Channel Modeling in 5G Wireless Communication Systems [1st ed. 2020] 978-3-030-32868-9, 978-3-030-32869-6

Table of contents :
Front Matter ....Pages i-xvi
Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems (Hao Jiang, Guan Gui)....Pages 1-14
Geometry-Based Statistical MIMO Channel Modeling (Hao Jiang, Guan Gui)....Pages 15-39
3D Scattering Channel Modeling for Microcell Communication Environments (Hao Jiang, Guan Gui)....Pages 41-64
Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street Environments (Hao Jiang, Guan Gui)....Pages 65-86
A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication Environments (Hao Jiang, Guan Gui)....Pages 87-113
A 3D Non-stationaryWideband Channel Model for MIMO V2V Tunnel Communications (Hao Jiang, Guan Gui)....Pages 115-150
An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation Algorithm (Hao Jiang, Guan Gui)....Pages 151-167
3D Non-stationary Wideband UAV Channel Model for A2G Communications (Hao Jiang, Guan Gui)....Pages 169-183
Summary (Hao Jiang, Guan Gui)....Pages 185-188
Back Matter ....Pages 189-194

Citation preview

Wireless Networks

Hao Jiang Guan Gui

Channel Modeling in 5G Wireless Communication Systems

Wireless Networks Series editor Xuemin Sherman Shen University of Waterloo, Waterloo, Ontario, Canada

More information about this series at http://www.springer.com/series/14180

Hao Jiang • Guan Gui

Channel Modeling in 5G Wireless Communication Systems

Hao Jiang College of Electronic and Information Engineering Nanjing University of Information Science and Technology Nanjing, China

Guan Gui College of Telecommunications and Information Engineering Nanjing University of Posts and Telecommunications Nanjing, China

ISSN 2366-1186 ISSN 2366-1445 (electronic) Wireless Networks ISBN 978-3-030-32868-9 ISBN 978-3-030-32869-6 (eBook) https://doi.org/10.1007/978-3-030-32869-6 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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

Preface

The fifth generation (5G) wireless communication networks are being standardized and will be commercially available from 2020, which aims to minimize traffic accidents, improve traffic efficiency, and foster the development of new applications such as mobile infotainment. In addition, multiple-input and multiple-output (MIMO) and massive MIMO technologies are becoming widely used for V2V communications, because the required large-scale antennas can easily be mounted on vehicular surfaces. The successful design and analysis of 5G wireless communication systems require investigation on the propagation characteristics between a mobile transmitter and a mobile receiver into the emerging B5G communication scenarios, satellite-to-satellite communications, unmanned aerial vehicle (UAV) communications, diverse vehicle-to-vehicle (V2V) communications, etc. For facilitating the design and analysis of the performance of 5G wireless communication systems, it is important to have reliable statistical channel models to capture the propagation properties between transmitters and receivers. Regarding the approach of channel modeling, the models can be categorized into deterministic models (e.g., ray-tracing method that requires a detailed description of communication environments) and stochastic models, while the latter rely on large measurement data sets that contain the underlying statistical properties of wireless channels obtained from a variety of mobile communication environments. Here, faced with the big datasets generated in the emerging 5G wireless communication networks, it is important to introduce artificial intelligence (AI) technologies, e.g., deep learning (DL) and deep reinforcement learning (DRL), into 5G communication networks to have better network management. Meanwhile, signal detection algorithms should be assisted to improve the 5G communication system performance. Therefore, the applications of AI to 5G channel modeling should be fully explored. In Chap. 1, we present an overview of current practices in V2V channel modeling, including spatial correlation functions (CFs) and Doppler power spectrum densities (PSDs). Moreover, we compare V2V channel models with conventional fixed-to-mobile (F2M) cellular channel models to determine their fundamental distinctions. Then, recent developments on V2V channel models for 5G systems

