
3.3.1 Statistical Properties
3.3.1.1 Space-Time Correlation Function
3.3.1.2 Space Cross-Correlation Function
3.3.1.3 Time Auto-Correlation Function
3.3.1.4 Wigner-Ville Spectrum
3.3.2 Model Simulation
3.3.3 Model Validation
3.4 Summary
References
4 A 3D IS-GBSM for Massive MIMO V2V Channels
4.1 Framework of Massive MIMO V2V IS-GBSM
4.1.1 Introduction and Contributions of Proposed IS-GBSM
4.1.2 Channel Impulse Response of Proposed Cluster-Based IS-GBSM
4.1.2.1 Complex Channel Gain of LoS Component
4.1.2.2 Complex Channel Gain of NLoS Component
4.2 Space-Time Non-stationary Modeling with VehicularTraffic Density
4.2.1 VTD-Combined Time Cluster Evolution Calculation
4.2.2 VTD-Combined Array Cluster Evolution Calculation
4.2.3 Steps of VTD-Combined Time-Array Cluster Evolution Algorithm
4.3 Simulations and Discussions
4.3.1 Statistical Properties
4.3.1.1 Space–Time–Frequency Correlation Function
4.3.1.2 Doppler Power Spectral Density
4.3.2 Model Simulation
4.3.3 Model Validation
4.4 Summary
References
5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive MIMO V2V Channels
5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM
5.1.1 Introduction and Contributions of Proposed IS-GBSM with Continuously Arbitrary Trajectory
5.1.2 Channel Impulse Response of Proposed Channel Model
5.1.2.1 Calculation of Transmission with Continuously Arbitrary Trajectory
5.1.2.2 Complex Channel Gain of LoS Component
5.1.2.3 Complex Channel Gain of NLoS Component Resulting from Dynamic Clusters
5.1.2.4 Complex Channel Gain of NLoS Component Resulting from Static Clusters
5.2 Space–Time–Frequency Non-stationary Modeling with Continuously Arbitrary Trajectory
5.2.1 Selective Cluster Evolution Based Space–Time–Frequency Non-stationary Modeling Method
5.2.1.1 Selective Cluster Evolution
5.2.1.2 Array-Time Evolution of Static Clusters
5.2.1.3 Array-Time Evolution of Dynamic Clusters
5.2.1.4 Frequency-Dependent Path Gain
5.3 Simulations and Discussions
5.3.1 Statistical Properties
5.3.1.1 Space–Time–Frequency Correlation Function
5.3.1.2 Doppler Power Spectral Density
5.3.1.3 Time Stationary Interval
5.3.2 Model Simulation
5.3.2.1 Model Validation
5.4 Summary
References
6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave Massive MIMO V2V Channels
6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM
6.1.1 Introduction and Contributions of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency
6.1.2 Channel Impulse Response of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency
6.1.2.1 Complex Channel Gain of LoS Component
6.1.2.2 Complex Channel Gain of NLoS Component via Ground Reflection
6.1.2.3 Complex Channel Gain of NLoS Components via Static Single-Clusters and Twin-Clusters
6.1.2.4 Complex Channel Gain of NLoS Component via Dynamic Single-Clusters and Twin-Clusters
6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space Consistency
6.2.1 Modeling of Space Non-stationarity and Consistency by Observable Semi-spheres Assigned to Antennas
6.2.1.1 Conditions of Array Observable Static/Dynamic Single-Clusters
6.2.1.2 Conditions of Array Observable Static/Dynamic Twin-Clusters
6.2.2 Modeling of Time Non-stationarity and Consistency by Observable Spheres Assigned to Clusters
6.2.2.1 Conditions of Time Observable Static Single/Twin-Clusters
6.2.2.2 Conditions of Time Observable Dynamic Single/Twin-Clusters
6.2.3 Soft Transition Factor
6.2.4 Frequency-Dependent Factor
6.3 Simulations and Discussions
6.3.1 Statistical Properties
6.3.1.1 Space–Time–Frequency Correlation Function
6.3.1.2 Power Delay Profile
6.3.1.3 Time Stationary Interval
6.3.1.4 Doppler Power Spectral Density
6.3.2 Model Simulation
6.3.3 Model Validation by Measurement and RT-Based Results
6.4 Summary
References
7 Conclusions and Future Research Directions
7.1 Conclusions
7.1.1 Discussions and Summary of Chap.1
7.1.2 Discussions and Summary of Chap.2
7.1.3 Discussions and Summary of Chap.3
7.1.4 Discussions and Summary of Chap.4
7.1.5 Discussions and Summary of Chap.5
7.1.6 Discussions and Summary of Chap.6
7.2 Future Research Directions
7.2.1 Channel Measurement Perspective
7.2.1.1 Measurement Platform Establishment
7.2.1.2 Measurement of Complicated Channel Non-stationarity and Consistency
7.2.2 Channel Modeling Perspective
7.2.2.1 Capturing Non-stationarity and Consistency of mmWave-THz Ultra-Massive MIMO V2V Channels
7.2.2.2 Developing Comprehensive and Efficient Hybrid Modeling Methods
7.2.2.3 Machine Learning Enabled Channel Non-stationarity and Consistency Capture
7.2.3 Channel Application Perspective
References
Index

Citation preview

Wireless Networks

Xiang Cheng Ziwei Huang Lu Bai

Channel Characterization and Modeling for Vehicular Communications

Wireless Networks Series Editor Xuemin Sherman Shen, University of Waterloo, Waterloo, ON, Canada

The purpose of Springer’s Wireless Networks book series is to establish the state of the art and set the course for future research and development in wireless communication networks. The scope of this series includes not only all aspects of wireless networks (including cellular networks, WiFi, sensor networks, and vehicular networks), but related areas such as cloud computing and big data. The series serves as a central source of references for wireless networks research and development. It aims to publish thorough and cohesive overviews on specific topics in wireless networks, as well as works that are larger in scope than survey articles and that contain more detailed background information. The series also provides coverage of advanced and timely topics worthy of monographs, contributed volumes, textbooks and handbooks.

Xiang Cheng • Ziwei Huang • Lu Bai

Channel Characterization and Modeling for Vehicular Communications

Xiang Cheng State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Peking University Beijing, China

Ziwei Huang State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Peking University Beijing, China

Lu Bai Shandong Research Institute of Industrial Technology Jinan, Shandong, China Joint SDU-NTU Centre for Artificial Intelligence Research (C-FAIR) Shandong University Jinan, Shandong, China

ISSN 2366-1186 ISSN 2366-1445 (electronic) Wireless Networks ISBN 978-3-031-47439-2 ISBN 978-3-031-47440-8 (eBook) https://doi.org/10.1007/978-3-031-47440-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

Since the end of the last century, the vehicular communication, which is one of the most significant technologies of intelligent transportation systems (ITSs), has received extensive attention from academia and industry. In the upcoming sixth generation (6G) era, wireless communication networks aim to construct space-airground-sea integrated networks (SAGSINs), where the vehicular communication is an indispensable component. It is certain that vehicular communications have the benefits of reducing the traffic accident as well as ameliorating the traffic efficiency. Therefore, vehicular communications can essentially facilitate future potential applications, including autonomous and intelligent vehicular applications. Aiming at supporting more advanced applications, vehicular communication systems are required to have a proper design to meet the demands of extremely low latency, high throughput, as well as high reliability. To facilitate the successful design and performance evaluation of vehicular communication systems, the accurate characterization and modeling are indispensable, which are also regarded as the enabling foundations of each communication system. To the best of our knowledge, there is currently no monograph that can adequately present the channel characterization and modeling for complex high-mobility vehicular communications. This monograph is intended to fill this gap and provide academia and industry with comprehensive coverage of channel characterization and modeling for highly dynamic and complicated vehicular communications. The monograph is organized as follows. Chapter 1 overviews vehicular communications and further presents the unique physical features and distinctive scattering environment of vehicular communications, followed by the typical vehicular channel features and characteristics. In Chap. 2, a time-frequency non-stationary single-input single-output (SISO) vehicle-to-vehicle (V2V) non-geometry stochastic model (NGSM) is introduced. To consider the impact of scattering geometry, Chap. 3 gives a massive multiple-input multiple-output (MIMO) V2V regularshaped geometry-based stochastic model (RS-GBSM), with the space-time nonstationarity and the effect of uniform planar antenna (UPA) taken into account. In Chap. 4, a V2V twin-cluster-based irregular-shaped geometry-based stochastic model (IS-GBSM) is presented, which captures the space-time non-stationarity of v

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Preface

vehicular channels in consideration of vehicular traffic density (VTD). To support the design of millimeter wave (mmWave) massive MIMO V2V communication systems, Chap. 5 elaborates on a space-time-frequency (S-T-F) non-stationary mmWave massive MIMO V2V IS-GBSM, which further mimics continuously arbitrary trajectories of transceivers. In Chap. 6, a mixed-bouncing based V2V ISGBSM, which is capable of simultaneously modeling the S-T-F non-stationarity and time-space (T-S) consistency, is given for mmWave massive MIMO V2V communication systems. Finally, Chap. 7 discusses conclusions and promising directions, hoping to stimulate future research outcomes in the field of vehicular channels from three different perspectives, including channel measurements, modeling, and applications. This monograph is dominantly written for researchers and professionals whose focus is wireless communications, while advanced-level students majoring in computer science and/or electrical engineering can also find the valuable content and suggestion. We would like to thank Mingran Sun and Mengyuan Lu for their inspiring discussions on the research work presented in this monograph. Finally, we would like to thank the continued support from the National Key R&D Program of China (Grant No. 2021ZD0112700), the National Natural Science Foundation of China (Grants No. 62125101, 62341101, 62001018, and 62371273), Shandong Natural Science Foundation (Grant No. ZR2023YQ058), the New Cornerstone Science Foundation through the XPLORER PRIZE, and Taishan Scholars Program. Beijing, China Beijing, China Jinan, China

Xiang Cheng Ziwei Huang Lu Bai

Contents

1

2

3

Introduction of Vehicular Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of Vehicular Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Propagation Characteristics of Vehicular Communication Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classification of Vehicular Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

A NGSM for SISO V2V Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Framework of SISO V2V NGSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Introduction and Contributions of Proposed SISO V2V NGSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Capturing Severe Fading Characteristic and Including LoS Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modeling of Time-Frequency Non-stationarity . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Modeling of Time Non-stationarity by Markov Chains. . . . . . . 2.2.2 Modeling of Frequency Non-stationarity by Generating Correlated Taps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Comparisons of Conventional NGSMs and Proposed Model via Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11

A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Framework of Massive MIMO Vehicular RS-GBSM . . . . . . . . . . . . . . . . . 3.1.1 Introduction and Contributions of Proposed RS-GBSM . . . . . . 3.1.2 Geometrical Representation and Channel Impulse Response of Proposed RS-GBSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 7 9

11 13 14 14 15 18 18 19 23 25 25 29 29 29 32 vii

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3.2 Space-Time Non-stationary Modeling with Uniform Planar Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

A 3D IS-GBSM for Massive MIMO V2V Channels. . . . . . . . . . . . . . . . . . . . . . 4.1 Framework of Massive MIMO V2V IS-GBSM . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Introduction and Contributions of Proposed IS-GBSM . . . . . . . 4.1.2 Channel Impulse Response of Proposed Cluster-Based IS-GBSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Space-Time Non-stationary Modeling with Vehicular Traffic Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 VTD-Combined Time Cluster Evolution Calculation . . . . . . . . 4.2.2 VTD-Combined Array Cluster Evolution Calculation . . . . . . . . 4.2.3 Steps of VTD-Combined Time-Array Cluster Evolution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive MIMO V2V Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM . . . . . . . 5.1.1 Introduction and Contributions of Proposed IS-GBSM with Continuously Arbitrary Trajectory . . . . . . . . . . . 5.1.2 Channel Impulse Response of Proposed Channel Model . . . . . 5.2 Space–Time–Frequency Non-stationary Modeling with Continuously Arbitrary Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Selective Cluster Evolution Based Space–Time–Frequency Non-stationary Modeling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 49 49 50 54 58 59 63 64 64 67 75 75 76 77 79 79 81 87 90 90 93 93 93 96 103

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6

7

A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave Massive MIMO V2V Channels . . . . . . . . . . . . . 6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Introduction and Contributions of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency . . 6.1.2 Channel Impulse Response of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency . . 6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Modeling of Space Non-stationarity and Consistency by Observable Semi-spheres Assigned to Antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Modeling of Time Non-stationarity and Consistency by Observable Spheres Assigned to Clusters . . . . . . . . . . . . . . . . . . 6.2.3 Soft Transition Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Frequency-Dependent Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Simulations and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Model Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Model Validation by Measurement and RT-Based Results . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Discussions and Summary of Chap. 1 . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Discussions and Summary of Chap. 2 . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Discussions and Summary of Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Discussions and Summary of Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Discussions and Summary of Chap. 5 . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.6 Discussions and Summary of Chap. 6 . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Channel Measurement Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Channel Modeling Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Channel Application Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

125 126 126 129 137

138 144 150 150 151 151 155 159 163 164 167 167 167 168 168 170 171 173 174 174 176 180 181

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Acronyms

1D 1G 2D 3D 3GPP 4G 5G 6G AAoA AAoD ACF AI AoA AoD B5G BD BS CATT CDF CEA CIR COST C-V2X DPSD EAoA EAoD FCF GBDM GBSM GCS

One-Dimensional First Generation Two-Dimensional Three-Dimensional Third-Generation Partnership Project Fourth Generation Fifth Generation Sixth Generation Azimuth Angle of Arrival Azimuth Angle of Departure Auto-Correlation Function Artificial Intelligence Angle of Arrival Angle of Departure Beyond Fifth Generation Birth-Death Base Station China Academy of Telecommunication Technology Cumulative Distribution Function Cluster Evolution Area Channel Impulse Response European COoperation in the field of Scientific and Technical research Cellular Vehicle-to-Everything Doppler Power Spectral Density Elevation Angle of Arrival Elevation Angle of Departure Frequency Correlation Function Geometry-Based Deterministic Model Geometry-Based Stochastic Model Global Coordinate System xi

xii

GR HMIMO IEEE ISAC IS-GBSM ITS K-M LCS LoS LTE LTE-V MIMO ML mmWave MPC NGSM NLoS NR OD OHT OLT PDF PDP PHY RAA RF RIS RMS RT RS-GBSM RSS Rx SAGSIN SCCF SD SE SISO SNR S-T CF S-T-F STF-CF TACF T-S TVTF Tx

Acronyms

Ground Reflection Holographic Multiple-Input Multiple-Output Institute of Electrical and Electronics Engineers Integrated Sensing and Communication Irregular-Shaped GBSM Intelligent Transportation System Kuhn-Munkres Local Coordinate System Line-of-Sight Long-Term Evolution LTE for Vehicle Multiple-Input Multiple-Output Machine Learning Millimeter Wave Multipath Component Non-Geometry Stochastic Model Non-Line-of-Sight New Radio Opposite Direction Open High Tran Open Low Tran Probability Density Function Power Delay Profile Physical Layer Real Antenna Array Radio Frequency Reconfigurable Intelligent Surface Root-Mean Square Ray Tracing Regular-Shaped GBSM R2V-Suburban Street Receiver Space-Air-Ground-Sea Integrated Network Space Cross-Correlation Function Same Direction Scattering Environment Single-Input Single-Output Signal-to-Noise Ratio Space-Time Correlation Function Space-Time-Frequency Space-Time-Frequency Correlation Function Temporal Auto-Correlation Function Time-Space Time-Varying Transfer Function Transmitter

Acronyms

ULA UOC UPA URA US V2I V2N V2P V2V V2X VANET VAA VEO VMT VSDW VTD WSS WSSUS

xiii

Uniform Linear Array Urban Outside Car Uniform Planar Antenna Uniform Rectangular Array Uncorrelated Scattering Vehicle-to-Infrastructure Vehicle-to-Network Vehicle-to-Pedestrian Vehicle-to-Vehicle Vehicle-to-Everything Vehicular Ad Hoc Network Virtual Antenna Array V2V-Expressway Oncoming Vehicular Movement Trajectory V2V-Same Direction with Wall Vehicular Traffic Density Wide-Sense Stationary Wide-Sense Stationary Uncorrelated Scattering

Chapter 1

Introduction of Vehicular Communications

Since the end of the last century, the vehicular communication, which is an important technology of the next generation intelligent transportation system (ITS), has received extensive attention. With the help of vehicular communications, many potential advanced applications associated with future vehicles, such as autonomous vehicles as well as intelligent vehicles, can be facilitated and promoted. In this chapter, the overview of vehicular communications is first presented, including introductions of vehicular ad hoc network (VANET) together with cellular vehicle-to-everything (C-V2X). Then, four physical features of actual vehicular communication scenarios are given, and unique vehicular channel characteristics are elaborated. Next, existing vehicular channel models are properly classified according to the modeling methods. Finally, the organization of the monograph is discussed.

1.1 Overview of Vehicular Communications The vehicle, which is an important mean of transportation in modern society, brings comfort and convenience to human beings. However, the rapid growth in the number of vehicles leads to prominent problems, such as traffic safety, urban congestion, environmental pollution, and carbon emissions. The vehicle industry has shown the development trend of electrification, intelligence, as well as networking. At the same time, the field of intelligent transportation systems has also proposed development directions, including digitization, networking, intelligence, and automation, all of which urgently need the ability of the vehicular communications to provide fundamental communications and connection support for them [1, 2]. With the increasing development of vehicular communications, according to [3], the complete selfdriving intelligent vehicle, i.e., level 4, will be put into utilization in the next decade, as shown in Fig. 1.1. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_1

1

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1 Introduction of Vehicular Communications

Fig. 1.1 Five development levels of the intelligent vehicle

Aiming at meeting the aforementioned demands, vehicular communications, i.e., vehicle-to-everything (V2X) communications, came into being, which can realize vehicle-to-vehicle (V2V), vehicle-to-pedestrian (V2P), vehicle-to-infrastructure (V2I), vehicle-to-network (V2N), and other communication connections and efficient information exchange. Actually, since the end of the last century, vehicular communications have received more and more attention. To facilitate a variety of applications, such as vehicle safety, entertainment, as well as transportation efficiency, vehicular communications need to satisfy the demands of high reliability and low latency. Based on the mobile ad hoc architecture, the VANET first attracted extensive research and standardization efforts. As an international standard of VANET, Institute of Electrical and Electronics Engineers (IEEE) 802.11p has been supported and promoted by the federal government of the United States. Moreover, systems based on IEEE 802.11p together with alternative systems have been constructed and standardized in Japan, China, as well as European Union, etc. Nevertheless, from the industrial perspective, VANET requires extensive investments in network infrastructure [4]. Unlike VANET, cellular network possesses sufficient development, a mature business model, and comprehensive standardization progress. As the first generation (1G) cellular systems had been developed and standardized in 1983, a new generation of the cellular systems is rolled out every decade. Up to now, the fourth generation (4G) long-term evolution (LTE) system has been extensively constructed and commercialized. Vehicular communications supported by the mature LTE, i.e., LTE for vehicle (LTE-V), was first proposed by the China Academy of Telecommunication Technology (CATT)/Datang in May 2013. LTE-V has the ability to achieve the significant operational benefit together with spectrum efficiency. The standardization of LTE-V has been developed actively in the 4G era. Note that, in March 2017, LTE-V completed the standardization as LTE-V2X in the ThirdGeneration Partnership Project (3GPP) Rel-14. As the cellular systems evolve

1.2 Propagation Characteristics of Vehicular Communication Channels

3

from 4G LTE to fifth generation (5G), C-V2X evolves from LTE-V2X to new radio (NR)-V2X. NR-V2X systems can support some advanced V2X services with demanding requirements, including high throughput, high system capacity, large coverage, high reliability, as well as low latency. The standardization of NRV2X is also being developed actively, and as planned, 3GPP Rel-16 and 3GPP Rel-17 can be completed in December 2019 and June 2021, respectively. In the coming beyond fifth generation (B5G) and sixth generation (6G) era, vehicular communication systems are expected to support more advanced V2X services with more demanding requirements. Specifically, aiming at supporting more potential applications associated with future vehicles, including autonomous and intelligent vehicles, vehicular communications for B5G and 6G require extremely low latency for the extremely high amount of data transmission together with extremely high reliability, as shown in Fig. 1.2. Consequently, it is significantly challenging to research and design the vehicular communication system for B5G and 6G in the future, which is drawing extensive attention.

1.2 Propagation Characteristics of Vehicular Communication Channels Detailed channel knowledge and realistic channel models play an important role in the proper design of communication systems, particularly for vehicular communication systems under high-mobility dynamic scenarios [5–7]. Different from conventional cellular communications, vehicular communications have some unique physical features. First, both the transmitter (Tx) vehicle and the receiver (Rx) vehicle are equipped with low-elevation antennas [8]. Second, transceivers and their surrounding scattering clusters, hereinafter referred to as clusters, are mobile at high speeds with a variety of vehicular movement trajectories (VMTs), including the quarter turn, U-turn, curve driving, etc. Third, in the propagation environment, the ratio of the number of mobile vehicles, i.e., dynamic clusters, to roadside buildings and trees, i.e., static clusters, is different as the vehicular traffic density (VTD) varies in various vehicular communication scenarios [9, 10]. Generally, vehicular communication scenarios with fewer and more moving vehicles are regarded as low and high VTD communication scenarios, respectively. Fourth, attributed to the restricted road topology, the communication distance of the vehicular communication is short, and there is generally a line-of-sight (LoS) component [11, 12]. In Fig. 1.3, a typical two-way two-lane vehicular communication scenario is depicted, where the wave at the Tx side passes through several scattering objects in the propagation environment and reaches the Rx side. These scattering objects contain dynamic clusters, e.g., cars and buses, as well as static clusters, e.g., buildings and trees. Therefore, the vehicular communication scenario is essentially complex. According to the vehicular measurement campaigns [13–16], the aforementioned four physical features significantly lead to unique vehicular channel characteristics

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1 Introduction of Vehicular Communications

Fig. 1.2 Vehicular communications for B5G and 6G

and further affect characteristics of vehicular communication channels. First, the high mobility of transceivers and their surrounding clusters result in the rapid appearance and disappearance of multipath components (MPCs) in a short time segment [17]. In this case, vehicular channels exhibit the apparent non-stationarity in the time domain, i.e., time non-stationarity. As a consequence, the statistical properties of vehicular channels will vary in the time domain. In addition, because of severer delay dispersion and Doppler dispersion, the fading of MPCs in fastchanging vehicular channels is generally worse than Rayleigh fading, resulting in the severe fading characteristic. Furthermore, VTDs and various VMTs of transceivers and dynamic clusters have an obvious influence on the vehicular

1.3 Classification of Vehicular Channel Models

5

Fig. 1.3 The representation of a typical two-way two-lane vehicular communication scenario

channel statistical properties. In general, vehicular channels with higher VTD under more complex VMTs of transceivers and dynamic clusters lead to a lower temporal correlation [18, 19].

1.3 Classification of Vehicular Channel Models In consideration of the aforementioned unique physical features and characteristics of vehicular communication channels, extensive vehicular channel models have been developed to characterize the vehicular communication scenarios [20–23]. In light of modeling methods, existing vehicular channel models are classified into the geometry-based deterministic model (GBDM), non-geometry stochastic model (NGSM), as well as geometry-based stochastic model (GBSM) [24–26]. Generally, vehicular GBSMs are further divided into regular-shaped GBSM (RS-GBSM) and irregular-shaped GBSM (IS-GBSM), depending on whether clusters are located in regular shapes, such as cylinders, ellipsoids, and spheres, or the irregular shape. For clarity, Fig. 1.4 summarizes the classification of existing vehicular channel modeling methods and their complexity and accuracy.

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1 Introduction of Vehicular Communications

Fig. 1.4 The classification of existing vehicular channel modeling methods and their complexity and accuracy

For the deterministic modeling method, i.e., GBDM, it is worth mentioning that the detailed and comprehensive procedure of physical radio propagation in a specific environment is mimicked based on the theory of electromagnetic field [27–30]. Aiming at developing a vehicular GBDM, it is essential to perform comprehensive measurements of vehicular channels. In such a case, the time-consuming and precise representation of the site-specific vehicular communication scenarios is required, resulting in high accuracy together with complexity. Unlike the deterministic one, in stochastic modeling methods, i.e., NGSM and GBSM, vehicular channel-related parameters are determined in a stochastic manner. For vehicular NGSMs, they characterize vehicular communication channels stochastically and do not assume any underlying scattering geometry [31–33]. More specifically, the construction of vehicular NGSMs depends totally on channel statistical properties together with probability density functions (PDFs) of empirical parameters, which should be acquired from the vehicular measurement. Because of the ignorance of underlying scattering geometry, the precision of NGSMs is lower than that of GBDMs. Nonetheless, based on stochastic modeling methods, the complexity of NGSMs is lower than that of GBDMs. To consider the effect of underlying scattering geometry on the vehicular channel modeling, a lot of vehicular GBSMs have been proposed. By exploiting simplified fundamental laws of wave propagation, GBSMs can be constructed from the predefined stochastic distribution of the cluster in geometrical propagation environments [34–39]. Vehicular RS-GBSMs assume that the cluster is stochastically distributed

1.4 Organization of the Monograph

7

according to a specific regular shape geometry. This simplified assumption leads to the fact that vehicular RS-GBSMs are low-complexity. However, as the cluster is simply assumed to be placed on the surface of objects with regular geometry, vehicular RS-GBSMs are also low accuracy. Different from vehicular RS-GBSMs, vehicular IS-GBSMs assume that locations of clusters in the propagation environment obey a specific statistical distribution. Note that the assumed statistical distribution is properly determined according to accurate measurement results of vehicular channels. Because of the requirement for the comprehensive vehicular channel measurement, the precision of IS-GBSMs is higher than that of RS-GBSMs, while the complication of IS-GBSMs is higher than that of RS-GBSMs.

1.4 Organization of the Monograph As shown in Fig. 1.5, the monograph is organized as follows: 1. In Chap. 1, vehicular communications are outlined and four unique physical features and distinctive scattering environment (SE) of vehicular communications are presented, followed by the typical vehicular channel physical features and characteristics. 2. In Chap. 2, a time–frequency non-stationary single-input single-output (SISO) V2V NGSM is introduced, where time–frequency non-stationarity, severe fading, the existence of LoS component, and significant Doppler effect are simultaneously captured. 3. In Chap. 3, to consider the impact of scattering geometry, a massive multipleinput multiple-output (MIMO) V2V RS-GBSM is given, which combines threedimensional (3D) multi-confocal semi-ellipsoids and semi-spheres. The proposed RS-GBSM mimics the space-time non-stationarity in consideration of the effect of the uniform planar antenna (UPA). 4. In Chap. 4, a twin-cluster-based IS-GBSM for V2V massive MIMO communication systems is presented, which captures the space-time non-stationarity of vehicular channels in consideration of VTD via the birth–death (BD) process method. 5. In Chap. 5, to support the design of millimeter wave (mmWave) massive MIMO V2V communication systems, a space–time–frequency non-stationary mmWave massive MIMO V2V IS-GBSM is discussed, which further mimics continuously arbitrary trajectories of transceivers. 6. In Chap. 6, a mixed-bouncing based V2V IS-GBSM, which is capable of simultaneously modeling the space–time–frequency non-stationarity and time– space consistency, is given for mmWave massive MIMO V2V communication systems. 7. In Chap. 7, conclusions and promising directions are given, hoping to stimulate future research outcomes in the field of vehicular channels from three different perspectives, including channel measurements, modeling, and applications.

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1 Introduction of Vehicular Communications

Chapter 1. Introduction of Vehicular Communications NGSM

Chapter 2. A NGSM for SISO V2V channels

Model description

SISO

NGSMoRS-GBSM SISOoMassive MIMO

RS-GBSM Chapter 3. A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V Channels

Model description

Massive MIMO

Space-time non-stationarity RS-GBSMoIS-GBSM

IS-GBSM

Chapter 4. A 3D IS-GBSM for Massive MIMO V2V Channels

Model description

Massive MIMO

Space-time non-stationarity Sub-6 GHzommWave Unifrom linear motionoContinuously arbitrary trajectory

Chapter 5. A 3D IS-GBSM with Model description Continously Arbitrary Trajectory for mmWave Massive MIMO V2V Channels

IS-GBSM

Massive MIMO/mmWave

Continuously arbitrary trajectory Space-time-frequency non-stationarity

Multi-bouncingoMixed-bouncing Time-space inconsistencyoTimespace consistency

IS-GBSM

Massive MIMO/mmWave Chapter 6. A 3D Mixed-Bouncing IS- Model description GBSM with Time-Space Consistency for mmWave Massive MIMO V2V Channels

Mixed-bouncing

Space-time-frequency non-stationarity

Time-space consistency

Chapter 7. Conclusions and Future Research Directions

Fig. 1.5 Organization of this monograph

References

9

References 1. X. Cheng, R. Zhang, L. Yang, 5G-Enabled Vehicular Communications and Networking (Springer, Cham, 2019) 2. S. Chen et al., Vehicle-to-everything (V2X) services supported by LTE-Based systems and 5G. IEEE Commun. Stand. Mag. 1(2), 70–76 (2017) 3. X. Cheng, R. Zhang, L. Yang, 5G-Enabled Vehicular Communications and Networking, 1st edn. (Springer, Cham, 2019) 4. S. Chen, J. Hu, Y. Shi, L. Zhao, LTE-V: A TD-LTE-based V2X solution for future vehicular network. IEEE Int. Things. J. 3(6), 997–1005 (2016) 5. J. Zhou, Z. Chen, H. Jiang, H. Kikuchi, Channel modelling for vehicle-to-vehicle MIMO communications in geometrical rectangular tunnel scenarios. IET Commun. 14(19), 3420– 3427 (2020) 6. C.-X. Wang, J. Huang, H. Wang, X. Gao, X. You, Y. Hao, 6G wireless channel measurements and models: trends and challenges. IEEE Veh. Technol. Mag. 15(4), 22–32 (2020) 7. W. Khawaja, I. Guvenc, D.W. Matolak, U.-C. Fiebig, N. Schneckenburger, A survey of air-toground propagation channel modeling for unmanned aerial vehicles. IEEE Commun. Surveys Tuts. 21(3), 2361–2391 (2019) 8. X. Cheng, Z. Huang, S. Chen, Vehicular communication channel measurement, modelling, and application for beyond 5G and 6G. IET Commun. 14(19), 3303–3311 (2020) 9. W. Viriyasitavat, M. Boban, H. Tsai, A. Vasilakos, Vehicular communications: survey and challenges of channel and propagation models. IEEE Veh. Technol. Mag. 10(2), 55–66 (2015) 10. Z. Huang, L. Bai, M. Sun, X. Cheng, A 3D non-stationarity and consistency model for cooperative multi-vehicle channels. IEEE Trans. Veh. Technol. 72, 11095–11110 (2023). https://doi.org/10.1109/TVT.2023.3268664 11. G. Acosta-Marum, M.A. Ingram, Model development for the wideband expressway vehicleto-vehicle 2.4 GHz channel, in IEEE Wireless Communications and Networking Conference, 2006. WCNC 2006, Las Vegas, USA (2006), pp. 1283–1288 12. Z. Huang, X. Cheng, N. Zhang, An improved non-geometrical stochastic model for nonWSSUS vehicle-to-vehicle channels. ZTE Commun. 17(4), 62–71 (2019) 13. G. Acosta-Marum, M.A. Ingram, Six time- and frequency- selective empirical channel models for vehicular wireless LANs. IEEE Veh. Technol. Mag. 2(4), 4–11 (2007) 14. I. Sen, D.W. Matolak, Vehicle-vehicle channel models for the 5-GHz band. IEEE Trans. Intell. Trans. Syst. 9(2), 235–245 (2008) 15. M. Yang et al., A cluster-based three-dimensional channel model for vehicle-to-vehicle communications. IEEE Trans. Veh. Technol. 68(6), 5208–5220 (2019) 16. X. Cai, B. Peng, X. Yin, A. Yuste, Hough-transform-based cluster identification and modeling for V2V channels based on measurements. IEEE Trans. Veh. Technol. 67(5), 3838–3852 (2018) 17. R. He et al., High-speed railway communications: From GSM-R to LTE-R. IEEE Veh. Technol. Mag. 11(3), 49–58 (2016) 18. A.G. Zajic, G.L. Stuber, T.G. Pratt, S.T. Nguyen, Wideband MIMO mobile-to-mobile channels: geometry-based statistical modeling with experimental verification. IEEE Trans. Veh. Technol. 58(2), 517–534 (2009) 19. Z. Huang et al., A non-stationary 6G V2V channel model with continuously arbitrary trajectory. IEEE Trans. Veh. Technol. 72, 4–19 (2022) 20. D.W. Matolak, Q. Wu, Markov models for vehicle-to-vehicle channel multipath persistence processes, in Proc. IEEE Veh. Tech. Society Wireless Access in Veh. Env. (WAVE), Dearborn, MI, USA (2008), pp. 8–9 21. X. Cheng, C.-X. Wang, B. Ai, H. Aggoune, Envelope level crossing rate and average fade duration of non-isotropic vehicle-to-vehicle Ricean fading channels. IEEE Trans. Intell. Transp. Syst. 15(1), 62–72 (2014)

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22. J. Maurer, W. Wiesbeck, A ray-optical channel model for vehicular Ad-Hoc networks, in Proceedings of the European Wireless, Nicosia, Cyprus (2006), pp. 1–7 23. Z. Huang et al., A mixed-bouncing based non-stationarity and consistency 6G V2V channel model with continuously arbitrary trajectory. IEEE Trans. Wireless Commun., to be published, (2023). https://doi.org/10.1109/TWC.2023.3293024 24. C.-X. Wang, J. Bian, J. Sun, W. Zhang, M. Zhang, A survey of 5G channel measurements and models. IEEE Commun. Surveys Tutor. 20(4), 3142–3168 (2018) 25. R. He et al., Propagation channels of 5G millimeter-wave vehicle-to-vehicle communications: recent advances and future challenges. IEEE Veh. Technol. Mag. 15(1), 16–26 (2020) 26. X. Cheng, Z. Huang, L. Bai, Channel nonstationarity and consistency for beyond 5G and 6G: a survey. IEEE Commun. Surveys Tutor. 24(3), 1634-1669 (2022) 27. M. Mbeutcha, W. Fan, J. Hejsclbæck, G.F. Pedersen, Evaluation of massive MIMO systems using time-reversal beamforming technique, in Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (IEEE PIMRC), Valencia, Spain (2016), pp. 1–6 28. O. Renaudin, V. Kolmonen, P. Vainikainen, C. Oestges, Non-stationary narrowband MIMO inter-vehicle channel characterization in the 5-GHz band. IEEE Trans. Veh. Technol. 59(4), 2007–2015 (2010) 29. S. Knörzer, M.A. Baldauf, T. Fugen, W. Wiesbeck, Channel analysis for an OFDM-MISO train communications system using different antennas, in Proceedings of the IEEE Vehicular Technology Conference (VTC-Fall), Baltimore, USA (2007), pp. 809–813 30. M. Elamassie, M. Karbalayghareh, F. Miramirkhani, R.C. Kizilirmak, M. Uysal, Effect of fog and rain on the performance of vehicular visible light communications, in 2018 IEEE 87th Vehicular Technology Conference (VTC Spring), Porto, Portugal (2018), pp. 1–6 31. D.W. Matolak, Channel modeling for vehicle-to-vehicle communications. IEEE Commun. Mag. 46(5), 76–83 (2008) 32. C. Li, L. Liu, J. Xie, Finite-state Markov wireless channel modeling for railway tunnel environments. China Commun. 17(2), 30–39 (2020) 33. Z. Huang, X. Zhang, X. Cheng, Non-geometrical stochastic model for non-stationary wideband vehicular communication channels. IET Commun. 14(1), 54–62 (2020) 34. A. Chelli, M. Patzold, A non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical street model, in 2011 IEEE Vehicular Technology Conference (VTC Fall), San Francisco, CA, USA (2011), pp. 1–6 35. E. Michailidis, N. Nomikos, P. Trakadas, A.G. Kanatas, Three-dimensional modeling of mmWave doubly massive MIMO aerial fading channels. IEEE Trans. Veh. Technol. 69(2), 1190–1202 (2020) 36. Y. Yuan, C.-X. Wang, Y. He, M.M. Alwakeel, e.M. Aggoune, 3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channels. IEEE Trans. Wirel. Commun. 14(12), 6883–6895 (2015) 37. Aalto University, AT&T, BUPT, CMCC, Ericsson, Huawei, Intel, KT Corporation, Nokia, NTT DOCOMO, New York University, Qualcomm, Samsung, University of Bristol, and University of Southern California. 5G Channel Model for Bands Up to 100 GHz (2016). [Online]. Available: http://www.5gworkshops.com/5GCM.html 38. Measurement results and final mmMAGIC channel models, mmMAGIC, Report. H2020-ICT671650-mmMAGIC/D2.2 (2017) 39. S. Jaeckel, L. Raschkowski, K. Börner, L. Thiele, QuaDRiGa-Quasi Deterministic Radio Channel Generator, User Manual and Documentation, Fraunhofer Heinrich Hertz Institute, Technical Report. v2.0.0 (2017)

Chapter 2

A NGSM for SISO V2V Channels

To successfully support the design and performance of V2V communication systems, a proper V2V channel model is essential. In this chapter, a new NGSM for time-frequency non-stationary wideband V2V communication channels is proposed. To include the LoS component, the proposed model first generates a non-uniformly distributed tap phase, which can be obtained from the widely used uniformly distributed tap phase. Moreover, the proposed model can practically experience variable types of Doppler spectra for different delays by modifying the auto-correlation function (ACF) of channel impulse response (CIR). To model the time non-stationarity, the Markov chains are leveraged to capture the appearance and disappearance of paths. In consideration of the non-stationarity in frequency the domain of vehicular communication channels, we further consider that the amplitude and phase of different taps are correlated. To evaluate the performance of the proposed model, some significant statistical properties in terms of the power delay profile (PDP), tap correlation coefficient matrix, and Doppler power spectral density (DPSD) are derived. It is demonstrated that, in comparison with the existing NGSMs, the proposed model possesses the ability to accurately mimic characteristics of real vehicular channels. Finally, the excellent agreement is achieved between the simulation results and the corresponding measurement data, confirming the accuracy of the proposed model.

2.1 Framework of SISO V2V NGSM 2.1.1 Introduction and Contributions of Proposed SISO V2V NGSM Since the Tx and Rx are moving in vehicular communication, the characteristics of vehicular channels significantly differ from those of conventional mobile cellular © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_2

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2 A NGSM for SISO V2V Channels

channels, where only one terminal is in motion [1, 2]. Furthermore, Doppler spectrum is the key factor that distinguishes vehicular channels from conventional cellular channels [3, 4]. As mentioned in [5], there are four unique and significant characteristics existing in vehicular channels, including the existence of LoS component, variable types of Doppler spectra for different delays, channel nonstationarity, and severe fading. Specifically, due to the characteristics of vehicular communication at short distance, the LoS component often exists, particularly in the situation of low VTD. Moreover, variable types of Doppler spectra for different delays can be experienced due to the frequency selectivity of vehicular channels. In general, the Tx, Rx, and some of the effective scatterers are all mobile, which results in the fact that the non-stationary characteristics often exist in vehicular channels. Finally, there are the Tx and Rx with low height and many fast moving scatterers, and thus MPCs tend to experience severe fading, which is worse than the conventional Rayleigh fading. To the best of the authors’ knowledge, there is no vehicular channel model having the ability to mimic the aforementioned four characteristics. Therefore, a vehicular channel model, which can capture the abovementioned characteristics of vehicular channels, is desirable to be developed. According to [6], vehicular channel models available in the literature can be categorized as deterministic models [7–9], NGSMs [10–15], and GBSMs [16–20]. Among all vehicular channel models, NGSMs are widely used due to advantages of low complexity and acceptable accuracy. In this case, based on the NGSM modeling approach, a new vehicular channel model is proposed. The major contribution and novelty of the proposed model are outlined as follows. 1. A new NGSM for non-stationary wideband vehicular channels. By taking the advantages of the existing NGSMs, the proposed model is the first NGSM that has the ability to consider and model the existence of LoS component, variable types of Doppler spectra for different delays, channel non-stationarity, and severe fading. 2. Different from the NGSM in [12], the proposed model generates a non-uniformly distributed tap phase to include the LoS component for the first time. The ACF is properly calculated, resulting in variable types of Doppler spectra for different delays. 3. In the proposed model, tap amplitude statistics are modeled as the Weibull distribution to practically mimic the severe fading. Also, the use of Markov chains and the generation of correlated stochastic variables describe the nonstationarity both in the time and frequency domains, respectively. To better mimic the non-stationarity in frequency domain, the complex correlation between the amplitude and phase of different taps is further modeled. 4. Several statistical properties are derived and analyzed, including the PDP, tap correlation coefficient matrix, as well as DPSD. The excellent agreement between the simulation and measured results verifies the utility of the proposed model.

2.1 Framework of SISO V2V NGSM

13

2.1.2 Capturing Severe Fading Characteristic and Including LoS Component In complex high-mobility vehicular channels, the transceiver and the surrounding scatterers move rapidly [21, 22]. In such a condition, MPCs will experience the severe fading, which is worse than the Rayleigh fading. Therefore, it is necessary to model the severe fading characteristic in the proposed model. Toward this aim, the Weibull distributed tap amplitude is generated. The Weibull distribution can be given as fW (x) =

.

β β−1 −x −β x e  , 

(2.1)

where x is the amplitude of the Gaussian stochastic variable, . is the energy parameter, and .β is a fading parameter to determine the fading severity of channels. In particular, the channel quality becomes better with .β being increased. Specially, when .β = 2, the Weibull distribution can be transformed to extensively used Rayleigh distribution. Note that, as mentioned in [23], compared to the widely utilized Nakagami distribution, the Weibull distribution is a better method over all the empirical data due to the capacity of modeling the severe fading characteristic. Then, the LoS component is included in the proposed NGSM. In the NGSM [12], a uniformly distributed phase is generated, which results in the absence of LoS component. To include the LoS component, a non-uniformly distributed phase is generated. Unlike the uniformly distributed phase, the non-uniformly distributed in the proposed model is flexibly combined with the fading parameter .β employed in the Weibull distribution. By changing the value of fading parameter .β, the proposed NGSM has the ability to mimic diverse vehicular communication scenarios with and without LoS components. In the modeling of severe fading, the NGSM in [12] generates the complex correlated Gaussian stochastic variable V from the independent Gaussian stochastic variables X and Y , and the complex correlated Gaussian stochastic variable V can be expressed as Vl = Xl + j Yl = Rl eφl , l ∈ {0, 1, . . . , L − 1},

.

(2.2)

where R and .φ are the amplitude and phase of the complex correlated Gaussian stochastic variable, respectively. In addition, the amplitude statistic R is modeled as the Weibull distribution via the transformation .|W | = R 2/β . Nevertheless, the phase statistics .φ still follow the uniform distribution in the interval .(−π, π ). Mathematically, the mean of the complex correlated Gaussian stochastic variable V with uniformly distributed phase can be calculated as 1 .E(Vl ) = 2π



π

−π

2/βl j φl

Rl

e

dφl = 0.

(2.3)

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2 A NGSM for SISO V2V Channels

Considering that the uniformly distributed phase implies the zero-mean quadrature and in-phase components, which causes the absence of LoS component. Therefore, a non-uniformly distributed phase to contain the LoS component is developed. The non-uniformly distributed is derived from the Weibull fading parameter .β and can be written as θl = φl · 2/βl , φl ∈ (−π, π ),

.

(2.4)

where .θ is the phase of complex correlated Gaussian stochastic variable in the proposed model. As .β increases, .θ concentrates within a smaller phase range. Mathematically, the complex correlated Gaussian stochastic variable .Vˆ with nonuniformly distributed phase can be expressed as Vˆl = Xˆ l + j Yˆl = Rl eθl

.

= Rl eφl ·2/βl , φl ∈ (−π, π ).

(2.5)

Similarly, the mean of the complex correlated Gaussian stochastic variable .Vˆ with non-uniformly distributed phase can be computed as E(Vˆl ) =

1 2π

=

1 2π

.

=



2π/βl

−2π/βl



2π/βl

−2π/βl

2/βl j θl

Rl

e

dθl |βl =2

2/βl j (φl ·2/βl )

Rl

e

d(φl · 2/βl )|βl =2

  4π 1 2/βl 1 − cos ej 4π/βl |βl =2 = 0. Rl π βl

(2.6)

As can be readily observed in (2.6), the complex correlated Gaussian stochastic variable .Vˆ no longer denotes the zero-mean quadrature and in-phase components. Consequently, the LoS component can be successfully contained in the proposed model.

2.2 Modeling of Time-Frequency Non-stationarity 2.2.1 Modeling of Time Non-stationarity by Markov Chains Since the propagation environment of vehicular channels changes dynamically, the statistics of channels alter over moderate time scales, resulting in non-wide-sense stationary (non-WSS) in the time domain, also named as non-stationarity in the time domain or time non-stationarity. The proposed model describes the non-WSS characteristics of vehicular channels by modeling the MPC persistence. The CIR of this model can be derived from the complex CIR of the NGSM in [24] through

2.2 Modeling of Time-Frequency Non-stationarity

15

adding an additional term, namely the “birth/death” process. This process can be further described as a switching function .zl (t) = {0, 1}, which represents the finite lifetime of the l-th resolvable path. To limit the number of taps with the non-negligible energy, the model employs the widely used thresholding methods [25], in which the tap “death” means below 25-dB threshold from the main tap. Additionally, the state transition process of the “birth/death” process can be modeled by the two-state first-order Markov chains, and the state transition matrix .(TS) and the steady-state matrix .(SS) can be expressed as  TS =

.