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are discussed. Finally, we detail scarcely explored aspects and technical challenges for V2V channel modeling in future communication networks. In Chap. 2, we developed a geometry-based statistical channel model for radio propagation environments, which provides the statistics of the angle of arrival and time of arrival of the multipath components. This channel model assumes that each multipath component of the propagating signal undergoes only one bounce traveling from the transmitter to the receiver and that scattering objects are located according to Gaussian and exponential spatial distributions, and a new scatterer distribution is proposed as a trade-off between the outdoor and the indoor propagation environments. Using the channel model, we analyze the effects of directional antennas at the base station on the Doppler spectrum of a mobile station due to its motion and the performance of its MIMO systems. In Chap. 3, we proposed a generalised three-dimensional (3D) scattering channel model for land mobile systems to simultaneously describe the angular arrival of multipath signals in the azimuth and elevation planes. The model considers a base station (BS) located at the center of a 3D semi-spheroid-shaped scattering region and a mobile station (MS) located within the region. Using this channel model, the authors first derive the closed-form expression for the joint and marginal probability density functions (PDFs) of the angle of arrival and time of arrival measured at the MS corresponding to the azimuth and elevation angles. Next, we derive an expression for the Doppler spectra distribution due to the motion of the MSs. In Chap. 4, we present a generalized visual scattering channel model for car-tocar (C2C) mobile radio environments, in which an asymmetric directional antenna is deployed at the mobile transmitter (MT). The signals received at the mobile receiver (MR) from the MT are assumed to experience multi-bounced propagation paths. More importantly, the proposed model first separates the multi-bounced propagation paths into odd- and even-numbered-bounced propagation conditions. General formulations of the marginal PDFs of the angle of departure (AoD) at the transmitter and angle of arrival (AoA) at the receiver have been derived for the above two conditions, respectively. From the proposed model, we derive an expression for the Doppler frequency due to the relative motion between the MT and MR, which broadens the research of the proposed visual street scattering channel model. In Chap. 5, we presents a 3D vehicle massive MIMO antenna array model for V2V communication environments. A spherical wavefront is assumed in the proposed model instead of the plane wavefront assumption used in the conventional MIMO channel model. Using the proposed V2V channel model, we first derive the closed-form expressions for the joint and marginal PDFs of the AoD and AoA in the azimuth and elevation planes. We additionally analyze the time and frequency cross-correlation functions (CCFs) for different propagation paths. In the proposed model, we derive the expression of the Doppler spectrum due to the relative motion between the mobile transmitter and mobile receiver. In Chap. 6, we present a 3D wideband geometry-based channel model for MIMO V2V communications in tunnel environments. We introduce a two-cylinder model to describe moving vehicles, as well as multiple confocal semi-ellipsoid models to depict internal surfaces of tunnel walls. The received signal is constructed as a sum

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of direct line-of-sight (LoS) propagations, rays with single and double interactions. The movement between the MT and MR results in time-varying geometric statistics that make our channel model non-stationary. Using this channel model, the proposed channel characteristics are studied for different V2V scenarios. In Chap. 7, we propose an estimated wideband geometry-based channel model for V2V communication environments, which is based on an AoD and AoA estimation algorithm, to determine the ellipse scattering region and to efficiently study the V2V channel characteristics for different propagation delays, i.e., pertap channel statistics. In the first stage, we estimate the AoD and AoA for the first tap. In this case, the ellipse-scattering region for the first tap can be determined. Then, we estimate the ellipse channel models for other taps based on the estimated model parameters for the first tap. Furthermore, the spatial CCFs are derived and thoroughly investigated for different propagation delays. In Chap. 8, we propose a novel 3D MIMO channel model to describe the airto-ground (A2G) communication environments. The model introduces the UAV transmitter and ground MR located at the foci points of the boundary ellipsoid, while different ellipsoids represent the propagation properties for different time delays. In light of this, we are able to investigate the propagation properties of the A2G channel model for different time delays. Furthermore, the time-varying parameters of azimuth angle of departure (AAoD), elevation angle of departure (EAoD), azimuth angle of arrival (AAoA), and elevation angle of arrival (EAoA) are derived to properly describe the channel non-stationarity, which is caused by the motion of the UAV transmitter, cluster, and MR. The impacts of the movement properties of the cluster in both the azimuth and elevation planes are investigated on the channel characteristics, i.e., spatial CCFs, temporal autocorrelation functions (ACFs), Doppler power spectrum density (PSD), and power delay profiles (PDPs). Finally, Chap. 9 summarizes this book and discusses the future research directions. Nanjing, China Nanjing, China September 2019

Hao Jiang Guan Gui

Acknowledgment

This work was partly supported by the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology.

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Contents

1

2

Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 V2V Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 MIMO V2V Channel Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Key Differences Between V2V and Conventional F2M Channel Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Recent Developments in V2V Channel Modeling. . . . . . . . . . . . . . . . . . . . . 1.3.1 V2V Communication Environments and GBSMs . . . . . . . . . . . . 1.3.2 Requirements for MIMO V2V Channel Modeling. . . . . . . . . . . . 1.3.3 Narrowband and Wideband V2V Channel Models . . . . . . . . . . . 1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 9 10 11 11 12 13

Geometry-Based Statistical MIMO Channel Modeling . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 System Model and Joint PDFs of AoA/ToA . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Geometrical Model and PDFs of AoA/ToA . . . . . . . . . . . . . . . . . . . 2.2.2 Scatterer Distributed Density and ES/RP . . . . . . . . . . . . . . . . . . . . . . 2.3 Marginal PDFs of AoA and ToA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Marginal PDFs of AoA at BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Marginal PDFs of AoA at MS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Marginal PDFs of ToA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Doppler Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 MIMO Performance Over the Channel Model . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 SFC of MIMO Multiple Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Capacity Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Joint and Marginal PDFs of AoA/ToA . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Doppler Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Performance of MIMO Multiple Antennas . . . . . . . . . . . . . . . . . . . .

15 15 16 16 19 22 22 22 24 26 27 27 29 30 30 33 35

1 1 2 2

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2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 38

3D Scattering Channel Modeling for Microcell Communication Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 System Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spatial Characteristics of the 3D Scattering Channel Model . . . . . . . . . 3.3.1 Probability Density Functions of the AoA . . . . . . . . . . . . . . . . . . . . 3.3.2 Probability Density Functions of the TOA . . . . . . . . . . . . . . . . . . . . 3.4 Performance of the MIMO Antenna Receiving Systems. . . . . . . . . . . . . . 3.4.1 Doppler Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 MIMO ULA and UCA Antenna Receiving Systems . . . . . . . . . 3.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 PDFs of the AoA and ToA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Doppler Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Performance of MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 44 46 47 48 50 51 52 55 55 58 60 62 62

Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Generalized Visual Street Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Virtual Model Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Odd-Numbered-Bounced Propagation Paths . . . . . . . . . . . . . . . . . . 4.2.3 Even-Numbered-Bounced Propagation Paths . . . . . . . . . . . . . . . . . 4.3 Spatial Characteristics of the Street Channel Model . . . . . . . . . . . . . . . . . . 4.3.1 Scattering Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Marginal PDF of the AoD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Marginal PDF of the AoA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Doppler Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Distribution of the AoD and AoA PDFs . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Distribution of the Doppler Frequency . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 68 68 71 73 74 74 75 76 78 80 80 83 85 85

A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proposed 3D Vehicle Massive MIMO Array Model . . . . . . . . . . . . . . . . . . 5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model . . . . . . . . 5.3.1 Geometric Properties of V2V Communication Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Spatial Characteristics of the Radio Channel . . . . . . . . . . . . . . . . . .

87 87 90 93 93 96

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5.3.3 Time Cross-Correlation Function Analysis. . . . . . . . . . . . . . . . . . . . 5.3.4 Frequency Cross-Correlation Function Analysis . . . . . . . . . . . . . . 5.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 AoD and AoA Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Time and Frequency Cross-Correlation Analysis . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 99 102 103 104 107 111 111

A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 System Channel Model for Flat Communications. . . . . . . . . . . . . . . . . . . . . 6.2.1 Descriptions of the Proposed V2V Channel Model . . . . . . . . . . . 6.2.2 Complex Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Non-stationary Time-Varying Parameters . . . . . . . . . . . . . . . . . . . . . 6.3 Proposed Theoretical Channel Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Spatial Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Frequency CFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Doppler PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Power Delay Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Stationary Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 System Channel Model for Slope Communications. . . . . . . . . . . . . . . . . . . 6.5 Moving Vehicles on the Channel Characteristics . . . . . . . . . . . . . . . . . . . . . . 6.6 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 AAoA Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Spatial and Frequency CFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Doppler PSDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 PDPs and Stationary Intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 118 120 121 125 127 127 130 131 131 132 132 133 136 138 139 144 145 147 148

An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Wideband Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Description of the Proposed Channel Model . . . . . . . . . . . . . . . . . . 7.3 AoD and AoA Estimation for the First Tap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Channel Characteristics of the Proposed Model . . . . . . . . . . . . . . . . . . . . . . . 7.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Estimated Angular Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Spatial CCFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 153 153 154 155 160 161 161 164 166 166