P00 P01 P10 P11



 , SS =

S0 S1

 ,

(2.7)

where .Pij in the matrix .TS denotes the state transition probability from state i to state j . For the matrix .SS, the steady state probability of the state u can be described as .Pu . Then, the CIR of the NGSM [12] can be derived as h(t, τ ) =

L−1 

.

zl (t)cl (t)δ[(τ − τl (t))]

l=0

ej [ωD,l (t)(t−τl (t))−ωc (t)τl (t)] ,

(2.8)

where l is the index of taps, .zl (t) = {0, 1} is a switching function of the l-th tap, .cl (t) is the amplitude of the l-th tap, .δ(·) is impulse response function, and .τl (t) denotes the delay of the l-th tap. We have .ωD,l (t) = 2πfD,l (t), where .fD,l (t) is the Doppler shift, and .ωc (t) = 2πfc (t) is the carrier frequency. As a result, by exploiting the two-state first-order Markov chain, the birth and death behavior of taps is modeled to capture the non-stationarity in the time domain, i.e., time non-stationarity.

2.2.2 Modeling of Frequency Non-stationarity by Generating Correlated Taps In order to obtain variable types of DPSD for different delays, an accurate ACF of CIR, which considers the different time correlation of each path of taps, is developed. Moreover, the modeling of complex correlation between the amplitude and phase of different taps is used to properly mimic the non-uncorrelated scattering (non-US), also named as the non-stationarity in frequency domain or frequency non-stationarity. Since the non-WSS means that the channel statistical property is time-variant and can be practically described by the switching function .zl (t), the assumption of WSS can be used when models the frequency non-stationarity. Consequently, the ACF of CIR can be written as rhh (τ1 , τ2 ; t, t + τi ) = rhh (τ1 , τ2 ; τi ),

.

(2.9)

16

2 A NGSM for SISO V2V Channels

where .τl is the delay of the l-th path, and .τi is the time interval. Note that the ACF of CIR contains the time domain and the delay domain. Additionally, we still assume that the correlated scattering characteristics of channels do not change with time, and thus the joint ACF of CIR is independent in time and delay domains and can be derived as rhh (τ1 , τ2 ; τi ) = ρ(τ1 , τ2 )rhh (t, t + τi ),

.

(2.10)

where .ρ(τ1 , τ2 ) and .rhh (t, t + τi ) are the ACF in the delay and time domains, respectively. Furthermore, considering that the amplitude of each path follows the Rayleigh distribution, the CIR can be obtained by two independent linear correlation operations. Specifically, the Doppler filter convolution is employed to describe the time domain of CIR, while the linear operation is utilized to represent the delay domain of CIR. Consequently, the CIR can be properly given as h(t, τ ) = σ (τ )



.

hus (t, τi )L(t, τi ),

(2.11)

i

where .hus (t, τ ) follows a complex Gaussian distribution .N (0, 1), and L can be obtained by Cholesky decomposition .LLH = ρ. Through the substitution formula (2.11), the ACF of CIR in the time domain can be calculated as rhh (t, t + τi ) = E[h(t, τ )h∗ (t + τi , τ )]  L(t, τ )LH (t + τi , τ ) = σ 2 (τ )

.

i

= σ (τ )ρ(t, t + τi ). 2

(2.12)

Consequently, each path no longer has the same correlation function in the time domain. On the contrary, the ACF of two paths varies with the time interval .τi between two paths. Furthermore, we substitute (2.12) into (2.10) to derive the ACF of CIR, which can be expressed as rhh (τ1 , τ2 ; τi ) = ρ(τ1 , τ2 )rhh (t, t + τi )

.

= σ 2 (τ )ρ(τ1 , τ2 )ρ(t, t + τi ).

(2.13)

In addition, the DPSD by the ACF can be obtained, which can be given as PSD (fD ) = F[rhh (Δf = 0; τ )]  +∞ = rhh (τi )e−j 2πfD τi dτi ,

.

−∞

(2.14)

2.2 Modeling of Time-Frequency Non-stationarity

17

where .F(·) denotes the Fourier transform, and .fD is the Doppler frequency. Therefore, we change the assumption that time correlation of each path is the same and further modifies the ACF used in NGSM [12]. However, since it is no longer assumed that each path has the same time correlation, there is a significant decrease in the correlation between different taps. Consequently, the frequency non-stationarity cannot be practically described. To overcome this limitation, the complex correlation between different taps both in amplitude and phase is further considered. Based on the NGSM in [12], the correlation coefficients of stochastic variables of two taps can be given as (V )

ρi,j =

.

E[Vi Vj ] − E[Vi ]E[Vj ]  , var[Vi ]var[Vj ]

(2.15)

where .Vi and .Vj are the stochastic variables of the i-th and j -th taps with complex Gaussian distribution, and .E(·) and .var(·) designate the statistical expectation and variance, respectively. As mentioned in [26], the amplitude and phase of the complex Gaussian stochastic variable V are independent. As a consequence, .E[Vi Vj ] can be expressed as E[Vi Vj ] = E[Ri Rj ]E[ej φi ej φj ],

.

(2.16)

where R is the amplitude of the complex Gaussian stochastic variable, which follows the Rayleigh distribution. While .φ is the phase of the complex Gaussian stochastic variable, which follows the uniform distribution. Based on the method of the literature [27], the expected value .E[Ri Rj ] can be obtained, which can be written as  +∞  +∞ β /2 β /2 .E[Ri Rj ] = ωi i ωj j fωi ,ωj (ωi , ωj )dωi dωj , (2.17) 0

0

where .fωi ,ωj (ωi , ωj ) is the joint probability density function of the bivariate Weibull distribution, which is provided in [28] and can be given as β −1 β −1

βi βj ωi i ωj j   .fωi ,ωj (ωi , ωj ) = (W ) i j 1 − ρi,j ⎡ ⎤  βj β ωj i ωi 1 ⎦  ×exp ⎣−  + (W ) i j 1 − ρi,j ⎡  ⎤ (W ) βi /2 βj /2 2 ρi,j ωi ωj ⎦, ×I0 ⎣ (W ) √ 1 − ρi,j i j

(2.18)

18

2 A NGSM for SISO V2V Channels

where .I0 (x) is zeroth-order modified Bessel function of the first kind. Similarly, E[ej φi ej φj ] can be computed as

.

 E[e

.

j φi j φj

e

2π

]= 0



2π 0

ej φi ej φj fφi ,φj (φi , φj )dφi dφj ,

(2.19)

where .fφi ,φj (φi , φj ) is the joint probability density function of the bivariate uniform distribution, which is provided in [25] and can be expressed as fφi ,φj (φi , φj ) =

.

    φj φi 1 (φ) 1 + 3ρ 1 − 1 − . i,j π π 4π 2

(2.20)

However, the proposed model generates the non-uniformly distributed phase, which is derived from the conventional uniformly distributed phase by the linear transform (φ) (θ) .θl = φl · 2/βl , and thus .ρ i,j = ρi,j . In summary, .E[Ri Rj ] can be obtained by substituting (2.18) into (2.17), and jφ jφ .E[e i e j ] can be derived by substituting (2.20) into (2.19). Therefore, the (V ) correlation coefficients of stochastic variables of two taps .ρi,j can be computed as      3 3 3  (W ) 2 (θ) (W ) (V ) , , 1, ρi,j , (2.21) i j 1 − ρi,j ρi,j × H .ρ i,j = − 8π 2 2 where .H (a, d, z) is the conventional Gaussian hypergeometric function, and .i and j are the energy parameters of i-th and j -th taps, respectively. Therefore, different from the NGSM [12], we further consider the correlation between the amplitude and phase of different taps, and thus the frequency nonstationarity can be better modeled.

.

2.3 Simulations and Discussions To evaluate the characteristics and performance of the proposed model, typical channel statistical properties, including PDP, tap correlation coefficient matrix, as well as DPSD are presented.

2.3.1 Parameter Setting In the simulation, unless otherwise specified, all the channel-related parameters used are listed in Table 2.1. In addition, the basic parameters can be set as the carrier frequency .fc = 5.9 GHz, and the bandwidth .B = 10 MHz. As for the duration of each “birth/death” process state .Tc , .0.005 s to .0.01 s is considered as a pragmatic

2.3 Simulations and Discussions

19

Table 2.1 Channel-related parameters used in the simulation All scenarios UOC (V2V-SD)

.β

Weibull factor = [3.19, 1.61, 1.63, 1.73]

OLT (V2V-SD)

.β

OHT (V2V-SD)

.β

VSDW (V2V-SD) .β

VEO (V2V-OD)

.β

RSS (R2V)

.β

Parameters = 24 m/s .fLoS = [−55, −20, −56, 0] .P00 = [0.271, 0.442, 0.557] .P11 = [0.915, 0.817, 0.748] = [5.15, 1.63] .v = 52 m/s .fLoS = [−237, 169] Hz .P00 = [0.3836] .P11 = [0.8525] = [4.3, 1.64, 1.89] .v = 52 m/s .fLoS = [405, 467, 261] Hz .P00 = [0.3625, 0.5999] .P11 = [0.8366, 0.6973] = [2.6, 2.7, 2.1, 2, 1.8, 1.9] .v = 24 m/s .fLoS = [−55, −20, −56, 0, −8, −99] Hz .P00 = [0, 0, 0, 0, 0, 0] .P11 = [1, 1, 1, 1, 1, 1] = [2.59, 2.21, 1.91] .v = 40 m/s .fLoS = [431, 204, 265] Hz .P00 = [0, 0, 0] .P11 = [1, 1, 1] = [2.95, 2.80, 2.75, 2.02, 1.95] .v = 24 m/s .fLoS = [428, 224, 582, −99, 527] Hz .P00 = [0, 0, 0, 0, 0] .P11 = [1, 1, 1, 1, 1] .v

range. Moreover, the proposed model is based on a comprehensive measurement campaign, including six different and typical vehicular communication scenarios, e.g., Urban Outside Car (UOC), Open Low Tran (OLT), Open High Tran (OHT), V2V-Same Direction with Wall (VSDW), V2V-Expressway Oncoming (VEO), and R2V-Suburban Street (RSS). It is important to emphasize that scenarios contain V2V and R2V communications and the situation that vehicles move in same directions (SDs) and opposite directions (ODs).

2.3.2 Simulation Results 2.3.2.1

Power Delay Profiles

Figure 2.1 compares PDPs of the proposed model with those of the conventional NGSMs in [12, 24]. To be more specific, the PDPs of the proposed model are simulated and compared with the measured PDPs of the NGSM [12] for the

20

2 A NGSM for SISO V2V Channels 0 Conventional NGSM, UOC scenario Proposed model, UOC scenario Conventional NGSM, OHT scenario Proposed model, OHT scenario Conventional NGSM, OLT scenario Proposed model, OLT scenario

Power (dB)

-5

-10

-15

-20

-25 0

50

100

150

200

250

300

350

Delay (ns)

(a) 5 Conventional NGSM, VEO scenario Proposed model, VEO scenario Conventional NGSM, RSS scenario Proposed model, RSS scenario Conventional NGSM, VSDW scenario Proposed model, VSDW scenario

0

Power (dB)

-5

-10

-15

-20

-25

-30 0

100

200

300

400

500

600

700

Delay (ns)

(b)

Fig. 2.1 Comparisons of the PDP of the conventional NGSMs in [12, 24] with the proposed model for different communication scenarios. (a) Comparisons of the PDP of the conventional NGSM in [24] with the proposed model. (b) Comparisons of the PDP of the conventional NGSM in [12] with the proposed model

UOC, OHT, and OLT scenarios in Fig. 2.1b. It is observed from Fig. 2.1b that, in both models, the spread delay in OLT scenario is shorter than that of others. The underlying physical reason is that the reflection and scattering caused by mobile vehicles are less than other scenarios. Moreover, the delay spread in UOC scenario

2.3 Simulations and Discussions

21

is higher than that of others, which is consistent with the measurement result in [29]. Another important phenomenon worth noting is that the energy of the proposed model for each scenario is more centered in the first path compared with the NGSM in [12]. In other words, the energy of other paths in the NGSM [12] is less than that of the proposed model. This is because the LoS component is successfully added into the first path of the proposed model, resulting in a visible decline of the proportion of energy in other paths after normalization. To validate the accuracy of the proposed model, the comparisons between the PDPs of the proposed model and those of the NGSM [12] for the VSDW, VEO, and RSS scenarios are shown in Fig. 2.1a. It is obvious that the simulated PDPs of the proposed model closely match with those of the NGSM [24]. This is due to the fact that both models have the ability to include the LoS component. This excellent agreement demonstrates that the LoS component is properly included in the proposed model and proves the usefulness and practicality of the proposed model.

2.3.2.2

Tap Correlation Coefficient Matrix

Since the frequency non-stationarity cannot be described in the NGSM [24], it can be concluded that the tap correlation coefficient matrix of the NGSM in [24] is a null matrix. As a result, we compare the tap correlation coefficient matrix of the proposed model with those of the NGSMs [12, 27] in Tables 2.2 and 2.3, respectively. Since the correlation coefficient matrix is symmetric about the diagonal, the table can be simplified as follows. The lower triangular part corresponds to the tap correlation coefficient matrix of the proposed model, while the upper triangular part corresponds to the tap correlation coefficient matrix of the conventional NGSMs. Compared with the NGSM [12] in Table 2.2, the tap correlation coefficients of

Table 2.2 Correlation coefficient matrix of the NGSM in [12] and the proposed model for the UOC scenario (lower/upper triangular part: proposed model/NGSM in [12]) i, j 1 2 3 4

1 1.0000 0.8411 0.7023 0.7270

2 0.4628 1.0000 0.6739 0.4961

3 0.3579 0.2752 1.0000 0.9123

4 0.3579 0.2354 0.3035 1.0000

Table 2.3 Correlation matrix coefficient of the NGSM in [27] and the proposed model for the UOC scenario (lower/upper triangular part: proposed model/NGSM in [27]) i, j 1 2 3 4

1 1.0000 0.8411 0.7023 0.7270

2 0.6898 1.0000 0.6739 0.4961

3 0.6518 0.4922 1.0000 0.9123

4 0.5772 0.5142 0.8479 1.0000

22

2 A NGSM for SISO V2V Channels

the proposed model are obviously larger. This is reasonable due to the fact that the proposed model not only generates complex correlated Gaussian stochastic variables but also models the complex correlation between different taps both in the amplitude and phase part. For verification purposes, Table 2.3 presents the comparisons of the obtained theoretical tap correlation coefficient matrix with the measurement data in [27]. It can be observed that the corresponding parameters of the proposed model only fluctuate with a few correlation coefficient values compared with those of the NGSM [27], while most of them remain consistent. This is because in both models, the complex correlation between amplitude and phase of different taps is taken into account. This excellent agreement between these allows us to conclude that the proposed model can properly consider the complex correlation between different taps both in the amplitude and phase part, and thus the frequency non-stationarity can be better mimicked.

2.3.2.3

Doppler Power Spectral Density

Figure 2.2 presents the DPSDs of the NGSM in [12] and the proposed model under different scenarios, including UOC and OHT. First, it can be observed from Fig. 2.2 that the DPSD of the proposed model has a dominant narrow peak, which is characteristic to communications in presence of the LoS component [30]. Specially, it can be noticed that, in the proposed model, when .β > 2, a dominant LoS component is successfully added in the DPSD. On the contrary, when .β < 2, the DPSD cannot include the LoS component. As previously mentioned, .β is the fading parameter. When .β < 2, vehicular channels experience the severe fading characteristic, which is worse than the Rayleigh fading. This observation is fundamentally different from that in the NGSM [12], where the generation of the optimal uniformly distributed phase results in the absence of LoS components. Therefore, the DPSD of the NGSM [12] does not have a dominant narrow peak. To further validate the accuracy of the proposed model, Fig. 2.3 shows the DPSD of the proposed model for the first and second taps. In the proposed model, it is worth noting that, when two vehicles move in the SD, the Doppler shift of the LoS component is significantly large. In contrast, when two vehicles move in the OD, the Doppler shift of the LoS component is close to zero, and thus the LoS component appears in the middle of the DPSD. The underlying physical reason is that the Doppler shift of LoS component is proportional to the relative speed of two vehicles. When two vehicles move in the OD at almost the same speed, the relative speed is close to zero. Nonetheless, in the SD situation, the relative speed of two vehicles is relatively large. This phenomenon is also consistent with the measured DPSD in [24], where the position of the LoS component also varies flexibly with the change of motion direction of two vehicles. Therefore, according to Fig. 2.3, the aforementioned comparison also demonstrates that the LoS component is properly included in the proposed model.

2.3 Simulations and Discussions

23

Fig. 2.2 DPSDs of the NGSM in [12] and the proposed model for different scenarios. (a) DPSD of the NGSM in [12] for the UOC scenario. (b) DPSD of the proposed model for the UOC scenario. (c) DPSD of the NGSM in [12] for the OHT scenario. (d) DPSD of the proposed model for the OHT scenario

2.3.3 Comparisons of Conventional NGSMs and Proposed Model via Simulation In Table 2.4, to clearly present the advantage of the proposed model, the proposed model is compared with the NGSMs in [12, 24]. Based on the aforementioned analysis, there are four unique and significant characteristics in vehicular channels, i.e., the existence of LoS component, variable types of Doppler spectra for different delays, non-stationarity, and severe fading. The NGSM in [24] can mimic the aforementioned first two channel characteristics, while the NGSM in [12] can model the last two channel characteristics. Different from the NGSMs in [12, 24], based on the aforementioned simulation result, it is certainly clear that the proposed model has the ability to describe all these four characteristics. Moreover, the proposed model also models the complex correlation between the amplitude and phase of different taps, and thus the time-frequency non-stationarity can be modeled more accurately. It is important to emphasize that, for simplicity, the proposed model is compared with the performance of conventional NGSMs to validate the

24

2 A NGSM for SISO V2V Channels

Fig. 2.3 DPSDs of the first and second tap of the proposed model for different scenarios. (a) DPSD of the first tap for the VSDW scenario. (b) DPSD of the second tap for the VSDW scenario. (c) DPSD of the first tap for the RSS scenario. (d) DPSD of the second tap for the RSS scenario

accuracy of the proposed model. It is reasonable because other kinds of models have significantly similar performance to these conventional NGSMs. Specifically, similar to the NGSM in [24], it can be seen from Doppler spectra presented in [10, 31] that, the energy of each tap for each scenario is almost the same. This is due to the fact that the severe fading is not taken into account in these models [10, 24, 31]. As with the conventional NGSM in [12], the models in [22, 32] only possess one type of the Doppler spectra, while the proposed model has variable types of Doppler spectra for different delays. Furthermore, different from the model in [33], the proposed model can properly mimic the time-frequency non-stationarity in vehicular channels. Finally, it is worth noting that the proposed model and all the aforementioned models in [10, 22, 31–33] can explicitly describe the existence of LoS component. Therefore, it can be properly concluded that, compared with the conventional NGSMs and other kinds of channel models, the proposed model is more suitable for vehicular channels.

References

25

Table 2.4 Comparisons of the conventional NGSMs in [12, 24] and the proposed model (WSSUS: wide-sense stationary uncorrelated scattering) NGSM in [24] WSSUS With LoS components DPSD with variable types Stationarity Rayleigh/Ricean fading

NGSM in [12] non-WSSUS Without LoS components DPSD with a single type Non-stationarity Severe fading

The proposed model non-WSSUS With LoS components DPSD with variable types Non-stationarity Severe fading

2.4 Summary In this chapter, a new NGSM for time-frequency non-stationary wideband vehicular communication channels has been developed. Based on the measurement data, there are four unique and significant characteristics existing in vehicular channels, i.e., the existence of LoS component, variable types of Doppler spectra for different delays, time-frequency non-stationarity, and severe fading. By taking the advantages of the existing NGSMs in [12, 24], the proposed model has the ability to practically mimic the aforementioned four characteristics. Specifically, the Weibull distributed tap amplitude has been generated to properly mimic the severe fading characteristic. Furthermore, the proposed model has employed the Markov chains and generated the correlated stochastic variables to describe the non-stationarity both in the time and frequency domains, respectively. Different from the NGSM in [12], the proposed model has generated a non-uniformly distributed tap phase to contain the LoS component. Also, the proposed model has modified the assumption used in the existing NGSM and has further modified the ACF of CIR to obtain variable types of Doppler spectra for different delays. Moreover, the complex correlation between different taps both in the amplitude part and phase part has been properly modeled. Thus, the non-stationarity in the frequency domain can be mimicked more accurately. Finally, based on the proposed model, several statistical properties have been derived and analyzed, including the PDP, tap correlation coefficient matrix, and DPSD. Finally, it has been observed that the excellent agreement has been achieved between the simulation results and measurement data, which demonstrates the utility of the proposed model.

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25. N. Johnson, S. Kotz, Distributions in Statistics: Continuous Multivariate Distributions (John Wiley & Sons, Inc., New York, 1972) 26. J.I.-Z. Chen, C.-C. Yu, Y.-F. Chung, S.-H. Yan, On the impact of CFO for OFDM systems with un-equal gain diversity schemes over small-term fading, in Proceedings of the 2008 3rd International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CrownCom 2008), Singapore (2008), pp. 1–5 27. Y. Li et al., A TDL based non-WSSUS vehicle-to-vehicle channel model. Int. J. Antennas Propag. 2013, 103461-1–103461-8 (2013) 28. N.C. Sagias, G.K. Karagiannidis, Gaussian class multivariate Weibull distributions: theory and applications in fading channels. IEEE Trans. Inf. Theory 51(10), 3608–3619 (2005) 29. O. Renaudin, V. Kolmonen, P. Vainikainen, C. Oestges, Wideband MIMO car-to-car radio channel measurements at 5.3 GHz, in Proceedings of the 2008 IEEE 68th Vehicular Technology Conference (VTC) (2008), pp. 1–5 30. D.W. Matolak, Channel modeling for vehicle-to-vehicle communications. IEEE Commun. Mag. 46(5), 76–83 (2008) 31. G. Acosta-Marum, M.A. Ingram, A BER-based partitioned model for a 2.4 GHz vehicle-tovehicle expressway channel. Wireless Pers. Commun. 37, 421–446 (2006) 32. I. Ivan, P. Besnier, X. Bunion, L. Le Danvic, M. Drissi, On the simulation of Weibull fading for V2X communications, in Proceedings of the 2011 11th International Conference on ITS Telecommunications (2011), pp. 86–91 33. G. Acosta, M.A. Ingram, Model development for the wideband expressway vehicle-to-vehicle 2.4 GHz channel, in IEEE Wireless Communications and Networking Conference (WCNC) (2006), pp. 1283–1288

Chapter 3

A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V Channels

With the 5G wireless communication networks being at the stage of commercial deployment, the upcoming B5G/6G wireless communication network has attracted more and more attention, where the V2V is a typical communication scenario. Furthermore, massive MIMO has been considered as one promising technology for V2V communications. To support the design of massive MIMO V2V communication systems, a novel 3D non-stationary wideband V2V RS-GBSM with uniform planar antenna arrays (UPAs) for massive MIMO wireless communication systems is proposed. In the proposed RS-GBSM, a novel method, so-called BD process and seed algorithm based selective cluster evolution, is developed to capture the space non-stationarity of massive MIMO V2V with UPA channels. The time non-stationarity is further mimicked by employing this novel method over the entire timeline. In addition, the proposed RS-GBSM divides clusters into static and dynamic clusters to investigate the impact of VTD on channel statistics. Important statistical properties, such as the space-time correlation function (S-T CF), space cross-correlation function (SCCF), time auto-correlation function (TACF), and DPSD are derived and thoroughly investigated. Simulation results show that the space-time non-stationarity is successfully mimicked and the VTD has a significant impact on channel statistics. Finally, an excellent agreement is achieved between simulation results and measurements, validating the accuracy of the proposed RSGBSM.

3.1 Framework of Massive MIMO Vehicular RS-GBSM 3.1.1 Introduction and Contributions of Proposed RS-GBSM It is well known that the V2V wireless communication scenario is one of the most important B5G/6G communication scenarios [1–3]. Nevertheless, the high-mobility © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_3

29

30

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

characteristic of V2V communications brings huge challenges on the improvement of communication performance, e.g., increasing the transmission latency, making the data connections unreliable, and so on. To properly solve these problems, massive MIMO, which is extensively employed in B5G/6G [4, 5], has been considered as one important technology for V2V communications. Clearly, massive MIMO can achieve a descent signal-to-noise ratio (SNR) through sharpening directional transmission largely, mitigate multipath fading due to spatial diversity substantially, and increase the channel capacity because of spatial multiplexing [6, 7]. To successfully design B5G/6G massive MIMO V2V systems, it is necessary to establish a general, accurate, as well as easy-to-use channel model, which can adequately mimic and model the underlying characteristics of B5G/6G massive MIMO V2V channels [8–11]. The V2V wireless communication system, as its name indicates, is the wireless system equipped with vehicles as the Tx and Rx. In general, the Tx, Rx, and clusters in the propagation environment are moving at a high speed. Unlike cellular communications, the V2V channel has its unique properties. First, there are two kinds of clusters in the propagation environment, including static clusters on behalf of roadside static obstacles and dynamic clusters on behalf of moving vehicles around the transceiver [12, 13]. Second, the ground reflection (GR) becomes a key component in the V2V channel modeling due to low-elevation antennas at the transceiver [14]. Third, the influence of VTD, including the rural street scenarios with few vehicles (low VTD) and the highway street scenarios with many vehicles (high VTD), on channel statistical properties needs to be explored [15, 16]. Finally, due to the rapid movement of the transceiver and clusters, key channel parameters, such as power, delay, Doppler frequency, angles of departure (AoDs), and angles of arrival (AoAs), are time-variant, resulting in the time non-stationary V2V channels [15, 17]. In the coming B5G/6G era, it can be expected that massive MIMO will be extensively adopted in V2V communications [18, 19]. The use of massive MIMO results in the channel space non-stationarity. Furthermore, V2V channels typically exhibit the time non-stationarity. As a result, for massive MIMO V2V channels, the non-stationarity will occur in both the space domain and the time domain, i.e., space-time non-stationarity. To properly support the design of massive MIMO high-mobility V2V communication systems, a massive MIMO V2V channel model, which has the capability to simultaneously consider the GR component, model the space-time non-stationarity, and explore the effects of VTDs on channel statistics, is indispensable. Toward this end, a novel 3D space-time non-stationary wideband V2V RS-GBSM with UPAs for massive MIMO communication systems is proposed. Note that the RS-GBSM modeling approach employs the regular geometry to describe the distribution of the cluster and, hence, possesses a low complexity. Consequently, the RS-GBSM modeling approach has been extensively exploited in theoretical research of current V2V channel modeling. Furthermore, since UPAs have a smaller physical size than ULAs, it is natural that they are more practical for massive MIMO V2V communications. The main contributions and novelties are summarized as follows.

3.1 Framework of Massive MIMO Vehicular RS-GBSM

31

1. A novel space-time 3D non-stationary wideband V2V RS-GBSM with UPAs for massive MIMO wireless communication systems is proposed. The proposed RS-GBSM combines 3D multi-confocal semi-ellipsoids and semi-spheres, where the CIR is sum of the LoS component, the GR component, and the doublebounced components. By dividing clusters into static clusters and dynamic clusters, the impact of VTDs on channel statistics can be modeled properly and explored sufficiently. Furthermore, the double-bounced components between dynamic clusters and static clusters with different delays for wideband systems are considered. 2. The proposed RS-GBSM is modeled in a 3D coordinate system and channel parameters are calculated by 3D vectors. In the proposed RS-GBSM, no inequality is leveraged to simplify the transmission path calculation, and thus the spherical wavefront propagation is modeled accurately. Furthermore, obtaining the geometric relationship between the clusters and the antenna by vector calculation can significantly reduce the computational complexity. Moreover, the antenna vector in local coordinate system (LCS) is converted to the antenna vector in global coordinate system (GCS) through the rotation matrix, and thus the rotations of UPAs are properly taken into account. 3. A novel method, so-called BD and seed algorithm based selective cluster evolution, is developed to model the space non-stationarity of massive MIMO V2V with UPA channels. In the selective cluster evolution, it is assumed that only clusters in cluster evolution areas (CEAs), named as the array non-stationary clusters, experience the array evolution because of near-field effect, and other clusters, named as the array stationary clusters, can be observed by all antennas. Then, by combining the BD process with the seed algorithm, the appearance and disappearance of array non-stationary clusters in two-dimensional (2D) UPAs can be consistently and jointly modeled. By applying this developed method over the entire timeline, the time non-stationarity is further modeled. Therefore, the proposed RS-GBSM can jointly model the space-time non-stationarity. 4. The channel parameters in this model, including delay, Doppler frequency, AoD, and AoA, are time-variant and are considered in both the azimuth direction and elevation direction. The channel characteristics, such as the S-T CF, SCCF, TACF, and DPSD, are derived and investigated. Some interesting observations and conclusions are presented. 5. The simulation results show that the space-time non-stationarity is captured and the impact of VTDs on channel statistics are investigated. Moreover, the utility of the developed method is also validated by the simulation result. Finally, the simulated TACF, DPSDs, as well as SCCFs are compared with the available measurement. The close agreement is achieved between the simulation results and measurements, demonstrating the utility of proposed RS-GBSM.

32

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

3.1.2 Geometrical Representation and Channel Impulse Response of Proposed RS-GBSM In the theoretical reference model, transceivers are equipped with UPAs with .MT (m row .× n column) and .MR (p row .× q column) omni-directional antennas, respectively. The central points of UPAs are located at two focal points of the multiconfocal ellipsoids with major axis 2f . The antenna spacing of the Tx/Rx is .δT/R . Note that .ATuw (.AR lk ) represents the vector of the u (l)-th row and w (k)-th column antenna in Tx (Rx) antenna array, and .ATuw (.AR lk ) represents the u (l)-th row and w (k)-th column antenna in Tx (Rx) antenna array. In the proposed model, the clusters in the surroundings are divided into static clusters and dynamic clusters to model the impact of VTDs. In Fig. 3.1, static clusters and dynamic clusters are located on semi-ellipsoids and semi-spheres, respectively. In the low VTD scenarios, the number of static clusters is larger than that of dynamic clusters, e.g., the rural street scenarios with few vehicles. In the high VTD scenarios, the number of static clusters is smaller than that of dynamic clusters, e.g., the highway street scenarios with many vehicles. The coordinates of antennas in the GCS and LCS are different. The scattering environment is defined in the GCS. The LCS is defined for UPAs of Tx and Rx.  T T T   In the LCS, the antenna vector in the Tx is .A uw = x uw , y  Tuw , z Tuw , while the T   R , y  R , z R antenna vector in the Rx is .A R = x lk lk lk lk , where

Fig. 3.1 The NLoS component of dynamic clusters and static clusters

3.1 Framework of Massive MIMO Vehicular RS-GBSM

x  uw T

.

⎧ m − 2u + 1 ⎪ ⎪ δT , u < m+1 ⎨− 2 , 2 = ⎪ m − 2u + 1 ⎪ ⎩ δT , u ≥ m+1 2 2

y  uw = T

.

33

⎧ n − 2w + 1 ⎪ ⎪ δT , w < ⎨− 2 ⎪ ⎪ ⎩ n − 2w + 1 δT , w ≥ 2

n+1 2 ,

(3.2) n+1 2

z uw = 0 T

.

x  lk =

.

R

y  lk

.

R

(3.3)

⎧ p − 2l + 1 ⎪ ⎪ δR , l < ⎨− 2 ⎪ ⎪ ⎩ p − 2l + 1 δR , l ≥ 2

p+1 2 ,

(3.4) p+1 2

⎧ q − 2k + 1 ⎪ ⎪ δR , k < q+1 ⎨− 2 , 2 = ⎪ q − 2k + 1 ⎪ ⎩ δR , k ≥ q+1 2 2 z lk = 0.

.

(3.1)

R

(3.5)

(3.6)

The placement of the UPAs in the GCS is defined by the transformation from the LCS to the GCS. In Fig. 3.2, we describe an arbitrary 3D rotation of the LCS with respect to the GCS by the angles .α, .β, and .γ . The orientation of the UPAs with respect to the GCS is defined by a sequence of rotations in [20]. Note that R = RZ (α)RY (β)RX (γ ) ⎛ ⎞⎛ ⎞⎛ ⎞ +cosα −sinα 0 +cosβ 0 +sinβ 1 0 0 = ⎝ +sinα +cosα 0 ⎠⎝ 0 1 0 ⎠⎝ 0 +cosγ −sinγ ⎠ 0 0 1 −sinβ 0 +cosβ 0 +sinγ +cosγ

.

(3.7)

ATuw = RA uw

(3.8)

 AR lk = RA lk .

(3.9)

T

.

.

R

34

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

Fig. 3.2 Orienting the LCS with respect to the GCS by a sequence of 3 rotations: .α, .β, .γ

.

The massive MIMO CIR can be characterized by an .MR × MT [(p × q) × (m × n)] complex matrix [21, 22]. Note that ⎡

h1,1 h1,2 ⎢ h2,1 h2,2 ⎢ .H(t, τ ) = ⎢ . .. ⎣ .. . hp,1 hp,2

⎤ · · · h1,m · · · h2,m ⎥ ⎥ . ⎥, .. . .. ⎦ · · · hp,m

(3.10)

⎤ · · · hl1,un · · · hl2,un ⎥ ⎥ .. ⎥ , .. . . ⎦ · · · hlq,un

(3.11)

where ⎡

hl,u

.

hl1,u1 hl1,u2 ⎢ hl2,u1 hl2,u2 ⎢ =⎢ . .. ⎣ .. . hlq,u1 hlq,u2

where .u = 1, 2, . . . , m, .w = 1, 2, . . . , n, .l = 1, 2, . . . , p, and .k = 1, 2, . . . , q. The complex tap coefficient for the first tap from .ATuw to .AR lk is a superposition of the LoS component, the GR component, and the double-bounced component between two dynamic clusters and can be expressed as

3.1 Framework of Massive MIMO Vehicular RS-GBSM

h0lk,uw (t, τ ) .    DB GR = PLoS hLoS lk,uw (t) + PGR hlk,uw (t, τGR ) + PDB hlk,uw (t, τDB ),

35

(3.12)

GR DB where .hLoS lk,uw (t), .hlk,uw (t, τGR ), and .hlk,uw (t, τDB ) are the complex channel gains of the LoS component, the GR component, and the double-bounced component between two dynamic clusters, respectively. .PLoS , .PGR , and .PDB are the pathpowers of the LoS component, the GR component, and the double-bounced component between two dynamic clusters, which can be calculated as

PLoS + PGR =

.

PDB =

.

PLoS

.

K K +1

1 K +1

  K G2 = 1− 2 K +1

PGR =

.

G2 K , 2 K +1

(3.13)

(3.14)

(3.15)

(3.16)

where K and G are the Ricean K-factor and the reflection coefficient which varies depending on the polarization of the incident wave and the electromagnetic properties of the ground. The reflection coefficient is a function of the electromagnetic properties of a material [23], and it can be calculated as  G=

.

 2 0.5 G  + 0.5 |G⊥ |2

(3.17)

 sin θ 2 − Z  sin θ 2 + Z

(3.18)

sin θ 2 − Z sin θ 2 + Z

(3.19)

 − cos2 θ r ,

(3.20)

G =

.

G⊥ =

.

Z=

.



36

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

where .θ r and . are the angle between the ground and the reflected path (see Fig. 3.3), and the complex-valued relative permittivity, ., can be expressed as .

 = r − j

17.98σ σ ≈ r − j [GHz] , 2πfc 0 fc

(3.21)

where .0 , .r , and .σ are the permittivity parameter in vacuum, the relative permittivity parameter, and the conductivity of the material. They are frequency-dependent, which can be expressed as .

r = A(fc[GHz] )B

(3.22)

σ = C(fc[GHz] )D ,

(3.23)

.

where A, B, C, and D are the parameters model this dependency and have been provided by the general guidelines in [24] and [25]. The complex tap coefficient for the second tap from .ATuw to .AR lk is a superposition of the double-bounced component between the dynamic cluster around Tx and the first static cluster (DBT-1) and the double-bounced component between the dynamic cluster around Rx and the first static cluster (DBR-1) and can be expressed as DBR h1lk,uw (t, τ ) = ηDBT,1 hDBT lk,uw (t, τDBT,1 ) + ηDBR,1 hlk,uw (t, τDBR,1 ),

.

(3.24)

DBR where .hDBT lk,uw (t, τDBT,1 )/.hlk,uw (t, τDBR,1 )) is the complex channel gain for the first tap of DBT-1/DBR-1. .ηDBT,1 and .ηDBR,1 are the energy-related parameters, which present the ratio of DBT and DBR and satisfy .ηDBT,1 + ηDBR,1 = 1. The complex tap coefficient for the (.o + 1)-th tap from .ATuw to .AR lk is a superposition of the double-bounced component between the dynamic cluster around Tx and the o-th static cluster (DBT-o) and the double-bounced component between the dynamic cluster around Rx and the o-th static cluster (DBR-o) and can be expressed as .

DBR holk,uw (t, τ ) = ηDBT,o hDBT lk,uw (t, τDBT,o ) + ηDBR,o hlk,uw (t, τDBR,o ),

(3.25)

DBR where .hDBT lk,uw (t, τDBTo )/.hlk,uw (t, τDBR,o ) is the complex channel gain for the (.o + 1)-th tap of DBT-o/DBR-o. .ηDBT,o and .ηDBR,o are the energy-related parameters, which present the ratio of DBT-o and DBR-o and satisfy .ηDBT,o + ηDBR,o = 1.

3.1.2.1

For LoS Component

The complex gain of the LoS component from .ATuw to .AR lk can be expressed as

3.1 Framework of Massive MIMO Vehicular RS-GBSM

LoS .hlk,uw (t)

  LoS (t)t+φ LoS (t) j 2πflk,uw lk,uw

=e

37

(3.26)

,

LoS (t) and .φ LoS (t) denote the Doppler frequency and the received phase where .flk,uw lk,uw of LoS component from the antenna .ATuw to the antenna .AR lk , respectively. The LoS (t), is expressed as T from .Auw , .f Doppler frequency of .AR lk lk,uw

LoS LoS flk,uw (t) = fmax

.

 DLoS lk,uw (t), vR (t) − vT (t)  DLoS lk,uw (t) vR (t) − vT (t)

,

(3.27)

LoS , .DLoS (t), .v (t), and .v (t) are the maximum Doppler frequency of where .fmax R T lk,uw LoS component, the vector of LoS path from the antenna .ATuw to the antenna .AR lk , the velocity vector of Rx, and the velocity vector of Tx, respectively. The received LoS T phase of .AR lk from .Auw , .φlk,uw (t), can be expressed as LoS φlk,uw (t) = φ0 +

.

2π LoS Dlk,uw (t), λ

(3.28)

where .φ0 and .λ are the initial phase of the signal at the Tx and the carrier wavelength. The distance vector between the antenna .ATuw and the antenna .AR lk , LoS (t), can be computed by the antenna vectors .AT (t), .AR (t), and the distance .D uw lk,uw lk vector between the Tx and Rx along the y axis .D = (0,2f, 0) LoS T DLoS lk,uw (t) = Dlk,0 (t) − Auw

(3.29)

R DLoS lk,0 (t) = D(t) + Alk

(3.30)

D(t) = D + (vR (t) − vT (t))t.

(3.31)

.

.

.

3.1.2.2

For Ground Reflection Component

As Fig. 3.3 shows, the GR is modeled on a plane. Here, the difference of the heights between the Tx and Rx is considered. The complex gains of the GR component, GR (t, τ (t)), can be presented as .h GR lk,uw hGR lk,uw (t, τGR (t)) .

  GR,T GR (t) j 2πfuw (t)t+2πflkGR,R (t)t+φlk,uw

=e

(3.32) δ(τ − τGR (t)).

38

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

Fig. 3.3 The GR component in the proposed RS-GBSM

GR,T The Doppler frequency of the antenna .ATuw via the GR component, .fuw (t), is expressed as

GR,T T fuw (t) = fmax

.

 DGR,T uw (t), vT (t)  DGR,T uw (t) vT (t)

,

(3.33)

T where .fmax and .DGR,T uw (t) are the maximum Doppler frequency of Tx and the distance vector from the antenna .ATuw to the reflection point on the ground. The GR,R calculation of Doppler frequency of .AR (t), is similar lk via the GR component, .flk GR (t), R T T with .Auw . The received phase of .Alk from .Auw via the GR component, .φlk,uw can be computed as GR φlk,uw (t) = φ0 +

.

 2π  GR,T Duw (t) + DGR,R (t) . lk λ

(3.34)

The relationship calculations of the distance vectors for the GR component are given as follows. The absolute distance between the Tx and Rx on the ground, .dG (t), can be derived as

3.1 Framework of Massive MIMO Vehicular RS-GBSM

dG (t) =

.



D(t)2 − (hT − hR )2 .

39

(3.35)

The absolute distance between the Tx and the reflection point on the ground, .dT (t), is derived by dT (t) =

.

dG (t)hT . hT + hR

(3.36)

The absolute values of the distance vector from Tx/Rx to the reflection point on the ground can be calculated as DGR,T (t) =

.

D

.

GR,R



(t) =

 dT2 (t) + h2T

dG2 (t) + (hT + hR )2 − DGR,T (t).

(3.37)

(3.38)

Then, their distance vectors can be written as ⎛

⎞ cosψ GR,T (t) cosθ GR,T (t) GR,T .D (t) = DGR,T (t) ⎝ sinψ GR,T (t) cosθ GR,T (t) ⎠ sinθ GR,T (t)

(3.39)

⎛

⎞ cosψ GR,R (t) cosθ GR,R (t) GR,R .D (t) = DGR,R (t) ⎝ sinψ GR,R (t) cosθ GR,R (t) ⎠ , sinθ GR,R (t)

(3.40)

where .ψ GR,T (t)/.ψ GR,R (t) and .θ GR,T (t)/.θ GR,R (t) are the azimuth angle and elevation angle of the distance vector .DGR,T (t)/.DGR,R (t). Since the azimuth angles of the GR path are identical with the LoS path, and the sum-power of the LoS path and the GR path does not change, only the elevation angles of the GR path need to be considered here [23], which can be calculated as θ GR,T (t) = −(θr (t) − θl (t))

(3.41)

θ GR,R (t) = (θr (t) + θl (t)) − π,

(3.42)

.

.

where θr (t) = arctan

.

hT + hR dG (t)

(3.43)

40

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

θl (t) = arctan

.

hT − hR . dG (t)

(3.44)

Based on the distance vectors from Tx/Rx to the reflection point on the ground DGR,T (t)/.DGR,R (t) and antenna vectors, we can get

.

GR,T DGR,T (t) − ATuw uw (t) = D

(3.45)

DGR,R (t) = DGR,R (t) − AR lk . lk

(3.46)

.

.

3.1.2.3

For the Double-Bounced Component Through Two Dynamic Clusters

As shown in Fig. 3.1, the dynamic clusters around the Tx and the Rx, for example, the vehicles or pedestrians, are modeled on the semi-spheres. The radius of dynamic clusters around the Tx and the Rx are .RT and .RR . Furthermore, there are .Ot /.Or dynamic clusters around Tx/Rx. There are S rays in the transmission through two dynamic clusters. Note that the .ot -th and .or -th dynamic clusters are presented as T R .Clustero and .Clustero . t r The complex gains of the double-bounced component through two dynamic clusters of the theoretical model .(S −→ ∞) from the antenna .ATuw to the antenna R at the delay .τ (t), .hDB (t, τ (t)), can be presented as .A DB DB lk lk,uw hDB lk,uw (t, τDB (t)) =

Ot  Or 

.

ot =1 or =1

hDB lk,uw,ot or (t)δ(τ − τDB,ot ,or (t)),

(3.47)

where .hDB lk,uw,ot or (t) and .τDB,ot ,or (t) are the channel complex gain and delay of double-bounced component through the .ot -th and .or -th dynamic clusters, respectively. R – if .ATuw ∈ ATClusterT and .AR lk ∈ AClusterR ot

or

hDB lk,uw,ot ,or (t)    S . DBlk,uw,ot ,or ,s DB,T DB,R 1  j 2πfuw,o (t) t ,s (t)t+2πflk,or ,s (t)t+φ = lim √ e S→∞ S s=1

(3.48)

– otherwise hDB lk,uw,ot ,or (t) = 0,

.

(3.49)

3.1 Framework of Massive MIMO Vehicular RS-GBSM

where .AT

ClusterTot

and .AR

ClusterR or

41

are the Tx antenna set that can observe the .ot -

th dynamic cluster around Tx and the Rx antenna set that can observe the .or -th dynamic cluster around Rx, respectively. The Doppler frequency of the antenna T .Auw via the s-th ray in the double-bounced component through the .ot -th and .or DB,T th dynamic clusters, .fuw,o t ,s (t), can be expressed as DB,T T fuw,o (t) = fmax t ,s

.

T  DDBD,T uw,ot ,s (t), vT (t) − vot (t)  T DDBD,T uw,ot ,s (t) vT (t) − vot (t)

(3.50)

,

T T where .DDBD,T uw,ot ,s (t) and .vot (t) are the distance vectors from the antenna .Auw to the .ot -th dynamic cluster around Tx via the s-th ray and the velocity vector of the .ot th dynamic cluster around Tx, respectively. The calculation of Doppler frequency of .AR lk via the s-th ray in the double-bounced component through the .ot -th and .or DB,R DB,T th dynamic clusters, .flk,o (t), is similar to .fuw,o t ,s (t). The received phase of the r ,s T antenna .AR from the antenna .Auw via the s-th ray of the double-bounced component lk DB through the .ot -th and .or -th dynamic clusters .φlk,uw,o (t) can be computed as t or ,s DB φlk,uw,o (t) t or ,s .