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Contents

3D Non-stationary Wideband UAV Channel Model for A2G Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Channel Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Time-Varying Angular Parameters and Propagation Paths . . . 8.3 Channel Statistics of the Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 171 171 173 175 177 178 180 182

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Summary of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 187 188

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Acronyms

2D 3D 3GPP 4G 5G A2G AAoA AAoD ACF AI AoA AoD AS BER BS C2C DL DRL EAoA EAoD ES F2M GBSMs IoT LoS M2M MIMO MR MS MT NGBSMs

Two-dimensional Three-dimensional 3rd Generation Partnership Project Fourth generation Fifth generation Air-to-ground Azimuth angle of arrival Azimuth angle of departure Auto-correlation function Artificial intelligence Angle of arrival Angle of departure Angle spread Bit error ratio Base station Car to car Deep learning Deep reinforcement learning Elevation angle of arrival Elevation angle of departure Effective scatterers Fixed-to-mobile Geometry-based stochastic models Internet of things Line-of-sight Mobile-to-mobile Multiple-input and multiple-output Mobile receiver Mobile station Mobile transmitter Nonregular-shaped geometry-based stochastic models xv

xvi

NLoS V2V VTD PAP PDFs PDPs PSD RP RS-GBSMs SFC SISO ToA UAV UCA UCCA ULA URA WSS WSSUS

Acronyms

Non-line-of-sight Vehicle-to-vehicle Vehicle traffic density Power azimuth profile Probability density functions Power delay profiles Power spectrum density Reflection probability Regular-shaped geometry-based stochastic models Spatial fading correlation Single-input and single-output Time of arrival Unmanned aerial vehicle Uniform circular array Uniform concentric circular array Uniform linear array Uniform rectangular array Wide-sense stationary Wide-sense stationary uncorrelated scattering

Chapter 1

Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

1.1 Introduction V2V communications is a major research topic in the context of intelligent transportation systems of 5G, which aims to minimize traffic accidents, improve traffic efficiency, and foster the development of new applications such as mobile infotainment [1, 2]. In addition, multiple-input and multiple-output (MIMO) and massive MIMO technologies are becoming widely used for V2V communications, because the required large-scale antennas can easily be mounted on vehicular surfaces [3]. The successful design and analysis of V2V communication systems require investigation on the propagation characteristics between an MT and an MR in V2V channels. To this end, realistic channel models provide effective means to approximate the propagation characteristics and serve as the basis for performance evaluation in general communication systems [4]. Thus far, there have been a variety of literatures to study the F2M cellular channels between a static base station (BS) and a mobile station (MS). However, in V2V scenarios, the propagation characteristics are significantly different from those of the above conventional F2M channels. To be specific, the MR antenna in V2V channels is generally placed at low heights between 1 and 2.5 m, thus limiting its coverage below that of common cellular systems. Furthermore, massive dense scatterers around the MT and MR result in different propagation characteristics from those of the conventional F2M cellular environments. Likewise, the stationary interval of V2V channels is short given the mobility of both ends, and thus non-stationarity should be reflected in the channel model. These key features distinguish V2V channels from conventional F2M channels. In addition, early cellular models only accounted for static propagation characteristics impacted by roadside environments. Subsequently, these models were enhanced by considering moving vehicles around the transmitter/receiver and key technologies specific to V2V communications, for describing the dynamic properties of the channels. In previous studies, many channel measurement campaigns © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_1

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1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

have been carried out in V2V communication environments; meanwhile, a series of preliminary V2V channel models have been provided. In recent years, research on V2V channel models has been carried out by several research groups worldwide. For instance, Andreas F. Molisch et al. have conducted various V2V channel measurement activities in different environments at the Lund University since 2005, claiming the importance of using well characterized measurement-based V2V channel models to describe vehicular communications [5, 6]. Likewise, Jianhua Zhang et al. have collected big data from channel measurement to investigate the dynamic properties of mobile radio communication environments [7, 8]. Furthermore, Cheng-Xiang Wang et al. have proposed a variety of simulation models to describe different V2V channels [9, 10], thus greatly improving the developments of V2V channel models for 5G communications. Moreover, Zaichen Zhang et al. have proposed a series of geometric models to describe radio communications in V2V environments [11, 12]. However, no study has systematically reviewed V2V channel models for 5G communication systems so far. Therefore, we address recent developments and potential challenges of V2V channel modeling in this paper, and provide guidelines for the theoretical analysis of diverse factors in vehicular communication environments from both the MT and MR perspectives. The remainder of this paper is summarized as follows. In the following, we briefly explain V2V channel models and thoroughly discuss the differences between V2V and conventional F2M channel models. Then, we address recent developments on V2V channels for 5G communication systems and discuss some future challenges and potential solutions for V2V channel modeling. The final section draws the conclusions of this article.

1.2 V2V Channel Models For the developments of V2V communications in 5G systems, it is essential to study the channel characteristics in advance. In this section, we detail V2V channel modeling and the differences between V2V and conventional F2M channel models.

1.2.1 MIMO V2V Channel Modeling In early developments, classical models (e.g., Rayleigh and Ricean processes) provided information on received signals containing power level distributions and Doppler shifts. However, in reality, these models are only applicable to narrowband cellular radio communications, which assume that the propagation of waves experience similar delays. Subsequently, based on the classic understanding of fading and Doppler spread, spatial channel models incorporated the concepts of propagation delays, angle of departure (AoD), angle of arrival (AoA), and array antenna geometries.

1.2 V2V Channel Models

3

More recently, the increasing complexity of cellular communications demanded researchers to consider aspects such as reflections from large obstacles, diffraction of electromagnetic waves around objects, and signal scattering, especially in V2V channels, as illustrated in Fig. 1.1. These complex interactions generate multipath signal components between the MT and MR. Consequently, more accurate models can be derived by incorporating concepts that reflect the multiple factors of real environments. Figure 1.2a illustrates a simplified geometric MIMO channel model

Fig. 1.1 A typical vehicular communication environment between MTs and MRs

Fig. 1.2 Diagrams of geometric channel models in V2V propagation environments: (a) geometric update of V2V channels; (b) MIMO V2V channels

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1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

with MT transmit and MR receive omnidirectional antenna elements for V2V communication environments.   The MIMO V2V channel, in reality, can be described by matrix H(t) = hpq (t, τ ) M ×M of dimension MR × MT , where subscripts p R T (p = 1, . . . , MT ) and q (q = 1, . . . , MR ) denote the MIMO antenna elements. Here, the complex impulse response (CIR) between the p-th antenna of the transmit array and the q-th antenna of the receive array in MIMO V2V channels can be L(t) expressed as hpq (t, τ ) = l=1 Al hl,pq (t)δ(τ −τl (t)), where Al and τl (t) represent the gain and delay of the l-th (l = 1, . . . , L(t)) propagation path, respectively, L(t) denotes the number of multipath components, and δ(·) is the Dirac delta function. Here, the complex fading envelope, hl,pq (t), between the p-th transmit and q-th receive antenna of the l-th path at carrier frequency fc is given by   N   j ψl,pq,n −2πfc Dl,pq,n (t)/c hl,pq t = al,pq,n (t)e n=1

× ej 2πfd,l,pq,n (t) , where al,pq,n (t) (n = 1, 2, . . . , N ), ψl,pq,n , and Dl,pq,n (t) denote, respectively, the amplitude, phase, and propagation distance of the l-th path in the n-th cluster, with N representing the total number of clusters and c is the speed of light. The term ej 2πfd,l,pq,n (t) is the Doppler frequency component caused by the motion of the MT and MR, such that   ej 2πfd,l,pq,n (t) = ej 2π tvT /λ cos αTp,l,n (t)−ηT cos βTp,l,n (t)   × ej 2π tvR /λ cos αRq,l,n (t)−ηR cos βRq,l,n (t) , where αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), and βRq,l,n (t) represent, respectively, the time-varying azimuth angle of departure (AAoD), elevation angle of departure (EAoD), azimuth angle of arrival (AAoA), and elevation angle of arrival (EAoA) of the l-th path in the n-th cluster, as depicted in Fig. 1.2a, λ is the carrier wavelength, vT and ηT denote the moving velocity and direction of the MT, respectively, whereas vR and ηR denote the moving velocity and direction of the MR, respectively.