= φ0 +

 2π  DBD,T DBV,M Duw,ot ,s (t) + DDBD,R (t) + D (t) , ot or ,s lk,or ,s λ

(3.51)

where .DDBV,M ot or ,s (t) is the distance vector from the .ot -th dynamic cluster around Tx to the .or -th dynamic cluster around Rx via the s-th ray. T The delay of the antenna .AR lk from the antenna .Auw via the s-th ray of the doublebounced component through the .ot -th and .or -th dynamic clusters, .τDB,ot ,or (t), is assumed to be the sum of three components τDB (t) =

.

DBD,R DBV,M DDBD,T uw,ot ,s (t) + Dlk,or ,s (t) + Dot or ,s (t)

c

,

(3.52)

where c is the speed of light. The relationship calculations of the distance vectors in the double-bounced component through the .ot -th dynamic cluster around Tx and the .or -th dynamic cluster around Rx are presented as follows. Based on the angular relationships, we can get the distance vector between Tx and the .ot -th dynamic cluster around Tx via (t), can be presented as the s-th ray, .DDBD,T ot ,s ⎞ (t) cosθoDBD,T (t) cosψoDBD,T t ,s t ,s ⎟ ⎜ DBD,T .Do ,s (t) = RT  ⎝ sinψoDBD,T (t) cosθoDBD,T (t) ⎠ , t ,s t ,s t sinθoDBD,T (t) t ,s ⎛

(3.53)

42

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

where .ψoDBD,T (t) and .θoDBD,T (t) are azimuth angle and elevation angle of AoD, i.e., t ,s t ,s azimuth angle of departure (AAOD) and elevation angle of departure (EAOD), of the distance vector from Rx to the .ot -th dynamic cluster around Tx via the s-th ray, (t), the distance vector between Rx and the .or -th respectively. Similar to .DDBD,T ot ,s dynamic cluster around Rx via the s-th ray, .DDBD,R (t), can be presented as or ,s ⎞ (t) cosθoDBD,R (t) cosψoDBD,R r ,s r ,s ⎟ ⎜ DBD,R .Do ,s (t) = RR  ⎝ sinψoDBD,R (t) cosθoDBD,R (t) ⎠ , r ,s r ,s r DBD,R sinθor ,s (t) ⎛

(3.54)

where .ψoDBD,R (t) and .θoDBD,R (t) are the azimuth angle of arrival (AAoA) and eler ,s r ,s vation angle of arrival (EAoA) of the distance vector from Rx to the .or -th dynamic cluster around Rx via the s-th ray, respectively. Based on the aforementioned equations, we have DBD,T DDBV,M (t) + DDBD,R (t) ot or ,s (t) = D(t) − Dot ,s or ,s

(3.55)

DBD,T DDBD,T (t) − ATuw uw,ot ,s (t) = Dot ,s

(3.56)

DBD,R DDBD,R (t) − AR lk . lk,or ,s (t) = Dor ,s

(3.57)

.

.

.

3.1.2.4

For Double-Bounced Component Through Dynamic Clusters and Static Clusters

As shown in Fig. 3.1, the static clusters in the surroundings, such as the buildings and trees, are modeled on the multi-confocal semi-ellipsoids. Each semi-ellipsoid is the rotation of an ellipse with respect to the y axis. The o-th static cluster is located on the o-th ellipsoid with major axis .2ao . The number of static clusters is O. The complex gains of the double-bounced component through the dynamic cluster around Tx and the static cluster of the theoretical model .(S −→ ∞) from R at delay .τ DBT T .Auw to .A DBTo (t), .hlk,uw,o (t, τDBTo (t)), can be presented as lk hDBT lk,uw,o (t, τDBT,o (t)) =

Ot 

.

ot =1

hDBT lk,uw,ot o (t)δ(τ − τDBT,ot o (t)),

(3.58)

where .hDBT lk,uw,ot o (t) and .τDBT,ot o (t) are the complex gain and delay of the doublebounced component through the .ot -th dynamic cluster and o-th static cluster, respectively.

3.1 Framework of Massive MIMO Vehicular RS-GBSM

– if .ATuw ∈ AT

ClusterTot

.

hDBT lk,uw,ot o (t)  = lim

S→∞

43

R and .AR lk ∈ A

ClusterSo

  S DBT,T DBT,R DBT 1  j 2πfuw,o t ,s (t)t+2πflk,o,s (t)t+φlk,uw,ot o,s (t) e √ S s=1



(3.59)

– otherwise hDBT lk,uw,ot o (t) = 0,

(3.60)

.

where .AR

ClusterSo

is the Rx antenna set that can observe the o-th static cluster. The

Doppler frequency of the antenna .ATuw via the s-th ray in the double-bounced DBT,T component through the .ot -th dynamic cluster and o-th static cluster, .fuw,o t ,s (t), can be expressed as DBT,T T fuw,o (t) = fmax t ,s

.

T  DDBD,T uw,ot ,s (t), vT (t) − vot (t)  T DDBD,T uw,ot ,s (t) vT (t) − vot (t)

(3.61)

,

where .DDBD,T uw,ot ,s (t) is the .ot -th dynamic cluster around Tx via the s-th ray. The calculation of Doppler frequency of .AR lk via the s-th ray in the double-bounced DBT,R component through the .ot -th dynamic cluster and o-th static cluster, .flk,o,s (t), is DBT,T R similar with .fuw,o t ,s (t). The received phase of the antenna .Alk from the antenna T .Auw via the s-th ray of the double-bounced component through the .ot -th dynamic DBT cluster and o-th static cluster, .φlk,uw,o (t), can be computed as t o,s DBT φlk,uw,o (t) t o,s .

= φ0 +

 2π  DBD,T DBS,R Duw,ot ,s (t) + DDBV,T ot o,s (t) + Dlk,o,s (t) , λ

(3.62)

where .DDBV,T ot o,s (t) is the distance vector from the .ot -th dynamic cluster around Tx to the o-th static cluster via the s-th ray. .DDBS,R lk,o,s (t) is the distance vector from the R o-th static cluster to the antenna .Alk via the s-th ray. The delay of the antenna .AR lk from the antenna .ATuw via the s-th ray of the double-bounced component through the .ot -th dynamic cluster and o-th static cluster, .τDBT,ot o (t), is assumed to be the sum of three components τDBT,ot o (t) =

.

DBV,T DBS,R DDBD,T uw,ot ,s (t) + Dot o,s (t) + Dlk,o,s (t)

c

.

(3.63)

44

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

The calculation of complex gains of the double-bounced component through the dynamic cluster around Rx and the static cluster of the theoretical model .(S −→ ∞) DBR from the antenna .ATuw to the antenna .AR lk at the delay .τDBRo (t), .hlk,uw,o (t, τDBRo (t)), DBT is similar with .hlk,uw,o (t, τDBTo (t)). Note that, to avoid repeated description, only DBT (t, τ DBR .h DBTo (t)) is given, .hlk,uw,o (t, τDBRo (t)) follows the same procedure. lk,uw,o The relationship calculations of the distance vectors in the double-bounced component through the dynamic clusters and static clusters are given as follows. The absolute distance between the Rx and the o-th static cluster via the s-th ray can be derived according to their geometrical relationships DDBS,R (t) = o,s

.

T 2ao sinαo,s ,  T + sin π − α R sinαo,s o,s

(3.64)

T , and .α R mean the semi-long axis of the o-th semi-ellipsoid, AoD where .ao , .αo,s o,s DBS,T and y axis from Tx to the o-th cluster via the s-th ray, and AoA between .Do,s and y axis from Rx to the o-th cluster via the s-th ray, respectively. between .DDBS,R o,s (t) between With the angular relationships, we can get the distance vector .DDBS,R o,s the Rx and the o-th static cluster via the s-th ray

⎞ DBS,R DBS,R (t) cosθo,s (t) cosψo,s ⎟ ⎜ DBS,R DBS,R DBS,R .Do,s (t) = DDBS,R (t) ⎝ sinψo,s (t) cosθo,s (t) ⎠ , o,s DBS,R sinθo,s (t) ⎛

(3.65)

DBS,R DBS,R where .ψo,s (t) and .θo,s (t) are the AAoA and EAoA of the distance vector DBS,R .Do,s , respectively. Based on the aforementioned equations, we have

DDBS,T (t) = D(t) − DDBS,R (t) o,s o,s

(3.66)

DBS,R DDBS,R (t) − AR lk . lk,o,s (t) = Do,s

(3.67)

.

.

For the NLoS static cluster components in the multi-confocal semi-ellipsoids, R and the AoD .α T are not independent in an ellipsoid model. Their the AoA .αo,s o,s relationship can be expressed as

T .αo,s

⎧ ⎨

R ) R ≤α if 0 < αo,s g(αo,s 0 R R = g(αo,s ) + π if α0 < αo,s ≤ 2π − α0 ⎩ R ) + 2π if 2π − α < α R ≤ 2π g(αo,s 0 o,s

 R .g(αo,s )

= arctan

R (k02 − 1)sinαo,s R 2k0 + (k02 + 1)cosαo,s

(3.68)  (3.69)

3.2 Space-Time Non-stationary Modeling with Uniform Planar Antenna

R .αo,s

 DBS,R DBS,R DBS,R + cos2 θo,s sin2 ψo,s = arcsin sin2 θo,s 

k02 − 1 .α0 = π − arctan 2k0 k0 =

.

45

(3.70)



ao . f

(3.71)

(3.72)

Given the semi-major axis f of the first semi-ellipsoid, the semi-long axis of the o-th semi-ellipsoid, .ao , can be determined relative to the first semi-ellipsoid as ao (t) = (DDBS,R (t) + DDBS,T (t)) + a1 . o,s o,s

.

(3.73)

Based on the aforementioned distance vectors, we have .

DBD,T DDBV,T (t) + DDBS,R (t) ot o,s (t) = D(t) − Dot ,s o,s

(3.74)

DBS,T DDBV,R (t) + DDBD,R (t). oor ,s (t) = D(t) − Do,s or ,s

(3.75)

.

The channel parameters of each sub-channel and the calculation with no inequality ensure the modeling of spherical wavefront propagation accurately.

3.2 Space-Time Non-stationary Modeling with Uniform Planar Antenna In the proposed massive MIMO V2V RS-GBSM with UPAs, a novel BD and seed algorithm based selective cluster evolution method is developed. In massive MIMO channel models, non-stationary channel characteristics in array/space domain and spherical wavefront should be considered because of the large scale of the antenna array and near-field effect [26–30]. Note that the massive MIMO V2V in [31– 33] solely consider ULAs. Compared to the ULA, the physical size of the UPA is smaller, which is more practical for massive MIMO communications. However, the application of massive MIMO with UPAs will bring two challenges to the modeling of space non-stationarity. The first challenge is that the small size of the antenna array with UPAs leads to the small Rayleigh distance. When the distances among the transceiver and clusters are much larger than Rayleigh distance, the space nonstationarity may no longer exist. The second challenge is that it is unreasonable to directly use the BD process to model the space non-stationarity. Obviously, in T the BD process, the survival probabilities of clusters at the Tx .Psurvival and the Rx R .P are given by Wu et al. [27] survival

46

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . . δT

T Psurvival = e−λ Ds

.

δR

R Psurvival = e−λ Ds .

.

(3.76)

(3.77)

When clusters evolve on the one-dimensional (1D) ULA, the evolution path of clusters from the n-th antenna to the m-th antenna is unique. The unique evolution path leads to the unique visible state of clusters, i.e., the survival state or not, for the m-th antenna. As a result, when the BD process is exploited to characterize the array evolution in ULAs, the visible state of clusters is unique for the antenna. Unlike a 1D ULA, in a 2D UPA, there are many evolution paths from the n-th antenna to the m-th antenna, and different evolution paths result in different states of the cluster. In this case, for the m-th antenna, the cluster has both the survival state and nonsurvival state due to the different evolution paths. Therefore, when the BD process is solely employed in UPAs, it leads to the inconsistent visible state of clusters for the antenna, which is not in line with reality. To meet the aforementioned two challenges, a novel BD and seed algorithm based selective cluster evolution method is developed in the proposed RS-GBSM for massive MIMO with UPA channels. The steps of this novel method are described in detail below. Step 1 Aiming at solving the first challenge mentioned above, CEAs used to define the evolution area of clusters are presented in Fig. 3.1. Note that, only clusters in CEAs, named as array non-stationary clusters, experience the array evolution because of near-field effect, and other clusters, named as array stationary clusters, can be observed by all antennas. Furthermore, the CEAs are defined as semi-spheres with radius of .ΓT and .ΓR at the Tx and Rx sides, which can be written by   2δT2 m2 + n2 .T = λ   2δR2 p2 + q 2 , .R = λ

(3.78)

(3.79)

where m/p and n/q are the numbers of row and column of UPA in Tx/Rx. As shown in Fig. 3.1, .ClusterTot is in the CEA of Tx and it is an array non-stationary cluster on the Tx planar antenna array. As a result, its visible state needs to be determined. In contrast, .ClusterR or is not in the CEA of the Rx side. Therefore, it is an array stationary cluster and visible to all antennas in the Rx antenna array. Step 2 Taking the static clusters placed on the multi-confocal semi-ellipsoids at the Tx side as an example, the flowchart of the second step of the novel method is presented in Fig. 3.4. Note that the dynamic cluster follows the same procedure.

3.2 Space-Time Non-stationary Modeling with Uniform Planar Antenna

47

Fig. 3.4 The flowchart of second step of the novel BD and seed algorithm based selective cluster evolution method

48

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

In the beginning, the .ΓT according to the configuration of the Tx antenna array is calculated. In Fig. 3.1, the red line sphere at the Tx side with the radius .ΓT is the CEA for distinguishing between array stationary clusters and array non-stationary clusters. Then, the location information of Tx and the o-th static cluster is given (t), is and the distance vector between the Tx and the o-th static cluster, .DDBS,T o computed. After acquiring the radius of the CEA and aforementioned distance vectors, DBS,T .Do (t) and .ΓT are compared. If .DDBS,T (t) is longer than .ΓT , all antennas at the o Tx side can observe the o-th static cluster. In this case, the o-th static cluster is an array stationary cluster. If not, the array evolution will be carried out. To consistently model the array evolution of clusters in UPAs, a new approach is proposed, which combines the BD process and the seed algorithm to mimic the consistent visible state of clusters for the antenna. First, randomly select an antenna on the UPA at the Tx side, .ATuw , which is used as the seed antenna. This is reasonable because, in high-mobility V2V communication scenarios, which antennas are able to observe clusters are essentially random. To model this phenomenon, the probability of the selection of each antenna is assumed to follow the uniform distribution. Assuming the o-th static cluster is visible to this seed antenna .ATuw . Second, the visible region on UPA of the o-th static cluster, i.e., the o-th static cluster is visible to the antennas within this visible region on the UPA, starts to grow based on the survival probability T .P survival . Specifically, if the o-th static cluster survives in the array evolution based on the survival probability .PTsurvival in BD processes, the circular visible region of the o-th static cluster grows on UPAs with the seed antenna .ATuw as the center and the .δT as the radius. Third, once the o-th static cluster is dead in the array evolution, the circular visible region of the o-th static cluster immediately stops growing, and this visible region is taken as the final visible region. Through the aforementioned three procedures, which antennas can observe the o-th static cluster, i.e., visible region, can be properly acquired. For example, if the o-th static cluster survives continuously for n times and dies at the (.n + 1)-th time, the radius of the visible region of the o-th static cluster is .n × δT . Note that the circular visible regions of all clusters can be computed and acquired by the aforementioned three procedures. Step 3 Typically, the V2V channel model needs to mimic the channel time non-stationarity caused by the rapid movement of Tx, Rx, and clusters in the environment. By operating this novel method recursively with respect to time, the proposed RS-GBSM further models the non-stationarity in the time domain, i.e., time non-stationarity. Based on the aforementioned analysis, the first challenge and second challenge of modeling space non-stationarity of massive MIMO with UPA channels can be sufficiently solved by the first step and second step of the novel method, respectively. In the step 3, the developed novel is adopted over the entire timeline to capture the channel time non-stationarity. Therefore, for massive MIMO V2V with UPA channels, both the space non-stationarity and the time non-stationarity are properly modeled in the proposed massive MIMO V2V RS-GBSM.

3.3 Simulations and Discussions

49

3.3 Simulations and Discussions 3.3.1 Statistical Properties From the proposed massive MIMO V2V RS-GBSM, the corresponding statistical properties are derived.

3.3.1.1

Space-Time Correlation Function

The S-T CF between the channel gains .hlk,uw (t) and .hl  k  ,u w (t) is defined as ρlk,uw,l  k  ,u w (δT , δR , t; t) = E

.

3.3.1.2

! h∗lk,uw (t)hl  k  ,u w (t + t) . |h∗lk,uw (t)||hl  k  ,u w (t + t)|

(3.80)

Space Cross-Correlation Function

The SCCF .ρlk,uw,l  k  ,u w (δT , δR ; t) can be obtained from the S-T CF by setting .t = 0 ! h∗lk,uw (t)hl  k  ,u w (t) . (3.81) .ρlk,uw,l  k  ,u w  (δT , δR ; t) = E |h∗lk,uw (t)||hl  k  ,u w (t)| 3.3.1.3

Time Auto-Correlation Function

By setting .u = u , w = w  and .l = l  , k = k  , the S-T CF reduces to the TACF .ρlk,uw (t; t) ρlk,uw (t; t) = E

.

3.3.1.4

! h∗lk,uw (t)hlk,uw (t + t) . |h∗lk,uw (t)||hlk,uw (t + t)|

(3.82)

Wigner-Ville Spectrum

The Wigner-Ville distribution, which is also named as DPSD, is presented as [34] " W (t, f ) =

+∞

.

−∞

h∗lk,uw (t −

 τ  −j 2πf τ τ )hlk,uw t + e dτ. 2 2

(3.83)

The Wigner-Ville spectrum is the expectation value of the Wigner-Ville distribution, which is the Fourier transform of the local TACF .ρlk,uw (t, τ )

50

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . .

#" Slk,uw (t, f ) = E "

.

=

+∞ −∞

h∗lk,uw

+∞ −∞

$   τ τ  −j 2πf τ hlk,uw t + e t− dτ 2 2 −j 2πf τ

ρlk,uw (t, τ )e

(3.84)

dτ.

3.3.2 Model Simulation In Fig. 3.5, the normalized absolute SCCFs of the LoS component, the GR component, and the double-bounced component through two dynamic clusters are compared. Clearly, in comparison to the components with few bounces, e.g., the LoS component with zero bounce and the GR component with one bounce, the SCCFs of the double-bounced component are lower. This is because, with the increase of the antenna spacing, reflections/scatterings increase the transmission difference among the sub-paths transmission and reduce the spatial correlation. Furthermore, the SCCFs of the double-bounced component through two dynamic clusters under high VTDs are lower than those under low VTDs. This phenomenon can be explained that, the spatial diversity of channels increases as VTDs increase, resulting in a decrease in the spatial correlation.

1

Normalized absolute SCCF

0.95

0.9

0.85

0.8 LoS Ground reflection Double bounce, low VTD Double bounce, high VTD

0.75

0.7 0

0.01

0.02

0.03

0.04

0.05

Antenna spacing of Rx, R (m)

Fig. 3.5 The comparison of the SCCFs among the LoS component, the GR component, and the double-bounced component through two dynamic clusters (.MT = 12 .× 12, .MR = 12 .× 12, .u = 1,     .w = 1, .l = 2, .k = 2, .u = 1, .w = 1, .l = 12, .k = 11, .D = 15 m, .T0 = 1 s, .vT (t) = 7 m/s, T R .vR (t) = 6.8 m/s, .vo (t) = 2 m/s, .vo (t) = 2 m/s, Low VTD: .Ot = 1, .Or = 1, .O = 6, High VTD: t r .Ot = 6, .Or = 6, .O = 2)

3.3 Simulations and Discussions

51

1 0.95

Normalized absolute SCCF

0.9 0.85 0.8 0.75 0.7 Tap 1, Low VTD Tap 2, Low VTD Tap 3, Low VTD Tap 1, High VTD Tap 2, High VTD Tap 3, High VTD

0.65 0.6 0.55 0.5 0

0.005

0.01

0.015

0.02

0.025

Antenna spacings of Rx,

R

0.03

0.035

0.04

(m)

Fig. 3.6 The comparison of the SCCFs among tap 1, tap 2, and tap 3 under the low VTD and high VTD (.MT = 12 .× 12, .MR = 12 .× 12, .u = 1, .w = 1, .l = 2, .k = 2, .u = 1, .w  = 1, .l  = 12,  T R .k = 11, .D = 15 m, .T0 = 1 s, .vT (t) = 7 m/s, .vR (t) = 6.8 m/s, .vo (t) = 2 m/s, .vo (t) = 2 m/s, t r Low VTD: .Ot = 1, .Or = 1, .O = 6, High VTD: .Ot = 6, .Or = 6, .O = 2)

Figure 3.6 presents the normalized absolute SCCFs of the LoS component, the GR component, and the double-bounced component through two dynamic clusters. Three interesting observations are given as follows. First, the effects of VTDs on the SCCFs of tap 2 and tap 3 are larger than those of tap 1. This is reasonable because taps 2 and 3 are composed of double-bounced components through dynamic clusters and static clusters, and thus the VTD has a significant influence on their SCCFs. Different from taps 2 and 3, the dominant LoS component in tap 1 makes the impact of VTDs on its SCCFs not obvious. Second, compared to low VTDs, SCCFs of taps 2 and 3 in high VTDs are higher. This phenomenon can be explained that, due to the close location and a large number of dynamic clusters in high VTDs, sub-paths near the Tx and Rx caused by dynamic clusters have similar angular parameters, resulting in a higher SCCFs. Third, under the low VTD, the SCCF of tap 3 is lower than that of tap 2. This is because, compared with tap 2, the transmission distance between dynamic and static clusters in tap 3 is longer, and thus parameters of the received signal are more different, leading to a lower SCCF. However, this phenomenon under high VTDs is not as obvious as under low VTDs. The underlying physical reason is that the high VTD is a more clustered environment, where the number of dynamic clusters is far more than that of the static clusters. In this case, the impact of the dynamic clusters on the signal transmission is larger. As a result, the influence of the distance between the static clusters and the dynamic clusters significantly decreases.

52

3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . . 1 0.9

Normalized absolute SCCF

0.8 0.7 0.6 0.5 0.4

M =7 T

0.3

7, M =7 R

7

M =11

11, M =11

11

0.2

M T=14

14, M R=14

14

0.1

M T=18

18, M R=18

18

T

R

0 0

0.002

0.004

0.006

0.008

0.01

0.012

Antenna spacing of Rx, R (m)

Fig. 3.7 The comparison of the SCCFs with different numbers of antennas under the high VTD (.u = 1, .w = 1, .l = 2, .k = 2, .u = 1, .w  = 1, .l  = 5, .k  = 5, .D = 35 m, .T0 = 0 s, .vT (t) = 25 m/s, T R .vR (t) = 25 m/s, .vo (t) = 12 m/s, .vo (t) = 10 m/s, High VTD: .Ot = 6, .Or = 6, .O = 3) t r

In order to show the influence of the number of antennas on the channel statistical properties, the comparison of SCCFs with different numbers of antennas is given in Fig. 3.7. It can be readily observed from Fig. 3.7 that, as the number of antennas increases, the simulated SCCF shows a downward trend. More specifically, compared with .MT = 18 × 18 and .MR = 18 × 18, the SCCF with .MT = 7 × 7 and .MR = 7 × 7 is larger. The underlying physical reason is that the spatial diversity of channels increases as the number of antennas increases, resulting in a decrease in the spatial correlation. In Fig. 3.8, the normalized DPSDs of tap 1, tap 2, and tap 3 under different VTDs are given. Also, the dominant LoS component in tap 1 reduces the effects of VTDs on the DPSDs. The DPSDs of the LoS component and the GR component are solely influenced by the movement of transceivers. Moreover, the moving speed of the Tx is not dispersed by the moving speed of the dynamic clusters in the same direction, and thus the Doppler frequency shift is more obvious. In addition, since there are more dynamic clusters in high VTDs, the Doppler frequency shift under the high VTD is higher than that under the low VTD, which is consistent with the derivations related to the Doppler frequency. Furthermore, the dominant LoS component also leads to the steep DPSDs of tap 1. Another phenomenon worth noting is that, compared to high VTDs, DPSDs under low VTDs are more steeply distributed. This is because the received power under low VTDs mainly concentrates on several Doppler frequencies, resulting in steep distributions of DPSDs. In contrast, under high VTDs, the received power tends to come from dynamic vehicles over all directions, and thus the DPSDs are flatly distributed.

3.3 Simulations and Discussions

53

0.3

Tap 1, Low VTD Tap 2, Low VTD Tap 3, Low VTD Tap 1, High VTD Tap 2, High VTD Tap 3, High VTD

Normalized DPSD (dB)

0.25

0.2

0.15

0.1

0.05

0 -60

-50

-40

-30

-20

-10

0

10

20

Frequency, f (Hz)

Fig. 3.8 The comparison of the DPSDs among tap 1, tap 2, and tap 3 under the low VTD and high VTD (.MT = 12 .× 12, .MR = 12 .× 12, .u = 2, .w = 2, .l = 3, .k = 3, .D = 15 m, .T0 = 1 s, T R .vT (t) = 3 m/s, .vR (t) = 3 m/s, .vo (t) = 6 m/s, .vo (t) = 6 m/s, Low VTD: .Ot = 1, .Or = 1, .O = 6, t r v,T v,T v,R High VTD: .Ot = 6, .Or = 6, .O = 2, .ψot ,s (t) = π/4, .ψov,R r ,s (t) = π/2, .θot ,s (t) = 0, .θor ,s (t) = 0, v,T (t) = 0, .ψ v,R (t) = π/4, .θ v,T (t) = 0, .θ v,R (t) = 0) .ψ

Figure 3.9 depicts the normalized DPSDs of tap 1, tap 2, and tap 3 with different directions. From Fig. 3.9, no matter moving in the same direction or the opposite directions, the Doppler frequency shifts of tap 2 and tap 3 are smaller than those of tap 1. Additionally, similar to Fig. 3.8, the distributions of DPSDs of tap 1 are steeper than those of DPSDs of taps 2 and 3. Finally, it can be seen from Fig. 3.9 that the Doppler frequency shifts of moving in the opposite directions are larger than those in the same direction. This is because that the relative speed of moving in the opposite directions is larger. Aiming at demonstrating that both the space non-stationarity and the time nonstationarity are embedded in the proposed RS-GBSM, Fig. 3.10a and b give the SCCFs of .h33,11 , .h66,11 , and .h1212,11 and the TACFs at .t = 1 s, 5 s, and 10 s, respectively. In Fig. 3.10a, a decreasing trend of the SCCFs can be readily seen as the antenna spacing .δR enlarges, and SCCFs are also dependable on the reference antennas. Consequently, the proposed RS-GBSM has the ability to capture the channel space non-stationarity. Similarly, from Fig. 3.10b, the simulated TACFs decrease as the time separation .t enlarges, and TACFs are also different at the different time instants, e.g., .t = 1 s, 5 s, and 10 s. As a result, it can be concluded that the proposed RS-GBSM is essentially time non-stationary. In Fig. 3.11, the x axis is the row index of antenna and the y axis is the column index of antenna. It shows the evolution of different clusters on the UPA. From

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3 A 3D RS-GBSM with Uniform Planar Antenna Array for Massive MIMO V2V. . . 0.3 Tap 1, opposite Tap 2, opposite Tap 3, opposite Tap 1, same Tap 2, same Tap 3, same

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Fig. 3.9 The comparison of the Doppler PSDs among tap 1, tap 2, and tap 3 with same directions and opposite directions under the high VTD (.MT = 12 .× 12, .MR = 12 .× 12, .u = 2, .w = 2, T R .l = 3, .k = 3, .D = 15 m, .T0 = 1 s, .vT (t) = 5 m/s, .vR (t) = 5 m/s, .vo (t) = 5 m/s, .vo (t) = 5 m/s, t r v,T v,R v,T v,R .Ot = 6, .Or = 6, .O = 2, Same direction: .ψot ,s (t) = π/4, .ψor ,s (t) = π/2, .θot ,s (t) = 0, .θor ,s (t) = v,T 0, .ψ v,T (t) = 0, .ψ v,R (t) = π/4, .θ v,T (t) = 0, .θ v,R (t) = 0, Opposite direction: .ψot ,s (t) = 4π/3, v,R v,T v,R v,T (t) = 0, .ψ v,R (t) = π/3, .θ v,T (t) = 0, .θ v,R (t) = 0) .ψor ,s (t) = π , .θot ,s (t) = 0, .θor ,s (t) = 0, .ψ

Fig. 3.11, it can be observed that, for the cluster 2, as an example, the seed antenna ATuw is .AT72 . Furthermore, it is clear that the cluster 2 survives continuously for three times and dies at the fourth time. As a result, the radius of circular visible region is .3 × δT . Similarly, circular visible regions of clusters 1, 3, and 4 can be readily obtained from Fig. 3.11. Therefore, the appearance and disappearance of clusters on the UPA are consistently modeled in the proposed RS-GBSM, which demonstrates the utility of the developed method, i.e., BD and seed algorithm based selective cluster evolution. Furthermore, it is clear that different antennas on the UPA observe different clusters, and thus the space non-stationarity of massive MIMO V2V with UPA channels is captured sufficiently.

.

3.3.3 Model Validation Figure 3.12 compares the simulated normalized absolute TACFs under low VTDs and the measurement data in [35]. For a fair comparison, the model-related parameters are properly set according to the measurement, which is a MIMO vehicular channel measurement at .5.8 GHz with a speed of 60 km/h. Based on the description of the measurement scenario, it is reasonable to assume that this is a

3.3 Simulations and Discussions

55

1

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h33,11, t=1 s h66,11, t=1 s h1212,11, t=1 s

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h44,22, t=1 s h44,22, t=5 s h44,22, t=10 s

0.3 0.2 0

0.005

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0.015

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t (s)

(b)

Fig. 3.10 Verification of space-time non-stationarity. (a) The comparison of the DPSDs among tap 1, tap 2, and tap 3 with same directions and opposite directions under the high VTD (.MT = 12 .× 12, .MR = 12 .× 12, .u = 2, .w = 2, .l = 3, .k = 3, .D = 15 m, .T0 = 1 s, .vT (t) = 5 m/s, T R .vR (t) = 5 m/s, .vo (t) = 5 m/s, .vo (t) = 5 m/s, .Ot = 6, .Or = 6, .O = 2, Same direction: t r v,T v,R v,T v,R v,T (t) = 0, .ψ v,R (t) = π/4, .ψot ,s (t) = π/4, .ψor ,s (t) = π/2, .θot ,s (t) = 0, .θor ,s (t) = 0, .ψ v,T v,T v,T v,R .θ (t) = 0, .θ (t) = 0, Opposite direction: .ψot ,s (t) = 4π/3, .ψov,R r ,s (t) = π , .θot ,s (t) = 0, v,R v,T v,R v,T v,R .θor ,s (t) = 0, .ψ (t) = 0, .ψ (t) = π/3, .θ (t) = 0, .θ (t) = 0. (b) The comparison of the SCCFs of .h33,11 , .h66,11 , and .h1212,11 under the low VTD (.MT = 12 .× 12, .MR = 12 .× 12,     .u = 1, .w = 1, .l = 2, .k = 2, .u = 1, .w = 1, .l = 3/6/12, .k = 3/6/12, .D = 15 m, .T0 = 1 s, T R .vT (t) = 7 m/s, .vR (t) = 6.8 m/s, .vo (t) = 2 m/s, .vo (t) = 2 m/s, .Ot = 1, .Or = 1, .O = 6 t r

low VTD scenario. From Fig. 3.12, it can be observed that the simulated TACFs can match well with the measurement, verifying the utility of the proposed RS-GSBM. In addition, the simulated TACFs under the high VTD is depicted. Also, compared with the low VTD, the simulated TACF under the high VTD is lower. To the best

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cluster1 cluster2 cluster3 cluster4

10 8 6 4 2 0 0

2

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Fig. 3.11 The evolution of different clusters on the Tx antenna array

1 Simulation results, Low VTD Measurement data, Low VTD Simulation results, High VTD

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Fig. 3.12 The fitting of the TACF of .h33,11 with the measurement in [35] (.MT = 1 .× 4, .MR = 1 .× 4, .fc = 5.8 GHz, .δT = δR = 2 .λ, .u = 1, .w = 2, .l = 1, .k = 3, .D = 15 m, .vT (t) = 0 m/s, T R .vR (t) = 16.6 m/s, .vo (t) = 2 m/s, .vo (t) = 1.3 m/s, Low VTD: .Ot = 2, .Or = 2, .O = 5, High t r VTD: .Ot = 10, .Or = 10, .O = 4)

3.3 Simulations and Discussions

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Fig. 3.13 The fitting of the DPSD of .h33,11 with the measurement in [36] (.fc = 2.435 GHz, = δR = 2.943.λ, .T0 = 1 s, Low VTD: .MT = 1 .× 2, .MR = 1 .× 2, .u = 1, .w = 2, .l = 1, .k = 2, .D = 300 m, .Ot = 2, .Or = 2, .O = 2, .vT (t) = vR (t) = 11 m/s, High VTD: .MT = 1 .× 2, .MR = 1 .× 4, .u = 1, .w = 2, .l = 1, .k = 4, .D = 180 m, .Ot = 6, .Or = 6, .O = 2, .vT (t) = vR (t) = 22 m/s, v,T v,R v,T v,R v,T (t) = 0, .ψ v,R (t) = 0, .θ v,T (t) = 0, .ψot ,s (t) = 0, .ψor ,s (t) = 0, .θot ,s (t) = 0, .θor ,s (t) = 0, .ψ v,R .θ (t) = 0) .δT

of our knowledge, there is currently no measurement data analyzing of TACF under the high VTD in the measurement [35], and thus we cannot compare the simulated TACF under the high VTD with the corresponding measurement data in Fig. 3.12. Figure 3.13 compares the simulated DPSDs of the proposed RS-GBSM and the measurements of vehicular MIMO channels in [36], which were carried out in the urban surface street area and Interstate highway at .2.435 GHz. Based on the description of the measurement scenario, it can be assumed that the urban scenario and highway scenario are the low VTD scenario and high VTD scenario, respectively. In Fig. 3.13, the excellent agreement between the simulation results and the measurement demonstrates the accuracy of the proposed RS-GBSM. Finally, it is worth mentioning that, compared to the low VTD, the DPSD under the high VTD exhibits a flatter distribution, which is in correspondence with the phenomenon in Fig. 3.8. In Fig. 3.14, the simulated SCCFs are compared with the measurement data in [37]. Unfortunately, since there is no available measurement result related to the massive MIMO V2V channel measurement, we solely compare the measurement data in a static massive MIMO scenario, which was performed at .2.6 GHz with 128 element array. Also, for a fair comparison, the typical channel parameters are set as the measurement campaign. It can be readily observed from Fig. 3.14 that both the simulated and measured SCCFs gradually decrease when the normalized

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0.9

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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

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2 R

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/

Fig. 3.14 The fitting of the SCCF with the measurement in [37] (.fc = 2.6 GHz, .MT = 1 .× 128, = 1 .× 128, .u = 1, .w = 1, .l = 1, .k = 2, .u = 1, .w  = 1, .l  = 1, .k  = 3, .D = 35 m, .T0 = 1 s, T R .vT (t) = 0 m/s, .vR (t) = 0 m/s, .vo (t) = 0 m/s, .vo (t) = 0 m/s t r .MR

antenna spacing .δR /λ enlarges. Furthermore, the proposed RS-GBSM explicitly fits the measurement well, demonstrating the utility of the proposed RS-GBSM. To show the competitive benefit of the proposed model, the model in [33], the measurement data in [37], as well as the proposed model have been compared. Note that the algorithm of modeling space-time non-stationarity in [33] has been widely used in the existing channel models, i.e., [28, 29, 31, 32, 38]. More importantly, the authors in [39] claimed that the model developed in [33] is a 5G standard channel model, which has received a lot of recognition. From Fig. 3.14, compared to the proposed model, it is clear that the agreement between the simulation result in [33] and measurement data is worse. This is because the algorithm of modeling spacetime non-stationarity in [33] did not consider the selection of massive MIMO space non-stationarity and the model in [33] also did not consider the GR. Therefore, the competitive benefit of our proposed model can be verified by comparing the proposed model and the model in [33] during the model validation.

3.4 Summary In Chap. 3, a 3D space-time non-stationary wideband RS-GBSM for massive MIMO V2V with UPA channels has been proposed, where the GR component has been considered. Furthermore, channel parameters have been calculated by 3D vectors

References

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and clusters have been distinguished by static clusters and dynamic clusters to explore the impact of VTDs on channel statistics. Furthermore, the method of combining the BD process and the seed algorithm based selective cluster evolution has been developed to mimic the space non-stationarity of massive MIMO V2V with UPA channels for the first time. Also, the time non-stationarity has been modeled by using the developed method over the entire timeline. From the proposed RSGBSM, important channel statistical properties have been derived and investigated sufficiently. Simulated SCCFs and TACFs have demonstrated that the proposed RSGBSM has the ability to mimic space-time non-stationarity. Moreover, the evolution of different clusters on the UPA has shown that the appearance and disappearance of clusters can be characterized sufficiently. Simulation results have further shown that the proposed RS-GBSM under low VTDs has lower SCCFs, higher TACFs, and more steeply distributed DPSDs than that under high VTDs. Finally, close agreements between simulation results and measurement data on SCCFs, TACFs, and DPSDs have verified the utility of the proposed RS-GBSM.

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Chapter 4

A 3D IS-GBSM for Massive MIMO V2V Channels

Nowadays, the upcoming B5G and 6G wireless communication network has attracted the increasing attention. It is well known that one of the most important areas in B5G/6G wireless communications is the V2V communication, where the massive MIMO technology is expected to be leveraged to achieve a descent SNR, mitigate multipath fading, and increase the channel capacity. In this chapter, a 3D cluster-based model for B5G/6G massive MIMO V2V wideband channels is proposed. It is the first cluster-based IS-GBSM to distinguish the dynamic clusters and static clusters in vehicular massive MIMO communication scenarios. This model not only considers the high time variance, the time non-stationarity, and the VTD of V2V channels, but also models the massive MIMO channel characteristics, such as spherical wavefront by 3D vector calculation and space non-stationarity. Meanwhile, the proposed IS-GBSM integrates the VTD into BD process to model the massive MIMO V2V channel characteristics jointly and deeply, where a novel VTD-combined time-array cluster evolution algorithm for B5G/6G massive MIMO V2V channel model is developed. Based on the proposed ISGBSM, some expressions of channel statistical properties, including STF-CF and DPSD, are derived. The influence of several parameters, e.g., VTDs, vehicular moving directions, and antenna spacing, on the channel statistical properties is explored, which can provide assistance for the design of vehicular massive MIMO communication systems. Finally, the utility of the proposed IS-GBSM is verified by the close agreement between simulation results and measurement data.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_4

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4.1 Framework of Massive MIMO V2V IS-GBSM 4.1.1 Introduction and Contributions of Proposed IS-GBSM In the upcoming B5G/6G era, the goal is to further increase the communication rate, expand the communication space, and explore the diversified application of related technologies in vertical industries [1]. Massive MIMO has been regarded as one of the most essential technologies for B5G/6G wireless communication systems [2, 3]. Furthermore, B5G/6G communications focus on more dynamic and various communication scenarios, where the V2V communication is an indispensable component. Under the high-mobility communication scenario, the goal of V2V communication system is to improve the system performance, reduce latency, and provide users with reliable data connections [4–6]. The advantages of the combination of massive MIMO technologies and the V2V communication include expanding communication applications, enhancing user experience, and improving spectrum efficiency. In order to analyze and design B5G/6G vehicular communication systems, the basic research on corresponding channel models and propagation characteristics is strongly important [7, 8]. Compared with traditional cellular communications, the V2V communication has several unique characteristics, e.g., (1) vehicles at Tx and Rx are equipped with low-elevation antennas, and (2) Tx, Rx, and some surrounding clusters in the environment are moving at high speed. Figure 4.1 shows a simple and typical vehicular communication scenario with moving vehicles and roadside static environments. The waves at the Tx pass through different obstacles in the environment and reach the Rx. In order to reflect a more practical V2V communication channel, it is necessary to distinguish between moving and static obstacles in the environment [9, 10]. At the same time, the influence of VTD on channel characteristics should be considered in the modeling process [11, 12], such as rural street scenarios with low VTDs and fewer buildings, while urban street scenarios with more cars and buildings [13]. In addition, due to the existence of high-speed moving vehicles, some important parameters, e.g., the power, delay, angle, of the received signal are time-variant, and thus the channel properties will fluctuate significantly over time [14, 15]. The fast time-variant property in the V2V scenario results in the channel non-stationarity in the time domain. Note that channel non-stationarity in a certain domain means that the channel statistical characteristics vary in this domain. In this case, the V2V channel statistics are time-variant essentially. Compared with the MIMO, the massive MIMO system is equipped with a large number of antennas at the Tx and Rx, which results in its unique channel properties, such as non-stationarity in the space (array) domain [16] and spherical wavefront propagation [17]. The authors in [18] measured the channel statistical properties in the massive MIMO scenario, and the measurement results showed that there were some unobservable clusters on the space domain, and new clusters appeared in some sliding windows. Therefore, aiming at promoting the development of B5G/6G massive MIMO V2V communication, the channel model needs to give consideration

4.1 Framework of Massive MIMO V2V IS-GBSM

65

Fig. 4.1 A common vehicular communication scenario

to both V2V channel characteristics and massive MIMO channel characteristics. This means that the channel model not only needs to model the high time variance, the influence of VTD, and spherical wavefront characteristic of massive MIMO V2V channels, but also needs to consider the modeling of the combination between the space-time non-stationarity and the VTD [19–21]. At present, extensive literature have focused on the V2V communication channel modeling [22–38], where these proposed models can be classified as deterministic models and stochastic models. The deterministic model needs to perform detailed measurements on actual V2V scenarios to obtain corresponding channel parameters. The authors in [22, 23] proved that the deterministic model established by ray tracing (RT) can analyze wideband and narrowband channels effectively, and these models can obtain specific CIRs. Based on the highway scenario [24], a deterministic model was proposed to explore the impact of different antenna positions on vehicular communication performance. The deterministic model has the advantages of high accuracy and practicality, and it considers a large number of situations that may occur in the realistic modeling process. However, it requires a detailed description of a specific propagation environment, which requires a large amount of computation time, and is difficult to promote. Compared with deterministic models, stochastic models have become popular in the field of V2V channel modeling due to its higher generality and lower complexity, which can be further classified into non-geometrical stochastic models (NGSMs) and geometry-

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

based stochastic models (GBSMs). In the early days of vehicular channel modeling, some researchers focused on the development of NGSMs. The proposed model mainly serves the scenario based on the IEEE 802.11p standard [39]. The authors proposed three types of V2V NGSM and three types of roadside-to-vehicle NGSM [25], whereas the V2V channel non-stationarity in the time domain, i.e., time non-stationarity, was disregarded. To capture the channel time non-stationarity, a non-stationary NGSM for wideband vehicular communication channels was proposed in [26]. Unlike the NGSM, the GBSM can directly analyze the effective scatterers in the vehicular environment through geometric relationships. It can be further divided into the RS-GBSM and the IS-GBSM. The RS-GBSM uses a regular geometry to represent the distribution of effective scatterers or clusters and thus has low complexity. As a result, the RS-GBSMs have been widely used in the theoretical study of V2V channel modeling. The existing RS-GBSMs [27– 33] cover a variety of shapes, i.e., from 2D models to 3D models, for different communication channels, i.e., from narrowband channels to wideband channels. Note that the disadvantage of RS-GBSM is its low accuracy. With the advent of B5G/6G commercialization, the demand for high-precision standardized models has become increasingly prominent. Compared with the RS-GBSM, the IS-GBSM has higher accuracy and is the recommended model for standardized channel models [7]. Meanwhile, to properly model high time-varying characteristics of V2V channels and sufficiently explore the impact of VTDs on V2V channel statistics, the IS-GBSMs are more suitable than the RS-GBSMs [40]. This is because the clusters in the IS-GBSM modeling approach are not on certain shapes and the ISGBSM can more flexibly model dynamic clusters and static clusters separately. Therefore, the IS-GBSM has become a hot research topic in vehicular channel modeling. Based on the analysis of ten up-to-date standardized 5G channel models in [41], there is no standardized channel model that can model dual mobility and high mobility and can further investigate the impact of VTDs on channel statistics. Therefore, these standardized 5G channel models cannot sufficiently capture the channel characteristics in B5G/6G vehicular communication scenarios. Considering the aforementioned shortcomings, it is necessary to establish a channel model for B5G/6G massive MIMO V2V wideband channels that is easy to analyze and has limited complexity. This model needs to capture space-time non-stationarity and distinguish dynamic clusters and static clusters and needs to further consider the impacts of VTDs on both the channel statistics and the space-time non-stationarity. To fill the above gaps, a 3D cluster-based IS-GBSM is developed for the B5G/6G massive MIMO V2V wideband channel. The proposed IS-GBSM considers the high time variance of V2V channels, the separation between dynamic clusters and static clusters, and the spherical wavefront characteristic by 3D vector calculation. Through the combination between VTD and appearance and disappearance of the clusters on both the time axis and array axis, the proposed IS-GBSM can be exploited to describe the channel space-time non-stationarity in consideration of VTDs and to explore the impact of VTD on channel statistical properties. The proposed IS-GSBM can provide some suggestions for the establishment of B5G/6G

4.1 Framework of Massive MIMO V2V IS-GBSM

67

V2V standardized channel models. Main contributions and novelties of this chapter can be summarized as follows: 1. The proposed IS-GBSM is the first B5G/6G massive MIMO V2V IS-GBSM that distinguishes the dynamic and static clusters, considers the difference of spacetime non-stationary modeling for dynamic and static clusters, and investigates the impacts of VTDs on channel statistics in massive MIMO V2V scenarios. By adjusting relevant parameters, the proposed IS-GBSM can be adapted to different vehicular communication scenarios. 2. From the proposed IS-GBSM, a novel VTD-combined time-array cluster evolution algorithm is developed to model the joint channel space-time nonstationarity for massive MIMO V2V scenarios. In the developed algorithm, the clusters are properly divided into dynamic clusters and static clusters, and their impact on space-time non-stationarity is considered adequately. Some important expressions of dynamic clusters and static clusters are distinguished, such as the expression of generation/recombination probability in the BD process, and the position distribution of clusters in different VTD scenarios. 3. The channel parameters in this model, including delay, Doppler frequency, angle of departure (AoD), and angle of arrival (AoA), are time-variant and are considered in both the azimuth direction and elevation direction. Furthermore, the key model-related parameters are computed by 3D vectors, and thus the spherical wavefront propagation is accurately described. From the proposed ISGBSM, the expressions of some channel statistical properties, i.e., STF-CF and DPSD, are derived. 4. Through the simulation, the influences of some important parameters, e.g., VTDs, vehicular moving directions, and antenna spacing, on channel statistical properties, are investigated. Additionally, simulation results demonstrate that the proposed IS-GBSM has the ability to describe the non-stationarity of the massive MIMO V2V wideband channel in both the time domain and the space domain effectively, i.e., space-time non-stationarity. At the same time, the excellent agreement between simulation results and measurement data verifies the utility of the proposed IS-GBSM.

4.1.2 Channel Impulse Response of Proposed Cluster-Based IS-GBSM The proposed 3D cluster-based IS-GBSM for B5G/6G massive MIMO V2V wideband channels is shown in Fig. 4.2. The initial horizontal distance between Tx and Rx is D. The Tx and Rx are equipped with .MT and .MR omni-directional antennas. The antenna spacing of the Tx and Rx are .δT and .δR , respectively. Parameters .θT , φT and .θR , φR represent the azimuth angle and elevation angle at Tx and Rx antenna arrays, respectively. The Tx and Rx move with .vT and .vR , respectively. Considering that the proposed IS-GBSM needs to have the ability

68

4 A 3D IS-GBSM for Massive MIMO V2V Channels

Fig. 4.2 The proposed 3D cluster-based model for B5G/6G massive MIMO V2V wideband channels

to explore the impact of different VTDs on channel statistics, we distinguish the position distribution of dynamic clusters and static clusters in different scenarios. In high VTD scenarios, the proportion of dynamic clusters is large, and the distribution of dynamic clusters nearby the Tx and Rx is dense. In addition, the dynamic clusters in the V2V environment are mainly low vehicles, while the static clusters are mainly tall buildings on the roadside. In such a case, it is essential to distinguish the azimuth and elevation angles of the dynamic clusters’ and static clusters’ position distribution. In low VTD scenarios, static clusters mainly play a dominant role, and dynamic clusters are sparsely distributed. The numbers of static clusters and dynamic clusters are .M(t) and .N(t) at the time instant t, respectively. The velocity of the n-th dynamic cluster is .vn . The cluster-based massive MIMO V2V CIR can be characterized by a .MR ×  MT complex matrix .H(t, τ ) = hqp (t, τ ) M ×M , where .p = 1, 2, . . . , MT and R T .q = 1, 2, . . . , MR . The complex coefficient CIR from the p-th Tx antenna to the q-th Rx antenna can be divided into the LoS and the NLoS components, which can be further divided into links passing through dynamic clusters (i.e., NLoS,n) and passing through static clusters (i.e., NLoS,m). Overall, the CIR can be expressed by 

  K LoS hLoS qp δ τ − τqp (t) K +1  M(t) Im   η1   NLoS,s s hqp,m,i (t)δ τ − τqp,m (t) + m K +1

hqp (t, τ ) =

.

m=1 im =1

4.1 Framework of Massive MIMO V2V IS-GBSM

 +

69

N (t) In   η2   NLoS,d d hqp,n,i (t)δ τ − τ (t) . qp,n n K +1

(4.1)

n=1 in =1

As shown in (4.1), .hLoS qp (t) means the complex channel gain of the LoS component, NLoS,d hqp,n,i (t) indicates the complex channel gain of the NLoS component through the n

.

NLoS,s in -th ray within the n-th dynamic cluster, and .hqp,m,i (t) indicates the complex m channel gain of the NLoS component through the .im -th ray within the m-th static cluster. The parameter K denotes the Ricean factor, which assumes a Ricean distribution of the signal. Also, the Ricean factor K represents the ratio of the received power of LoS component to the received power of NLoS component [42]. Furthermore, .τn/m (t) denotes the delay of rays passing through the .Clustern/m . There are .Im and .In rays in the m-th static cluster and the n-th dynamic cluster, respectively. Note that .η1 and .η2 are the parameters to model the power ratio of static clusters and dynamic clusters and satisfy .η1 + η2 = 1.

.

4.1.2.1

Complex Channel Gain of LoS Component

The complex channel gain of the LoS component from the p-th Tx antenna to the q-th Rx antenna can be expressed as .

LoS (t)t

j 2πfqp hLoS qp (t) = e

e j 0 e j

2π LoS λ Dqp (t)

,

(4.2)

LoS (t) is the Doppler frequency of the LoS component from the p-th Tx where .fqp antenna to the q-th Rx antenna. .0 is the initial phase. .DLoS qp (t) is the distance vector of LoS path from the p-th Tx antenna to the q-th Rx antenna. The Doppler frequency of the LoS component can be calculated by the maximum Doppler frequency of the LoS , the distance vector, and the velocity vectors of transceiver LoS component .fmax

LoS LoS fqp (t) = fmax

.

 DLoS qp (t), vR − vT  DLoS qp (t) vR − vT 

,

(4.3)

where .·, · is defined as the inner product operator. Furthermore, we have T R DLoS qp (t) = dp (t) − dq (t)

.

⎡ ⎤ cos φR (t) cos θR (t) − 2q + 1 M R R δR ⎣ cos φR (t) sin θR (t) ⎦ .dq (t) = 2 sin φR (t)

(4.4)

(4.5)

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

⎡ ⎤ cos φT (t) cos θT (t) − 2p + 1 M T T δT ⎣ cos φT (t) sin θT (t) ⎦ .dp (t) = 2 sin φT (t)

(4.6)

D(t) = D + (vR − vT )t,

(4.7)

.

T where .D = [D, 0, 0] is the distance vector between the Tx and Rx. .dR q (t) and .dp (t) are the location vectors of the q-th Rx antenna and the p-th Tx antenna. The delay of the LoS component from the p-th Tx antenna to the q-th Rx antenna can be expressed as

LoS τqp (t) =

.

DLoS qp (t) c

,

(4.8)

where c is the speed of light.

4.1.2.2

Complex Channel Gain of NLoS Component

The complex channel gain of the NLoS component passing through dynamic clusters from the p-th Tx antenna to the q-th Rx antenna can be expressed as NLoS,d hqp,n,i (t) n 

. 2π R T R T

n j 2π tfqn,i (t) j 2π tfpn,i (t) j Φ0 j λ Dqn,in (t)+Dpn,in (t) n n e e e e , = In

(4.9)

R (t) and .f T where .fqn,i pn,in (t) are the Doppler frequencies of NLoS component n passing through the n-th dynamic cluster at the q-th Rx antenna and the p-th Tx T antenna, respectively. .DR qn,in (t)/.Dpn,in (t) is the distance vector between the n-th dynamic cluster and the q-th Rx antenna/the p-th Tx antenna. The complex channel gain of the NLoS component passing through static clusters from the p-th Tx antenna to the q-th Rx antenna can be expressed as NLoS,s hqp,m,i (t) m 

. 2π R T R T

m j 2π tfqm,i (t) j 2π tfpm,i (t) j Φ0 j λ Dqm,im (t)+Dpm,im (t) m m e e e e , = Im

(4.10)

R T where .fqm,i (t) and .fpm,i (t) are the Doppler frequencies of NLoS component m m passing through the m-th static clusters at the q-th Rx antenna and the p-th Tx T antenna, respectively. .DR qm,im (t)/.Dpm,im (t) is the distance vector between the m-th static cluster and the q-th Rx antenna/the p-th Tx antenna.

4.1 Framework of Massive MIMO V2V IS-GBSM

71

Channel measurements demonstrated that the power of clusters can be modeled by exponential PDP, and the normalization of power is widely exploited to achieve the same average power in different modulation methods [43, 44]. Therefore, the generation of the normalized power of the n-th dynamic cluster .Clusterdn , . n , is given as   −Zn rτ − 1 d · 10 10 ,

n = exp −τqp,n rτ τDS

.

(4.11)

where parameter .Zn ∼ N(0, ζ 2 ) represents the shadowing term of .Clustern . Further, normalize the power, which can be given as



n = n N n

.

 n=1 n

,

(4.12)

where . m (t) is the normalized power of the m-th static cluster .Clustersm . The generation of . m (t) is similar with that of . n (t). The Doppler frequencies of NLoS components passing through dynamic clusters are expressed as 

 DR (t), v − v R n qn,in R R  .fqn,i (t) = fmax  n   R Dqn,in (t) vR − vn   DTpn,in (t), vT − vn T T  .fpn,i (t) = fmax  , n  T  Dpn,in (t) vT − vn 

(4.13)



(4.14)

T and .f R are the maximum Doppler frequencies of Tx and Rx. The where .fmax max velocity vectors are calculated as

⎡

⎤ cos β T (t) cos α T (t) T T .vT = vT ⎣ cos β (t) sin α (t) ⎦ T sin β (t)

(4.15)

⎡

⎤ cos β R (t) cos α R (t) R R .vR = vR ⎣ cos β (t) sin α (t) ⎦ R sin β (t)

(4.16)

⎡

⎤ cos βn (t) cos αn (t) .vn = vn ⎣ cos βn (t) sin αn (t) ⎦ , sin βn (t)

(4.17)

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

where .α T and .β T indicate the azimuth angle and elevation angle of the speed of Tx. Similarly, .α R and .β R indicate the azimuth angle and elevation angle of the speed of Rx. .αn and .βn indicate the azimuth angle and elevation angle of the speed of the n-th dynamic cluster. The distance vectors from the n-th dynamic cluster to the q-th Rx antenna and from the p-th Tx antenna to the n-th dynamic cluster via the .in -th ray are expressed as .

R R DR qn,in (t) = dn,in (t) + dq (t)

(4.18)

DTpn,in (t) = dTn,in (t) − dTp (t),

(4.19)

.

R/T

where the .dn,in is the distance vector between the .in -th ray within n-th dynamic cluster and the Rx/Tx center, which can be presented as ⎤ EoA (t) cos α AoA (t) cos βn,i n,in n ⎥ ⎢ R R EoA AoA .dn,i (t) = −Dn (t) ⎣ cos β n,in (t) sin αn,in (t) ⎦ n EoA (t) sin βn,i n

(4.20)

⎤ EoD (t) cos α AoD (t) cos βn,i n,i n n ⎥ ⎢ T T EoD AoD .dn,i (t) = −Dn (t) ⎣ cos β n,in (t) sin αn,in (t) ⎦ . n EoD (t) sin βn,i n

(4.21)

⎡

⎡

The delay of the NLoS component from the p-th Tx antenna to the q-th Rx antenna through n-th dynamic cluster can be expressed as d τqp,n (t) =

.

T DR qn,in (t) + Dpn,in (t)

c

.

(4.22)

The Doppler frequencies of NLoS component passing through the m-th static cluster are expressed as 

 DR (t), v R qm,im R R  .fqm,i (t) = fmax  m   R Dqm,im (t) vR   DTpm,im (t), vT T T  .fpm,i (t) = fmax  . m  T  Dpm,im (t) vT 

(4.23)



(4.24)

The distance vectors from the m-th static cluster to the q-th Rx antenna and from the p-th Tx antenna to the m-th static cluster via the .im -th ray are expressed as

4.1 Framework of Massive MIMO V2V IS-GBSM

73

.

R R DR qm,im (t) = dm,im (t) + dq (t)

(4.25)

DTpm,im (t) = dTm,im (t) − dTp (t),

(4.26)

.

R/T

where the .dm,im is the distance vector between the .im -th ray within the m-th static cluster and the Rx/Tx center, which can be presented as ⎤ EoA (t) cos α AoA (t) cos βm,i m,im m ⎥ ⎢ R R EoA AoA .dm,i (t) = −Dm (t) ⎣ cos β m,im (t) sin αm,im (t) ⎦ m EoA (t) sin βm,i m

(4.27)

⎤ EoD (t) cos α AoD (t) cos βm,i m,im m ⎥ ⎢ T T EoD AoD .dm,i (t) = −Dm (t) ⎣ cos β m,im (t) sin αm,im (t) ⎦ . m EoD (t) sin βm,i m

(4.28)

⎡

⎡

The delay of the NLoS component from the p-th Tx antenna to the q-th Rx antenna through m-th static cluster can be expressed as s τqp,m (t) =

.

T DR qm,im (t) + Dpm,im (t)

c

.

(4.29)

T (t), D R (t) indicate the distance between clusters and Tx/Rx The parameters .Dn/m n/m and follow an exponential distribution [43]. For the LoS component, parameters AoA EoA AoD EoD .α LoS and .βLoS indicate the AAoA and EAoA, and .αLoS and  .βLoS indicate the 

AoA , β EoA and . α AoD , β EoD AAoD and EAoD. Similarly, for the NLoS path, . αn,i n,in n,in n,in n indicate arrival angles and departure angles of the .in -th (.in = 1, 2, . . . , In) ray AoA , β EoA and passing through the n-th (.n = 1, 2, . . . , N(t)) dynamic cluster. . αm,i m,im m   AoD EoD . α m,im , βm,im represent arrival angles and departure angles of the .im -th (.im = 1, 2, . . . , Im ) ray passing through the m-th (.m = 1, 2, . . . , M(t)) static cluster. The aforementioned angular parameters are modeled to obey the wrapped Gaussian distributions, and the corresponding mean value and standard deviation are denoted by .μ and .σ . It is noticed that the following expressions use dynamic clusters as an example. AoA/AoD

= αn

EoA/EoD

= βn

αn,in

.

βn,in

.

AoA/AoD

EoA/EoD

AoA/AoD

+ Δαn,in

AoA/AoD

EoA/EoD

+ Δβn,in

EoA/EoD

,

(4.30) (4.31)

where .αn,in and .βn,in are the offset angles, respectively. The key model-related parameters are given in Table 4.1.

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

Table 4.1 The meaning of key parameters used in the proposed IS-GBSM Symbol

Definition Total number of Tx/Rx antennas Antenna spacing of Tx/Rx antennas Azimuth angle and elevation angle at Tx/Rx antenna array Velocity vectors of Tx/Rx and the n-th dynamic cluster Number of dynamic/static clusters

.MT /.MR .δT /.δR .θT , φT /.θR , φR .vT /.vR , .vn .N (t)/.M(t)

LoS (t)

CIR of the LoS component

.hqp

NLoS,d

.hqp,n,i

n

(t)/.hNLoS,s qp,m,im (t)

LoS (t)

Doppler frequency of LoS component of the .p/q-th Tx/Rx antenna Distance vector of LoS path of the .p/q-th Tx/Rx antenna Location vector of the q-th Rx antenna/p-th Tx antenna Doppler frequency of NLoS component related to dynamic clusters Distance vector related to dynamic cluster and Tx/Rx antenna Azimuth angle and elevation angle of the speed of Tx/Rx Azimuth angle and elevation angle of the speed of dynamic clusters Distance vector between clusters and Tx/Rx

.fqp

LoS (t)

.Dqp

R T .dq (t)/.dp (t) R T .fqn,i (t)/.fpn,i (t) n n R

.Dqn,i

n

.α

(t)/.DTpn,in (t)

T , .β T /.α R , .β R

.αn , .βn

T

R

.Dn (t)/.Dn (t)

AoA

EoA

AoD

CIR of the NLoS component of the dynamic/static cluster

EoD

.αLoS /.βLoS / .αLoS /.βLoS

AAoA/EAoA/AAoD/EAoD of LoS path

AoA /β EoA /α AoD , β EoD n,in n,in n,in n

AAoA/EAoA/AAoD/EAoD related to the dynamic cluster

AoA /β EoA /α AoD /β EoD m,im m,im m,im m

AAoA/EAoA/AAoD/EAoD related to the static cluster

.αn,i

.αm,i

R T .fqm,i (t)/.fpm,i (t) m m

Doppler frequency of NLoS component related to static clusters R T Distance vector related to static clusters and the Tx/Rx .Dqm,i (t)/.Dpm,i (t) m m antenna LoS d s Delay of LoS component/NLoS component of .τqp /τqp,n /τqp,m dynamic/static cluster Initial number of dynamic/static clusters at time t .N (t)/.M(t) .λG,n/m , .λR,n/m Generation and recombination rates of dynamic/static clusters Time,d (t)/.P Time,s (t) Time survival probability of the dynamic cluster and static .P clusters t t Coefficient of dynamic/static clusters to describe the spatial .Dc,n /.Dc,m correlation .E [Nnew (t + t)]/.E [Mnew (t + t)] Mean number of newly generated dynamic/static clusters p p Number of dynamic/static clusters that is observed by the .N (t)/.M (t) Tx antenna Array,d /.P Array,s .P Array survival probability of dynamic/static clusters a a .Dc,n /.Dc,m Antenna spacing correlation coefficient of dynamic/static clusters survival,d /.P survival,s .P Time-array survival probability of dynamic/static clusters

4.2 Space-Time Non-stationary Modeling with Vehicular Traffic Density

75

4.2 Space-Time Non-stationary Modeling with Vehicular Traffic Density Attributed to the high-speed mobility in V2V communication scenarios, the corresponding parameters of the proposed channel model will change over time, which leads to the channel non-stationarity in the time domain, i.e., time non-stationarity. Furthermore, the large-scale antenna array in massive MIMO further brings the channel non-stationarity in the space domain, i.e., space non-stationarity. To model the space-time non-stationarity, the BD process has been extensively employed, which is a special case of continuous-time Markov process where the state transitions are of only two types, i.e., birth and death. For modeling the time nonstationarity, observing temporal fluctuations of the propagation channel, clusters appear and disappear after a certain time. Such a behavior can be modeled by the BD process. Similarly, for modeling the space non-stationarity, the appearance and disappearance of clusters on the array evolution can also be characterized by the BD process. Also, the space non-stationarity and time non-stationarity in massive MIMO V2V channel model were modeled by the BD processes jointly in [35]. However, the authors in [35] did not consider the influence of VTD on channel statistical properties, especially on the space-time non-stationarity. The VTD not only has a significant influence on channel statistical properties in V2V communication scenarios, but also affects the space-time non-stationarity. This is because the static clusters and dynamic clusters have different evolution behaviors on the space-time non-stationarity. Therefore, it is necessary to model the space-time non-stationarity with VTD jointly. To this end, a novel VTD-combined time-array cluster evolution algorithm is developed based on the BD processes for the proposed 3D cluster-based massive MIMO V2V channel model. The developed algorithm considers the influence of VTD on the space-time non-stationarity and models the time-array cluster evolution on static clusters and dynamic clusters separately. Next, the VTD-combined time cluster evolution calculation, the array cluster evolution calculation, and the flowchart of VTD-combined time-array cluster evolution algorithm are presented, respectively.

4.2.1 VTD-Combined Time Cluster Evolution Calculation In order to represent the non-stationarity in the time domain, i.e., time nonstationarity, the BD process is adopted to describe the cluster evolution on the time axis, i.e., the appearance and disappearance of clusters at different times. The “birth state/death state” in the BD process corresponds to the cluster that can/cannot be observed. The evolution of “birth state/death state” of clusters depends on the last state and the survival probability. Due to the movement of dynamic clusters, the time cluster evolution for static clusters and dynamic clusters are different, and the corresponding model-related parameters also change.

76

4 A 3D IS-GBSM for Massive MIMO V2V Channels

The initial numbers of dynamic clusters and static clusters at the initial time instant .t0 are expressed by N(t0 ) =

λG,n λR,n

(4.32)

M(t0 ) =

λG,m , λR,m

(4.33)

.

.

where parameters .λG,n/m and .λR,n/m indicate the generation rate and the recombination rate of dynamic/static clusters. At the next time instant .t + t, the numbers of clusters change over time, i.e., .N(t) → N(t + t) and .M(t) → M(t + t). The survival of the dynamic and static clusters in the time domain is determined by the time survival probabilities .P Time,d (t) and .P Time,s (t), which can be expressed as P Time,d (t) = e

−λR,n

(vT −vn +vR −vn )t

.

P Time,s (t) = e

.

−λR,m

t Dc,n

(4.34)

(vR +vT )t t Dc,m

,

(4.35)

t and .D t where .Dc,n c,m indicate two time-correlated distance coefficients that affect the time non-stationary modeling of dynamic clusters and static clusters, respectively, and typical values include 10 m, 30 m, 50 m, and 100 m [43]. At the same time, the numbers of newly generated dynamic and static clusters are given by

E [Nnew (t + t)] =

 λG,n  1 − P Time,d (t) λR,n

(4.36)

E [Mnew (t + t)] =

 λG,m  1 − P Time,s (t) . λR,m

(4.37)

.

.

The model-related parameters of the surviving clusters and newly generated clusters are also updated.

4.2.2 VTD-Combined Array Cluster Evolution Calculation Considering the huge number of antennas in the massive MIMO technology, the channel non-stationarity in the space domain, i.e., space non-stationarity, is reflected that different antennas can observe different dynamic and static clusters, i.e., p p p p .N (t) → N (t) and .M (t) → M (t). In order to explore the evolution of the dynamic/static clusters on the array axis, taking the Tx side as an example, it is assumed that the newly generated dynamic/static clusters can be observed by the p-th receiving antenna. Whether the dynamic and static clusters are observed by the

4.2 Space-Time Non-stationary Modeling with Vehicular Traffic Density

77

p -th receiving antenna is determined by the array survival probabilities .PT Array,s and .PT . If it cannot be observed, the next antenna is judged. The parameters Array,d Array,s .P and .PT can be expressed as T Array,d

.

Array,d

=e

PT

.

Array,s

=e

PT

.

   T T  d −d   p p  −λR,n D a c,n

   T T  d −d   p p  −λR,m D a c,m

(4.38)

(4.39)

,

a and .D a are defined as array-correlated distance coefficients where parameters .Dc,n c,m that affect the array (space) non-stationary modeling of dynamic clusters and static clusters, respectively [43]. After considering the cluster evolution in both the time domain and the space   domain, i.e., .N p (t) → N p (t + t) and .M p (t) → M p (t + t), the survival probabilities of the dynamic clusters and static clusters can be, respectively, expressed as 

P survival,d (t) =e

−λR,n

(vT −vn +vR −vn )t



t Dc,n

⎤      dT −dT +dR −dR    p p   q q ⎦ ⎣ −λR,n a Dc,n

⎡ 

.

×e



P survival,s (t) =e

−λR,m

(vR +vT )t



t Dc,m

⎤      dT −dT +dR −dR    p p   q q ⎦ ⎣ −λR,m a Dc,m

⎡ 

.

×e

(4.40)

(4.41) .

4.2.3 Steps of VTD-Combined Time-Array Cluster Evolution Algorithm Based on the VTD-combined time cluster evolution calculation and the array cluster evolution calculation, the VTD-combined time-array cluster evolution algorithm is proposed in Fig. 4.3. For clarity, the steps of the developed algorithm are given as follows: Step 1 According to (4.32) and (4.33), the numbers of dynamic clusters and static clusters at the initial time instant .t0 , .N(t0 ) and .M(t0 ), are obtained.

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

Fig. 4.3 The flowchart of VTD-combined time-array cluster evolution algorithm

Step 2 Taking the static clusters as an example, for each static cluster (.m = 1, 2, . . . , M(t0 )), the Tx and Rx antenna sets, which can observe the m-th static cluster, .ATClusters and .AR Clusters , are generated based on the BD process according to m

m

Array,s

the array survival probabilities of static clusters at Tx and Rx, .PT The dynamic clusters also take this step at the same time.

Array,s

and .PR

.

NLoS,s Step 3 For the static clusters, if .p ∈ ATClusters and .q ∈ AR Clusters , the CIR .hqp,m,im (t) m

m

NLoS,s is obtained as (4.10). Otherwise, the CIR .hqp,m,i (t) = 0. Similarly, for the dynamic m

clusters, if .p ∈ AT

Clusterdn

and .q ∈ AR

Clusterdn

NLoS,d , the CIR .hqp,n,i (t) is obtained as n

NLoS,d (4.9). Otherwise, the CIR .hqp,n,i (t) = 0. Then, combining the LoS component, the n NLoS component passing through static clusters, and the NLoS component passing through dynamic clusters, the CIR from the p-th Tx antenna to the q-th Rx antenna, i.e., .hqp (t, τ ), can be acquired.

4.3 Simulations and Discussions

79

Step 4 Furthermore, taking .t = t + t, the newly generated dynamic clusters and static clusters with mean numbers .E [Nnew (t)] and .E [Mnew (t)] are generated. Finally, Step 2 and Step 3 are operated recursively with respect to time.

4.3 Simulations and Discussions Based on the proposed massive MIMO V2V IS-GBSM, key channel statistical properties are derived and analyzed thoroughly. Then, these derived channel statistical properties are simulated, where the valuable observation can be obtained. Finally, the utility of the proposed massive MIMO V2V IS-GBSM can be properly validated by comparing measurement results and simulation results.

4.3.1 Statistical Properties As previously mentioned, a space-time non-stationary IS-GBSM for massive MIMO V2V channels is developed. Based on the proposed IS-GBSM, typical channel statistics, including STF-CF and DPSD, are obtained and analyzed adequately.

4.3.1.1

Space–Time–Frequency Correlation Function

In order to analyze the channel properties, the STF-CFs of different links are investigated. Here, a time-variant transfer function .Tqp (t, f ) is used to derive the STF-CF, which can be obtained by performing a Fourier transform on the CIR .hqp (t)  Tqp (t, f ) =

.

LoS (t) K −j 2πf τqp hLoS qp (t)e K +1  N (t) In d η1   NLoS,d + hqp,n,i (t)e−j 2πf τqp,n (t) n K +1

n=1 in =1

 +

(4.42)

M(t) Im s η2   NLoS,s hqp,m,i (t)e−j 2πf τqp,m (t) . m K +1 m=1 im =1

For the proposed IS-GBSM, the STF-CF of the two links .Tp − Rq and .Tp − Rq  can be expressed as ∗ Rqp,q  p (δT , δR , t, f, t, f ) = E[Tqp (t, f )Tq  p (t + t, f + f )],

.

(4.43)

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

where .(·)∗ and .E[·] denote the complex conjugate operation and statistical expectation operator, respectively. The corresponding expressions are given by LoS Rqp,q  p (δT , δR , t, f, t, f ) =Rqp,q  p  (δT , δR , t, f, t, f ) NLoS,d + Rqp,q  p  (δT , δR , t, f, t, f )

.

(4.44)

NLoS,s + Rqp,q  p  (δT , δR , t, f, t, f ) LoS Rqp,q  p  (δT , δR , t, f, t, f ) = .

K hLoS (t)hLoS q  p (t + t) K + 1 qp ×e

(4.45)

LoS (t)−(f +f )τ LoS (t+t)] j 2π [f τqp q  p

NLoS,d Rqp,q  p  (δT , δR , t, f, t, f ) =

η1 P survival,d K +1 ⎡ In N (t) N (t+t) In  .    NLoS,n hqp,n,i (t)hqNLoS,d × E⎣  p  ,n ,i  (t + t) n n=1

×e

n =1

(4.46)

n

in =1 in =1

d (t)−(f +f )τ d j 2π [f τqp,n (t+t)] q  p ,n



NLoS,s Rqp,q  p  (δT , δR , t, f, t, f ) =

η2 P survival,s K +1 ⎡ Im . M(t) Im   M(t+t)   NLoS,s hNLoS,s × E⎣ qp,m,im (t)hq  p ,m ,i  (t + t) m=1

×e

4.3.1.2

m =1

im =1 im =1

s j 2π [f τqp,m (t)−(f +f )τqs p ,m (t+t)]

(4.47)

m

 .

Doppler Power Spectral Density

When .δT = δR = 0, the STF-CF .Rqp,q  p (δT , δR , t, f, t, f ) can be simplified into the TACF .Rqp,qp (t, t). The corresponding DPSD of the proposed model can be seen as a Fourier transform of the derived TACF .Rqp,qp (t, t), which can be written as

4.3 Simulations and Discussions

81

 S(t, fD ) =

+∞

.

−∞

Rqp,qp (t, t)e−j 2πfD t dt,

(4.48)

where .fD represents the Doppler frequency. It can be observed that the derived DPSD .S(t, fD ) is time-varying, and thus the proposed IS-GBSM is also timevarying.

4.3.2 Model Simulation According to the derived channel statistical properties, the simulation results of the proposed IS-GBSM are presented. Meanwhile, the influence of important parameters on the proposed model are analyzed and some interesting observations are given. The utility and accuracy of the proposed model are sufficiently verified by comparing the simulation results with measurement data. It is noteworthy that basic parameters used for numerical analysis are shown in Table 4.2. Figure 4.4 compares the normalized absolute TACFs under different VTDs at .t = 0.5 s and .t = 1 s. It can be seen that the proposed IS-GBSM shows the obvious non-stationarity in the time domain, i.e., the curve of TACF at different times has different fluctuations because of the movement of Tx, Rx, and clusters, and the birth and death behavior of clusters in the time domain. At the same time, it shows that the TACF at high VTD scenarios is much lower than that at low VTD scenarios. This is because, under the high VTD, there are a large number of dynamic clusters and diverse scattering sub-paths near the transceiver. In this case, the communication scenario is significantly complicated, leading to greater changes in the propagation environment compared to that of the low VTD. Therefore, at a certain speed, it can be concluded that the TACFs under high VTDs are lower than the TACFs under low VTDs. Figure 4.5 shows the normalized absolute TACFs with different VTDs under different vehicular moving directions. It can be readily observed that, when the

Table 4.2 Parameters of the dynamic clusters and static clusters Parameter Antenna correlation coefficient Mean value of .DnR and .DnT .μAAoA , σAAoA .μAAoD , σAAoD .μEAoA , σEAoA .μEAoD , σEAoD .MT , .MR .θT , .φT .θR , .φR

Dynamic clusters a = 10 m 20 m ◦ ◦ .180 , 20 ◦ ◦ .10 , 20 ◦ ◦ .10 , 5 ◦ ◦ .10 , 5 32, 32 ◦ ◦ .60 , .45 ◦ ◦ .135 , .45

.Dc,n

Static clusters a = 20 m 150 m ◦ ◦ .180 , 20 ◦ ◦ .10 , 20 ◦ ◦ .30 , 10 ◦ ◦ .30 , 10

.Dc,m

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4 A 3D IS-GBSM for Massive MIMO V2V Channels 1

Low VTD, t=0.5 s Low VTD, t=1 s High VTD, t=0.5 s High VTD, t=1 s

qp

Normalized absolute TACF, |R |

0.9 0.8 0.7

Low VTD

0.6 0.5 0.4 High VTD 0.3 0.2 0.1 0 0

0.005

0.01

Time difference,

0.015

0.02

t (s)

Fig. 4.4 Normalized absolute TACFs of the proposed model with different VTDs at .t = 0.5 s and t = 50 m, .D t = 1 s (.D = 200 m, .vn = 10 m/s, .vT = vR = 10 m/s, .Dc,n c,m = 100 m, Low VTD: .λR,n = λR,m = 4/m, .λG,n = 20/m, .λG,m = 60/m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 120/m, .λG,m = 60/m) .t

1 0.9 qp

Normalized absolute TACF, |R |

Same direction 0.8 0.7 0.6 0.5 0.4

Any direction

0.3 0.2

Low VTD, Same direction Low VTD, Any direction High VTD, Same direction High VTD, Any direction

0.1 0 0

0.005

0.01

Time difference,

0.015

0.02

t (s)

Fig. 4.5 Normalized absolute TACFs of the proposed model with different VTDs for the different directions of movement of the Tx and Rx (.t = 0 s, .D = 200 m, .vn = 15 m/s, .vT = vR = 20 m/s, t t .Dc,n = 10 m, .Dc,m = 30 m, Low VTD: .λR,n = λR,m = 4/m, .λG,n = 20/m, .λG,m = 60/m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 80/m, .λG,m = 60/m)

4.3 Simulations and Discussions

83

0 Any direction, Any direction,

-10

R R

Normalized DPSD, |S| (dB)

Same direction, Same direction,

-20

=0 =4 R R

=0 =4

-30

-40

-50

-60

-70 -1000

-500

0

500

1000

Frequency, f (Hz)

Fig. 4.6 Normalized DPSDs of the proposed model for different moving directions of the Tx and Rx under low VTDs with antenna spacing .δT =.δR =0 and .δT =.δR =.4 λ (.t = 1 s, .D = 200 m, t t .vn = 10 m/s, .vT = vR = 10 m/s, .Dc,n = 50 m, .Dc,m = 100 m, Low VTD: .λR,n = λR,m = 4/m, .λG,n = 20/m, .λG,m = 60/m)

vehicle is moving in the same direction, the TACF is higher than that when the vehicle is moving in any direction. This is also related to the actual vehicular scenarios, i.e., when the vehicle is driving in any direction, the change of observable clusters in the environment is significantly obvious, leading to a decrease in the time correlation. Also, compared to low VTDs, the TACFs under high VTDs exhibit a lower correlation, which is consistent with the simulation results in Fig. 4.4. Normalized DPSDs under low VTDs for different vehicular movement directions with different antenna spacing are described in Fig. 4.6. It is clear that the distribution of DPSD is steeper when the vehicle is moving in the same direction, and the distribution of the DPSD is more uniform when the vehicle is moving in any direction. This phenomenon can be properly explained that, when the vehicle moves in any direction, the received power mainly comes from the clusters distributed in various directions, which results in a more dispersed distribution of the DPSDs. Moreover, it can be readily observed that the antenna spacing also has a significant impact on the simulated DPSDs. Figures 4.7 and 4.8 give the normalized absolute Rx SCCFs of the proposed model, i.e., .Rqp,q  p (δT = 0, δR , τ = 0). Figure 4.7 shows the SCCFs at the Rx side with the normalized Rx antenna spacing .δR /λ under different VTDs. For links of .T1 − R2 and .T1 − R3 as an example, the channel SCCFs gradually decrease as the normalized Rx antenna spacing .δR /λ increases. Furthermore, Fig. 4.7 demonstrates that VTDs have a significant impact on the channel spatial correlation. Compared

84

4 A 3D IS-GBSM for Massive MIMO V2V Channels

Normalized absolute SCCF at Rx side, |R

qp,q'p

|

1

High VTD Low VTD

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.4

0.8

1.2

1.6

Normalized antenna spacing, R/

Fig. 4.7 Normalized absolute Rx SCCFs of the proposed model for different VTDs (.t = 0 s, t t = 200 m, .vn = 10 m/s, .vT = vR = 10 m/s, .Dc,n = 50 m, .Dc,m = 100 m, Low VTD: .λR,n = λR,m = 4/m, .λG,n = 20/m, .λG,m = 60/m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 120/m, .λG,m = 60/m) .D

Normalized absolute SCCF at Rx side, |R qp,q'p|

1 0.9 0.8 0.7 0.6 0.5 0.4

The index of reference of antenna=1 The index of reference of antenna=3

0.3 0.2 0

1

2

3

4

5

6

7

Absolute difference between antenna indices, |q-q'|

Fig. 4.8 Normalized absolute Rx SCCFs of the proposed model under high VTDs for different t reference antennas (.t = 1 s, .D = 300 m, .vn = 25 m/s, .vT = vR = 30 m/s, .Dc,n = 10 m, t .Dc,m = 30 m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 32/m, .λG,m = 32/m)

4.3 Simulations and Discussions

85

with the low VTD, the spatial diversity in high VTD scenarios is larger, leading to a lower spatial correlation. In addition to exploring the changes in channel spatial correlation with the normalized Rx antenna spacing .δR /λ, it can also be seen that, for the fixed transmitting antenna and receiving antenna distance of two rays, the spatial correlation with different receiving reference antennas is also different. For example, the SCCFs between .T1 − R1 link and .T1 − R3 link are different from that between .T1 − R3 link and .T1 − R5 , i.e., receiving antenna distances of two links are the same. Figure 4.8 shows an image in which the Rx space correlation changes with the receiving antenna position. In other words, the SCCF of the proposed model depends not only on the antenna spacing, but also on the specific position of the antenna. This is because observable dynamic and static clusters are different for each antenna, which demonstrates that the proposed model is space non-stationary essentially. Figure 4.9 illustrates the cluster evolution on the Rx array axis for dynamic clusters under high VTDs at .t = 1 s and .t = 2 s. It can be seen that the observable dynamic clusters change at different times and the dynamic clusters observed by different antennas are different. This means that the proposed model has the ability to describe the non-stationarity in both the space domain and the time domain and provides further reference for the exploration of vehicular channel models. To show the cluster evolution on the time axis, an instance of dynamic and static cluster sets at different time instants is presented in Fig. 4.10. There are 15 observable clusters in the environment at the initial time instant .t0 . From Fig. 4.10a, the change of the number of dynamic clusters and static clusters over

20

20

16

16

14

14

12 10 8

12 10 8

6

6

4

4

2

2 10

t=2 s

18

t=1 s

Dynamic cluster index

Dynamic cluster index

18

20

Antenna index

30

10

20

30

Antenna index

Fig. 4.9 Cluster evolution in the proposed model along the array axis under high VTDs at .t = 1 s t = 50 m, .D t and .t = 2 s (.D = 200 m, .vn = 10 m/s, .vT = vR = 10 m/s, .Dc,n c,m = 100 m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 60/m, .λG,m = 60/m)

4 A 3D IS-GBSM for Massive MIMO V2V Channels 20

20

18

18

16

16

Number of static clusters

Number of dynamic clusters

86

14 12 10 8

14 12 10 8

6

6

4

4

2

2 0

1

2

3

0

4

1

2

3

4

3

4

Time, t (s)

Time, t (s)

(a) 25

25

Newly generated static clusters

Newly generated dynamic clusters 20

Cluster index

Cluster index

20

15

15

10

10

5

5

0

1

2

3

0

4

1

2

Time, t (s)

Time, t (s)

(b) Fig. 4.10 Cluster evolution in the proposed model along the time axis under high VTDs (.D = t = 10 m, .D t 300 m, .vn = 15 m/s, .vT = vR = 15 m/s, .Dc,n c,m = 30 m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 60/m, .λG,m = 60/m)

time can be readily observed. In Fig. 4.10b, the specific state of each cluster, i.e., observable/unobservable state, in the environment at each time instant can be seen. It can be observed from Fig. 4.10a and b that the appearance and disappearance of

4.3 Simulations and Discussions

87

clusters along the time axis are characterized sufficiently. Therefore, the channel time non-stationarity of the proposed model is captured. Furthermore, it can be seen from Fig. 4.10a that, after .t = 2 s, the number of dynamic and static clusters is greater than the initial number of clusters and gradually increases. This is reasonable because, based on Eqs. (4.36)–(4.37), longer observation time will result in more newly generated clusters. This phenomenon can be readily observed in Fig. 4.10b. As a result, the increase in the number of newly generated clusters leads to an increase in the total number of observable clusters in the environment.

4.3.3 Model Validation In order to validate the accuracy of the proposed model, the simulated channel statistical properties, such as DPSD, TACF, as well as SCCF, are compared with the available measurement data. Figure 4.11 compares normalized DPSDs of the proposed model with the measurement data in [45], which was collected on an interstate highway scenario

0 Proposed model, High VTD Measurement, Highway, High VTD Proposed model, Low VTD Measurement, Urban, Low VTD

Normalized DPSD, |S| (dB)

-10

-20

-30

-40

-50

-60 -200

-150

-100

-50

0

50

100

150

200

Frequency, f (Hz)

Fig. 4.11 Comparison of the normalized DPSDs of the proposed model and measurement campaign under high VTDs and low VTDs (.fc = 2.435 GHz [45], .δT = δR = 2.943 .λ [45], T R T R .θT = θR = φT = φR = 0 [45], .α = α = π/2 [45], .αn = π/2, .β = β = 0 [45], .βn = 0, t t .Dc,n = 30 m, .Dc,m = 50 m, Low VTD: .D = 300 m [45], .MT = MR = 4 [45], .vn = 11 m/s, .vT = vR = 11 m/s [45] .λR,n = λR,m = 4/m, .λG,n = 8/m, .λG,m = 24/m, High VTD: .D = 180 m [45], .MT = 2 [45], .MR = 4 [45], .vn = 22 m/s, .vT = vR = 22 m/s [45] .λR,n = λR,m = 4/m, .λG,n = 24/m, .λG,m = 12/m)

88

4 A 3D IS-GBSM for Massive MIMO V2V Channels

and an urban surface street scenario. Based on the description of measured scenarios in [45], it is reasonable to assume that the interstate highway scenario is a high VTD scenario and the urban surface street scenario is a low VTD scenario. Note that the corresponding model-related parameters are selected to match the aforementioned measurement condition. It can be readily seen that the simulated DPSD has an obvious peak, which is mainly caused by the dominant LoS component due to its high power. As the velocity vectors of Tx and Rx are set to the same, i.e., .vT = vR , the Doppler frequency of LoS component is 0 Hz, which can be calculated based on (4.3). This is reasonable because the Tx and Rx are relatively static and the communication distance between them remains unchanged. Meanwhile, the velocity vectors of clusters and transceivers are set to the same, i.e., .vT = vR = vn , which further causes the peak for DPSD to be located at the 0 Hz. Also, the simulation results match well with the measurement, which validates the accuracy of the proposed model. Additionally, comparing normalized DPSDs with different VTDs, the DPSD under low VTD scenarios exhibits a peakier distribution due to a dominate LoS path. In contrast, the DPSD with high VTD shows a more Jake’salike spectrum, i.e., a flatter distribution. This is because the received power under high VTDs tends to come from dynamic vehicles over all directions. Figure 4.12 shows the normalized absolute TACFs between the simulation results and the available measurement data in [46]. For a fair comparison, key model-related parameters are set based on the measurement, where the .4 × 4 MIMO vehicular 1

qp

Normalized absolute TACF, |R |

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Simulated TACF, low VTD Measured TACF, low VTD Simulated TACF, high VTD

0.1 0 0

0.005

0.01

0.015

Time difference,

0.02

0.025

t (s)

Fig. 4.12 Normalized absolute TACFs between the proposed model and measurement in [46] (.fc = 5.8 GHz [46], .MT = MR = 4 [46], .D = 400 m, .δT = δR = 2 .λ [46], .vT = 0 m/s t t [46], .vR = 16.667 m/s [46], .Dc,n = 30 m, .Dc,m = 50 m, Low VTD: .λR,n = λR,m = 4/m, .λG,n = 8/m, .λG,m = 12/m, High VTD: .λR,n = λR,m = 4/m, .λG,n = 32/m, .λG,m = 12/m)

4.3 Simulations and Discussions

89

1

Normalized absolute SCCF at Rx side, |R

qp,q'p

|

Proposed model Measurement data

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

5

6

7

The absolute difference between antenna indices, |q-q'|

Fig. 4.13 Normalized absolute SCCFs between the proposed model and measurement in [18] (.fc = 2.6 GHz [18], .MT = MR = 128, .D = 10 m, .δT = δR = 0.5 .λ [18], .vT = vR = 0 m/s t = 50 m, .D t [18], .vn = 0 m/s [18], .λR,n = λR,m = 4/m, .λG,n = λG,m = 48/m, .Dc,n c,m = 100 m)

channel measurement was performed at .5.8 GHz with a speed of 60 km/h [46]. According to the description of the measurement scenario, it can be reasonably assumed that this is a low VTD scenario. As shown in Fig. 4.12, the descent match between simulation results and measurement data confirms the utility of the proposed model. Furthermore, the simulated TACF under the high VTD is depicted. Compared with low VTDs, the fluctuations of TACF under the high VTD are more obvious. This is because the communication environment of the high VTDs changes faster than that of the low VTD. More importantly, the impact of VTDs on TACFs is in agreement with Figs. 4.4 and 4.5, i.e., the TACFs under high VTDs are lower. Figure 4.13 compares the simulated normalized absolute SCCFs with the measurement in [18]. To the best of our knowledge, there is currently no massive MIMO V2V channel measurement. Note that the measurement in [18] was conducted in a static massive MIMO scenario at .2.6 GHz with a 128-element linear array. It is obvious that the SCCFs decrease as the absolute difference between antenna indices  .|q − q | increase. Moreover, the simulated SCCF is in descent agreement with the measurement data, which verifies the utility of the proposed model. It should be mentioned here that the proposed model is a statistical model, and its purpose is to analyze the channel statistical characteristics. At the same time, only effective clusters in the environment are considered in the model, not all clusters. Therefore, the proposed model may not completely match the measured data of the actual scenario. However, the proposed massive MIMO V2V IS-GBSM can achieve a higher match between simulation results and the measured data.