1.2.2 Key Differences Between V2V and Conventional F2M Channel Models The spatial CIR of the V2V channel model, which can be used to characterize the physical properties of wireless communication environments, is a superposition of multipath components with different amplitudes al,pq,n (t), phases ψl,pq,n , frequency shifts fd,1,pq,n (t), AAoDs αTp,l,n (t), EAoDs βTp,l,n (t), AAoAs αRq,l,n (t), and EAoAs βRq,l,n (t), as shown in Fig. 1.2a. The key differences between V2V and

1.2 V2V Channel Models

5

conventional F2M cellular radio systems are mainly caused by the antenna heights and the mobility of the MT and MR, as detailed in the sequel.

1.2.2.1

Non-Stationarity

One of the main characteristics of V2V channels that are not found in conventional cellular channels is the non-stationarity. Here, it is worth mentioning that most of conventional models assumed that the channels are wide-sense stationary (WSS), which only adopted time-invariant model parameters to characterize short-term variations of wireless channels, such as F2M channels, as illustrated in Fig. 1.2b. However, in reality, this assumption does not accurately reflect the dynamic channel properties in V2V communication environments. To improve accuracy, the models must consider the non-stationary characteristics of vehicular channels. Therefore, it is essential to derive time-varying propagation path and angular parameters to characterize the channel properties of V2V channel models. In general, the non-stationarity of V2V channels has several effects. For instance, Fig. 1.2c shows a time-varying V2V channel as either the MT moves from position P1 to P2 or the MR moves from position P3 to P4 . Hence, different time-varying AAoD αTp,l,n (t), EAoD βTp,l,n (t), AAoA αRq,l,n (t), and EAoA βRq,l,n (t) should be derived to capture the V2V statistical properties over time. Here, the parameters αTp,l,n (t) and βTp,1,n (t) are derived based on the MT motion (i.e., direction ηT , velocity vT , and time t) and the initial angular parameters (αTp,l,n (t0 ), βTp,l,n (t0 ), αRq,l,n (t0 ), and βRq,l,n (t0 )) before the MT and MR move (i.e., t = t0 ), whereas αRq,l,n (t) and βRq,l,n (t) are derived based on the MR motion (i.e., direction ηR , velocity vR , and time t) and the initial angular parameters [11]. The derivation of the angular parameters and propagation path lengths are based on the specific geometric channel models, as discussed in the following.

1.2.2.2

Moving Vehicles Surrounding the MT and MR

In V2V channels, waves from the MT impinge on obstacles (clusters) before reaching the MR. To be specific, when the transmitter and receiver are in conventional F2M communication environments, the received signals are mainly reflected by roadside environments, as shown in Fig. 1.2b. Here, it is appropriate to use an ellipse to describe the distribution of roadside environments, as illustrated in Fig. 1.3a. This is mainly due to the fact that, the ellipse model with an MT and MR located at the foci has the ability to determine the waves of the same propagation path lengths. In this case, time-varying model parameters αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), βRq,l,n (t), and Dl,pq,n (t) can be derived by applying standard mathematical operations according to the ellipse model. Jianhua Zhang [13] has conducted a series of measurement campaigns to demonstrate that, the assumption of twodimensional (2D) propagation may lead to an inaccurate performance estimation of V2V communication systems. Subsequently, many kinds of three-dimensional (3D)

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1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

Fig. 1.3 Top view of geometric channel models in different propagation environments: (a) conventional F2M channels; (b) narrowband V2V channels, corresponding to Fig. 1.2b; (c) wideband V2V channels

geometric channel models (including azimuth and elevation angles) can be used to represent roadside environments, such as the cylinder [14] and semi-ellipsoid models [11], which form an ellipse when viewed from above. On the other hand, in V2V communication environments, the clusters in Fig. 1.2c can be classified into obstacles corresponding to roadside environments and moving vehicles surrounding the MT and MR, as shown in Fig. 1.2d. This reflects that the rays of non-line-of-sight (NLoS) propagations in V2V channel models is usually erroneously considered to be similar to that in conventional F2M channel models (i.e., considering only single-bounced waves scattered from roadside environments, as shown in Fig. 1.3a). Instead, the NLoS propagation of waves in V2V channels corresponds to superpositions of single- and double-bounced components scattered from roadside environments and/or moving vehicles (normally described by a tworing model [15]), as shown in Fig. 1.3b. Hence, the propagation distance Dl,pq,n (t) represents the propagation path lengths from the transceiver to either the boundary of the two-ring model (solid lines in Fig. 1.3b) or the ellipse model (dashed lines in Fig. 1.3b), and path lengths from the transceiver to the combined two-ring and ellipse model (pecked lines in Fig. 1.3b). In this case, the CIRs corresponding to the various path lengths, Dl,pq,n (t), should be determined to describe the physical properties of the different propagating rays. To study the impacts of moving vehicles and roadside environments on the V2V channel characteristics, the corresponding spatial correlation functions (CFs) should be investigated, as illustrated in Fig. 1.4. Some studies have proven that these functions are related to the angular parameters and path lengths of V2V channels shown in Fig. 1.2c, and thus can be used to characterize the variation of V2V channel with respect to time, motion, or frequency. Based on the complex fading envelope,