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4 A 3D IS-GBSM for Massive MIMO V2V Channels

4.4 Summary For the B5G/6G massive MIMO V2V wideband channel, a 3D cluster-based channel model has been proposed. This is the first IS-GBSM for massive MIMO V2V wideband channels with the ability to model the impact of VTDs on channel statistical properties, and it can be regarded as a reference for the establishment of the B5G/6G V2V standardized channel model. In order to model the massive MIMO V2V channel space-time non-stationarity, a VTD-combined time-array cluster evolution algorithm to consider the influence of dynamic and static clusters has been developed, where their expressions of the BD processes have been distinguished. At the same time, the expressions of STF-CF and DPSD have been derived. Furthermore, it has been found out that different VTDs and vehicular movement directions have a significant impact on TACFs and the distribution of DPSDs. Additionally, the simulated TACF, SCCF, and observable cluster map have proved that the proposed model can effectively describe the non-stationarity of the massive MIMO V2V wideband channel in the space-time domain. It is worth mentioning that the aforementioned observations can provide some valuable suggestions for obtaining more stable channel characteristics. Finally, the descent agreement between simulation results and example measurement data has demonstrated the utility of the proposed model.

References 1. X. You et al., Towards 6G wireless communication networks: vision, enabling technologies, and new paradigm shifts. Sci. China Inf. Sci. 64(1), 1–74 (2021) 2. X. Liu, Y. Li, L. Xiao, J. Wang, Performance analysis and power control for multi-antenna V2V underlay massive MIMO. IEEE Trans. Wirel. Commun. 17(7), 4374–4387 (2018) 3. Z. Huang, X. Cheng, A general 3D space-time-frequency non-stationary model for 6G channels. IEEE Trans. Wirel. Commun. 20(1), 535–548 (2021) 4. F. Yang, S. Wang, J. Li, Z. Liu, Q. Sun, An overview of internet of vehicles. China Commun. 11(10), 1–15 (2014) 5. D. Kombate, Wanglina, The internet of vehicles based on 5G communications, in Proceedings of the IEEE iThings, GreenCom, CPSCom, and SmartData’16, Chengdu, China (2016), pp. 445–448 6. X. Cheng, R. Zhang, L. Yang, Wireless towards the era of intelligent vehicles. IEEE Int. Things J. 6(1), 188–202 (2019) 7. X. Cheng, Z. Huang, L. Bai, Channel nonstationarity and consistency for beyond 5G and 6G: a survey. IEEE Commun. Surveys Tutor. 24(3), 1634–1669 (2022) 8. X. Cheng, S. Gao, L. Yang, mmWave Massive MIMO Vehicular Communications (Springer Nature, Switzerland, 2022) 9. W. Viriyasitavat, M. Boban, H. Tsai, A. Vasilakos, Vehicular communications: Survey and challenges of channel and propagation models. IEEE Veh. Technol. Mag. 10(2), 55–66 (2015) 10. Z. Huang, L. Bai, M. Sun, X. Cheng, A 3D non-stationarity and consistency model for cooperative multi-vehicle channels. IEEE Trans. Veh. Technol. 72, 11095–11110 (2023). https://doi.org/10.1109/TVT.2023.3268664

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11. I. Sen, D.W. Matolak, Vehicle-vehicle channel models for the 5-GHz band. IEEE Trans. Intell. Transp. Syst. 9(2), 235–245 (2008) 12. Z. Huang et al., A mixed-bouncing based non-stationarity and consistency 6G V2V channel model with continuously arbitrary trajectory. IEEE Trans. Wirel. Commun., to be published (2023). https://doi.org/10.1109/TWC.2023.3293024 13. C.-X. Wang, X. Cheng, D.I. Laurenson, Vehicle-to-vehicle channel modeling and measurements: recent advances and future challenges. IEEE Commun. Mag. 47(11), 96–103 (2009) 14. R. He, B. Ai, G.L. Stüber, G. Wang, Z. Zhong, Geometrical-based modeling for millimeterwave MIMO mobile-to-mobile channels. IEEE Trans. Veh. Tech. 67(4), 2848–2863 (2018) 15. R. He et al., Propagation channels of 5G millimeter-wave vehicle-to-vehicle communications: recent advances and future challenges. IEEE Veh. Technol. Mag. 15(1), 16–26 (2020) 16. X. Gao, O. Edfors, F. Tufvesson, E.G. Larsson, Massive MIMO in real propagation environments: do all antennas contribute equally? IEEE Trans. Commun. 63(11), 3917–3928 (2015) 17. J.-S. Jiang, M.A. Ingram, Spherical-wave model for short-range MIMO. IEEE Trans. Commun. 53(9), 1534–1541 ( 2005) 18. S. Payami, F. Tufvesson, Channel measurements and analysis for very large array systems at 2.6 GHz, in 2012 6th European Conference on Antennas and Propagation (EUCAP), Prague, Czech Republic (2012), pp. 433–437 19. X. Cai, B. Peng, X. Yin, A. Yuste, Hough-transform-based cluster identification and modeling for V2V channels based on measurements. IEEE Trans. Veh. Technol. 67(5), 3838–3852 (2018) 20. J. Zhou, Z. Chen, H. Jiang, H. Kikuchi, Channel modelling for vehicle-to-vehicle MIMO communications in geometrical rectangular tunnel scenarios. IET Commun. 14(19), 3420– 3427 (2020) 21. J. Karedal et al., A geometry-based stochastic MIMO model for vehicle-to-vehicle communications. IEEE Trans. Wirel. Commun. 8(7), 3646–3657 (2009) 22. J. Maurer, T. Fugen, T. Schafer, W. Wiesbeck, A new inter-vehicle communications (IVC) channel model, in Proceedings of the IEEE VTC’04-Fall, LoS Angeles, CA, USA (2004), pp. 9–13 23. W. Wiesbeck, S. Knorzer, Characteristics of the mobile channel for high velocities, in 2007 International Conference on Electromagnetics in Advanced Applications, Torino, Italy (2007), pp. 116–120 24. L. Reichardt, T. Fugen, T. Zwick, Influence of antennas placement on car to car communications channel, in 2009 3rd European Conference on Antennas and Propagation, Berlin, Germany (2009), pp. 630–634 25. G. Acosta-Marum, M.A. Ingram, Six time- and frequency- selective empirical channel models for vehicular wireless LANs. IEEE Veh. Technol. Mag. 2(4), 4–11 (2007) 26. Z. Huang, X. Zhang, X. Cheng, Non-geometrical stochastic model for non-stationary wideband vehicular communication channels. IET Commun. 14(1), 54–62 (2020) 27. A.S. Akki, F. Haber, A statistical model of mobile-to-mobile land communication channel. IEEE Trans. Veh. Technol. 35(1), 2–7 (1986) 28. A.G. Zajic, G.L. Stuber, Time-space correlated mobile-to-mobile channels: modelling and simulation. IEEE Trans. Veh. Technol. 57(2), 715–726 (2008) 29. X. Cheng, C. Wang, D.I. Laurenson, S. Salous, A.V. Vasilakos, An adaptive geometry-based stochastic model for non-isotropic MIMO mobile-to-mobile channels. IEEE Trans. Wirel. Commun. 8(9), 4824–4835 (2009) 30. X. Cheng, C.X. Wang, B. Ai, H. Aggoune, Envelope level crossing rate and average fade duration of non-isotropic vehicle-to-vehicle Ricean fading channels. IEEE Trans. Intell. Transpor. Syst. 15(1), 62–72 (2014) 31. A.G. Zajic, G.L. Stuber, Three-dimensional modeling, simulation, and capacity analysis of space-time correlated mobile-to-mobile channels. IEEE Trans. Veh. Technol. 57(4), 2042– 2054 (2008) 32. Y. Li, X. Cheng, N. Zhang, Deterministic and stochastic simulators for non-isotropic V2VMIMO wideband channels. China Commun. 15(7), 18–29 (2018)

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33. A.G. Zajic, G.L. Stuber, Three-dimensional modeling and simulation of wideband MIMO mobile-to-mobile channels. IEEE Trans. Wirel. Commun. 8(3), 1260–1275 (2009) 34. A. Ghazal, Y. Yuan, C. Wang, Y. Zhang, Q. Yao, H. Zhou, W. Duan, A non-stationary IMTadvanced MIMO channel model for high-mobility wireless communication systems. IEEE Trans. Wirel. Commun. 16(4), 2057–2068 (2017) 35. S. Wu, C.-X Wang, e.M. Aggoune, M.M. Alwakeel, X. You, A general 3D non-stationary 5G wireless channel model. IEEE Trans. Commun. 66(7), 3065–3078 (2018) 36. M. Wang, N. Ma, J. Chen, B. Liu, A novel geometry-based MIMO channel model for vehicleto-vehicle communication systems, in 2019 IEEE 5th International Conference on Computer and Communications (ICCC), Chengdu, China (2019), pp. 762–767 37. H. Jiang, Z. Chen, J. Zhou, J. Dang, L. Wu, A general 3D non-stationary wideband twin-cluster channel model for 5G V2V tunnel communication environments. IEEE Access 7, 137744– 137751 (2019) 38. H. Jiang, W. Ying, J. Zhou, G. Shao, A 3D wideband two-cluster channel model for massive MIMO vehicle-to-vehicle communications in semi-ellipsoid environments. IEEE Access 8, 23594–23600 (2020) 39. IEEE P802.11p/D2.01, Standard for Wireless Local Area Networks Providing Wireless Communications While in Vehicular Environment (2007) 40. X. Cheng, Z. Huang, S. Chen, Vehicular communication channel measurement, modelling, and application for beyond 5G and 6G. IET Commun. 14(19), 3303–3311 (2020) 41. C.-X. Wang, J. Bian, J. Sun, W. Zhang, M. Zhang, A survey of 5G channel measurements and models. IEEE Commun. Surveys Tutor. 20(4), 3142–3168 (2018) 42. A. Abdi, C. Tepedelenlioglu, M. Kaveh, G. Giannakis, On the estimation of the K parameter for the Rice fading distribution. IEEE Commun. Lett. 5(3), 92–94 (2001) 43. Study on channel model for frequencies from 0.5 to 100 GHz, Version 16.1.0, Document 3GPP T.R. 38.901 (2019) 44. P. Kyostiet et al., WINNER II Channel Models, Version 1.1. (2007). [Online]. Available: http:// www.ist-winner.org/WINNER2-Deliverables/D1.1.2v1.1.pdf 45. A.G. Zajic, G.L. Stuber, T.G. Pratt, S.T. Nguyen, Wideband MIMO mobile-to-mobile channels: geometry-based statistical modeling with experimental verification. IEEE Trans. Veh. Technol. 58(2), 517–534 (2009) 46. A. Fayziyev, M. Pätzold, E. Masson, Y. Cocheril, T. Berbineau, A measurement-based channel model for vehicular communications in tunnels, in 2014 IEEE Wireless Communications and Networking Conference (WCNC), Istanbul, Turkey (2014), pp. 116–121

Chapter 5

A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive MIMO V2V Channels

In this chapter, a novel 3D mmWave massive MIMO IS-GBSM is proposed for 6G V2V channels. To support the high delay resolution in mmWave communications, rays within each cluster are resolvable, where rays also have different propagation delays. Furthermore, in the proposed IS-GBSM, clusters in the environment are properly divided into static clusters and dynamic clusters. The time-variant acceleration together with the integration of time during the transmission distance update are exploited. As a result, the continuously arbitrary trajectory of the transceiver and dynamic clusters is successfully captured. To jointly model space–time–frequency (S-T-F) non-stationarity of 6G V2V channels, a new method, which properly integrates the frequency-dependent factor, BD process, and selective evolution of static and dynamic clusters, is developed. Key channel statistics, including the space–time–frequency correlation function (STF-CF), time stationary interval, and DPSD, are obtained. Simulation results demonstrate that S-T-F non-stationarity is modeled and the impacts of VTD and vehicular movement trajectory (VMT) on channel statistics are analyzed thoroughly. Meanwhile, the effects of typical channel parameters in V2V communication scenarios on channel statistics are also explored. Finally, the generality and accuracy of the proposed IS-GBSM are validated through the comparison of simulation results and available measurement data.

5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM 5.1.1 Introduction and Contributions of Proposed IS-GBSM with Continuously Arbitrary Trajectory As we all know, the V2V communication has the benefits of minimizing traffic accidents, improving traffic efficiency, and enabling some new applications. In the future © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_5

93

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5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

6G era, the research on V2V communication systems will be increasingly essential and challenging. To support more potential applications such as autonomous and intelligent vehicles, the 6G V2V communication system should have a flexible design to fulfill the requirements of extremely low latency, high throughput, as well as high reliability. Aiming at fulfilling these requirements, mmWave communication and massive MIMO technology are expected to be simultaneously employed in 6G V2V communication systems. Clearly, mmWave communications with ultra-large bandwidth can support the transmission of massive amounts of data in real time [1]. Meanwhile, massive MIMO achieves an excellent SNR, mitigates multipath fading, as well as increases the channel capacity owing to the spatial multiplexing. It is worth mentioning that mmWave communication and massive MIMO technology exhibit an inherent symbiotic relationship [2, 3]. As the enabled foundation of any communication system design, an accurate and easy-to-use model, which has the capability to capture the underlying channel characteristics adequately, for 6G V2V mmWave massive MIMO channels is indispensable. As compared to mobile cellular communications, V2V communications are more dynamic and complicated [4–6]. There are two unique features in the V2V communication scenario, named as the VTD and VMT. For the former, the ratio of the number of moving vehicles, i.e., dynamic clusters, to roadside buildings and trees, i.e., static clusters, varies in diverse V2V communication scenarios [7]. In general, the scenario with fewer and more vehicles can be regarded as low and high VTD scenarios, respectively [8]. For the VMT, it is certain that vehicles have various movement trajectories, such as quarter turn, U-turn, and curve driving. According to the vehicular channel measurements in [9], [10], both the VTD and the VMT have distinct impacts on channel statistical properties and thus need to be modeled and analyzed in V2V communication channels. To describe unique V2V communication scenarios, extensive V2V channel models have been proposed. These models can be divided into the deterministic model [11], NGSM [12], and GBSM [13]. Note that, due to the consideration of the geometric propagation environment, the GBSM modeling method can be applied to a variety of complex scenarios, e.g., holographic MIMO (HMIMO) and reconfigurable intelligent surface (RIS)-based scenarios [14–16]. Generally, GBSMs are further divided into the RS-GBSM and the IS-GBSM, depending upon whether the effective clusters are placed on the regular shape, such as cylinder [17], ellipsoid [18], and sphere [19], or the irregular shape [20]. In RS-GBSMs, the cluster is assumed to be stochastically distributed according to a specific geometry with a regular shape [21]. By leveraging this assumption, the RS-GBSM modeling approach has low complexity. In such a condition, it is unnecessary to conduct a comprehensive and complicated measurement for developing a proper RS-GBSM. The RS-GBSM has the ability to adapt to multiple frequency bands by fine-tuning typical model-related parameters. However, the RS-GBSM modeling approach by simply placing the cluster on the regular geometry leads to its low accuracy. Different from the RS-GBSM modeling approach, in IS-GBSMs, the position of the cluster obeys a specific statistical distribution, which should be determined from the complicated channel measurement [22]. Through the comprehensive channel

5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM

95

measurement, the precision of the IS-GBSM modeling approach is higher than that of the RS-GBSM modeling approach. Nonetheless, the complexity of IS-GBSM modeling approach is higher than that of RS-GBSM modeling approach. Since the IS-GBSM modeling approach has a decent trade-off between complexity and accuracy, it is widely exploited in the existing models, including standardized models [23, 24]. Therefore, the proposed channel model exploits the IS-GBSM modeling approach. To integrate unique features of V2V communication scenarios into the modeling of S-T-F non-stationary characteristics, it is necessary to capture the frequent and complicated appearance and disappearance of static and dynamic clusters due to continuous and various VMTs of transceivers and dynamic clusters. Since the indepth integration of unique features and characteristics of V2V communication scenarios is significantly challenging, there is currently no V2V IS-GBSM that has the ability to mimic the S-T-F non-stationarity in consideration of VTDs and VMTs and efficiently support the design of 6G mmWave massive MIMO V2V communication systems. To fill the aforementioned gaps, a novel 3D mmWave massive MIMO V2V GBSM is proposed. The proposed IS-GBSM integrates the modeling of S-T-F non-stationarity and the capturing of VTDs and VMTs and thus can be considered as a 6G V2V channel model. Based on the proposed IS-GBSM, the understanding of the S-T-F non-stationarity can provide useful guidance for the design of 6G V2V systems. Meanwhile, the proposed IS-GBSM can be utilized as a simulation and validation platform to facilitate the development of efficient algorithms for 6G V2V systems. The major contributions and novelties of this paper are summarized as follows. 1. A novel IS-GBSM for 6G V2V mmWave massive MIMO channels is proposed. The proposed IS-GBSM is the first V2V channel model that has the capability to properly integrate the modeling of VTD, VMT, and S-T-F non-stationarity. Specifically, clusters are divided into static and dynamic clusters, and continuously arbitrary trajectories of transceivers and dynamic clusters are captured. The effects of VTD and continuously arbitrary VMTs on the S-T-F non-stationary modeling of 6G V2V channels are adequately considered. 2. The proposed IS-GBSM has the ability to imitate continuously arbitrary transceivers and dynamic clusters, which is different from GBSMs in [13, 25, 26] that simply assumed uniform rectilinear motion and/or discontinuous trajectories. To capture continuous trajectories, the integration of time during the transmission distance update is exploited. Meanwhile, unlike the IS-GBSM in [27], time-varying accelerations of both transceivers and dynamic clusters are considered to model arbitrary trajectories, including quarter turn, U-turn, and curve driving. 3. A new method, named as the selective cluster evolution based S-T-F nonstationary modeling method, is developed. The impacts of VTD and continuously arbitrary VMT on the S-T-F non-stationary modeling are simultaneously considered for the first time. Specifically, both the selective evolution of static/dynamic clusters and the BD process are exploited to model space-time non-stationarity.

96

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

For the frequency non-stationary modeling, a frequency-related factor is introduced to capture the frequency-dependent path gain. 4. According to the proposed IS-GBSM, important channel statistics are derived. Simulation results show that the proposed IS-GBSM can mimic the S-T-F nonstationarity. The influences of VTD and VMT on channel statistics are explored. To adequately verify the utility of the proposed IS-GBSM, two vehicular channel measurement campaigns were carried out. A close agreement between simulation results and measurement data is achieved.

5.1.2 Channel Impulse Response of Proposed Channel Model In the V2V communication system, the Tx/Rx is equipped with a uniform linear antenna (ULA) array, which is composed of .MT /.MR antennas with the antenna spacing .δT /.δR . The brief geometric representation of the proposed IS-GBSM between the p-th Tx antenna and q-th Rx antenna at the time instant t is presented in Fig. 5.1. The p-th/q-th antenna in the Tx/Rx array is .ATp /.AR q and its corresponding position vector is .ATp (t)/.AR (t). The velocity vectors of Tx and Rx are .vT (t) and q .vR (t). The distance vector between the Tx center and the Rx center is .Dcen (t). In the proposed IS-GBSM, the scattering objects between the Tx and Rx sides in the propagation environment are represented as effective twin-clusters. The twinclusters are composed of the cluster nearby the Tx side and the cluster nearby the Rx side [28]. The propagation between twin-clusters nearby Tx and Rx is abstracted

Fig. 5.1 The proposed 3D twin-cluster model for 6G V2V mmWave massive MIMO channels

5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM

97

by a virtual link, where other clusters may exist and introduce second-order and beyond reflections/interactions. By setting the delay of the virtual link to zero, the twin-clusters overlap, where the first-order reflection is captured [29]. Considering the unique feature of V2V communication scenarios, i.e., VTD, the clusters are separated as static clusters and dynamic clusters, which can represent the static buildings/vegetation on the roadside and moving vehicles on the road, respectively. The numbers of static and dynamic clusters in the environment are given as .S(t) and .D(t). The s-th static cluster and d-th dynamic cluster nearby the Tx/Rx side dyn,T/R sta,T/R and .Cd . The dynamic clusters nearby Tx and Rx are presented as .Cs dyn,T dyn,R move with the velocity vectors .vd (t) and .vd (t) at the time instant t. Here, a parameter is further introduced to represent the ratio of the number of dynamic clusters to static clusters, i.e., .ζ (t) = D(t)/S(t). Therefore, the VTD is time-variant. To support the high delay resolution in mmWave channels, rays within clusters can be resolvable and there are .Ns (t) and .Nd (t) rays within the s-th static cluster and the d-th dynamic cluster. Note that there may be different numbers of rays within different static/dynamic clusters. The position vectors of the .ns -th ray within the sth static cluster and the .nd -th ray within the d-th dynamic cluster nearby the Tx/Rx dyn,T/R sta,T/R side are .Ss,ns and .Sd,nd (t) as shown in Fig. 5.2a, b, respectively. 5.1.2.1

Calculation of Transmission with Continuously Arbitrary Trajectory

To capture the spherical wavefront in massive MIMO channels, distance parameters of each transmission via the ray within cluster are calculated in the proposed model. Specifically, the distance vector from the antenna .ATp /.AR q to the dynamic cluster dyn,T/R

Cd

.

dyn,T/R

via the .nd -th ray at time t, .Dp/q,d,nd (t), is given as dyn,T/R

dyn,T/R

Dp/q,d,nd (t) = Dd,nd

.

dyn,T/R

where .Dd,nd

T/R

(t) − Ap/q (t),

(5.1)

(t) is the distance vector from the Tx/Rx center to the dynamic

dyn,T/R .C d

cluster via the .nd -th ray. As shown in Fig. 5.2b, based on the geometric dyn,T/R relationship, the distance vector .Dd,nd (t) is properly calculated as dyn,T/R

Dd,nd .

dyn,T/R

(t) = Dd,nd

dyn,T/R

(0) + rd (t) − rT/R (t)  t  t dyn,T/R dyn,T/R vd (t)dt − vT/R (t)dt, = Dd,nd (0) + 0

dyn,T/R

(5.2)

0

where .rd (t) and .rT/R (t) denote the movement distance vectors of dynamic clusters and transceivers, respectively, which are shown in Fig. 5.2. Furthermore, dyn,T/R the distance vector at the initial time instant, .Dd,nd (0), is generated by

98

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

Fig. 5.2 Geometric relationships related to the antennas .ATp and .AR q and (a) the .ns -th ray within the s-th static cluster (b) the .nd -th ray within the d-th dynamic cluster

⎡ .

dyn,T/R

Dd,nd

dyn,T/R

(0) = Dd,nd

dyn,T/R

where .αd,nd

dyn,T/R

cosαd,nd

dyn,T/R

(0) cosβd,nd

(0)

⎤

⎥ ⎢ dyn,T/R (0) ⎣ sinαdyn,T/R (0) cosβd,nd (0) ⎦ , d,nd dyn,T/R sinβd,nd (0)

dyn,T/R

(0) and .βd,nd

(5.3)

(0) denote the azimuth and elevation angles of the

dyn,T/R cluster .Cd

via the .nd -th ray at the initial time instant, respectively. dynamic sta,T/R The distance vector from the antenna .ATp /.AR via the q to the static cluster .Cs sta,T/R

ns -th ray at time t, .Dp/q,d,nd (t), is computed as

.

sta,T/R

sta,T/R

Dp/q,s,ns (t) = Ds,ns

.

sta,T/R

T/R

(t) − Ap/q (t),

(5.4)

where .Ds,ns (t) represents the distance vector from the Tx/Rx center to the static sta,T/R cluster .Cs via the .ns -th ray. Based on the geometric relationship between the continuous movement distances of the Tx/Rx and the distance vector at the initial

5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM sta,T/R

time instant, the distance vector .Ds,ns sta,T/R

Ds,ns

99

(t) is properly expressed by

sta,T/R

(t) = Ds,ns

(0) − rT/R (t)  t sta,T/R = Ds,ns (0) − vT/R (t)dt,

.

(5.5)

0

sta,T/R

where the distance vector at the initial time .Ds,ns

(0) is generated by

⎡

⎤ sta,T/R sta,T/R cosαs,ns (0) cosβs,ns (0) ⎢ ⎥ sta,T/R sta,T/R sta,T/R .Ds,ns (0) = Ds,ns (0) ⎣ sinαsta,T/R (0) cosβs,ns (0) ⎦ , s,ns sta,T/R sinβs,ns (0) sta,T/R

(5.6)

sta,T/R

where .αs,ns (0) and .βs,ns (0) denote the azimuth and elevation angles of the sta,T/R static cluster .Cs via the .ns -th ray at the initial time instant, respectively. dyn,T/R dyn,T/R sta,T/R sta,T/R , .Dp/q,d , .Ds , and .Dp/q,s related Similarly, the distance vectors .Dd to dynamic and static clusters can be computed. According to the geometry relationship, the transmission path related to each antenna and each ray within each cluster is calculated. As a consequence, the spherical wavefront propagation in massive MIMO channels is captured. To mimic the arbitrary trajectory of transceivers and dynamic clusters, the dyn,T/R (t) = accelerations of them .aT/R (t) = [aT/R,x (t), aT/R,y (t), 0]T and .ad dyn,T/R dyn,T/R T [ad,x (t), ad,y (t), 0] are time-varying. By characterizing the arbitrary trajectory of them, the unique feature of V2V communication scenarios, i.e., various VMTs, including quarter turn, U-turn, and curve driving, can be modeled. In addition, the velocity vector of Tx/Rx can be presented as .vT/R (t) = [vT/R,x (t), vT/R,y (t), 0]T , where 

t

vT/R,x (t) = vT/R,x (0) +

aT/R,x (t)dt

(5.7)

aT/R,y (t)dt.

(5.8)

.

0



t

vT/R,y (t) = vT/R,y (0) +

.

0

Similarly, the velocity vector of the d-th dynamic cluster can be presented as dyn,T/R dyn,T/R dyn,T/R vd (t) = [vd,x (t), vd,y (t), 0]T , where

.

dyn,T/R .v d,x/y (t)

=

dyn,T/R vd,x/y (0) +



t 0

dyn,T/R

ad,x/y

(t)dt.

(5.9)

Based on Fig. 5.2a, b, the LoS transmission distance vector from the Tx antenna .ATp LoS to the Rx antenna .AR q at the time instant t, .Dqp (t), can be expressed by

100

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . . LoS R T DLoS qp (t) = Dqp (0) + r (t) − r (t)  t  t . LoS vR (t)dt − vT (t)dt. = Dqp (0) + 0

5.1.2.2

(5.10)

0

Complex Channel Gain of LoS Component

For the LoS component, the complex channel gain of the sub-channel between antennas .ATp and .AR q is given as hLoS qp (t) =

.

  t LoS LoS (t)exp j 2π fqp (t)dt + j ϕqp (t) ,

(5.11)

0

T0

where .T0 is the observation time interval and function [30], which is expressed as



.

T0 (t)

is a rectangular window

1, 0  t  T0 , . (t) = 0, otherwise.

(5.12)

T0

LoS (t), phase shift .ϕ LoS (t), and delay .τ LoS (t) of the LoS The Doppler frequency .fqp qp qp component can be computed by

  (t), vR (t) − vT (t) DLoS qp 1 LoS   .fqp (t) =  LoS  λ D (t)

(5.13)

qp

 2π   LoS  Dqp (t) λ    LoS  Dqp (t) LoS , .τqp (t) = c

LoS ϕqp (t) = ϕ0 +

.

(5.14)

(5.15)

where .·, · and .· are inner product and Frobenius norm. .ϕ0 is the initial phase shift, .λ is the wavelength corresponding to the center frequency .fc , and c is the speed of light.

5.1.2.3

Complex Channel Gain of NLoS Component Resulting from Dynamic Clusters

For the non-LoS (NLoS) component resulting from dynamic clusters, the complex channel gain of the sub-channel between the antennas .ATp and .AR q through the

5.1 Framework of mmWave Massive MIMO Vehicular IS-GBSM dyn,T

dynamic twin-clusters .Cd

dyn,R

and .Cd

101

via the .nd -th ray is given as

dyn

hqp,d,nd (t) =

 t    t  . dyn dyn,T dyn,R dyn fp,d,nd (t)dt fq,d,nd (t)dt + j ϕqp,d,nd (t) , (t) Pd,nd (t)exp j 2π 0

T0

0

(5.16) dyn .P d,nd (t)

where represents the normalized power of the .nd -th ray within the d-th dynamic cluster. The complex channel gain related to dynamic clusters can be given by (5.16) only if the antennas satisfy .ATp ∈ AC dyn,T (t) and .AR q ∈ AC dyn,R (t), where d

d

AC dyn,T (t) and .AC dyn,R (t) are the Tx and Rx antenna sets that can observe the d-

.

d

d

dyn,T

dyn,R

and .Cd . Otherwise, the complex channel gain is 0. th dynamic clusters .Cd Therefore, only observable dynamic clusters to the antenna can contribute to the complex channel gain. Then, the Doppler frequencies related to dynamic clusters are written by 

dyn,T

dyn,T

1 Dp,d,nd (t), vT (t) − vd dyn,T   .f (t) = p,d,nd  dyn,T  λ Dp,d,nd (t) 

dyn,R

dyn,R

1 Dq,d,nd (t), vR (t) − vd dyn,R   .f (t) = q,d,nd  dyn,R  λ Dq,d,nd (t)

 (t) (5.17) 

(t) (5.18)

.

dyn

The delay .τqp,d,nd (t) at the time instant t can be given by dyn

τqp,d,nd (t) =

.

     dyn,T   dyn,R  Dp,d,nd (t) + Dq,d,nd (t) c

+ τ˜d (t),

(5.19)

where .τ˜d (t) denotes the abstracted delay of the virtual link between the dynamic dyn,T dyn,R and .Cd and can be assumed to follow the exponential twin-clusters .Cd distribution [31]. The phase shift of the component from the Tx antenna .ATp to dyn,T

the Rx antenna .AR q via the .nd -th ray within the dynamic twin-clusters .Cd dyn,R dyn .C , .ϕqp,d,nd (t), d dyn

is derived as

ϕqp,d,nd (t) = ϕ0 +

.

and

    2π   dyn,T   dyn,R  Dp,d,nd (t) + Dq,d,nd (t) + cτ˜d (t) . λ

(5.20)

102

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

5.1.2.4

Complex Channel Gain of NLoS Component Resulting from Static Clusters

For the NLoS component resulting from static clusters, the complex channel gain of the sub-channel between the antennas .ATp and .AR q through the static twin-clusters Cssta,T and .Cssta,R via the .ns -th ray is given as

.

hsta qp,s,ns (t) =

 t    t  . sta,T sta,R sta sta (t) Ps,ns (t)exp j 2π fp,s,ns (t)dt + fq,s,ns (t)dt + j ϕqp,s,ns (t) , 0

T0

0

(5.21) sta (t) where .Ps,n s

is the power of the .ns -th ray within the s-th static cluster. Also, the complex channel gain related to static clusters can be given by (5.21) only if the antennas satisfy .ATp ∈ AC sta,T (t) and .AR q ∈ ACssta,R (t), where .ACssta,T (t)/.ACssta,R (t) is s the Tx/Rx antenna set that can observe the s-th static clusters at the Tx and Rx sides sta,T .Cs and .Cssta,R . The Doppler frequencies related to static clusters are written as   Dsta,T p,s,ns (t), vT (t) 1 sta,T   .fp,s,n (t) = s  sta,T λ  Dp,s,ns (t)   Dsta,R q,s,ns (t), vR (t) 1 sta,R   . .fq,s,n (t) = s  sta,R λ  Dq,s,ns (t)

(5.22)

(5.23)

sta The delay .τqp,s,n (t) related to static twin-clusters at the time instant t can be s similarly given by

sta τqp,s,n (t) = s

.

     sta,T   sta,R  Dp,s,ns (t) + Dq,s,ns (t) c

+ τ˜s (t),

(5.24)

where .τ˜s (t) denotes the abstracted delay of the virtual link between the static twinclusters .Cssta,T and .Cssta,R and follows the exponential distribution. Also, the phase shift of the component from the Tx antenna .ATp to the Rx antenna .AR q via the .ns -th sta ray within the static twin-clusters .Cssta,T and .Cssta,R , .ϕqp,s,n (t), can be derived by s sta ϕqp,s,n (t) = ϕ0 + s

.

    2π   sta,T    Dp,s,ns (t) + Dsta,R q,s,ns (t) + cτ˜s (t) . λ

(5.25)

In summary, the CIR of sub-channel between the antennas .ATp and .AR q , .hqp (t, τ ), consists of three parts, i.e., the complex channel gain of LoS component .hLoS qp (t),

5.2 Space–Time–Frequency Non-stationary Modeling with Continuously. . .

103

the complex channel gain of NLoS component resulting from dynamic clusters dyn hqp,d,nd (t), and the complex channel gain of NLoS component resulting from static clusters .hsta qp,s,ns (t). Consequently, the CIR .hqp (t, τ ) is written as

.

 hqp (t, τ ) =

.

  K(t) LoS hLoS qp (t)δ τ − τqp (t) K(t) + 1  D(t) d (t)    N ηD (t) dyn dyn hqp,d,nd (t)δ τ − τqp,d,nd (t) + K(t) + 1 d=1 nd =1

+

S(t) N s (t)   s=1 ns =1



(5.26)

  ηS (t) sta sta hqp,s,ns (t)δ τ − τqp,s,n (t) , s K(t) + 1

where .K(t) is the Ricean factor. Then, .ηD (t) and .ηS (t) denote the time-variant power ratio of dynamic and static clusters and they satisfy .ηD (t) + ηS (t) = 1. Overall, the CIR of the proposed model for 6G mmWave massive MIMO V2V channels with delay .τ at time t can be characterized by an .MR × MT complex matrix, which is given by ⎡

⎤ h1,2 (t, τ ) · · · h1,MT (t, τ ) ⎢ h2,2 (t, τ ) · · · h2,MT (t, τ ) ⎥ ⎢ ⎥ .H(t, τ ) = ⎢ ⎥. .. .. .. ⎣ ⎦ . . . hMR ,1 (t, τ ) hMR ,2 (t, τ ) · · · hMR ,MT (t, τ ) h1,1 (t, τ ) h2,1 (t, τ ) .. .

(5.27)

5.2 Space–Time–Frequency Non-stationary Modeling with Continuously Arbitrary Trajectory 5.2.1 Selective Cluster Evolution Based Space–Time–Frequency Non-stationary Modeling Method As previously mentioned, there are two unique features in V2V communication scenarios, i.e., time-variant VTDs and various VMTs. To adequately model them, clusters are divided into static clusters and dynamic clusters, and the continuously arbitrary trajectories of transceivers and dynamic clusters are further captured. In the upcoming 6G era, the application of massive MIMO technology and mmWave technologies in high-mobility V2V communication scenarios will result in a typical channel characteristic, i.e., S-T-F non-stationarity. To properly consider the impact of VTD and VMT on S-T-F non-stationary modeling, a new method, named as the selective cluster evolution based S-T-F non-stationary modeling method, is devel-

104

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

oped. Here, the selective cluster evolution, array-time evolution of static/dynamic clusters, and modeling of frequency-dependent path gain are presented in order. Finally, the flowchart of the developed method is given.

5.2.1.1

Selective Cluster Evolution

As the number of antennas increases, the massive MIMO antenna array spans long distances. The large-scale antenna array results in new propagation effects, i.e., nearfield effect and cluster array evolution. The near-field effect is attributed to clusters located within the Fresnel region of the antenna array, which can be delimited by the Rayleigh distance. When the distance between the transceiver and clusters is smaller than the Rayleigh distance, the effect of spherical wavefront propagation is distinct [32]. In this case, massive MIMO channels are regarded as space non-stationary. As indicated in measurement [33], the appearance and disappearance of clusters are observed on the antenna array axis, where clusters are solely observable to a part of antennas in the array. This also leads to channel space non-stationarity. However, with the application of mmWave technology, the scale of massive MIMO antenna arrays will decrease. This further results in the decrease of Rayleigh distance [34]. For twin-clusters that are far away from the Tx/Rx, the distance between them and Tx/Rx is much larger than the Rayleigh distance. In this case, as indicated in [35], these clusters are observable to all antennas in the Tx/Rx array, named as the array stationary clusters in this paper. To model the aforementioned phenomenon, a concept of CEA is put forward. The CEA is exploited to determine the scope of cluster array evolution. When a cluster lies within the CEA, it needs to perform the cluster array evolution. Otherwise, it is observable to all antennas in the array, i.e., an array stationary cluster. Considering the impact of the near-field effect and the movement of vehicles on the ground, the CEAs at Tx and Rx are defined as semi-spheres centered at the center of the Tx and Rx arrays with the radii of .ΥT and .ΥR , respectively. Based on the calculation of Rayleigh distance, the radii .ΥT and .ΥR are given as .

ΥT =

2(MT − 1)2 δT2 2DT2 = λ λ

(5.28)

ΥR =

2(MR − 1)2 δR2 2DR2 . = λ λ

(5.29)

.

Since the determination of CEA is solely dependable on the antenna array, it can be reasonably assumed that static and dynamic clusters share the same CEA. In this case, the condition that the .s/d-th static/dynamic cluster nearby Tx/Rx is an array stationary static/dynamic cluster nearby Tx and Rx satisfies sta,T/R

Ds

.

(t) ≥ ΥT/R

(5.30)

5.2 Space–Time–Frequency Non-stationary Modeling with Continuously. . . dyn,T/R

Dd

.

(t) ≥ ΥT/R ,

105

(5.31)

dyn,T/R

sta,T/R

where .Ds (t) and .Dd (t) represent the distances from the Tx/Rx center dyn,T/R sta,T/R to the static cluster .Cs and the dynamic cluster .Cd , respectively. It dyn,T/R sta,T/R is noteworthy that distance parameters .Ds (t) and .Dd (t) are properly computed as sta,T/R

Ds

.

   t  sta,T/R   D (t) =  (0) − v (t)dt T/R  s 

(5.32)

0

dyn,T/R .D (t) d sta,T/R

   t  t  dyn,T/R  dyn,T/R  = Dd (0) + vd (t)dt − vT/R (t)dt  , 0

(5.33)

0

dyn,T/R

where .Ds (0) and .Dd (0) are the initial distance vectors from the Tx/Rx dyn,T/R sta,T/R center to the static cluster .Cs and dynamic cluster .Cd , respectively. In the selective cluster evolution, the impact of near-field effect and continuously arbitrary trajectories of transceivers and dynamic clusters on the determination of array stationary clusters is considered.

5.2.1.2

Array-Time Evolution of Static Clusters

In massive MIMO channels, clusters within the CEA need to perform cluster evolution on the array axis, i.e., cluster array evolution. In this case, each antenna has its own observable set of clusters. The high velocity of the transceiver further causes the rapid transition between the birth state and death state of clusters over short periods. Therefore, in 6G V2V channels, the appearance and disappearance of clusters on the array and time axes, i.e., cluster array-time evolution, distinctly exist. This is also the physical mechanism underlying channel space-time non-stationarity. Here, the array-time evolution of static clusters is characterized. It is certain that the BD process method is a special case of continuous Markov process, where the state transitions include two types, i.e., birth state together with death state. Attributed to its decent trade-off between complexity and accuracy, the BD process method is employed to model the cluster array-time evolution. The birth/death state of BD process method in the cluster array and time evolution determines the states where clusters are observable/unobservable to this antenna and this time instant, respectively. Take the s-th static cluster nearby Tx, .Cssta,T , as an example for analysis and the array-time evolution of other static clusters is similar to it. Assume that the static cluster .Cssta,T lies within the Tx CEA at the initial time. At the beginning, randomly select a Tx antenna .ATp (.1  p  MT ) and assume that the static cluster .Cssta,T is observable to the antenna .ATp at the initial time. Subsequently, the observable probability that the static cluster .Cssta,T is still

106

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

observable to the adjacent antennas of the antenna .ATp , i.e., .ATp−1 and .ATp+1 , is written as [32] sta,T .P observable

  sta δT = exp −λR a , Dc

(5.34)

a where .λsta R (per meter) is the recombination rate of static clusters and .Dc denotes the scenario-dependent correlation factor on the array axis. Each static cluster has the same observable probability during the cluster array evolution. Based on the probability in (5.34), the static cluster .Cssta,T evolves from the antenna .ATp−1 to the antenna .AT1 and from the antenna .ATp+1 to the antenna .ATMT at the same time. Note

that, when .p = 1 or .p = MT , the static cluster .Cssta,T solely needs one-sided array evolution, i.e., from .ATp+1 to .ATMT or from .ATp−1 to .AT1 . Therefore, the Tx antenna

set that can observe the static cluster .Cssta,T , .AC sta,T (t), is obtained. s After the cluster array evolution and before entering the next time instant, the static cluster .Cssta,T experiences the cluster time evolution. Based on the statement in [26] and further considering the continuously arbitrary trajectories of the transceiver, the survival probability that the static cluster .Cssta,T is survival at the time instant t and still survival at the next time instant .t + t is given by

sta  t+t  λ sta,T vT (t)dt , Psurvival (t, t) = exp − Rt Dc t

.

(5.35)

where .Dct denotes the scenario-dependent correlation factor on the time axis. Each static cluster has the same survival probability during the cluster time evolution. Additionally, the survival probability is time-variant and a higher velocity of Tx results in a lower survival probability of static clusters [36]. Note that, once the static cluster .Cssta,T is not surviving, it is unobservable to all antennas in the Tx array and is eliminated. By substituting .T by .R in (5.34) and (5.35), the observable and survival probabilities of the static cluster nearby Rx .Cssta,R are given as   sta δR = exp −λR a . Dc 

sta  t+t λ sta,R vR (t)dt . (t, t) = exp − Rt Psurvival Dc t sta,R .P observable

(5.36) (5.37)

Based on the aforementioned steps, the antenna set .AC sta,T/R (t), which can s observe static clusters, are essentially time-variant. Meanwhile, due to various VMTs of the transceiver, there is the phenomenon of the static cluster .Cssta,T entering and leaving the CEA over time. This also leads to the time-variant antenna set .A sta,T/R (t). Therefore, the cluster array evolution and cluster time evolution affect Cs each other and space-time non-stationarity is jointly modeled. For clarity, if the

5.2 Space–Time–Frequency Non-stationary Modeling with Continuously. . .

107

sta,T/R

static cluster .Cs is observable to a certain Tx/Rx antenna and further survives at time t, it is termed as an effective static cluster nearby Tx. Furthermore, effective static clusters nearby Tx and Rx are randomly paired. Only the effective clusters nearby Tx and Rx that are successfully paired can form effective static twin-clusters sta,T .Cs and .Cssta,R , which can contribute to the complex channel gain of NLoS component via static clusters. Attributed to the high-mobility characteristics of V2V communication scenarios, new clusters appear randomly in the propagation environment. The number of newly generated static clusters nearby Tx/Rx is modeled to obey the Poisson distribution with the mean   λsta   sta,T/R sta,T/R 1 − P E Nnew (t + Δt) = G (t, Δt) , survival λsta R

.

(5.38)

where .λsta G (per meter) represents the generation rate of static clusters. Mean λsta

numbers of static clusters nearby Tx and Rx are equal to . λGsta . It is worth mentioning R that, for generality, the determination of parameters and array-time evolution of newly generated static clusters are the same as those of other static clusters.

5.2.1.3

Array-Time Evolution of Dynamic Clusters

In addition to static clusters, dynamic clusters also experience the array-time dyn,T , as an example and evolution. Take the d-th dynamic cluster nearby Tx, .Cd assume that it is located within the Tx CEA at the initial time. Note that, the array evolution of dynamic clusters is analogous to that of static clusters and the observable probability of dynamic clusters can be properly given as   dyn,T dyn δT , Pobservable = exp −λR Dca

.

dyn

(5.39)

where .λR (per meter) is the recombination rate of dynamic clusters. After the array dyn,T evolution, the Tx antenna set that can observe the dynamic cluster .Cd , .AC dyn,T (t), d is determined. dyn,T For the dynamic cluster .Cd , its movement leads to its complex and rapidlychanging appearance and disappearance during the cluster time evolution. Considering the continuously arbitrary trajectories of the transceiver and dynamic clusters dyn,T simultaneously, the survival probability of the dynamic cluster .Cd is expressed as   dyn   t+t  λR dyn,T dyn,T vT (t) − vd (t) dt . (5.40) .P d,survival (t, t) = exp − Dct t

108

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

The time-variant velocities of the transceiver and dynamic clusters result in timevariant survival probabilities. Furthermore, the multi-velocity of dynamic clusters is considered. Unlike static clusters, dynamic clusters with different velocities have different survival probabilities. When the difference between velocity vectors of dyn,T the dynamic cluster .Cd and Tx is smaller, its survival probability is larger. Analogously, the observable and survival probabilities of the dynamic cluster nearby dyn,R Rx .Cd are given as   dyn,R dyn δR Pobservable = exp −λR Dca

(5.41)

.



dyn,R .P d,survival (t, t)

dyn

λ = exp − R t Dc



t+t

t

   dyn,R vR (t) − vd (t) dt .

(5.42)

Based on (5.39)–(5.42), the effective dynamic clusters nearby Tx and Rx can be obtained, respectively. The effective dynamic twin-clusters are further determined by randomly paring, which contribute to the complex channel gain of NLoS component via dynamic clusters. The number of newly generated dynamic clusters nearby Tx and Rx is based on the Poisson distribution with the mean   λdyn   dyn,T/R dyn,T/R G 1 − P¯survival (t, t) , E Nnew (t + t) = dyn λT

.

(5.43)

dyn

where .λG (per meter) represents the generation rate of dynamic clusters. dyn,T/R .P¯ survival (t, t) denotes the mean survival probability of dynamic clusters nearby Tx/Rx and is computed as D(t) dyn,T/R .P¯ survival (t, t)

5.2.1.4

=

d=1

dyn,T/R

Pd,survival (t, t) D(t)

.

(5.44)

Frequency-Dependent Path Gain

When the mmWave technology with ultra-large communication bandwidth is utilized to 6G V2V scenarios, different frequency components exhibit different transmission coefficients. Meanwhile, 6G V2V mmWave channels with ultra-large communication bandwidth need to be described by a correlated attenuation and phase shift of components with different delays [37]. Therefore, the uncorrelated scattering (US) assumption that is valid for sub-6 GHz no longer holds in mmWave. This results in frequency non-stationarity, where channel statistics vary on the frequency domain. An important channel statistical property that characterizes channels in the frequency domain is the time-varying transfer function (TVTF).

5.2 Space–Time–Frequency Non-stationary Modeling with Continuously. . .

109

Note that the TVTF can be obtained by using the Fourier transform to CIR .hqp (t, τ ) in respect of delay .τ and is given as  Hqp (t, f ) =



∞

.

−∞

hqp (t, τ )exp (−j 2πf τ ) dτ



  K(t) LoS (t)exp −j 2πf τ (t) hLoS qp K(t) + 1 qp  D(t) Nd (t)   ηD (t)   dyn dyn + hqp,d,nd (t)exp −j 2πf τqp,d,nd (t) K(t) + 1

=

 +

d=1 nd =1

S(t) Ns (t)   ηS (t)   sta hsta qp,s,ns (t)exp −j 2πf τqp,s,ns (t) . K(t) + 1

(5.45)

s=1 ns =1

As indicated in [38], one efficient way for modeling frequency non-stationary channels is to capture the frequency-dependent path gain. Toward this end, a frequency-dependent factor .( ffc )γ is introduced and applied to the NLoS component  (t, f ) in (5.45) is rewritten by of TVTF. Therefore, the TVTF .Hqp Hqp (t, f )    K(t) LoS hLoS = qp (t)exp −j 2πf τqp (t) K(t) + 1   γ D(t) d (t)    N . f ηD (t) dyn dyn + hqp,d,nd (t)exp −j 2πf τqp,d,nd (t) K(t) + 1 fc d=1 nd =1

 +

(5.46)

ηS (t) K(t) + 1



f fc

γ  S(t) N s (t) 

  sta hsta (t)exp −j 2πf τ (t) , qp,s,ns qp,s,ns

s=1 ns =1

where .γ is a frequency-dependent parameter, which depends on the V2V scenario [39]. Through the introduction of the frequency-dependent factor, the frequencydependent path gain is captured and the channel statistics are further modeled as frequency-variant. Therefore, frequency non-stationarity of channels is adequately mimicked via a low-complexity way. For clarity, the flowchart of the developed selective cluster evolution based ST-F non-stationary modeling method is depicted in Fig. 5.3. With the help of the developed method, the S-T-F non-stationarity in mmWave massive MIMO 6G V2V channels can be jointly modeled.

110

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

Fig. 5.3 Flowchart of the developed selective cluster evolution based S-T-F non-stationary modeling method

5.3 Simulations and Discussions 5.3.1 Statistical Properties According to the proposed mmWave massive MIMO IS-GBSM with continuously arbitrary trajectory, important channel statistical properties, including STF-CF, DPSD, as well as time stationary interval, are derived and analyzed adequately.

5.3 Simulations and Discussions

5.3.1.1

111

Space–Time–Frequency Correlation Function

The STF-CF can be calculated based on the TVTF as [40] ∗ ξqp,q  p (t, f ; t, f, δT , δR ) = E[Hqp (t, f )Hq  p (t + t, f + f )],

.

(5.47)

where .E[·] and .(·)∗ denote the expectation operation and complex conjugate operation, respectively. The STF-CFs of LoS components and NLoS components are assumed as independent of each other [26]. In this case, the STF-CF can be further computed as the sum of the STF-CFs of LoS components and NLoS components LoS ξqp,q  p (t, f ; t, f, δT , δR ) = ξqp,q  p  (t, f ; t, f, δT , δR ) .

NLoS + ξqp,q  p  (t, f ; t, f, δT , δR )

(5.48)

with LoS ξqp,q  p  (t, f ; t, f, δT , δR ) =  K(t)K(t + t) hLoS∗ (t)hLoS . qp,q  p (t + t) (K(t) + 1)(K(t + t) + 1) qp   LoS LoS (t) − (f + f )τqp (t + t) × exp j 2πf τqp

(5.49)

NLoS ξqp,q  p  (t, f ; t, f, δT , δR ) =  ηS (t)ηS (t + t) (K(t) + 1)(K(t + t) + 1) ⎡ ⎤ S(t) S(t+t) s (t) Ns (t+t)   N  sta ⎦ × E⎣ hsta∗ qp,s,ns (t)hq  p ,s  ,n (t + t)ξqp,s,ns (t) s=1

.



s  =1

ns =1

ns =1

s

ηD (t)ηD (t + t) (K(t) + 1)(K(t + t) + 1) ⎡ ⎤ D(t) (t+t) d (t) Nd   D(t+t)  N dyn∗ dyn × E⎣ hqp,d,nd (t)hq  p ,d  ,n (t + t)ξqp,d,nd (t)⎦ +

d=1

d  =1

nd =1

nd =1

d

(5.50) with   ξqp,s,ns (t) = exp j 2π τqp,s,ns (t)f − (f + f )τq  p ,s  ,ns (t + t)

.

(5.51)

112

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

  ξqp,d,nd (t) = exp j 2π τqp,d,nd (t)f − (f + f )τq  p ,d  ,nd (t + t) .

.

(5.52)

Based on the STF-CF, when .f = 0, p = p , q = q  , the STF-CF is simplified to the TACF. When .t = f = 0, by setting the same antenna on one side and different antennas on the other side, e.g., .p = p , q = q  or .p = p , q = q  , the STF-CF can be simplified to the SCCF at the Rx or Tx side. When .t = 0, p = p , q = q  , the STF-CF is simplified to the frequency correlation function (FCF).

5.3.1.2

Doppler Power Spectral Density

By taking the Fourier transfer of TACF, the DPSD can be given as  .

Sqp (t; fD ) =

+∞ −∞

ξqp (t; t)e−j 2π fD Δt d(t),

(5.53)

where .ξqp (t; Δt) represents the TACF of channels and .fD represents the Doppler frequency. Certainly, the derived DPSD is essentially time-varying, which demonstrates the time-varying characteristics of the proposed V2V channel model.

5.3.1.3

Time Stationary Interval

The time-variant PDP of channels can be computed by (t, τ ) =

S(t) N s (t)  

  sta sta Ps,n (t)δ τ − τs,n (t) s s

s=1 ns =1 .

+

D(t) d (t)  N d=1 nd =1

  dyn dyn Pd,nd (t)δ τ − τd,nd (t) .

(5.54)

In the time stationary interval, the channel amplitude response can be regarded as stationary. In general, the time stationary interval of channels is the maximum period when the ACF of the PDP exceeds a specific threshold, e.g., 80% in [41]. As a result, the time stationary interval is calculated as L(t) = inf{t|ξ T (t,t)≤0.8 },

.

(5.55)

where .ξ T (t, t) is the ACF of the PDP and is written as ξ T (t, t) =

.

(t, τ )(t + t, τ )dτ . max{ 2 (t, τ )dτ, 2 (t + t, τ )dτ }

(5.56)

5.3 Simulations and Discussions

113

It is worth mentioning that both the PDP and time stationary interval are timevariant, showing the time-variant characteristics of the proposed V2V channel model.

5.3.2 Model Simulation Important channel statistics are simulated and the effects of VTD and VMT on these channel statistics are explored. Since the proposed model captures the continuously arbitrary trajectory, it can mimic various VMTs in V2V communications. For generality, four typical trajectories of the transceiver, including the uniform rectilinear motion, quarter turn, U-turn, and curve driving, are considered for the first time. The time-varying accelerations and velocities of these four trajectories are listed in Table 5.1. For clarity, the quarter turn can be regarded as a .1/4 circular arc motion with a radius of R. The U-turn is regarded as a .1/2 elliptical arc motion with the semi-major axis .a = 6R 5 and the semi-minor axis .b = R. The curve driving consists of two same .1/2 circular arc motions with a radius of .R/4. For clarity, we take Tx as an example for analysis and Rx can follow the same procedure. For Tx, instances of these four trajectories with .R = 36/π m and .vT (0) = [vT,x (0), vT,y (0), vT,z (0)]T = [0, 6, 0]T m/s from the start time time instant .t0 = 0 s to the end time instant .tend = 3 s is depicted in Fig. 5.4. Simulated TACFs under the aforementioned four trajectories of transceivers in the low VTD are depicted in Fig. 5.5. As the time separation .t enlarges, the TACF shows a downward trend. In Trajectory IV, since the velocity and trajectory changes of transceivers are the most complicated, the TACF of the V2V channel is the lowest. Conversely, in Trajectory I, where transceivers are in uniform rectilinear motions with the acceleration .aT/R = 0 .m/s2 , the V2V channel exhibits the highest TACF. Compared to Trajectory II, the V2V channel under Trajectory III has a lower TACF attributed to the faster change in velocities of transceivers. In Fig. 5.6, simulated TACFs under uniform and accelerated motions of dynamic clusters in the high VTD at .t = 2 s and .t = 4 s are illustrated. Here, the initial dyn,T/R acceleration of dynamic clusters is set to .ad,x/y (0) = 1 m/s2 . In Fig. 5.6, the motion of dynamic clusters has a significant impact on the TACF. In comparison with the uniform motion, the TACF under the accelerated motion of dynamic clusters is lower. This is because dynamic clusters with higher mobility make them move in and out of CEA more frequently, leading to more rapid changes in the propagation environment. Additionally, when the time instant t is larger, the decorrelation of TACF is faster as the time separation .t increases. This is consistent with results in [27, 42]. Another important observation is that, when the time separation .t is the same, TACFs at the time instants .t = 2 s and .t = 4 s are also significantly different. Therefore, TACFs are not only dependable on the time

114

5 A 3D IS-GBSM with Continuously Arbitrary Trajectory for mmWave Massive. . .

Table 5.1 Parameters of four typical trajectories of the transceiver in the simulation Trajectories Trajectory I Uniform rectilinear motion

.aT/R,x (t)

.aT/R,y (t)

.vT/R,x (t)

.vT/R,y (t)

0

0

.vT/R,x (0)

.vT/R,y (0)

Trajectory II Quarter turn

.

2 (0) vT/R R

.

Trajectory III U-turn

Trajectory IV Curve driving



.

× cos

.

2 (0) 12vT/R 5R



.

× cos

.

2 (0) 4vT/R R

.

× cos



vT/R (0)t R



2vT/R (0)t R

2 (0) −vT/R R

.vT/R (0)

vT/R (0)t R

.

× sin

.

2 (0) −2vT/R R

 .

× sin

(i): . − 4vT/R (0)t R



 .

× sin

(ii): . .





2vT/R (0)t R



4vT/R (0)t R

× sin

4vT/R (0)t R

× sin

.

6vT/R (0) 5

.

× sin



.vT/R (0)

vT/R (0)t R

 .

.

× sin



× cos



vT/R (0)t R



.vT/R (0)

2vT/R (0)t R

 .

× cos



2vT/R (0)t R



(i): .vT/R (0)

.vT/R (0)



2 (0) 4vT/R R



.



2 (0) 4vT/R R



4vT/R (0)t R

 .

× cos



4vT/R (0)t R



(ii): . − vT/R (0)  .

× cos



4vT/R (0)t R



separation .t but also on the time instant t, demonstrating the time non-stationarity of the proposed IS-GBSM. In Fig. 5.7, cumulative distribution functions (CDFs) of time stationary intervals under the aforementioned four trajectories of transceivers in the low VTD are presented. Also, the threshold of the ACF of PDP is set to 80% [41]. It can be seen from Fig. 5.7 that the time stationary interval is significantly small because of the large Doppler spread of mmWave high-mobility V2V channels. Furthermore, it is noteworthy that CDFs of time stationary intervals under different trajectories of transceivers are distinctly different. Among these four trajectories, since Trajectory I simply assumes that transceivers perform the uniform rectilinear motion, the median of the time stationary interval is the largest. On the contrary, the median of the time stationary interval is the smallest when transceivers perform the most complicated trajectory, i.e., Trajectory IV. Figure 5.8 gives simulated FCFs with communication frequencies .f = 58 GHz, .f = 60 GHz, and .f = 62 GHz under Trajectory II of transceivers at the time instant .t = 1 s. Since the frequency-dependent path gain is modeled, the FCF is frequencyvarying and thus cannot be reduced to a function of frequency separation .f in the 4 GHz bandwidth. This shows that frequency non-stationarity is properly captured in the proposed IS-GBSM. Moreover, consistent with the result in [29], compared

5.3 Simulations and Discussions

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to those at frequencies .f = 58 GHz and .f = 60 GHz, the V2V channel at the frequency .f = 62 GHz has the highest FCF. R Figure 5.9 shows the simulated SCCFs at Rx reference antennas .AR 1 , .A20 , and R .A 40 under Trajectory III of transceivers at the time instant .t = 1.6 s. In Fig. 5.9, for the same normalized antenna spacing, SCCFs under different reference antennas are also different. Therefore, SCCFs depend on both the normalized antenna spacing and the reference antenna. This observation shows space non-stationarity of the proposed IS-GBSM.

5.3.2.1

Model Validation

Two vehicular channel measurements were carried out in [43, 44]. By comparing simulation results and measurement, the generality and accuracy of the proposed IS-GBSM are verified. Currently, due to hardware limitations, most of the existing mmWave massive MIMO channel measurements were based on virtual antenna arrays (VAAs) or directional scan sounding [45–47], which are only applicable for static channels, leading to the fact that there is NO proper vehicular mmWave massive MIMO channel measurement data in the open literature for the validation of the pro-

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Trajectory I Trajectory II Trajectory III Trajectory IV

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Fig. 5.5 Normalized absolute TACFs under four typical trajectories of the transceiver (.fc = 28 GHz, .MT = MR = 64, .ATp=2 , .AR q=64 , .Dcen (0) = 100 m, .vT (0) = 12 m/s, .vR (0) = 6 m/s, t .Dc

dyn

= 36/π m, .λsta G = 20/m, .λG = 50 m, .t = 1 s)

Fig. 5.6 Normalized absolute TACFs under the uniform and the accelerated motion of dynamic clusters at different time instants t (.fc = 28 GHz, .MT = 32, T R .MR = 64, .Ap=1 , .Aq=20 , .Dcen (0) = 150 m, .vT (0) = 20 m/s, sta .vR (0) = 15 m/s, .λG = 16/m, dyn dyn,T .λG = 24/m, .vmax (0) = dyn,R

vmax (0) = 16 m/s, a t .Dc = 30 m, .Dc = 50 m, .t = 2/4 s)

dyn,T

dyn,R

= 12/m, .vmax (0) = vmax (0) = 20 m/s, .Dca = 30 m,

1

Uniform motion of dyanmic clusters, t=2 s Uniform motion of dyanmic clusters, t=4 s Accelerated motion of dyanmic clusters, t=2 s Accelerated motion of dyanmic clusters, t=4 s

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posed IS-GBSM.1 Therefore, to the best of our available measurement ability, we carried out two measurement campaigns as described in [43] and [44] that are 1 Using real massive MIMO antenna arrays at mmWave bands requires a large number of radiofrequency (RF) chains including A/D converters, power amplifiers, etc., which are significantly expensive and power consuming. Moreover, it is challenging to coordinate/calibrate all RF chains.

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Fig. 5.7 CDFs of time stationary intervals under four typical trajectories of the transceiver (.fc = 28 GHz, .MT = MR = 64, .Dcen (0) = 120 m, .vT (0) = 14 m/s, .vR (0) = 12 m/s, .R = 72/π m, dyn dyn,T dyn,R sta .λG = 20/m, .λG = 16/m, .vmax (0) = 16 m/s, .vmax (0) = 24 m/s, .Dca = 50 m, .Dct = 50 m, .t = 0 s)

dyn,R

vmax (0) = 18 m/s, a = 30 m, .D t = 50 m, c .t = 1 s) .Dc

1

f=58 GHz f=60 GHz f=62 GHz

0.9

Normalized absolute FCF

Fig. 5.8 Normalized absolute FCFs at different communication frequencies (.fc = 60 GHz, T .MT = MR = 64, .Ap=1 , R .Aq=1 , .Dcen (0) = 80 m, .vT (0) = vR (0) = 12 m/s, sta .R = 72/π m, .λG = 20/m, dyn dyn,T .λG = 24/m, .vmax (0) =

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

5

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20

25

f (MHz)

complementary to each other. By comparing the simulation results and measurement data in [43] and [44], the utility of the proposed IS-GBSM is verified. In [43], a vehicular channel measurement was performed exploiting a real antenna array (RAA) consisting of sixteen antennas at a sub-6 GHz band. As a supplement, the

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Normalized absolute SCCF at Rx

1 0.9

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Fig. 5.9 Normalized absolute SCCFs at different reference antennas (.fc = 28 GHz, .MT = MR = 64, .ATp=1 , .Dcen (0) = 135 m, .vT (0) = 15 m/s, .vR (0) = 18 m/s, .R = 54/π m, .λsta G = 16/m, dyn

.λG

dyn,T

dyn,R

= 12/m, .vmax (0) = 30 m/s, .vmax (0) = 25 m/s, .Dca = 50 m, .Dct = 50 m, .t = 1.6 s)

measurement in [44] was a SISO vehicular channel measurement at a mmWave band. More specifically, the measurement in [43] was performed in a vicinity of Aalborg, Denmark at the center frequency .fc = 1.8 GHz. As shown in Fig. 5.10, it is clear that the RAA was placed on top of a van to record downlink signals of commercial LTE networks along several different measurement routes. The measurement in [44] was carried out on seaside roads in Zhujiajian Island, Zhoushan, China, and the center frequency was .fc = 39 GHz. As shown in Fig. 5.11, the Tx antenna was mounted on the back of one car roof, while the Rx antenna was mounted on the front of another car roof. Readers are referred to [43, 44] for more details of the two channel measurement campaigns. In Fig. 5.12, simulated TACFs in high and low VTDs are compared with measurement data in [43]. As there are few vehicles in the measurement, it can be regarded as a low VTD scenario. Note that the processed CIR was collected from the highway in Route 2 and the corresponding VMT was the blue line in Route 2 in [43]. Therefore, Trajectory I is utilized to obtain simulated TACFs. For a fair comparison, key parameters of the proposed IS-GBSM are set to be the same as the measurement setup in [43]. In Fig. 5.12, the simulated TACF in the low VTD fits well with measurement data. Nevertheless, the simulated TACF in the high VTD is lower than measurement data. This phenomenon can be explained that there are more highly dynamic vehicles in V2V communication scenarios under high VTDs. In such a case, the propagation environment is more rapidly-changing and

5.3 Simulations and Discussions

119

Fig. 5.10 The setup of the vehicular channel measurement campaign in [43]. (a) A schematic diagram. (b) A photo of the measurement vehicle

complex, leading to a lower TACF. Consequently, it is essential to distinguish static and dynamic clusters for accurate characterization of V2V scenarios and investigate effects of them on channel statistics. In Fig. 5.13, simulated SCCFs with a high number of clusters and a small number of clusters are compared with measurement data in [43]. In the measurement [43], there are a few dynamic vehicles and static roadside buildings and trees. Therefore, the simulated SCCF with a small number of clusters fits well with measurement data, which demonstrates the accuracy of the proposed IS-GBSM. However, the simulated SCCF with a high number of clusters is smaller than that observed in the measurement data. This is because a high number of clusters in the environment lead to larger channel space diversity and lower spatial correlation. In Fig. 5.14, the comparison of the measurement in [44] and simulated DPSDs under the aforementioned four trajectories of transceivers is given. U-turns of transceivers were considered and the measurement was repeated twice. Since there are few vehicles on seaside roads in the measurement, it can be regarded as a low VTD scenario. Also, key parameters of the proposed model are set to be the same

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Fig. 5.11 The setup of the vehicular channel measurement campaign in [44]. (a) A schematic diagram. (b) A photo taken during the measurement Fig. 5.12 Comparison of normalized absolute TACFs of the proposed IS-GBSM and measurement in [43] (.fc = 1.8 GHz [43], .MT = 1 [43], .MR = 16 [43], .AT1 [43], R .A12 [43], .Dcen (0) = 400 m [43], .vT (0) = 0 m/s [43], .vR (0) = 27.78 m/s [43], .δR = 0.475 λ [43], dyn,T dyn,R .vmax (0) = vmax (0) = a 16 m/s, .Dc = 30 m, t .Dc = 50 m, .t = 0.12 s [43], high VTD: .λsta G = 28/m, dyn .λG = 40/m; low VTD: dyn sta .λG = 48/m, .λG = 20/m)

as the measurement step in [44]. The close agreement between the measurement and simulated DPSD under U-turn, i.e., Trajectory III, of transceivers is achieved.

5.3 Simulations and Discussions

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Normalized absolute SCCF at Rx

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Fig. 5.13 Comparison of normalized absolute SCCFs of the proposed IS-GBSM and measurement in [43] (.fc = 1.8 GHz [43], .MT = 1 [43], .MR = 16 [43], .Dcen (0) = 400 m [43], dyn,T dyn,R .vT (0) = 0 m/s [43], .vR (0) = 27.78 m/s [43], .δR = 0.475 λ [43], .vmax (0) = vmax (0) = 12 m/s, dyn sta a t .Dc = 30 m, .Dc = 50 m, .t = 0 s [43], high number of clusters: .λG = 64/m, .λG = 48/m; small dyn number of clusters: .λsta G = 48/m, .λG = 20/m) 0

Trajectory I, simulation Trajectory II, simulation Trajectory III, simulation Trajectory IV, simulation Trajectory III, measurement

-5

Normalized DPSD (dB)

Fig. 5.14 Comparison of normalized DPSDs of the proposed IS-GBSM and measurement in [44] (.fc = 39 GHz [44], .MT = MR = 1 [44], .Dcen (0) = 15 m [44], .vT (0) = vR (0) = 8.33 m/s [44], .R = 50/π m, dyn sta .λG = 16/m, .λG = 12/m, dyn,T dyn,R .vmax (0) = vmax (0) = 16 m/s, .Dca = 30 m, t .Dc = 50 m, .t = 1 s [44])

-10 -15 -20 -25 -30 -35 300

350

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500

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However, simulated DPSDs under other trajectories of transceivers are significantly different from measurement data. In contrast, the V2V channel under Trajectory I, which is widely assumed and used in existing channel models [13, 25, 26], exhibits the steepest distribution of DPSD. Therefore, the simple assumption of

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uniform rectilinear motion of transceivers leads to a decrease in the accuracy of the channel model. By comparing the proposed model and existing models, both the superiority of the proposed model and the necessity of modeling various VMTs are demonstrated.

5.4 Summary This chapter has proposed a novel 6G IS-GBSM for V2V massive multiple-input multiple-output MIMO mmWave channels, where clusters have been divided into static and dynamic clusters. In the proposed IS-GBSM, continuously arbitrary trajectories of transceivers and dynamic clusters have been captured and a new selective cluster evolution based S-T-F non-stationary modeling method has been developed. From the proposed IS-GBSM, key channel statistics have been derived. The simulated SCCF, TACF, and FCF have shown that S-T-F non-stationarity has been captured. Simulation results have also demonstrated that VTD and VMT have significant effects on channel statistics. Compared to low VTDs, V2V channels in high VTDs exhibit lower TACF. When the acceleration of the transceiver is smaller, the V2V channel exhibits higher TACF, larger time stationary interval, and steeper distribution of DPSD. The comparison between simulation results and measurement has verified the accuracy and generality of the proposed IS-GBSM. Also, the comparison has indicated that the division of static and dynamic clusters as well as the modeling of various VMTs are indispensable for an accurate imitation of V2V scenarios. In the future, based on the GBSM method, the proposed channel model can be further extended to other complex communication scenarios, e.g., HMIMO and RIS-based communication scenarios.

References 1. S. Gao, X. Cheng, L. Yang, Estimating doubly-selective channels for hybrid mmWave massive MIMO systems: a doubly-sparse approach. IEEE Trans. Wireless Commun. 19(9), 5703–5715 (2020) 2. X. You et al., Towards 6G wireless communication networks: vision, enabling technologies, and new paradigm shifts. Sci. China Inf. Sci. 64(1), 1–74 (2021) 3. S.A. Busari, K.M.S. Huq, S. Mumtaz, L. Dai, J. Rodriguez, Millimeter-wave massive MIMO communication for future wireless systems: a survey. IEEE Commun. Surveys Tutor. 20(2), 836–869 (2018). Second quarter 4. H. Wu, J. Chen, C. Zhou, J. Li, X. Shen, Learning-based joint resource slicing and scheduling in space-terrestrial integrated vehicular networks. J. Commun. Inf. Netw 6(3), 208–223 (2021) 5. Z. Huang, L. Bai, M. Sun, X. Cheng, A 3D non-stationarity and consistency model for cooperative multi-vehicle channels. IEEE Trans. Veh. Technol. (2023). https://doi.org/10.1109/ TVT.2023.3268664 6. Z. Huang et al., A mixed-bouncing based non-stationarity and consistency 6G V2V channel model with continuously arbitrary trajectory. IEEE Trans. Wireless Commun. (2023). https:// doi.org/10.1109/TWC.2023.3293024

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Chapter 6

A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave Massive MIMO V2V Channels

In this chapter, a novel 3D IS-GBSM is proposed for 6G mmWave massive multiple-input multiple-output (MIMO) V2V channels. To investigate the impact of vehicular traffic density (VTD) on channel statistics, clusters are divided into static clusters and dynamic clusters, which are further distinguished into static/dynamic single/twin-clusters to capture the mixed-bouncing propagation. A new method, which integrates the visibility region and BD process methods, is developed to model S-T-F non-stationarity of V2V channels with time-space consistency. The continuously arbitrary VMT and soft cluster power handover are simultaneously modeled to further ensure channel time-space consistency. From the proposed model, key channel statistics, including the STF-CF, PDP, time stationary interval, and DPSD, are derived. Simulation results show that S-T-F non-stationarity of channels with time-space consistency is modeled and the impacts of VTD and VMT on channel statistics are analyzed. The generality and accuracy of the proposed model are validated by comparing simulation results and measurement/RT-based results. Meanwhile, through comparison, the necessity of dividing single- and twinclusters and capturing channel consistency is also demonstrated. Based upon the proposed IS-GBSM, the capturing and investigation of the S-T-F non-stationarity and time-space consistency can provide valuable suggestion for designing the 6G V2V system. Additionally, the proposed IS-GBSM is used as a simulation and validation platform to support the design of 6G V2V system algorithms.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_6

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6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM 6.1.1 Introduction and Contributions of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency One key technology of the ITS is the V2V communication, which can facilitate diverse applications. In the upcoming 6G era, V2V communication system needs to meet the demands of significantly low latency, high throughput, as well as high reliability. Toward this objective, massive MIMO and mmWave technologies should be employed in the 6G V2V system. For a proper design and performance evaluation of 6G V2V systems, realistic and easy-to-use 6G V2V channel models need to be developed [1]. It is worth mentioning that V2V communications are extremely dynamic, where Tx, Rx, and the surrounding vehicles are moving at high speed [2–4]. There are two unique features in the actual V2V propagation environment i.e., various VTDs and complex vehicular movement trajectories (VMTs), and a distinctive SE, i.e., the existence of mixed-bouncing propagation and GR, which distinctly affect the channel statistics according to the measurement and analysis [5–9]. First, V2V channels exhibit different channel statistics in high/low VTD, which represents the high/low ratio of the number of mobile vehicles, i.e., dynamic clusters, to the number of static buildings and trees, i.e., static clusters. Second, VMTs are diverse and channel statistics are different under different VMTs. Third, due to rapidlychanging and complicated V2V communication scenarios, both the single-bouncing and multi-bouncing, i.e., mixed-bouncing propagation, and the GR component exist. To capture the actual V2V propagation environment, many V2V GBSMs have been developed, which can be divided into the RS-GBSMs and IS-GBSMs. In RS-GBSMs, clusters are placed on regular shapes, e.g., ellipsoid, cylinder, semisphere, etc. [10]. In IS-GBSMs, the position of clusters follows a certain statistical distribution [11, 12]. It is noteworthy that, owing to the excellent trade-off between complexity and accuracy, the modeling approach of IS-GBSM has been widely adopted in the standardized channel modeling [13]. In the V2V IS-GBSM [14], the clusters were divided into static and dynamic clusters and the influence of VTD on channel statistics was analyzed. Nonetheless, the VMTs of transceivers were discontinuous, uniform, and rectilinear. To address this limitation, a V2V IS-GBSM in [15] modeled the continuously various VMTs by considering the acceleration vector and integral operation, while the values of transceiver velocities were constant. The authors in [16] developed a MIMO sub6 GHz vehicular IS-GBSM, which mimicked the arbitrary VMTs and dynamic clusters via the time-variant acceleration and the random walk process, respectively. However, the aforementioned IS-GBSMs in [14–16] cannot model the mixedbouncing propagation. The IS-GBSMs in [14] and [15] exploited the method of

6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM

127

twin-clusters to successfully model the multi-bouncing propagation. Nevertheless, the AoD via the twin-cluster nearby Tx and the AoA via the twin-cluster nearby Rx were uncorrelated, which cannot imitate the single-bouncing propagation. Conversely, the IS-GBSM in [16] solely captured the single-bouncing propagation by generating single reflection clusters, named as single-clusters in this paper. In [17], a general IS-GBSM was proposed to attempt to imitate the mixed-bouncing propagation, whereas the GR component was ignored. To overcome this limitation, the IS-GBSM in [18] considered the GR component and attempted to develop a mixed-bouncing based IS-GBSM. However, in the IS-GBSMs [17, 18], the effect of the ratio of single-bouncing via single-clusters to multi-bouncing via twin-clusters on channel statistics was ignored. Note that, the adequate modeling of mixed-bouncing propagation needs to simultaneously capture the single/multibouncing propagation and analyze the impact of the ratio between them on channel statistics. The unique features, i.e., various VTDs and complex VMTs, of V2V communications were also neglected in the IS-GBSMs [17, 18]. Therefore, a V2V channel model, which can capture the actual V2V propagation environment, including VTD, VMT, mixed-bouncing propagation, and GR component, is still lacking. In the upcoming 6G, when massive MIMO and mmWave technologies are applied to highly dynamic V2V scenarios, channels will exhibit space–time– frequency (S-T-F) non-stationarity. Channel non-stationarity is an essential channel characteristic and channel non-stationarity in a particular domain, such as space/time/frequency, represents that channel statistical properties vary in this domain [19]. As mentioned in [6, 7], channels under massive MIMO, high-mobility communication scenarios, and massive MIMO exhibit space, time, and frequency non-stationarity, respectively. To model S-T-F non-stationarity, a V2V IS-GBSM was developed in [20], which integrated the improved K-Means clustering algorithm and BD process method to model the correlated cluster based array-time evolution. However, the IS-GBSM in [20] ignored the capturing of time-space (T-S) consistency, which is an inherent channel physical feature. Channel consistency in the space/time domain means that the channel varies smoothly and consistently as the array/time evolves. When massive MIMO and mmWave technologies are utilized in high-mobility communication scenarios, it is exceedingly necessary to capture channel consistency [7]. The emergence of integrated sensing and communication (ISAC) systems further results in the necessity of capturing T-S consistency [21]. Unlike [13, 22], this paper makes a more strict distinction of channel consistency similarly to [21]. Specifically, since the space domain refers to the antenna array domain with the advent of MIMO technology, T-S consistency represents that the channel changes smoothly as time-array evolves. In [23], a general IS-GBSM was proposed to capture S-T-F non-stationarity and time consistency by exploiting the BD process and tracking the locations of transceivers and clusters. In the European COoperation in the field of Scientific and Technical research (COST) 2100 channel model [24], a different method, named as the visibility region, was proposed to

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6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

mimic channel non-stationarity and consistency. By modeling the smooth cluster evolution as clusters move in and out of visibility region, time non-stationarity and consistency were captured. However, both the IS-GBSMs in [23, 24] cannot model S-T-F non-stationarity with T-S consistency. To overcome this drawback, our previous work in [21] developed a general S-T-F non-stationary IS-GBSM with T-S consistency. In [21], the visibility region method was applied to the array-time axis and a frequency-dependent factor was developed. Nonetheless, the unique features, i.e., various VTDs and complex VMTs, and distinctive SE, i.e., mixed-bouncing propagation and GR component, of V2V communications were disregarded, which affect the modeling of S-T-F non-stationarity with T-S consistency. Currently, a method that has the capability to integrate the modeling of S-T-F non-stationarity with T-S consistency and the capturing of actual V2V propagation environment is still lacking. To fill the above gaps, a novel 6G V2V massive MIMO mmWave IS-GBSM is proposed. The proposed IS-GBSM can be regarded as a 6G channel model by modeling S-T-F non-stationarity of channels with T-S consistency in consideration of the impacts of VTD, continuously arbitrary VMTs, and mixed-bouncing propagation. Based upon the proposed IS-GBSM, the capturing and investigation of the S-T-F non-stationarity and T-S consistency can provide valuable suggestion for designing the 6G V2V system. Additionally, the proposed IS-GBSM is used as a simulation and validation platform to support the design of 6G V2V system algorithms. The major contributions and novelties of this paper are summarized as follows. 1. A new S-T-F non-stationarity and T-S consistency IS-GBSM with continuously arbitrary VMTs, GR component, and mixed-bouncing propagation for 6G massive MIMO mmWave V2V channels is proposed. From the proposed model, key channel statistics are obtained. 2. To model the mixed-bouncing propagation and describe the V2V scenarios with different VTDs, clusters are divided into static/dynamic clusters and single/twin-clusters. Two indexes are given to quantitatively characterize the ratios of the numbers of static/dynamic and single/twin-clusters. Continuously arbitrary VMTs of dynamic clusters and transceivers are further mimicked by the integration of time and time-varying acceleration. 3. A new method, which can model S-T-F non-stationarity with T-S consistency in consideration of the effects of VTD, continuously arbitrary VMT, and mixedbouncing propagation, is developed for the first time. The developed method integrates the visibility region and BD process methods. A soft transition factor and a frequency-dependent factor are proposed to capture the soft cluster power handover and frequency non-stationarity, respectively. 4. Simulation results show that the proposed model can mimic S-T-F nonstationarity of channels with T-S consistency, and the impacts of two indexes on channel statistics are explored. By comparing simulation results and measurement/RT-based results, the utility of the proposed model is verified.

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129

6.1.2 Channel Impulse Response of Proposed Mixed-Bouncing IS-GBSM with Time-Space Consistency The V2V communication system is equipped with ULA array with .MT /.MR antennas at Tx/Rx, where adjacent antenna spacing is .δT /.δR . In Fig. 6.1, the geometrical representation of proposed IS-GBSM is elaborated, where the angle information is omitted for clarity. The .p/q-th antenna in the Tx/Rx array is T R .Ap /.Aq . To mimic the continuously arbitrary VMT, the time-variant acceleration is introduced in the transceiver velocity. The corresponding velocity vector is T/R (t) = [v T/R (t), v T/R (t), 0]T with .v x y T/R .v x/y (t)

=

T/R vx/y (0) +



t 0

T/R

ax/y (t)dt,

(6.1)

T/R

where .vx/y (0) is the velocity of Tx/Rx at initial time. The distance between the center of transceivers is .Dcen (t). To consider the impact of VTD, clusters are divided into static/dynamic clusters according to whether the cluster velocity is zero. To mimic the mixed-bouncing propagation, clusters are further separated as single-clusters and twin-clusters. In Fig. 6.1, the link bounces from Tx to Rx once, i.e., single-bouncing propagation, via a single-cluster. Unlike single-clusters, the twin-cluster consists of a sub-cluster nearby Tx and a sub-cluster nearby Rx, which characterize the first bounce and the last bounce, respectively [25]. Bounces between the twin-clusters are abstracted as a virtual link. Therefore, twin-clusters can capture the multi-bouncing propagation.

Fig. 6.1 Geometrical representation of the proposed 6G V2V massive MIMO mmWave IS-GBSM

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6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

There are four types of clusters, including static single-clusters, static twin-clusters, dynamic single-clusters, and dynamic twin-clusters, and their numbers are .S(t), SS .I (t), .D(t), and .J (t), respectively. The .s/d-th static/dynamic single-cluster is .C s/d , DM,T/R

and the .i/j -th static/dynamic twin-cluster nearby Tx/Rx is .Ci/j . A scattering density index .SDI (t) is given to characterize the ratio of the numbers of singleclusters to twin-clusters, i.e., .SDI (t) = S(t)+D(t) I (t)+J (t) . In the sparse SE, the value of scattering density index is large. A vehicle density index .VDI (t) is given to represent the ratio of the numbers of static clusters to dynamic clusters, i.e., S(t)+I (t) .VDI (t) = D(t)+J (t) . In the low VTD scenario, the value of vehicle density index is large. In addition to modeling the impacts of VTD and SE, the proposed model also considers the mmWave communication. The mmWave communication can provide high data rate transmission and has been widely used [26]. To imitate the high delay resolution in mmWave communications, there are .Ns (t), .Ni (t), .Nd (t), and .Nj (t) SM,T/R DM,T/R rays within the clusters .CsSS , .Ci , .CdDS , and .Cj , respectively. To support the continuously arbitrary VMT, the time-varying acceleration is introduced into the DM,T/R DM,T/R velocity vectors of dynamic clusters .CdDS and .Cj , i.e., .vDS (t), d (t) and .vj which are calculated similarly to (6.1).

6.1.2.1

Complex Channel Gain of LoS Component

The complex channel gain of LoS component related to the sub-channel between the Tx antenna .ATp and the Rx antenna .AR q can be given by LoS .hqp (t)

   t LoS LoS = QTE (t)exp j 2π fqp (t)dt + j ϕqp (t) ,

(6.2)

0

where .TE denotes an observation time interval and .QTE (t) represents a rectangular window function, which is given as  1, 0  t  TE , .QTE (t) = 0, otherwise.

(6.3)

The delay, phase shift, as well as Doppler shift related to LoS component are

LoS τqp (t) =

.

LoS ϕqp (t) = ϕ0 +

.

   LoS  Dqp (t) c  2π   LoS  Dqp (t) , λ

 LoS (t), vR (t) − vT (t) D 1 qp LoS   , .fqp (t) =  LoS  λ Dqp (t)

(6.4) (6.5)



(6.6)

6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM

131

where c represents the speed of light, .λ is the carrier wavelength, and .ϕ0 denotes the initial phase shift. .·, · and .· denote the inner product and Frobenius norm, T R respectively. .DLoS qp (t) is the distance vector from antennas .Ap to .Aq and is written as  t  t LoS LoS T .D (6.7) v (t)dt + vR (t)dt, qp (t) = Dqp (0) − 0

0

T R where .DLoS qp (0) is the distance vector from the antennas .Ap to .Aq at initial time.

6.1.2.2

Complex Channel Gain of NLoS Component via Ground Reflection

The complex channel gain of GR component is written as  GR (t) hGR qp (t) =QTE (t) P

   t  t GR fpGR,T (t)dt + fqGR,R (t)dt + j ϕqp (t) , × exp j 2π

.

0

(6.8)

0

where .P GR (t) is the power of GR component. The delay, phase shift, and Doppler shift of GR component related to the Tx antenna .ATp and the Rx antenna .AR q can be given as GR τqp (t) =

.

(t) + DGR,R (t) DGR,T p q c

2π  GR,T Dp (t) + DGR,R (t) q λ   GR,T/R Dp/q (t), vT/R (t) 1 GR,T/R .f (t) = , p/q GR,T/R λ Dp/q (t)

GR ϕqp (t) = ϕ0 +

.

GR,T/R

(6.9) (6.10)

(6.11) T/R

where .Dp/q (t) denotes the distance vector from the Tx/Rx antenna .Ap/q to the GR point and its computation is presented below. First, based on Fig. 6.2, the azimuth distance between Tx and GR point is expressed by χT (t) =

.

ξT Dcen (t), ξT + ξR

(6.12)

where .ξT and .ξR are ground clearances of Tx and Rx. Second, based on the geometrical relationship, the distance from Tx to GR point and the distance from GR point to Rx can be written as

132

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

Fig. 6.2 Geometrical relationship related to the GR component

GR,T .Dcen (t)

= χT2 (t) + ξT2

(6.13)

GR,R .Dcen (t)

=

Dcen (t)2 + (ξT + ξR )2 − DGR,T cen (t). GR,T/R

Third, the distance vector .Dcen

(6.14)

(t) is expressed by ⎡

GR,T/R .Dcen (t)

=

GR,T/R Dcen (t)

GR (t) cosα GR (t) ⎤ cosβcen cen ⎣ cosβ GR (t) sinα GR (t) ⎦ , cen cen GR (t) sinβcen

(6.15)

GR (t) and .β GR (t) denote azimuth and elevation angles of the distance vector where .αcen cen GR,T/R .Dcen (t), respectively. Based on [27], only the elevation angle of distance vector GR .Dcen (t) needs to be taken into account and is given as GR βcen (t) = arctan

.

ξR . χR (t)

(6.16)

Fourth, the distance vectors from antennas .ATp and .AR q to the GR point are calculated by .

T DGR,T (t) = DGR,T p cen (t) − Ap

(6.17)

R DGR,R (t) = DGR,R q cen (t) − Aq .

(6.18)

.

6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM

6.1.2.3

133

Complex Channel Gain of NLoS Components via Static Single-Clusters and Twin-Clusters

The complex channel gain of NLoS components via static single/twin-clusters is derived. The complex channel gain of the sub-channel related to the Tx antenna .ATp and SS the Rx antenna .AR q via the .ns -th ray within the cluster .Cs is written by hSS qp,s,ns (t) .

  t

  t

SS,T SS,R SS SS (t)exp j 2π f (t)dt + f (t)dt + j ϕ (t) = QTE (t) Ps,n p,s,ns q,s,ns qp,s,ns s 0

0

(6.19)

SS (t) where .Ps,n s

is the power of the .ns -th ray within the cluster .CSS s . Also, the delay, and Doppler shift of NLoS component via the .ns -th ray within the

phase shift, cluster .CSS s are given as

SS τqp,s,n (t) = s

     SS,T   SS,R  Dp,s,ns (t) + Dq,s,ns (t)

.

c

   2π   SS,T    (t) Dp,s,ns (t) + DSS,R  q,s,ns λ   SS,T/R T/R (t) (t), v D 1 p/q,s,ns SS,T/R   .f p/q,s,ns (t) =   SS,T/R λ Dp/q,s,ns (t)

SS ϕqp,s,n (t) = ϕ0 + s

.

(6.20) (6.21)

(6.22)

SS,T/R

where .Dp/q,s,ns (t) is the distance vector between the .ns -th ray within the cluster T/R

SS,T CSS s and the antenna .Ap/q . Furthermore, the distance vector .Dp,s,ns (t) can be represented by

.

 SS,T SS,T Dp,s,n (t) = Ds,n (0) − s s

t

.

0

vT (t)dt − ATp

(6.23)

SS,T where .ATp is the vector of Tx antenna .ATp . .Ds,n s (0) is the distance vector between SS the .ns -th ray within the cluster .Cs and the Tx center at initial time and is given as

⎤ SS,T SS,T (0) cosβ (0) cosαs,n s,n s s ⎥ ⎢ SS,T SS,T SS,T SS,T .Ds,n (0) = Ds,n (0) ⎣ sinαs,n (0) cosβs,n (0) ⎦ , s s s s SS,T sinβs,ns (0) ⎡

(6.24)

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6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

SS,T SS,T where .αs,n s (0) and .βs,ns (0) are the azimuth and elevation angles of the distance SS,T vector .Ds,n s (0). Due to the single-bouncing propagation, the distance vectors SS,T SS,R .Dp,s,ns (t) and .Dq,s,ns (t) are correlated. Based on the geometrical relationship, the SS,R distance vector .Dq,s,n s (t) is computed as SS,R LoS Dq,s,n (t) = DSS,T p,s,ns (t) − Dqp (t). s

(6.25)

.

The complex channel gain of the sub-channel related to the Tx antenna .ATp and SM,T/R

the Rx antenna .AR q via the .ni -th ray within the cluster .Ci

is given as

hSM qp,i,ni (t) .

  t

  t

SM,T SM,R SM SM = QTE (t) Pi,ni (t)exp j 2π fp,i,ni (t)dt+ fq,i,ni (t)dt +j ϕqp,i,ni (t) , 0

0

(6.26)

SM (t) is the power of the .n -th ray within the cluster .CSM,T/R . Note that the where .Pi,n i i i phase shift and Doppler shift are derived similarly to the cluster .CSS s in (6.21) and

(6.22) and are written as     2π   SM,T   SM,R  SM (t) Dp,i,ni (t) + Dq,i,ni (t) + cτ˜qp,i,n i λ   SM,T/R T/R (t) (t), v D 1 p/q,i,ni SM,T/R   . .f p/q,i,ni (t) =  SM,T/R  λ Dp/q,i,ni (t)

SM ϕqp,i,n (t) = ϕ0 + i

.