1.2 V2V Channel Models

7

1.0 NLoS rays scattered by the moving vehicles

0.9 NLoS rays scattered by the roadside environments

Normalized spatial CF

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Normalized antenna spacing Fig. 1.4 Spatial correlation functions of NLoS rays scattered by moving vehicles and roadside environments in V2V channel models

the spatial CFs between two subchannels can be obtained, and it can be determined that the function for the NLoS rays scattered by moving vehicles is considerably higher than that scattered by roadside environments. This is mainly due to the fact that higher path lengths result in lower correlation [15].

1.2.2.3

Doppler Power Spectrum Density

For V2V scenarios, the Doppler power spectrum density (PSD), which provides statistical information on the variation of the frequency of a tone acquired by the receiver, has been widely studied in the existing literatures. For instance, in the classical Rayleigh and Ricean channels, the receiver uniformly collects signals from all directions, and the Doppler PSDs can be approximated as vR fc /c cos αn , with αn denoting the incidence angle scattered by the n-th cluster. In this case, the model is similar to the classical Clarke’s model with U -shaped Doppler spectrum (i.e., Jakes’ spectrum), shown as black lines in Fig. 1.5. However, in conventional F2M channels, the multipath components arriving at the MS from every direction are scattered using an ellipse model representing roadside environments, as shown in Figs. 1.2b and 1.3a. In this case, the angular power is non-uniformly distributed. The blue lines in Fig. 1.5 show that the Doppler PSDs in F2M channels under WSS assumption are positively (negatively) shifted as

1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

Normalized Doppler Power Spectrum Density (dB)

8 0

Clarke Model MR moves towards the MT (WSS) MR moves opposite to the MT (WSS)

-10

MR moves perpendicular to the rays of LoS propagations (WSS) MR moves perpendicular to the rays of LoS propagations (non-WSS)

Time-variant Time-invariant

-20

-30

-40

-50 -100

-80

-60

-40

-20

0

20

40

60

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Doppler frequency [Hz] Fig. 1.5 Doppler power spectrum densities for the classic Clarke’s model, F2M channel model, and V2V channel model with respect to the motion directions and velocities

the MR moves towards (away from) the MT. In addition, MR motion perpendicular to the direct line-of-sight (LoS) rays causes each curve of the Doppler PSDs in the stationary channel model to have a peak at zero [16]. The Doppler PSDs in V2V channels are different and can be obtained by applying the Fourier transform to the temporal auto-correlation functions in terms of propagation delay [15]. Some studies have demonstrated that the moving properties, including motion direction (i.e., ηT and ηR ), velocity (i.e., vT and vR ), and time t considerably impact the Doppler PSDs in the V2V channels shown in Fig. 1.2c. For instance, the red lines in Fig. 1.5 show that the Doppler PSDs in non-stationary V2V channels (i.e., non-WSS assumption) change over time, thus differing from the time-invariant distribution of the Doppler PSDs in F2M channels [17]. This variation in the Doppler PSDs is related to the geometric relations of the V2V channel model, which considers propagation path lengths (Dl,pq,n (t)) and angular parameters (αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), and βRq,l,n (t)) as time-varying and non-stationary V2V channels, as shown in Fig. 1.2c. Consequently, these can be considered as unique characteristics of V2V channels.

1.3 Recent Developments in V2V Channel Modeling

1.2.2.4

9

Channel Frequency

For 5G communication systems, the bandwidth will be much larger than that in fourth-generation (4G) systems. Accordingly, these requirements will apply to 5G V2V channel models. In fact, V2V channel models were proposed at 5.9 GHz [15, 18], and a geometric MIMO V2V channel model at 5.4 GHz [17]. These frequency characteristics for upcoming V2V channels notably differ from those of conventional cellular communications, whose carrier frequency usually ranges from 700 to 2100 MHz for communications between a BS and an MS.