(6.27)

(6.28)

SM,T However, unlike static single-cluster .CSS s , the distance vectors .Dp,i,ni (t) and SM,T/R

DSM,R q,i,ni (t) of static twin-clusters .Ci

.

are uncorrelated, which should be cal-

SM,R culated separately. The calculation of distance vectors .DSM,T p,i,ni (t) and .Dq,i,ni (t) is

similar to that of the distance vector .DSS,T p,s,ns (t) in (6.23) and (6.24). Furthermore, in the multi-bouncing propagation, the delay of virtual link between twin-clusters, named as the virtual delay, needs to be considered and the delay of NLoS component SM,T/R via the .ni -th ray within the cluster .Ci is expressed as SM τqp,i,n (t) = i

.

     SM,T   SM,R  Dp,i,ni (t) + Dq,i,ni (t) c

SM + τ˜qp,i,n (t), i

(6.29)

SM (t) is the virtual delay of the twin-clusters .CSM,T and .CSM,R and where .τ˜qp,i,n i i i follows the exponential distribution [28].

6.1 Framework of mmWave Massive MIMO V2V Mixed-Bouncing IS-GBSM

6.1.2.4

135

Complex Channel Gain of NLoS Component via Dynamic Single-Clusters and Twin-Clusters

Similar to static clusters, the complex channel gain of NLoS components via dynamic single/twin-clusters is obtained. The complex channel gain of the sub-channel related to the Tx and Rx antennas DS can be written by T R .Ap and .Aq via the .nd -th ray within the cluster .C d hDS qp,d,nd (t) =   t

  t

. DS,T DS,R DS DS QTE (t) Pd,nd (t)exp j 2π fp,d,nd (t)dt+ fq,d,nd (t)dt +j ϕqp,d,nd (t) , 0

0

(6.30)

DS (t) is the power of the .n -th ray within the cluster .CDS . Similar to where .Pd,n d d d the static single-cluster .CsSS , the delay and phase shift of NLoS component via the dynamic single-cluster .CDS d can be given by

DS τqp,d,n (t) = d

     DS,T   DS,R  Dp,d,nd (t) + Dq,d,nd (t)

.

DS ϕqp,d,n (t) = ϕ0 + d

.

c

   2π   DS,T   DS,R  Dp,d,nd (t) + Dq,d,nd (t) . λ

(6.31) (6.32)

Nonetheless, the Doppler shift of NLoS component via rays within dynamic clusters is different from that of NLoS component via rays within static clusters and is given as   DS,T/R T/R (t) − vDS (t) (t), v D d 1 p/q,d,nd DS,T/R   , (6.33) .f p/q,d,nd (t) =  DS,T/R  λ Dp/q,d,nd (t) DS,T/R

where .Dp/q,d,nd (t) is the distance vector between the .nd -th ray within the cluster T/R

DS,T CDS d and the antenna .Ap/q . The distance vector .Dp,d,nd (t) is expressed as

.

 DS,T .D p,d,nd (t)

=

DDS,T d,nd (0) −

0

t

 v (t)dt + T

0

t

T vDS d (t)dt − Ap ,

(6.34)

where .DDS,T d,nd (0) is the distance vector between the .nd -th ray within the cluster

DS,R CDS d and the Tx center at initial time. Also, the distance vector .Dq,d,nd (t) can be calculated by

.

DS,T LoS DDS,R q,d,nd (t) = Dp,d,nd (t) − Dqp (t).

.

(6.35)

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6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

The complex channel gain of the sub-channel related to the Tx and Rx antennas DM,T/R ATp and .AR is given by q via the .nj -th ray within the cluster .Cj

.

hDM qp,j,nj (t) .

  t

  t

DM,T DM DM DM = QTE (t) Pj,nj (t)exp j 2π fp,j,nj (t)dt+ fq,j,nj (t)dt +j ϕqp,j,nj (t) , 0

0

(6.36) DM,T/R .C . j

DM (t) is the power of the .n -th ray within the cluster where .Pj,n Based on j j the aforementioned analysis, the delay, phase shift, and Doppler shift of the NLoS DM,T/R component via the .nj -th ray within the cluster .Cj are analogously computed by

DM τqp,j,n (t) = j

.

     DM,T   DM,R  Dp,j,nj (t) + Dq,j,nj (t) c

DM + τ˜qp,j,n (t) j

    2π   DM,T   DM,R  DM (t) Dp,j,nj (t) + Dq,j,nj (t) + cτ˜qp,j,n j λ   DM,T/R T/R (t) − vDM,T/R (t) (t), v D j 1 p/q,j,nj DM,T/R   . .f p/q,j,nj (t) =  DM,T/R  λ Dp/q,j,nj (t)

DM ϕqp,j,n (t) = ϕ0 + j

.

(6.37) (6.38)

(6.39)

Based on the geometry relationship, transmission paths of antennas and rays within clusters are computed, where the different angular parameters of transmission paths are imitated. Hence, the spherical wavefront propagation in massive MIMO channels is mimicked sufficiently. The CIR of the developed IS-GBSM with delay .τ at time t can be represented as a matrix .H(t, τ ) = [hqp (t, τ )]MR ×MT with .q = 1, 2, ..., MR and .p = 1, 2, ..., MT . .H(t, τ ) contains the LoS, GR, as well as NLoS components. The CIR .hqp (t, τ ) related to the Tx and Rx antenna element pair p and q can be expressed as 

hqp (t, τ ) =

.

 K(t) LoS hLoS qp (t)δ τ − τqp (t) K(t) + 1   ηGR (t) GR GR hqp (t)δ τ − τqp + (t) K(t) + 1  S(t) Ns (t)  ηSS (t)   SS + hSS (t)δ τ − τ (t) qp,s,ns qp,s,ns K(t) + 1  +

s=1 ns =1

D(t) Nd (t)  ηDS (t)   DS hDS qp,d,nd (t)δ τ − τqp,d,nd (t) K(t) + 1 d=1 nd =1

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

 +  +

137

I (t) Ni (t)  ηSM (t)   SM hSM (t)δ τ − τ (t) qp,i,ni qp,i,ni K(t) + 1 i=1 ni =1

J (t) Nj (t)  ηDM (t)   DM DM hqp,j,nj (t)δ τ − τqp,j,n (t) , j K(t) + 1

(6.40)

j =1 nj =1

where .K(t) denotes the Ricean factor. .ηGR (t), .ηSS (t), .ηSM (t), .ηDS (t), and .ηDM (t) represent the power proportions of the GR component, component via static single-cluster, component via static twin-cluster, component via dynamic singlecluster, and component via dynamic twin-cluster. These power proportions have GR (t) + ηSS (t) + ηSM (t) + ηDS (t) + ηDM (t) = 1. Finally, it is noteworthy that .η the integration of time and time-variant acceleration are essential for the capturing of continuously arbitrary VMTs, where the numerical calculation of the integral can be efficiently computed by the method of superposition calculation. As a result, by modeling continuously arbitrary VMTs, it is possible to acquire the CIR and investigate channel properties under all positions on the track.

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space Consistency In the upcoming 6G, the conventional 5G technologies, i.e., massive MIMO and mmWave, still receive the extensive attention [29, 30]. When the massive MIMO and mmWave technologies are simultaneously employed in the V2V communication, S-T-F non-stationarity of V2V channels with T-S consistency needs to be modeled. In V2V communication scenarios, it is necessary to consider the effects of mixed-bouncing propagation, i.e., single/twin-clusters, VTD, i.e., static/dynamic clusters, and continuously arbitrary VMTs. In this section, a new method, named as the S-T-F non-stationarity of V2V channels with T-S consistency modeling method, is developed, which can model S-T-F non-stationarity with T-S consistency in consideration of mixed-bouncing propagation, VTD, and continuously arbitrary VMTs. Specifically, observable semi-spheres assigned to antennas are constructed to model space non-stationarity with space consistency. Observable spheres assigned to static/dynamic single/twin-clusters aim to model time nonstationarity with time consistency. A soft transition factor is further introduced to successfully model the soft cluster power handover. For the frequency nonstationary modeling, a frequency-dependent factor is developed.

138

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

6.2.1 Modeling of Space Non-stationarity and Consistency by Observable Semi-spheres Assigned to Antennas For channel models in [31, 32], the visibility region method is employed to model channel non-stationarity. In the visibility region method, only clusters in the visibility region are observable clusters. Since the models in [31, 32] are 2D, the visibility region is modeled as circles. Nonetheless, in the three-dimensional (3D) channel model, the visibility region needs to be modeled as spheres instead of circles. Further considering that vehicles move on the 2D ground, the visibility region is modeled as semi-spheres in the proposed 3D V2V channel model. The semi-sphere is constructed with each antenna as the center. By developing observable semi-spheres, different antennas with different observable semi-spheres have different sets of array observable clusters, hence modeling space non-stationarity. For adjacent antennas, they share many clusters, namely the shared array observable clusters, as their observable semi-spheres have the overlap area, thus capturing space consistency. To avoid the repetition, the Tx side is taken as an instance for analysis and the Rx side follows the same procedure. The number of shared array observable clusters strongly depends on the overlap area. The overlap area of observable semi-spheres of two adjacent antennas .ATp and T T .A p+1 with spacing .δT is a spherical crown, whose volume .Vshare can be calculated based on [21]

T Vshare =

.

 3 2 π 16raT − 12raT δT + δT3 24

,

(6.41)

where .raT denotes the radius of observable semi-sphere. At the Tx side, since antennas are uniformly distributed on the array with spacing .δT , the radius .raT of each observable semi-sphere is the same. To quantitatively calculate the radius .raT , T T a novel parameter .Γspace is given, which represents the ratio of volume .Vshare of T overlap area to volume .Vspace of semi-sphere T Γspace =

.

3 2 T T Vspace Vshare 16raT − 12raT δT + δT3 = . = T 3 2 T3 Vspace 16raT 3 π ra

(6.42)

Clusters in the observable semi-sphere are array observable clusters and only clusters in the overlap area are shared array observable clusters. Assumed that T clusters are uniformly distributed in the environment. Hence, the parameter .Γspace describes the probability that an array observable cluster is a shared array observable cluster, which is simultaneously observable to an antenna .ATp and its adjacent T antenna .ATp+1 . In the cluster array evolution, the parameter .Γspace also represents T the probability that a surviving cluster for an antenna .Ap is still surviving when it evolves to the adjacent antenna .ATp+1 . This is consistent with the spatial survival

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

139

probability. The spatial survival probability at Tx is given by [33]  T Psurvival = exp

.

−λR δT Dca

 (6.43)

,

where .λR is the recombination rate and .Dca denotes the scenario-dependent corT relation factor on the array axis. Make the parameter .Γspace and spatial survival T T T probability .Psurvival equal, i.e., .Γspace = Psurvival to solve for the radius .raT 3



2

16raT − 12raT δT + δT3

= exp

−λR δT Dca

 .

(6.44)

  δ 3 2 −λR DTa c 16 1 − e raT − 12δT raT + δT3 = 0.

(6.45)

.

16raT

3

Obviously, (6.44) can be written as .

Furthermore, the normalized radius at the Tx side, i.e., .r¯aT = raT /δT , is introduced. Consequently, (6.45) can be expressed by 

δ

16 1 − e

.

−λR DTa



c

3

2

r¯aT − 12¯raT + 1 = 0.

(6.46)

It is clear that (6.46) can be converted to solve the root of (6.47), which can be given by   δ −λR DTa c f (x) = 16 1 − e x 3 − 12x 2 + 1 = 0.

.

(6.47)

b , the form of function By employing the method of substitution, i.e., .x = y − 3a 3 2 3 .ax + bx + cx + d = 0 can be converted to the form of function .y + py + q = 0, where p and q are given by

p=

.

c b2 − 2 a 3a

(6.48)

bc d 2b3 − 2. (6.49) + a 27a 3 3a   δ −λR DTa c , .b = −12, .c = 0, and According to (6.47), it is obvious that .a = 16 1 − e q=

.

d = 1. Since .c = 0, p and q can be simplified as

.

140

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

p=−

.

q=

.

b2 3a 2

(6.50)

d 2b3 . + a 27a 3

(6.51)

The discriminant of the function .y 3 + py + q = 0 can be computed as =

 q 2 

2

=

+

 p 3 3

d b3 + 2a 27a 3

2

 −

b2 9a 2

3

b3 d d2 + 27a 4 4a 2 64 1 = 2− 4 4a a =

.

=

a 2 − 256 4a 4  2  δ −λR DTa c 16 1 − e − 256

=

(6.52)

 4  δ −λR DTa c 4 16 1 − e
0

c

(6.55)  < 0. δT

−λR D a c

From (6.55), it can be found that three real roots .x1 , .x2 , and .x3 consist of two positive real roots and one negative real root. In addition, if the overlapped area of the two spheres is not 0, the real root needs to satisfy the condition in .x > 1/2. When .x = 1/2 of (6.47), we have     δ 1 −λR DTa c x 3 − 12x 2 + 1 = 16 1 − e f 2 .

δ

= −2e

−λR DTa

(6.56)

c

< 0. Since .f (1/2) is less than 0, three real roots have and only one real root satisfies the condition of .x > 1/2. According to the properties of the three real roots, the real root that satisfies the condition of .x > 1/2 is given by

142

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

    2 3 q q 2  p 3 p b q q 3 3 T + + − − + − x1 = r¯a = − + 2 2 3 2 2 3 3a   .       2 3 3 3 b3 b a − 256  a 2 − 256 d b d  = − − , − − + + − − 2a 3a 2a 27a 3 4a 4 27a 3 4a 4 (6.57)   

δ

where .a = 16 1 − e

−λR DTa c

, .b = −12, and .d = 1. It is clear that the radius at the

Tx side .raT can be calculated by the normalized radius at the Tx side .r¯aT and can be expressed as raT = r¯aT δT = x1 δT .

(6.58)

.

As a result, the radius at the Tx side .raT can be given by ⎧ ⎫ ⎨ ⎬

1 T 3 T T T T  δT ,  ξspace + σspace + 3 ξspace − σspace +  . ra = ⎩ 4 1 − exp −λDRaδT ⎭ c (6.59) T T where .ξspace and .σspace are written as

T ξspace

.

  2 −2 1 − exp −λDRaδT +1 c = 3   64 1 − exp −λDRaδT

(6.60)

c

T σspace

.

     1 − exp −λR δT 2 − 1  a D c = 4 .    1024 1 − exp −λDRaδT

(6.61)

c

With the help of the derived closed-form solution, the impacts of the recombination rate .λR , adjacent antenna spacing .δT , and scenario-dependent correlation factor a T .Dc on the radius .ra can be investigated. By properly adjusting the recombination rate .λR and factor .Dca , the developed visibility region method in the space domain can be successfully applied to a variety of scenarios.

6.2.1.1

Conditions of Array Observable Static/Dynamic Single-Clusters

In the proposed mmWave V2V model, the adjacent antenna spacing is small. According to (6.59), the radius .raT is proportional to the adjacent antenna spacing T .δT , which leads to the small radius .ra . Consequently, it is necessary to introduce

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

143

parameters, namely shrinkage factors on the array axis, which can shrink the distance from the antenna .ATp to the cluster. Specifically, the distances between static/dynamic single-clusters and the antenna .ATp are shrunk by the shrinkage SS,T DS,T factors .εspace and .εspace , which can be expressed by

.

.

SS,T D˜ p,s (t) =

DS,T D˜ p,d (t) =

SS,T (t) Dp,s

(6.62)

SS,T εspace DS,T Dp,d (t) DS,T εspace

.

(6.63)

Based on the shrinkage distance and radius .raT , the clusters .CsSS and .CdDS are SS,T DS,T observable to the Tx antenna .ATp that need to satisfy .D˜ p,s (t) < raT and .D˜ p,d (t) < T ra , respectively. It is obvious that the values of shrinkage factors have great impacts on the array observable cluster condition. To properly set the value of shrinkage factors, randomly select a Tx antenna, e.g., the antenna .ATp , assuming that it can observe all static/dynamic single-clusters at initial time .t0 . This assumption is consistent with the standardized channel model in [34]. Under this assumption, SS,T DS,T shrinkage factors .εspace and .εspace are given by $

.

SS,T εspace

% SS,T Dp,s (t0 ) = max , p = 1, 2, · · · , MT , s = 1, 2, · · · , S (t0 ) raT $

DS,T .ε space

= max

DS,T Dp,d (t0 )

(6.64)

% , p = 1, 2, · · · , MT , d = 1, 2, · · · , D (t0 ) .

raT

(6.65)

Note that, the conditions of the array observable static and dynamic single-clusters CsSS and .CdDS are that clusters .CsSS and .CdDS are observable to both the antenna SS,T SS,R DS,T T R ˜ p,s .Ap and the Rx antenna .Aq , i.e., .D (t) < raT , D˜ q,s (t) < raR and .D˜ p,d (t) < DS,R T R ra , D˜ (t) < ra , respectively. .

q,d

6.2.1.2

Conditions of Array Observable Static/Dynamic Twin-Clusters

Analogous to single-clusters, the distances between static/dynamic twin-clusters SM,T DM,T nearby Tx and antenna .ATp are shrunk by the shrinkage factors .εspace and .εspace . The corresponding shrinkage distances are expressed by

.

SM,T (t) = D˜ p,i

SM,T Dp,i (t) SM,T εspace

(6.66)

144

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

˜ DM,T (t) = .D p,j

DM,T (t) Dp,j DM,T εspace

.

(6.67)

The twin-clusters nearby Tx .CiSM,T and .CjDM,T observable to the Tx antenna .ATp have .D˜ SM,T (t) < raT and .D˜ DM,T (t) < raT . Further assume that a randomly selected p,i

p,j

Tx antenna, e.g., .ATp , can observe all static/dynamic twin-clusters nearby Tx at SM,T DM,T initial time .t0 , and thus the shrinkage factors .εspace and .εspace are given by

$ .

SM,T εspace = max

SM,T Dp,i (t0 )

raT $

.

DM,T εspace = max

% , p = 1, 2, · · · , MT , i = 1, 2, · · · , I (t0 )

DM,R Dp,j (t0 )

raT

(6.68)

% , p = 1, 2, · · · , MT , j = 1, 2, · · · , J (t0 ) . (6.69)

Unlike single-clusters, the twin-clusters nearby Tx observable to the Tx antenna ATp and the twin-clusters nearby Rx observable to the Rx antenna .AR q need to be randomly paired. Only successfully paired static/dynamic twin-clusters nearby Tx and Rx, e.g., .CiSM,T /.CjDM,T and .CiSM,R /.CjDM,R , will form array observable static/dynamic twin-clusters. Therefore, the conditions of array observable static/dynamic twin-clusters are that clusters .CiSM,T /.CjDM,T and .CiSM,R /.CjDM,R are observable to the Tx antenna .ATp and the Rx antenna .AR q , respectively, and then

.

clusters .CiSM,T /.CjDM,T and .CiSM,R /.CjDM,R are paired.

6.2.2 Modeling of Time Non-stationarity and Consistency by Observable Spheres Assigned to Clusters To mimic time non-stationarity and consistency, the visibility region method in the time domain is developed. Unlike the 2D general COST 2100 channel model [24] that defined the visibility region as circles, the proposed model is 3D and further distinguishes static/dynamic single/twin-clusters. These clusters are distributed in 3D space, not limited to 2D ground. In this case, the visibility region in the proposed model is characterized as the sphere with each static/dynamic single/twin-cluster as the center. When the center of the transceiver antenna array is in a sphere of a static/dynamic single/twin-cluster at time t, the static/dynamic single/twin-cluster is observable to this time t. With the help of observable spheres, as the transceiver and dynamic cluster move, time observable cluster sets change over time, thus modeling time non-stationarity properly. Owing to their smooth and continuous movements, adjacent moments share many identical time observable clusters, thus capturing time consistency adequately. Also, take the Tx side as an example for analysis.

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

6.2.2.1

145

Conditions of Time Observable Static Single/Twin-Clusters

The velocity vectors of static single-cluster .CsSS and twin-cluster nearby Tx .CiSM,T SM,T relative to the Tx array are .vSS (t) = −vT (t). The Tx array can be s (t) = vs regarded as static for the cluster .CsSS /CiSM,T . As the cluster .CsSS /CiSM,T moves SM,T relative to the Tx array with the velocity vector .vSS (t), its observable s (t)/vi sphere also moves. At two adjacent moments t and .t + Δt, its observable sphere has the overlap area, which is dependable on the relative movement distance SS,T .Ds (t)/.DiSM,T (t) between this cluster .CsSS /.CiSM,T and Tx array in a time interval SS,T .Δt. Distance parameters .Ds (t) and .DiSM,T (t) are equal, which are unified as  D S,T (t) = DsSS,T (t) = DiSM,T (t) =

t+Δt

.

t

   T  −v (t) dt.

(6.70)

The time interval .Δt is short in mmWave high-mobility V2V communications, where channels need to be updated frequently [35]. The distance parameter S,T (t) can be approximated to be equal at two adjacent moments t and .t + t. .D Consequently, the overlap area of observable spheres assigned to static clusters can S,T (t) be regarded as consisting of two identical spherical crowns, whose volume .Vshare T is computed similarly to the volume .Vshare in (6.41) and is expressed by S,T Vshare (t) =

  3 2 3 π 16rtS,T (t) − 12rtS,T (t)D S,T (t) + D S,T (t)

.

12

(6.71)

,

where .rtS,T (t) denotes the time-variant radius of Tx observable sphere assigned to S,T the static cluster. Similarly, a new parameter .Γtime (t) is developed to denote the ratio of volume of overlap area to the volume of observable sphere and is given as

.

S,T Γtime (t) =

S,T (t) Vshare S,T Vtime (t)

3

=

2

3

16rtS,T (t) − 12rtS,T (t)D S,T (t) + D S,T (t) 3

16rtS,T (t)

.

(6.72)

Only when the center of array is located in an observable sphere of a cluster at two adjacent moments simultaneously, the cluster is a shared time observable S,T (t) represents the proportion of overlap area in the cluster. As the parameter .Γtime observable sphere, it also characterizes the probability that an observable cluster at time t will still be observable at the next time instant .t + Δt. This is in agreement with the definition of time survival probability in the BD process method from the perspective of cluster time evolution [34]. The time survival probability at Tx of static clusters is written as

146

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

& S,T .P survival (t)

= exp

−λR

' t+Δt  T  ( v (t) dt t , Dct

(6.73)

where .Dct denotes the scenario-dependent correlation factor on the time axis. Make S,T the proportion of overlap area equal to the time survival probability, i.e., .Γtime (t) = S,T Psurvival (t). Similarly, the radius of observable spheres assigned to static clusters is calculated as rtS,T (t) = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪

⎨ ⎬ 1 3 S,T 3 S,T S,T S,T

 ξtime (t)+σtime (t)+ ξtime (t)−σtime (t)+  ' t+Δt . ⎪ −λR t vT (t)dt ⎪ ⎪ ⎪ ⎩ ⎭ 4 1−exp Dt c

 ×

t+Δt

t

   T  v (t) dt (6.74)

with 



−2 1 − exp .

S,T ξtime (t) =

 64 1 − exp

−λR



' t+Δt  T  2 v (t)dt t Dct

−λR

' t+Δt t

vT (t)dt

+1

3

(6.75)

Dct



2  ' t+Δt T   1 − exp −λR t t v (t)dt −1 Dc  S,T  .σ (t) =

4 ,   time ' t+Δt  −λR t vT (t)dt 1024 1 − exp Dt

(6.76)

c

where the radius .rtS,T (t) is proportional to distance parameter .D S,T (t) and the effects of recombination rate .λR and velocity vector of Tx array .vT (t) on the radius .rtS,T (t) are explored. Note that the developed visibility region method in the time domain can support diverse scenarios by properly setting the recombination rate .λR and factor .Dct . The small time interval .Δt in mmWave V2V channels also results in the small radius .rtSS,T (t) of observable spheres based on (6.74)–(6.76). Similar to the array SS,T SM,T axis, shrinkage factors on the time axis .εtime and .εtime are introduced, which can shrink the distance from the center of Tx array to the static clusters .CsSS and .CiSM,T at time t, which can be given by

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

SS,T (t) = D˜ Os

.

SM,T (t) = D˜ Oi

.

SS,T DOs (t)

(6.77)

SS,T εtime SM,T DOi (t) SM,T εtime

147

.

(6.78)

The time observable static single/twin-cluster conditions at Tx, i.e., static clusSS,T ters .CsSS and .CiSM,T are time observable clusters, are .D˜ Os (t) < rtS,T (t) and SM,T S,T SS,T SM,T ˜ .D Oi (t) < rt (t). Also, the proper setting of shrinkage factors .εtime and .εtime is crucial. Similar to [33], assuming that all clusters are observable at initial time .t0 SS,T SM,T and the shrinkage factors .εtime and .εtime are determined by $ SS,T .ε time

= max

%

rtS,T (t0 ) $

SM,T .ε time

SS,T DOs (t0 )

= max

SM,T DOi (t0 )

rtS,T (t0 )

, s = 1, 2, 3, · · · , S (t0 )

(6.79)

, i = 1, 2, 3, · · · , I (t0 ) .

(6.80)

%

Therefore, the condition of the time observable static single-cluster is that the cluster SS,T SS,R CsSS simultaneously satisfies .D˜ Os (t) < rtS,T (t) and .D˜ Os (t) < rtS,R (t). Unlike static single-clusters, the static twin-clusters nearby Tx .CiSM,T and Rx .CiSM,R need SM,T SM,R to meet .D˜ Oi (t) < rtS,T (t) and .D˜ Oi (t) < rtS,R (t), respectively. Only when they are paired can they form time observable static twin-clusters.

.

6.2.2.2

Conditions of Time Observable Dynamic Single/Twin-Clusters

The velocity vectors of dynamic single-cluster .CdDS and twin-cluster nearby Tx DM,T .C relative to the Tx array are written as j DS T vDS d (t) = vd (t) − v (t)

(6.81)

vDM,T (t) = vDM,T (t) − vT (t). j j

(6.82)

.

.

The observable sphere of the dynamic cluster .CdDS /.CjDM,T at two adjacent moments t and .t + Δt has the overlap area, which also depends on the relative movement distance .DdDS,T (t)/.DjDM,T (t) between the cluster .CdDS /.CjDM,T and Tx array in a time interval .Δt. The distance parameters .DdDS,T (t) and .DjDM,T (t) can be written as

148

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

DS/DM,T

Dd/j

.

 (t) =

t+Δt

t

   DS/DM  (t) − vT (t) dt. vd/j

(6.83)

By considering the multi-velocity of dynamic clusters, the relative distances DdDS,T (t) and .DjDM,T (t) for different dynamic clusters are different. Similarly, the

.

DS,T DM,T (t)/.Vshare,j (t) of overlap area of observable sphere assigned to the volume .Vshare,d

dynamic cluster .CdDS /.CjDM,T can be properly obtained. Also, a new parameter DS,T DM,T Γtime (t)/.Γtime (t) is introduced to represent the ratio of volume of overlap area to the volume of observable sphere assigned to the cluster .CdDS /.CjDM,T

.

DS/DM,T 3

DS/DM,T .Γ time,d/j (t)

=

16rt,d/j

DS/DM 2

(t) − 12rt,d/j

DS/DM,T

(t)Dd/j

DS/DM,T 3

16rt,d/j

DS/DM,T 3

(t) + Dd/j

(t)

,

(t) (6.84)

DS,T DM,T where .rt,d (t)/.rt,j (t) represents the time-varying radius of Tx observable sphere DM,T DS . Analogously, combined with the BD process, the time of the cluster .Cd /.Cj survival probabilities at Tx of dynamic clusters .CdDS and .CjDM,T are given as

⎡ .

Psurvival,d/j (t) = exp ⎣ DS/DM,T

−λR

 ⎤ ' t+Δt   DS/DM,T  (t) − vT (t) dt vd/j t ⎦. Dct

(6.85)

Make the proportion of overlap area and the time survival probability equal, i.e., DS,T DS,T DM,T DM,T Γtime,d (t) = Psurvival,d (t) and .Γtime,j (t) = Psurvival,j (t), to solve the radius

.

DS/DM,T

rt,d/j

.

 

DS/DM,T DS/DM,T DS/DM,T 3 DS/DM,T ξtime,d/j (t) + σtime,d/j (t) + 3 ξtime,d/j (t) − σtime,d/j (t)

(t) =



t+Δt t

+

$

   DS/DM,T  (t) − vT (t) dt vd/j  ' t+Δt   DS/DM,T T (t) dt v (t) − v   d/j t &   (% '  DS/DM,T 

4 1 − exp

−λR

t+Δt vd/j t

(6.86)

(t)−vT (t)dt

Dct

with $

&

−2 1 − exp DS/DM,T .ξ time,d/j (t)

=

$ 64 1 − exp

−λR

 (%2 ' t+Δt   DS/DM,T  (t)−vT (t)dt vd/j t Dct

&

−λR

+1

 (%3 ' t+Δt   DS/DM,T  (t)−vT (t)dt vd/j t Dct

(6.87)

6.2 Space–Time–Frequency Non-stationary Modeling with Time-Space. . .

 & $  (%2 ' t+Δt   DS/DM,T   (t)−vT (t)dt  1 − exp −λR t vd/j −1  t Dc  DS/DM,T  .σ $ &  (%4 . time,d/j (t) =  ' t+Δt   DS/DM,T  T (t) −λ (t)−v v dt R d/j t  1024 1 − exp

149

(6.88)

Dct

The radii of observable spheres assigned to different dynamic clusters are different due to the multi-velocity of dynamic clusters. For dynamic clusters, shrinkage DS,T DM,T factors on the time axis .εtime and .εtime are introduced and can be given by DS,T (t) = D˜ Od

.

˜ DM,T (t) = .D Oj

DS,T DOd (t)

(6.89)

DS,T εtime DM,T (t) DOj DM,T εtime

.

(6.90)

The conditions that dynamic clusters .CdDS and .CjDM,T are time observable clusters are expressed as DS,T (t) < rtDS,T (t) D˜ Od

(6.91)

DM,T (t) < rtDM,T (t). D˜ Oj

(6.92)

.

.

Assuming that all dynamic clusters are observable at initial time .t0 and shrinkage factors are given by $ DS,T .ε time

= max

= max

%

rtDS,T (t0 ) $

DM,T .ε time

DS,T (t0 ) DOd

DM,T DOj (t0 )

rtDM,T (t0 )

, d = 1, 2, 3, · · · , D (t0 )

(6.93)

, j = 1, 2, 3, · · · , J (t0 ) .

(6.94)

%

Condition of the time observable dynamic single/twin-cluster is similar to that of time observable static single/twin-cluster. Therefore, the developed method considers the effects of VTD and continuously arbitrary trajectories. First, the developed method considers the difference in observable conditions caused by the difference in velocities between dynamic and static clusters. Second, continuous arbitrary trajectories of transceivers and dynamic clusters are characterized when computing time survival probabilities and radii of observable spheres assigned to static and dynamic clusters.

150

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

6.2.3 Soft Transition Factor The power of clusters is given as [13] type .Pn (t)

= exp

rτ − 1 type −τn (t) rτ DS



Zn

10− 10 ,

(6.95)

where .type means the type of clusters, i.e., .SS/DS/SM/DM, n is the n-th cluster, i.e., .s/d/i/j , .rτ is the delay scaling parameter, .DS is a random delay spread, and .Zn obeys the Gaussian distribution .N (0, 3). In the channel model [23], for each cluster, when it is switched from the observable state to the unobservable state, its power abruptly changes to 0, resulting in hard cluster power handover. On the contrary, when the cluster changes between observable and unobservable, its power changes continuously, i.e., soft cluster power handover. Meanwhile, when the center of transceiver antenna array is closer to the center of observable sphere, the power of the cluster assigned to this observable sphere is higher [24]. To mimic this phenomenon, a soft transition factor is introduced type,T/R

Pn

type,T/R

(t) = Ωn

.

= sin

2

$

π 2

type

(t)Pn & 1−

(t)

type,T/R (t) D˜ On type,T/R

rt

(t)

(%

Zn rτ − 1 type 10− 10 . exp −τn (t) rτ DS (6.96)

Since the soft transition factor obeys the squared sine function, which is a uniform continuity function, it can capture soft cluster power handover [18]. Only when the cluster is both an array observable cluster and a time observable cluster can it be an effective cluster, which contributes to the CIR. Note that the power of ineffective clusters is 0.

6.2.4 Frequency-Dependent Factor In mmWave channels, the US assumption, which is used in sub-6 GHz, no longer holds, leading to frequency non-stationarity [36]. To characterize channels in the frequency domain, TVTF is derived by adopting the Fourier transform to the CIR .hqp (t, τ ). To mimic frequency non-stationarity, the frequency-dependent path gain should be captured [37]. A frequency-dependent factor is employed in the TVTF to model frequency non-stationarity, which is given as  Hqp (t, f ) =

.

  K(t) LoS hLoS qp (t)exp −j 2πf τqp (t) K(t) + 1

6.3 Simulations and Discussions

 +  +  +  +  +

ηGR (t) K(t) + 1 ηSS (t) K(t) + 1 ηSM (t) K(t) + 1 ηDS (t) K(t) + 1 ηDM (t) K(t) + 1

151

 







f fc f fc f fc f fc f fc

ε

  GR hGR (t)exp −j 2πf τ (t) qp qp

ε  S(t) N s (t) 

  SS hSS qp,s,ns (t)exp −j 2πf τqp,s,ns (t)

s=1 ns =1

ε  I (t) N i (t)  i=1 ni =1

  SM hSM qp,i,ni (t)exp −j 2πf τqp,i,ni (t)

ε D(t) d (t)  N d=1 nd =1

ε  J (t) N j (t)  j =1 nj =1

  DS hDS (t)exp −j 2πf τ (t) qp,d,nd qp,d,nd   DM hDM qp,j,nj (t)exp −j 2πf τqp,j,nj (t) , (6.97)

where .ε denotes a frequency-dependent factor associated with vehicular scenarios [38]. Based on the measurement and analysis in [23, 39] the interaction between clusters and multipaths with different wavelengths results in the  frequency ε

dependence of rays. As a consequence, the frequency-dependence part . ffc has an impact on the NLoS component, which is generated by clusters and multipaths with different wavelengths. Similar to [21], the newly generated static/dynamic single/twin-clusters are taken into account. In summary, by using the visibility region-based method and constructing observable semi-spheres and observable spheres, the smooth cluster evolution in the space and time domains is modeled, thus mimicking space-time non-stationarity with T-S consistency. Due to the descent generality of the BD process method, it is embedded in the visibility region-based method. Soft transition factor and frequency-dependent factor are developed to model soft cluster power handover and frequency non-stationarity, respectively.

6.3 Simulations and Discussions 6.3.1 Statistical Properties Channel statistics, i.e., the STF-CF, PDP, time stationary interval, and DPSD, are derived.

152

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

6.3.1.1

Space–Time–Frequency Correlation Function

With the help of the derived TVTF in (6.97), the STF-CF is computed by [40] .

∗ ρqp,q  p (t, f ; Δt, Δf, δT , δR ) = E[Hqp (t, f )Hq  p (t + Δt, f + Δf )],

(6.98)

where .E[·] represents the expectation operation and .(·)∗ represents the complex conjugate operation. The LoS, GR, as well as static/dynamic single/multi-bouncing components are regarded as independent of each other [23]. Hence, the STF-CF can be expressed by ρqp,q  p (t, f ; Δt, Δf, δT , δR )

.

LoS GR = ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) + ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) SS DS + ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) + ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) SM DM + ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) + ρqp,q  p  (t, f ; Δt, Δf, δT , δR )

(6.99)

with LoS ρqp,q  p  (t, f ; Δt, Δf, δT , δR )  K(t)K(t + Δt) = (K(t) + 1)(K(t + Δt) + 1)

.

LoS × hLoS∗ qp (t)hq  p (t

+ Δt)e

  LoS (t)−(f +Δf )τ LoS (t+Δt) j 2π f τqp qp

(6.100)

GR ρqp,q  p  (t, f ; Δt, Δf, δT , δR )  ηGR (t)ηGR (t + Δt) = (K(t) + 1)(K(t + Δt) + 1)

.

GR × hGR∗ qp (t)hq  p (t

+ Δt)e

  GR (t)−(f +Δf )τ GR (t+Δt) j 2π f τqp qp

(6.101)

SS ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) =  ηSS (t)ηSS (t + Δt) (K(t) + 1)(K(t + Δt) + 1) . ⎡ ⎤ S(t) S(t+Δt) s (t) Ns (t+Δt)   N  SS SS ⎦ × E⎣ hSS∗ qp,s,ns (t)hq  p ,s  ,n (t + Δt)ξqp,s,ns (t) s=1

s  =1

ns =1

ns =1

s

(6.102)

6.3 Simulations and Discussions

153

DS ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) =  ηDS (t)ηDS (t + Δt) (K(t) + 1)(K(t + Δt) + 1) . ⎤ ⎡ D(t) (t+Δt) d (t) Nd   D(t+Δt)  N DS DS ⎦ hDS∗ × E⎣ qp,d,nd (t)hq  p ,d  ,n (t + Δt)ξqp,d,nd (t) d=1

d  =1

nd =1

d

nd =1

(6.103) SM ρqp,q  p  (t, f ; Δt, Δf, δT , δR ) =  ηSM (t)ηSM (t + Δt) (K(t) + 1)(K(t + Δt) + 1) . ⎤ ⎡ I (t) I (t+Δt) i (t) Ni (t+Δt)   N  SM SM ⎦ hSM∗ × E⎣ qp,i,ni (t)hq  p ,i  ,n (t + Δt)ξqp,i,ni (t) i=1

i  =1

ni =1

i

ni =1

(6.104) DM ρqp,q (t, f ; Δt, Δf, δ , δ ) =  p T R  ηDM (t)ηDM (t + Δt) (K(t) + 1)(K(t + Δt) + 1) . ⎡ ⎤ Nj (t) Nj (t+Δt) J (t) J (t+Δt)     ⎢ ⎥ DM DM × E⎣ hDM∗ qp,j,nj (t)hq  p ,j  ,n (t + Δt)ξqp,j,nj (t)⎦ j =1

j  =1

nj =1

j

nj =1

(6.105) with SS ξqp,s,n (t) = e s

.

DS .ξqp,d,n (t) d

SM .ξqp,i,n (t) i

=e

 (t+Δt)

q p ,s  ,ns

DS j 2π f τqp,d,n (t)−(f +Δf )τ DS  

=e

DM ξqp,j,n (t) = e j

.

SS SS j 2π f τqp,s,n s (t)−(f +Δf )τ  

d

i

q p ,i  ,ni

DM j 2π f τqp,j,n (t)−(f +Δf )τ DM   j



q p ,d  ,nd

SM j 2π f τqp,i,n (t)−(f +Δf )τ SM  

(6.106)

(t+Δt)

(6.107)

(t+Δt)

(t+Δt) q p ,j  ,nj

(6.108)

.

(6.109)

It is certain that the derived STF-CF is S-T-F-varying, which demonstrated S-T-F non-stationarity of the proposed IS-GBSM. The STF-CF is simplified to the SCCF at Tx/Rx through setting .Δt = 0, .Δf = 0, as well as .p = p /.p = p , .q = q  /.q = q  . The STF-CF can be simplified to the TACF through setting .Δf = 0, .p = p ,

154

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

as well as .q = q  . The STF-CF is simplified to the FCF through setting .Δt = 0,   .p = p , as well as .q = q .

6.3.1.2

Power Delay Profile

PDP shows the power and delay of MPCs and is given by S(t) N s (t)  

Θ(t, τ ) =

SS Ps,n (t)δ s

 τ

SS − τs,n (t) s



s=1 ns =1 .

+

I (t) N i (t)   i=1 ni =1

+

D(t) d (t)  N d=1 nd =1

 DS DS Pd,n (t)δ τ − τ (t) d,n d d

J (t) N j (t)     SM SM DM DM Pi,n (t)δ τ − τ (t) + P (t)δ τ − τ (t) . j,nj j,nj i,ni i j =1 nj =1

(6.110) Obviously, the derived PDP of the proposed IS-GBSM is time-varying.

6.3.1.3

Time Stationary Interval

The channel amplitude response is regarded as time stationary within the time stationary interval, which is the maximum period when the ACF of the PDP exceeds a threshold .threshold , e.g., 80% [41]. The time stationary interval is expressed by (t) = inf{Δt|Υ (t,Δt)≤threshold },

.

(6.111)

where .Υ (t, Δt) represents the auto-correlation function of the PDP '

Υ T (t, Δt) =

.

6.3.1.4

Θ(t, τ )Θ(t + Δt, τ )dτ ' . max{ Θ 2 (t, τ )dτ, Θ 2 (t + Δt, τ )dτ } '

(6.112)

Doppler Power Spectral Density

The DPSD is obtained through taking Fourier transfer of TACF and is expressed by  .

Ωqp (t; fd ) =

+∞ −∞

ρqp (t; Δt)e−j 2π fd Δt d(Δt),

where .ρqp (t; Δt) is the TACF and .fd is the Doppler shift.

(6.113)

6.3 Simulations and Discussions

155

6.3.2 Model Simulation Typical channel statistical properties are simulated and compared with the measurement/RT-based results. Key channel-related parameters are given below. The ULA array is utilized in the simulation and the adjacent antenna spacing is .δT = δR = 0.5 λ. The frequency-dependent factor is .ε = 1.45 [38]. Abstracted SM (t) and .τ˜ DM delays .τ˜qp,i,n qp,j,nj (t) obey the exponential distribution with the mean i and variance 80 ns and 15 ns [42]. The number of rays in clusters obeys Poisson distribution with mean and variance .λ˜ = 15 [43]. The heights of transceiver antenna are .ξT = ξR = 2 m. Other key channel-related parameters are listed in Table 6.1. In Fig. 6.3, the TACF under sparse/rich SE with the acceleration/deceleration VMT of transceivers is illustrated. In the rich SE, it is obvious that the number of twin-clusters is larger than that of single-clusters. In this case, compared to the sparse SE, the number of propagation paths is also larger in the rich SE. As a result, the rich SE is more complex and the corresponding TACF is lower than that under the sparse SE. TACF with deceleration VMTs is higher than that with acceleration VMTs. The underlying reason is that the deceleration VMTs of transceivers leads to more stable V2V communication environment. The DPSD under sparse/rich SE with high/low VTD is given in Fig. 6.4. Among these DPSDs, the DPSD under rich SE with high VTD exhibits the flattest distribution, while the DPSD under sparse SE with low VTD exhibits the steepest distribution. The philosophy is that the received power in V2V communications under rich SE with high VTD tends to come from clusters in all directions. Nonetheless, in the sparse SE with low VTD, the received power concentrates on several Doppler frequencies. Figure 6.5 presents the FCF under sparse/rich SE at different frequencies. Owing to the modeling of the frequency-dependent path gain, the FCF is frequency-variant, where the FCF at .f = 29 GHz is lower than that at .f = 27 GHz. The FCF is a function of frequency separation .Δf and frequency f , thus mimicking frequency non-stationarity and frequency selectivity. In contrast, the FCF under rich SE is lower than that under sparse SE. This is because in rich SE, the multi-bouncing propagation is more dominant and the delay spread is severer, resulting in more obvious channel frequency selectivity and lower FCF. In Fig. 6.6, the Rx SCCF under sparse/rich at different reference antennas is depicted. The numbers of Tx and Rx antennas are 32 and 40, respectively, which can be regarded as a massive MIMO scenario [34]. The SCCF depends on both the antenna spacing and reference antennas, and hence space non-stationarity brought by massive MIMO is imitated. In addition, compared to the rich SE, the SCCF under the sparse SE is higher. The physical reason is that more single-clusters lead to smaller number of propagation paths, smaller channel spatial diversity, and higher SCCF. Figure 6.7 gives the complementary cumulative distribution function (CCDF) of time stationary interval under sparse/rich SE with high/low VTD. Since there are more highly mobile vehicles in high VTD, the V2V communication scenario

2

.v ¯d

T

R

.Ap , Aq

.A2 , A31

T

R

.A20 , A1

t (s)

R

0

.VDI (0)

T

(a): .6/11 (b): .11/6 (a): .7/10 (b): .10/7 1

(a): .1/2 (b): 2 .4/5

.SDI (0)

.8/8

(0) (m/s)

3

Figure 6.4 28 2 .32, 32 .0.03 120 .14, 15

.6/7

.v ¯j

DM,T/R

DS (0)

(m/s)

Figure 6.3 28 2 .32, 32 .0.04 100 .15, 20

.fc

Symbol (GHz) .BW (GHz) .MT , MR K .Dcen (0) (m) T R .v (0), v (0) (m/s)

T

R

.A1 , A1

2

(a): .3/5 (b): .5/3 .7/9

.9/11

2

Figure 6.5 28 2 .32, 32 .0.05 70 .22, 17

Table 6.1 Key channel-related parameters in the simulation

(a): .AT1 , AR 8 (b): .AT1 , AR 13

.1.5

(a): .7/12 (b): .12/7 .7/12

.16/16

5

Figure 6.6 38 4 .32, 40 .0.1 130 .25, 24

T

R .A1 , A1

(a): .4/5 (b): .5/4 (a): .7/11 (b): .11/7 0

.12/12

.3.5

Figure 6.7 38 4 .40, 32 .0.09 150 .10, 12

(a): .0.03 (b): 0 (a): -, .AR 7 (b): -,-

1 (b): .11/6 1

.12/13

4

Figure 6.8 28 2 .32, 40 .0.1 125 .11, 9

Figure 6.10

.A1 , A2

.A1 , −

R

0 T

.7/10

(a): .2/3 (b): .3/2 0 T

(a): .6/11

.12/12

4

Figure 6.13 28 2 .32, 32 .0.06 85 .12, 10

.8/7

.8/8

2

.0, 16

500

.0.1

.1, 16

.0.015

.1.8

156 6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

6.3 Simulations and Discussions

157

1

Rich SE, acceleration Sparse SE, acceleration Rich SE, deceleration Sparse SE, deceleration

0.9

Normalized absolute TACF

0.8 0.7 0.6

Highest

0.5 0.4 0.3 0.2

Lowest 0.1 0 0

0.005

0.01

0.015

Time separation,

0.02

0.025

0.03

t (s)

Fig. 6.3 TACF under sparse/rich SE with the acceleration/deceleration VMT

Rich SE, high VTD Rich SE, low VTD Sparse SE, high VTD Sparse SE, low VTD

Steepest

Normalized DPSD (dB)

-10

-15

Flattest

-20

-25

-30

-35 200

250

300

350

400

450

Doppler shift, f (Hz) Fig. 6.4 DPSD under sparse/rich SE with high/low VTD

is more rapidly-changing, resulting in lower time stationary interval. Furthermore, attributed to more complex propagation in rich SE, the time stationary interval under rich SE is lower than that under sparse SE. Therefore, the V2V channel under rich SE with high VTD exhibits the smallest time stationary interval, whereas the largest time stationary interval under sparse SE with low VTD.

158

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . . 1

Highest

0.9

Normalized absolute FCF

0.8 0.7 0.6 0.5 0.4 0.3

Rich SE, 27 GHz Rich SE, 29 GHz Sparse SE, 27 GHz Sparse SE, 29 GHz

0.2 0.1 0 0

5

10

Lowest

15

20

Frequency separation,

25

30

35

f (MHz)

Fig. 6.5 FCF under sparse/rich SE at different frequencies

Normalized absolute SCCF

1 0.9

Rich SE, A R

0.8

Rich SE,

R

Sparse SE

0.7

8 AR 13

Sparse SE, A 8

Sparse SE, A R

0.6

13

0.5 0.4 0.3 0.2 0.1

Rich SE 0 0

1

2

3

4

Normalized antenna spacing,

5

6

/ R

Fig. 6.6 SCCF under sparse/rich SE at different frequencies

Cluster time and array evolution are shown in Fig. 6.8a, b, respectively. The numbers of static/dynamic single/twin-clusters are .S(0) = D(0) = I (0) = J (0) = 5 at initial time. In Fig. 6.8a, the simulation time is from .0.03 s to .0.28 s and the selected antenna is .AR 7 , where newly generated clusters can be observed. In

6.3 Simulations and Discussions

159

1

Rich SE, high VTD Rich SE, low VTD Sparse SE, high VTD Sparse SE, low VTD

0.9 0.8 0.7

CCDF

0.6 0.5

Largest

0.4 0.3 0.2 0.1

Smallest

0 0

20

40

60

80

100

120

140

160

Time stationary interval (ms)

Fig. 6.7 CCDF of TSI under sparse/rich SE with high/low VTD

Fig. 6.8b, the simulation time is the initial time, i.e., .t = 0 s. By exploiting the developed method, cluster evolution is smooth and consistent as time and array evolve. In this case, it is clear that adjacent time instants and antenna elements have similar observable cluster sets, and hence channel T-S consistency is embedded in the proposed IS-GBSM.

6.3.3 Model Validation by Measurement and RT-Based Results To validate the generality of the proposed model, we conducted a vehicular channel measurement in a vicinity of Aalborg, Denmark with the carrier frequency .fc = 1.8 GHz and bandwidth 15 MHz [44]. It is clear that .1.8 GHz is a typical frequency band of LTE V2X and is utilized for the carrier frequency of vehicular channel measurement. As shown in Fig. 6.9a, a RAA with sixteen antennas was equipped on the roof of a van, which can be regarded as a multi-antenna vehicular measurement. In the measurement, many routes were considered, and the dominant rural in Route 1, i.e., green dots in Fig. 6.9b, was chosen for comparison with the simulation results. The corresponding measured CIR is processed to obtain the measured SCCF, which is compared with the simulated SCCF in Fig. 6.10.1 Furthermore, since the

1 Although the proposed IS-GBSM is based on the ULA and the vehicular channel measurement utilizes the uniform circular array, the antenna array used in the simulation does not affect the output curve. This is because that power angular spectrum is a propagation channel characteristic, which is physically determined by the stochastic properties of the clusters exist in the scenario. Meanwhile, SCCF and power angular spectrum are Fourier transform pairs. Consequently, the comparison of simulated SCCF and measured SCCF is proper.

160

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

Static sinlge-clusters Dynamic single-clusters Static twin-clusters Dynamic twin-clusters

30

Cluster index

25

20

15

10

5

0 0

0.05

0.1

0.15

0.2

0.25

0.3

Time instant (s)

(a)

Static sinlge-clusters Dynamic single-clusters Static twin-clusters Dynamic twin-clusters

25

Cluster index

20

15

10

5

0 0

5

10

15

20

25

30

35

40

Antenna index

(b)

Fig. 6.8 Cluster evolution at Rx. (a) Time axis. (b) Array axis

measurement scenario was a dominant rural, it can be regarded as a low VTD scenario. The simulated SCCF with channel consistency at low VTD fits well with measurement. Nonetheless, as the spatial correlation is underestimated by randomly characterizing the cluster array evolution, the simulated SCCF without channel consistency is lower than the measurement. Also, the simulated SCCF with high VTD is lower than the measurement. This is because more vehicles, i.e., more

6.3 Simulations and Discussions

161

Fig. 6.9 Vehicular channel measurement campaign. (a) Measurement van with a RAA. (b) Measurement scenario 1 0.9

Fit well

Normalized absolute SCCF

0.8 0.7 0.6 0.5 0.4

Low VTD, consistency Low VTD, non-consistency High VTD, consistency Measurement resutls

0.3 0.2 0.1 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Antenna spacing (m)

Fig. 6.10 Comparison of the simulated SCCF and measured SCCF in [44]

dynamic clusters, around the transceiver lead to more propagation paths, larger channel spatial diversity, and smaller SCCF. As a result, by considering the effect of VTDs and capturing channel consistency, the proposed model can accurately reflect the reality. To validate the existence of frequency selectivity in the channel measurement scenario [44] with the carrier frequency .fc = 1.8 GHz and bandwidth 15 MHz, the measured CIR is processed to obtain the measured FCF. From Fig. 6.11, it can be observed that the measured FCF depends on the frequency separation .Δf and exhibits fluctuations in the frequency domain, which validates the existence of frequency selectivity in the channel measurement scenario. This is because that the multipath effect is obvious in complex vehicular communication scenarios, where the channel coherence bandwidth is small. In this case, the channel can exhibit frequency selectivity within the measurement bandwidth. Furthermore, the measured and simulated FCFs are compared. The close agreement between the measured FCF and the simulated FCF at low VTD is achieved. However, the FCF

162

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . . 1 0.9

Fit well

Normalized absolute FCF

0.8 0.7 0.6 0.5 0.4 0.3

Low VTD, simulation High VTD, simulation Measurement resutls

0.2 0.1 0 0

5

10

15

Frequency separation (MHz) Fig. 6.11 Comparison of the simulated FCF and the measured FCF in [44]

at high VTD is lower than that at low VTD as the V2V channel is more complex at high VTD. To further verify the accuracy of the developed IS-GBSM, the RT-based method is used. Since the RT-based method generates channel parameters according to the geometrical optics and uniform theory of diffraction, the RT-based result is of high fidelity and is widely utilized to validate the accuracy of the proposed channel models [45, 46]. Specifically, a massive MIMO mmWave V2V communication scenario is constructed by the RT tool, i.e., Wireless InSite [47]. In Fig. 6.12a, the scenario is given, which is in San Jose, California, the United States. In Fig. 6.12b, the propagation environment is depicted, where there are dynamic vehicles and static trees and buildings. In the Wireless InSite simulation platform, the carrier frequency is set to .fc = 28 GHz with 2 GHz bandwidth for mimicking the high frequency and wide band in mmWave communications. Furthermore, the numbers of antenna elements are .MT = MR = 32. The spacing of adjacent antennas is half wavelength. The simulated and RT-based DPSDs are compared in Fig. 6.13. The propagation environment can be regarded as a rich SE with high VTD according to Fig. 6.12b. The simulated DPSD under rich SE with channel consistency fits well with the RT-based DPSD. Nonetheless, when the BD process method used in [11, 20, 23] is exploited, the evolution of clusters is random rather than smooth, resulting in channel non-consistency. Under this condition, the DPSD distribution is flatter, attributed to more random and dispersed distribution of clusters in the environment.

6.4 Summary

163

Fig. 6.12 V2V communication scenario in Wireless Insite. (a) Scenario in San Jose, California, the United States. (b) Transceiver propagation environment

Also, compared to rich SE, the simulated DPSD under sparse SE has a steeper distribution. Consequently, it can be concluded that the necessities of capturing channel consistency and rich/sparse SE are demonstrated and the proposed model can accurately reflect the reality.

6.4 Summary This chapter has proposed a novel 3D 6G channel model for mmWave massive MIMO V2V channels, where the mixed-bouncing propagation, GR, V2V scenarios with different VTDs, and continuously arbitrary VMTs have been considered. A new method has been developed, which can simultaneously imitate S-T-F nonstationarity of V2V channels with T-S consistency. Key channel statistics have been derived. Simulation results have shown that S-T-F non-stationarity and T-S consistency have been captured. Compared to low VTD, channels in high VTD have flatter distribution of DPSD, smaller time stationary interval, and lower SCCF. Meanwhile, compared to rich SE, channels in sparse SE have higher TACF, steeper

164

6 A 3D Mixed-Bouncing IS-GBSM with Time-Space Consistency for mmWave. . .

Rich SE, consistency Rich SE, non-consistency Sparse SE, consistency RT-based results

Normalized DPSD (dB)

-10

-15

Fit well -20

-25

-30

-35 200

250

300

350

400

450

500

550

600

Doppler shift, f (Hz) Fig. 6.13 Comparison of the simulated DPSD and RT-based DPSD

distribution of DPSD, higher FCF, larger time stationary interval, and higher SCCF. The comparison between simulation results and measurement/RT-based results has verified the accuracy of the proposed model. The comparison has demonstrated that the capturing of channel consistency, high/low VTD, and rich/sparse SE is indispensable. In the future, with the development of measurement equipment, it is necessary to carry out the massive MIMO mmWave V2V channel measurement campaign to further validate the proposed IS-GBSM for 6G massive MIMO mmWave V2V channels.

References 1. X. Cheng, Z. Huang, S. Chen, Vehicular communication channel measurement, modelling, and application for beyond 5G and 6G. IET Commun. 14(19), 3303–3311 (2020) 2. Z. Huang, X. Cheng, N. Zhang, An improved non-geometrical stochastic model for nonWSSUS vehicle-to-vehicle channels. ZTE Commun. 17(4), 62–71 (2019) 3. J. Karedal et al., A geometry-based stochastic MIMO model for vehicle-to-vehicle communications. IEEE Trans. Wireless Commun. 8(7), 3646–3657 (2009) 4. C. Huang, R. Wang, C.-X. Wang, P. Tang, A.F. Molisch, A geometry-based stochastic model for truck communication channels in freeway scenarios. IEEE Trans. Wireless Commun. 70(8), 5572–5586 (2022) 5. I. Sen, D.W. Matolak, Vehicle-vehicle channel models for the 5-GHz band. IEEE Trans. Intell. Transp. Syst. 9(2), 235–245 (2008)

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Chapter 7

Conclusions and Future Research Directions

Since the end of last century, V2V communications have received the increasing attention. Aiming at properly and successfully supporting the design and performance evaluation of V2V wireless communication systems, both the comprehensive understanding of V2V wireless communication channel characteristics and the accurate and easy-to-use V2V wireless communication channel models are indispensable. In this chapter, the contents of Chaps. 1–6 are adequately summarized. Then, important research directions, which can be regarded as the guidelines for developing more precise and easy-to-use models for V2V wireless communication channels, are presented from the perspectives of the channel measurement, channel modeling, as well as channel application.

7.1 Conclusions For easy reading, the conclusions drawn by Chaps. 1–6 are presented as follows.

7.1.1 Discussions and Summary of Chap. 1 In Chap. 1, the overview of vehicular communications, such as the introduction VANET as well as the introduction of C-V2X, has been given. To support the vehicular communication system design, the complicated knowledge of vehicular communication channels is essential [1–3]. According to the vehicular channel measurements and analysis in [4–8], there are four unique physical features in V2V communications, including low-elevation antennas mounted on both the Tx and Rx vehicles, high-mobility characteristics of transceiver vehicles and clusters with complex vehicular movement trajectories (VMTs), different vehicular traffic © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8_7

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densities (VTDs) in different communication scenarios, as well as the existence of the LoS component. Note that the aforementioned four unique physical features in V2V communications lead to unique vehicular channel characteristics, such as the channel non-stationarity in the time domain, i.e., time non-stationary characteristic, the severe fading characteristic, and different channel statistical properties in V2V communication scenarios with different VTDs and VMTs [4, 6, 9]. In order to mimic the aforementioned unique physical features and characteristics of V2V communication channels, a large number of V2V channel models have been developed [10–14]. In light of modeling approaches, existing V2V channel models can be classified into the GBDM, NGSM, and GBSM [15–17]. It is noteworthy that GBSMs can be further divided into RS-GBSM and IS-GBSMs, depending on whether clusters are located on regular shapes or the irregular shape.

7.1.2 Discussions and Summary of Chap. 2 In Chap. 2, a time–frequency non-stationary NGSM for V2V wideband communication channels has been developed. To mimic the presence of the LoS component, a non-uniformly distributed tap phase has been generated. In addition, by modifying the ACF of CIR, the variable types of Doppler spectra for taps with different delays have been captured due to the high-mobility characteristics of V2V channels. To further model time–frequency non-stationarity, the Markov chains have been adopted to capture the appearance and disappearance of paths, and the complex correlation between amplitude and phase of different taps has been modeled via the Cholesky decomposition. Then, Weibull distributed tap amplitude has been generated to imitate the severe fading characteristics of V2V channels. Based on the developed NGSM, some significant channel statistical properties in terms of the PDP, tap correlation coefficient matrix, and DPSD have been derived. By simulating the derived PDP, tap correlation coefficient matrix, and DPSD, it has been demonstrated that, different from the existing NGSMs in [4, 18], the proposed NGSM has the ability to capture the presence of the LoS component, variable types of Doppler spectra, time–frequency non-stationarity, and severe fading characteristics. Finally, the excellent agreement has been achieved between the simulation results and the available measurement data, confirming the accuracy of the proposed NGSM.

7.1.3 Discussions and Summary of Chap. 3 In the future B5G/6G wireless communication network, more investigation needs to be given for V2V communications. As mentioned in [19], the massive MIMO technology is expected to be utilized in V2V communications. Attributed to the large-scale antenna array, the appearance and disappearance of clusters in the

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Fig. 7.1 Geometrical representation of the proposed massive MIMO V2V RS-GBSM in Chap. 3

environment on the array axis need to be mimicked [20, 21]. In such a condition, it is necessary to model the space non-stationarity in the massive MIMO channels [22–24]. Also, the rapidly changing V2V channels result in the significant time nonstationary characteristics of V2V channels. To sufficiently support the design of massive MIMO V2V communication system, a space-time non-stationary massive MIMO V2V channel model is essential. Considering the advantages of RS-GBSMs with low complexity, a spacetime non-stationary massive MIMO V2V channel model based on the RS-GBSM modeling approach with uniform planar antenna arrays (UPAs) has been developed. The developed RS-GBSM has combined the 3D multi-confocal semi-ellipsoids and semi-spheres, as shown in Fig. 7.1. In the developed RS-GBSM, the CIR consists of the LoS component, the GR component, and the double-bounced components. By dividing clusters in the propagation environment into static clusters and dynamic clusters, the effect of VTDs on channel statistical properties has been explored adequately. In the developed RS-GBSM, a novel method, the so-called BD process and seed algorithm based selective cluster evolution, has been proposed. Based on the developed method, both the channel space non-stationarity of massive MIMO with UPA and V2V channel time non-stationarity have been properly mimicked. From the proposed massive MIMO V2V RS-GBSM, typical channel statistical properties, including the S-T CF, SCCF, time TACF, as well as DPSD, have been derived and thoroughly investigated. Simulated SCCFs and TACFs have demonstrated that the proposed RS-GBSM has the ability to mimic the spacetime non-stationarity. Moreover, the evolution of different clusters on the UPA has shown the proper capturing of the appearance and disappearance of clusters

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in the environment. In addition, simulation results have demonstrated that the proposed RS-GBSM under low VTDs has lower SCCFs, higher TACFs, and more steeply distributed DPSDs than that under high VTDs. Finally, the close agreements between simulation results and available measurement data on SCCFs, TACFs, and DPSDs have verified the utility and generality of the proposed RS-GBSM.

7.1.4 Discussions and Summary of Chap. 4 In the B5G/6G V2V wireless communication system, the massive MIMO technology, which can achieve a descent SNR, mitigate multipath fading, and increase the channel capacity, is expected to be utilized. Therefore, a precise V2V channel model with low complexity is essential for designing B5G/6G massive MIMO V2V wireless communication systems. As previously mentioned, in the developed V2V channel model, the channel non-stationarity in the space-time domain, i.e., spacetime non-stationarity, needs to be characterized. In Chap. 4, a space-time non-stationary massive MIMO V2V channel model based on the IS-GBSM modeling approach has been proposed. The geometrical representation of the proposed IS-GBSM can be depicted in Fig. 7.2. In the ISGBSM modeling approach, positions of clusters in the environment are assumed to obey a specific statistical distribution, which is required to be obtained from detailed measurements. Because of the demand for carrying out channel measurements, the accuracy of IS-GBSM is higher than that of RS-GBSM. Considering the descent

Fig. 7.2 Geometrical representation of the proposed massive MIMO V2V IS-GBSM in Chap. 4

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trade-off between the accuracy and complexity, the IS-GBSM modeling approach has been extensively adopted in the current standardized channel models in [25– 28]. In the proposed massive MIMO V2V IS-GBSM, clusters in the propagation environment are distinguished into dynamic clusters and static clusters, where the impact of VTD on V2V channel statistics is investigated. To jointly model spacetime non-stationarity in massive MIMO V2V channels, a novel VTD-combined time-array cluster evolution algorithm has been proposed. Several expressions related to dynamic clusters and static clusters have been distinguished, including the expression of generation/recombination probability in the BD process, and the position distribution of clusters in different VTD scenarios. By computing key model-related parameters via 3D vectors, the spherical wavefront propagation has been described sufficiently. From the proposed massive MIMO V2V IS-GBSM, the expressions of some channel statistical properties, including the STF-CF and the DPSD, have been derived. Through the simulation, it has been found out that both VTDs and vehicular movement directions have significant impacts on TACFs and the distribution of DPSDs. In addition, the simulated TACF, SCCF, and observable cluster map have proved that the proposed IS-GBSM can effectively describe the space-time nonstationarity of the massive MIMO V2V wideband channels. It is worth mentioning that the aforementioned observations can provide some valuable suggestions for acquiring more stable channel characteristics. Finally, the excellent agreement between the simulation result and measurement data has demonstrated the accuracy of the proposed IS-GBSM.

7.1.5 Discussions and Summary of Chap. 5 In the future 6G era, V2V communications aim to meet the demand of extremely low latency, high throughput, as well as high reliability. Aiming at meeting these demands, the mmWave communication with ultra-wideband and massive MIMO technology with large-scale antenna array need to be simultaneously utilized in the 6G V2V wireless communication system. Note that, as the enabled foundation of any communication system design, precise and easy-to-use models, which can mimic the underlying V2V channel characteristics adequately, for 6G mmWave massive MIMO V2V channels are essential. In Chap. 5, a new 3D mmWave massive MIMO IS-GBSM has been proposed for 6G V2V channels and can be depicted in Fig. 7.3. In the proposed mmWave massive MIMO IS-GBSM, both the spherical wavefront propagation and high delay resolution have been modeled. Specifically, resolvable rays within each cluster have been captured, and geometrical propagation related to each ray within each cluster has been modeled. To investigate the effect of VTD on V2V channel statistics, clusters are distinguished into static and dynamic clusters. Furthermore, the time-variant acceleration and the integration of time during the transmission distance update have been exploited. Consequently, the continuously arbitrary

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Fig. 7.3 Geometrical representation of the proposed mmWave massive MIMO V2V IS-GBSM in Chap. 5

VMTs, including quarter turn, U-turn, and curve driving, of the transceiver and dynamic clusters have been captured. The 6G mmWave massive MIMO V2V channel is expected to exhibit the obvious S-T-F non-stationarity [15, 29–31]. In this case, the mmWave massive MIMO V2V channel statistics vary in the S-T-F domain. To jointly model the ST-F non-stationarity of 6G high-mobility mmWave massive MIMO V2V channels, a new method, named as the selective cluster evolution based S-T-F non-stationary modeling method, has been proposed. The impacts of VTD and continuously arbitrary VMT on the S-T-F non-stationary modeling have been taken into account. Specifically, both the selective evolution of static clusters and dynamic clusters together with the BD process have been utilized, thus capturing the space-time nonstationarity. Furthermore, a frequency-related factor has been introduced to mimic the frequency-dependent path gain, thus modeling the frequency non-stationarity. According to the proposed 6G high-mobility mmWave massive MIMO V2V ISGBSM, important channel statistical properties, including STF-CF, DPSD, as well as time stationary interval, have been derived. The simulated SCCF, TACF, and FCF have demonstrated that the channel S-T-F non-stationarity has been properly captured. Simulation results have also shown that both the VTD and the VMT have significant influences on channel statistical properties. Compared to low VTDs, V2V channels in high VTDs have exhibited lower TACF. When the acceleration of the transceiver is smaller, the V2V channel has exhibited higher TACF, larger time stationary interval, and steeper distribution of DPSD. The comparison between simulation results and measurement has verified the accuracy and generality of the proposed IS-GBSM. Also, the comparison has indicated that the division of

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Fig. 7.4 Geometrical representation of the proposed mixed-bouncing mmWave massive MIMO V2V IS-GBSM with time-space consistency in Chap. 6

static clusters and dynamic clusters together with the modeling of various VMTs is essential.

7.1.6 Discussions and Summary of Chap. 6 One key technology of the ITS is the V2V communication, which can facilitate diverse applications. In the upcoming 6G era, V2V communication system needs to meet the demands of significantly low latency, high throughput, as well as high reliability. Toward this objective, massive MIMO and mmWave technologies should be employed in the 6G V2V system. For a proper design and performance evaluation of 6G V2V systems, realistic and easy-to-use 6G V2V channel models need to be developed [3, 32]. Note that V2V communications are extremely dynamic, where transceiver and the surrounding vehicles are moving at high speed. The actual V2V propagation environment has two unique features, i.e., various VTDs and complex VMTs, and a distinctive SE, i.e., the existence of mixed-bouncing propagation and GR, which distinctly affect the channel statistical properties according to the channel measurement and analysis [4, 15, 33]. In Chap. 6, a novel 6G V2V massive MIMO mmWave IS-GBSM has been proposed, and the corresponding geometrical representation can be given in Fig. 7.4. The proposed IS-GBSM can be regarded as a 6G channel model by modeling ST-F non-stationarity of channels with time–space consistency in consideration of the impacts of VTD, continuously arbitrary VMTs, and mixed-bouncing propagation. To model the mixed-bouncing propagation and describe the V2V scenarios

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with different VTDs, clusters have been divided into static/dynamic clusters and single/twin-clusters. Two indexes have been given to quantitatively characterize the ratios of the numbers of static/dynamic and single/twin-clusters. Moreover, continuously arbitrary VMTs of dynamic clusters and transceivers are further mimicked by the integration of time and time-varying acceleration. In the proposed 6G V2V massive MIMO mmWave IS-GBSM, a new method, which can model S-T-F non-stationarity with time–space consistency in consideration of the effects of VTD, continuously arbitrary VMT, and mixed-bouncing propagation, has been developed for the first time. The developed method has properly integrated the two typical methods, i.e., visibility region methods and BD process methods. A soft transition factor and a frequency-dependent factor have been proposed to capture the soft cluster power handover and frequency non-stationarity, respectively. Based upon the proposed 6G V2V massive MIMO mmWave IS-GBSM, essential channel statistical properties, including STF-CF, PDP, time stationary interval, and DPSD, have been obtained and analyzed sufficiently. Simulation results have shown that S-T-F non-stationarity and time–space consistency have been captured. Compared to low VTD, channels in high VTD have flatter distribution of DPSD, smaller time stationary interval, and lower SCCF. Meanwhile, compared to rich SE, channels in sparse SE have higher TACF, steeper distribution of DPSD, higher FCF, larger time stationary interval, and higher SCCF. The comparison between simulation results and measurement/RT-based results has verified the accuracy of the proposed model. The comparison has demonstrated that the capturing of channel consistency, high/low VTD, and rich/sparse SE is essentially indispensable.

7.2 Future Research Directions The essential research direction, which can be regarded as the guideline for proposing more proper models for V2V wireless communication channels, is elaborated. The future research direction is given from the channel measurement perspective, the channel modeling perspective, and channel application perspective in the sequel.

7.2.1 Channel Measurement Perspective The V2V measurement campaign is beneficial for the construction and validation of an effective V2V channel model and the evaluation of communication system performance and network planning [34, 35]. However, due to the limited measurement equipment and platforms, current V2V channel measurement campaigns are not adequate to support the research of developing proper V2V channel models in B5G/6G. From the perspective of channel measurement, the main open issues are presented below.

7.2 Future Research Directions

7.2.1.1

175

Measurement Platform Establishment

The emergence of novel application scenarios and technologies will bring huge challenges for the V2V channel measurement in B5G/6G. Specifically, the method of VAAs cannot be adopted to V2V ultra-massive MIMO channel measurement campaigns. The philosophy is that the single antenna utilized for VAA measurements needs to be moved to all required locations within the channel coherence time. Attributed to the severe Doppler shift in mmWave-THz high-mobility V2V communication scenarios, the channel coherence time is significantly short [36]. In such a condition, it is remarkably difficult to implement the V2V ultra-massive MIMO measurement based on VAA method. Alternatively, RAA methods can be exploited to ultra-massive MIMO V2V communication measurement instead of VAA methods. Nevertheless, based upon RAA methods, the measurement cost will distinctly increase, and a complex and precise calibration for the V2V ultra-massive MIMO antenna array is indispensable. In addition, the consideration of mmWaveTHz communications makes the application of horn antenna RAA, which needs to be meticulously controlled electronically. As a result, the measurement cost as well as measurement complexity will further increase. Owing to the exceedingly huge difficulty of the establishment of proper measurement platforms, measurement campaigns related to the ultra-massive MIMO channels and mmWave-THz channels are rare. In [20], an ultra-massive MIMO channel measurement campaign was performed in the indoor office environment. Based on the VAA method at 11, 16, 28, and 38 GHz frequency bands, four numbers of antenna elements consisting of .51 × 51, .76 × 76, .91 × 91, and .121 × 121, with uniform rectangular array (URA) were considered and utilized. In the measurement campaign [20], the parameter drifting over the ultra-large-scale array was investigated. Furthermore, a measurement campaign of mmWave-THz channels at carrier frequency 300 GHz with the high directive antenna for two distinct indoor scenarios was carried out in [37]. The measurement in [37] investigated the channel transfer function, pathloss, as well as root-mean-square (RMS) delay spread. However, the aforementioned measurement campaigns were not carried out in high-mobility V2V communication scenarios. Note that ultra-massive MIMO and mmWave-THz channels in high-mobility V2V communication scenarios will experience the obvious S-T-F non-stationarity and time–space consistency. As the fundamental of investigating and capturing them, it is necessary and urgently required to establish a proper measurement platform related to ultra-massive MIMO mmWave-THz channels under high-mobility V2V communication scenarios.

7.2.1.2

Measurement of Complicated Channel Non-stationarity and Consistency

When ultra-massive MIMO and mmWave-THz technologies simultaneously utilized to highly dynamic V2V communication scenarios, channels will exhibit the obvious S-T-F non-stationarity and time–space consistency. It is noteworthy that

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the S-T-F non-stationarity is a typical channel characteristic and the time–space consistency is an inherent channel physical feature. The existence of them has been unveiled via channel measurement campaigns from diverse perspectives. A channel measurement campaign was conducted in [38], where the virtual planar antenna array with omni-directional antenna was exploited. The measurement result demonstrated the space non-stationarity as channel statistics changed in the aperture of antenna array. The authors in [20] conducted the channel measurement in various scenarios under multiple frequency bands. The measured time-variant channel statistics and smoothly varying LoS power indicated the time non-stationarity and time consistency, respectively. In [21], a measurement campaign utilizing a linear array base station (BS) and 26 different user positions under the LoS case and 10 different user positions under the NLoS case was carried out. The smooth variation of angular power spectrum over the antenna array was measured, where the space consistency can be observed. In [4], the authors measured channels in various V2V scenarios, where the correlated scattering in the environment was observed to demonstrate the existence of frequency non-stationarity. Although the aforementioned measurement campaigns can show the presence of S-T-F non-stationarity and time–space consistency, they cannot support the modeling of S-T-F non-stationarity of channels with time–space consistency. Specifically, the complicated evolution of clusters in both the time domain and space domain and the corresponding variation of channel statistical properties in the space, time, and frequency domains were not measured and analyzed thoroughly. However, the measurement of cluster evolution and variations of channel statistics can notably contribute to more accurate capture of S-T-F non-stationarity and time–space consistency. This further helps in establishing more realistic B5G/6G ultra-massive MIMO mmWave-THz V2V channel models. In the future, more channel measurement campaigns that have the ability to investigate the complicated cluster evolution in the environment and the exhaustive variation of ultra-massive MIMO mmWave-THz V2V channel statistical properties are exceedingly required.

7.2.2 Channel Modeling Perspective In the future B5G/6G era, the introduction of emerging application scenarios and technologies and the emergence of ISAC systems bring great challenges to the capture of channel non-stationarity and consistency in V2V communication channels [15]. The future challenges and potential research directions for the successful capture of channel non-stationarity and consistency and the development of accurate V2V channel models are discussed from the perspective of channel modeling as follows.

7.2 Future Research Directions

7.2.2.1

177

Capturing Non-stationarity and Consistency of mmWave-THz Ultra-Massive MIMO V2V Channels

In the upcoming B5G/6G era, the ultra-massive MIMO technology is predictable to be adopted [39]. By multiplexing a large number of parallel data streams on the same frequency channel, ultra-massive MIMO can achieve the ultra-high SE [40]. In addition, the energy efficiency can be enhanced, and latency can be reduced via the ultra-massive MIMO technology. Meanwhile, the mmWave-THz communication is celebrated as an important enabling technology for the B5G/6G communication system, which is expected to integrate the diverse data-demanding and delay-sensitive application [41, 42]. Note that the mmWave-THz communication has the capability to exploit the large available communication bandwidth to attain a terabit per second data rate without the supplementary spectrum efficiency enhancement technologies [43]. Fortunately, the ultra-massive MIMO and mmWave-THz technologies also exhibit a symbiotic relationship. An ultra-large number of antenna elements can be embedded in a few square millimeters at the mmWave-THz frequency, and ultra-massive MIMO in turn provides high beamforming gain to account for the increasing propagation loss in THz communications [44]. In the B5G/6G, highly dynamic V2V communication scenario is one of the most important communication scenarios and is a key component of the space–air–ground–sea integrated network (SAGSIN) [45]. It is worth mentioning that the application of ultra-massive MIMO and mmWaveTHz technologies in high-mobility V2V communication scenarios poses a larger challenge to the capture of channel non-stationarity and consistency and the development of the corresponding V2V communication channel models. To be specific, larger-scale antenna array, more dynamic communication scenarios, and wider bandwidth lead to more obvious appearance and disappearance of clusters and correlated scattering in the environment. In such a condition, the difficulty of channel S-T-F non-stationary modeling increases greatly. For the capture of channel consistency, shorter channel coherence time and smaller adjacent antenna spacing give rise to more frequent updates of channels in the time and space domains, which makes the channel consistency more apparent. In addition, since the number of clusters in mmWave-THz communications is significantly small [46], it is more essential to characterize the smooth and consistent evolution of each cluster and continuous changes of its parameters. This will distinctly increase the modeling complexity of channel consistency capture. Because of the huge modeling difficulty and complexity, to the best of our knowledge, there is only one general 3D mmWave-THz channel model for ultra-massive MIMO wireless communication systems that was proposed in [29]. The cluster evolution in the S-T-F domain was characterized by employing the BD process parametric method, and hence, the ST-F non-stationarity of channels was modeled. However, the proposed model in [29] ignored the modeling of correlated scattering and the capture of time–space consistency.

178

7 Conclusions and Future Research Directions

In the future, effective ultra-massive MIMO mmWave-THz high-mobility V2V communication channel models that can sufficiently capture 3D S-T-F nonstationarity of V2V communication channels with time–space consistency need to be developed.

7.2.2.2

Developing Comprehensive and Efficient Hybrid Modeling Methods

One of the future research directions is developing more comprehensive hybrid modeling methods, which properly combine parametric modeling methods together with geometric modeling methods, for a precise characterization of V2V communication channels. The parametric modeling method relies completely on the mathematical approach and thus is of low complexity. In addition, it can be exploited to diverse application scenarios by fine-tuning corresponding parameters and hence is of high scalability. However, it disregards the actual communication environment, leading to its relatively low accuracy. Different from the parametric modeling method, the geometric modeling method describes and imitates the underlying scattering geometry environment. This is the underlying physical mechanisms of channel non-stationarity and consistency in V2V communication channels, and thus the geometric modeling method is of high fidelity. However, the detailed description of scattering geometry environment results in the high computational complexity and low scalability. For the purpose of having an excellent compromise between complexity, accuracy, and scalability, the investigation and proposal of hybrid modeling methods are urgently required. Currently, some preliminary work has been carried out to capture the channel non-stationarity and consistency by hybrid modeling methods in the V2V communication channel models. A hybrid modeling method with the combination of BD process parametric method and correlated cluster geometric method was developed in [19], where the S-T-F non-stationarity was modeled. However, the time–space consistency of V2V communication channels was disregarded. The authors in [47] developed a different hybrid modeling method, which combined the visible factor parametric method, frequency-dependent path gain parametric method, and VR geometric method. This hybrid modeling method can mimic the S-T-F non-stationarity and time–space consistency of V2V channels, whereas it cannot jointly capture them. At present, the number of existing hybrid modeling methods is significantly limited due to the huge challenge of developing hybrid modeling methods. Furthermore, a hybrid modeling method that can jointly model the S-T-F non-stationarity of channels with time–space consistency is still an open topic. Therefore, how to successfully combine different types of parametric and geometric modeling methods and develop more comprehensive and efficient hybrid modeling methods, which take accuracy, complexity, as well as scalability into account, deserves further investigation.

7.2 Future Research Directions

7.2.2.3

179

Machine Learning Enabled Channel Non-stationarity and Consistency Capture

Machine learning (ML) is one of the fastest-growing technical fields, located at the intersection of computer science and statistics. Also, it is the core of artificial intelligence (AI) together with data science [48]. Because of the decent performance of ML methods for classification problems, some efficient ML-based methods have been employed to imitate the channel non-stationarity in the current V2V communication channel models [49, 50]. In developed ML-based methods, the conventional tracking methods [51], such as Kalman filters, particle filters, and extended Kalman filters, and the matching-based methods, such as Hungarian method [52] and Kuhn–Munkres (K-M) method [53], have been exploited to capture the appearance and disappearance of MPCs over time. Note that the accurate capture of the appearance and disappearance of MPCs can substantially help in modeling the time non-stationary property in high-mobility V2V communication channels. For the frequency non-stationary modeling, the authors in [19] exploited the K-Means clustering algorithm, an unsupervised learning technique in ML, to successfully obtain correlated clusters, which have similar positions in the environment. Attributed to the decent performance of K-Means clustering algorithm for classification problems, the correlated scattering was accurately captured, and frequency non-stationarity was properly modeled in the mmWave V2V communication channels. At present, the adoption of ML-based methods to model channel non-stationarity is only in its infancy and deserves further investigation. For the capture of channel consistency in the high-mobility V2V communication channels, the ML-based method can also provide a notable contribution. In the ISAC system for V2V scenarios, a large number of sensors enabled by the mmWaveTHz frequency band make it possible to attain sufficient knowledge related to the measured environment [54]. On the one hand, the ML-based method can abstract and extract the channel parameters from the channel measurement data accurately and precisely [49]. On the other hand, the ML-based method also has the ability to cluster the MPC in the scattering geometry environment under V2V scenarios efficiently [55, 56]. Therefore, with the help of ML-based methods, the obtained environmental information can be effectively processed to support a more realistic representation of the scattering geometry environment under V2V scenarios. Meanwhile, the channel consistency needs to be captured in V2V communication channels for ISAC systems. Currently, the ML-based algorithm has been widely used in the processing of sensing information, such as image, video, and laser radar information. Therefore, under the framework of unified ML-based methods, the sensing information can be processed and the channel consistency can be captured, which can significantly support the design of ISAC systems under V2V scenarios. This can naturally contribute to the accurate capture of channel consistency in V2V communication channels. Unfortunately, to the best of our knowledge, there is no ML-based method that can capture the channel consistency for V2V scenarios. For the future B5G/6G V2V channel modeling, an effective ML-based method to capture the channel consistency in V2V channels is strongly needed.

180

7 Conclusions and Future Research Directions

7.2.3 Channel Application Perspective To guide the investigation of V2V communication channels, extensive work related to V2V communication channel measurement has been carried out. This will lead to more realistic capture of channel characteristics and physical feature in the V2V communication channel modeling. Unfortunately, those that have the ability to provide some valuable suggestions for more efficient V2V communication channel modeling from the application perspectives are still at the preliminary stage. In the upcoming B5G/6G era, it is envisioned that the V2V communication channel application and channel modeling will be closely integrated [46]. For different V2V communication channel applications, the requirements of the V2V communication channel models between the accuracy and complexity are different [3, 57]. In such a case, for V2V communications, as a typical channel characteristic and an inherent channel physical feature, requirements for capturing channel nonstationarity and channel consistency are also significantly different under different channel applications. From the beginning of 5G to the future B5G/6G, the integration of applications and communications is closer attributed to the increasing emphasis of B5G/6G wireless communication on vertical application-oriented communications. Among these vertical applications, ISAC, high-precision localization, and digital twin systems are the three most typical applications [58–60]. Investigation related to these three applications has currently seen exponential growth, which is forecast to continue growing for the foreseeable future. By actively perceiving the physical environment with the help of sensors and exchanging observations mutually through wireless communications, the ISAC system enables some promising applications, including the self-driving, vehicular platooning, as well as Internet of vehicular services [61]. The high-precision localization system can precisely acquire the spatial location of the target and thus is a critical enabler for the location-aware application expected in the future communication network [62]. By creating the virtual twins of physical objects based on the digital twin technology, digital twin systems are beneficial for the real-time remote monitoring, risk assessment, as well as intelligent scheduling [63]. It can be predictable that future V2V channel models will serve the above three emerging applications, where a more accurate and comprehensive description of the V2V propagation environment is required. In this case, B5G/6G V2V channel models need to pay more attention to the capture of channel non-stationarity and consistency, and thus V2V communication channels should be modeled in a more precise and deterministic manner. Specifically, the underlying scattering communication environment should be more accurately described, where the movement, location, and distribution of clusters in the V2V propagation environment need to be precisely tracked and modeled in real time. This puts forward higher requirements for the precise capture of channel non-stationarity and consistency to support the ISAC, high-precision localization, and digital twin systems under V2V communication scenarios.

References

181

As we all know, the most intuitive way to construct a high-precision channel model is to use the deterministic modeling method, for example, RT technology. In the deterministic modeling method, a precise reconstruction of the environment will be performed. However, it is noteworthy that the above three applications are still based on communication networks. If a completely deterministic modeling method is adopted, the evolution of each cluster and changes of the V2V physical environment at each moment need to be captured. This will lead to remarkably huge computational complexity and thus makes it distinctly difficult to support systemlevel simulation. To reduce the complexity of system-level simulation in physical layer (PHY), while supporting the aforementioned three applications and precisely modeling the channel non-stationarity and consistency under V2V communication scenarios, the requirement of modeling complexity is further required to be considered. In the future, the closer integration of channel application and channel modeling will pose new challenges to the modeling of channel non-stationarity and consistency in V2V communication scenarios. More efforts should be provided to develop a novel V2V channel modeling method with a better trade-off between the accuracy and complexity from channel application perspectives.

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Index

B B5G/6G, 29, 30, 63, 64, 66–68, 90, 168, 170, 174–177, 179, 180 Birth-death (BD) process, 7, 15, 18, 31, 45, 46, 59, 67, 75, 78, 90, 93, 95, 105, 125, 127, 128, 145, 148, 162, 169, 171, 172, 174, 178

C Channel application, 167, 174, 180–181 Channel characteristics, 1, 2, 23, 31, 45, 63–66, 90, 94, 167, 168, 171, 180 Channel consistency, 125, 127, 160–164, 174, 177, 179, 180 Channel measurement, vi, 7, 54, 57, 71, 89, 94, 96, 115, 117–120, 159, 161, 164, 167, 170, 173–176, 179, 180 Channel modeling, 5, 6, 30, 65, 66, 126, 174, 176–181 Channel non-stationarity, 12, 64, 66, 75, 76, 127, 128, 138, 168, 170, 175–181 Channel statistical properties, 5, 18, 30, 59, 63, 64, 66, 67, 75, 81, 87, 90, 94, 110, 127, 155, 168, 169, 171–174, 176 Continuously arbitrary trajectory, vi, 7, 93–122, 149

D Doppler power spectral density (DPSD), 11, 12, 15, 16, 18, 22–25, 29, 31, 49, 52–55, 57, 59, 63, 67, 79–81, 83, 87, 88, 90,

93, 110, 113, 119–122, 125, 151, 154, 155, 157, 162, 163, 168–172, 174

G Geometry-based deterministic model (GBDM), 5, 6, 168

H Hybrid modeling method, 178

I Integration of time, 93, 95, 128, 137, 171, 174 Irregular-shaped geometry-based stochastic model (IS-GBSM), v, vi, 5, 7, 63–90, 93–122, 125–164, 168, 170–174

L Line-of-sight (LoS) component, 3, 7, 11–14, 21–25, 31, 34–37, 50–52, 68–74, 78, 88, 100, 102, 111, 130–131, 168, 169

M Machine learning (ML), 179 Markov chains, 11, 14–15, 25, 168 Massive multiple-input-multiple-output (MIMO), v, 7, 29–59, 63–90, 93–122, 125–164, 168–173

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 X. Cheng et al., Channel Characterization and Modeling for Vehicular Communications, Wireless Networks, https://doi.org/10.1007/978-3-031-47440-8

185

186 Millimeter wave (mmWave), vi, 7, 93–122, 125–164, 171–174, 179 Mixed-bouncing propagation, 125–129, 134, 137, 163, 173, 174

N Non-geometry stochastic model (NGSM), v, 5–7, 11–25, 65, 66, 94, 168

P Power delay profile (PDP), 11, 12, 18–21, 25, 71, 112–114, 125, 151, 154, 168, 174

R Ray-tracing (RT), 65, 125, 128, 155, 159–164, 174, 181 Regular-shaped geometry-based stochastic model (RS-GBSM), v, 5–7, 29–59, 66, 95, 126, 168–170

S Seed algorithm, 29, 31, 45–48, 54, 59, 169 Selective cluster evolution, 29, 31, 45–47, 59, 95, 103–110, 122, 169, 172 Severe fading, 4, 7, 12–14, 23–25, 168 Single-clusters, 125, 127–130, 133–137, 142–145, 147, 149, 151, 155, 158, 174 6G, v, 3, 4, 63, 93–96, 103, 105, 108, 109, 122, 125–129, 137, 163, 164, 171–174 Space-time-frequency (S-T-F) non-stationarity, vi, 7, 93, 95, 103–110, 122, 125, 127, 128, 137–151, 163, 172, 174–178 Space-time non-stationarity, v, 7, 29–31, 45–48, 55, 58, 59, 65–67, 75–79, 90, 95, 105, 106, 169–172 Spherical wavefront, 31, 45, 63–67, 97, 99, 104, 136, 171

Index T Tap correlation coefficient matrix, 11, 12, 18, 21–22, 25, 168 THz communications, 175, 177 Time-array cluster evolution, 63, 67, 75, 77–79, 171 Time-frequency non-stationarity, v, 7, 11, 14–18, 23–25, 103–110, 168 Time-space (T-S) consistency, vi, 127, 128, 137, 151, 159, 163 Time-variant acceleration, 126, 129, 137, 171 Twin-clusters, v, 7, 96, 97, 101, 102, 104, 107, 108, 125, 127–130, 133–137, 143–145, 147, 149, 151, 155, 158, 174

U Ultra-massive multiple-input–multiple-output (MIMO), 175–178 Uniform planar antenna arrays (UPAs), v, 7, 29–59, 169

V Vehicle-to-vehicle (V2V) communications, vi, 7, 11, 29, 30, 48, 63–65, 75, 93–97, 99, 103, 107, 113, 118, 126, 128, 129, 137, 145, 155, 162, 163, 167–169, 171, 173, 175–181 Vehicular communications, v, 1–8, 11, 13, 19, 65–67, 161, 167 Vehicular movement trajectory (VMT), 3–5, 93–96, 99, 103, 106, 113, 118, 122, 125–130, 137, 155, 157, 163, 167, 168, 172–174 Vehicular traffic density (VTD), 3, 12, 29, 63, 93, 125, 168 Visibility region (VR), 125, 127, 128, 138, 142, 144, 146, 151, 174, 178

W Wideband channel model, 90