1.2.2.5

Range of the Angular Parameters

In V2V channels, the model parameters depend on the communication environments. Studies have verified that the ranges of the angular parameters αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), and βRq,l,n (t) widely vary among different scales, such as macrocell, microcell, and picocell environments. For instance, the communication between a BS and MS in the macrocell scale requires the former to be deployed higher than the surrounding scatterers. Hence, the received signals at the BS result mainly from the scattering around the MS. Consequently, the AAoD of the signal is restricted to a small angular region, 0 < αT ,BW < 2π , as shown in Fig. 1.2b. However, V2V channels at the macrocell scale exhibit a considerably different statistical distribution of the AAoD. In this case, the scatterers surrounding the MR are assumed to be at the same height as the MT, that is, the signal at the MT arrives from every direction after bouncing in the surrounding scatterers. Thus, the AAoD αTp,l,n (t) is uniformly distributed over [0, 2π ], as shown in Figs. 1.2c and d.

1.3 Recent Developments in V2V Channel Modeling V2V channel models can be categorized into deterministic (e.g., ray-tracing method that requires a detailed description of V2V communication environment) and stochastic models. It is worth mentioning that, stochastic models rely on the large numbers of measurements that contain the underlying statistical properties of wireless channels obtained from a variety of V2V communication environments. From these models, propagation path parameters and angular parameters are derived to describe the properties of channel models. In reality, stochastic models can be separated into geometry-based stochastic models (GBSMs), correlationbased stochastic models (CBSMs), extended Saleh–Valenzuela stochastic models, and ray-tracing stochastic models. Likewise, GBSMs can be further divided into non-geometrical stochastic models (NGSMs) and regular-shape geometry-based stochastic models (RS-GBSMs), where the former are also known as parametric models, which are constructed based on channel measurements, whereas the latter are based on the scatter shape.

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1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

1.3.1 V2V Communication Environments and GBSMs The clusters in V2V communication environments mainly include moving vehicles and roadside environments randomly distributed between a transmitter and receiver, as shown in Fig. 1.2d. For simplifying the statistical analysis of propagation characteristics in V2V channels, it is worth mentioning that the path lengths (Dl,pq,n (t)) and angular parameters (αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), and βRq,l,n (t)) in different V2V scenarios can be derived according to specific geometric relations among the MT, MR, and scatterers. In previous studies, a series of geometric models are proposed to depict the scattering region of moving vehicles and roadside environments in V2V channels, as shown in Fig. 1.6. Among them, the one-ring (Fig. 1.6a), ellipse (Fig. 1.6b), and circle (Fig. 1.6c) models can describe macrocell, microcell, and picocell channels in mobile environments, respectively [19]. Moreover, V2V channels require the two-ring model (Fig. 1.6d) and either the twocylinder (Fig. 1.6e) [20] or two-sphere (Fig. 1.6f) [14, 21] model to describe moving vehicles around the transceiver in 2D and 3D space, respectively. Furthermore, the ellipse model (Fig. 1.6b) [22] and either the elliptic-cylinder (Fig. 1.6g) [14, 21] or semi-ellipsoid (Fig. 1.6h) [11, 17] model can be used to describe 2D and 3D roadside environments, respectively. For complicated 5G V2V communication environments, propagation characteristics can be described by combining several geometric shapes and adjusting the model parameters. Thus far, Cheng-Xiang Wang et al. have proposed various GBSMs for channels using this strategy. For instance, Xiang Cheng et al. proposed a combined two-ring and ellipse model to describe moving vehicles and roadside

Fig. 1.6 Classical regular-shape geometry-based stochastic models (RS-GBSMs): (a) one-ring model; (b) ellipse model; (c) circle model; (d) two-ring model; (e) two-cylinder model; (f) twosphere model; (g) elliptic-cylinder model; (h) semi-ellipsoid model

1.3 Recent Developments in V2V Channel Modeling

11

environments, respectively, and thus simulate propagation scenarios for V2V channels [15, 18]. Meanwhile, Yi Yuan et al. proposed a combined two-sphere and elliptic-cylinder model to reflect 3D V2V propagation environments [14, 21]. Furthermore, Zajic [20] presented a two-cylinder model to describe the distribution of moving and static vehicles in vicinity of the MT and MR. In addition, we adopted a combined two-cylinder and semi-ellipsoid model to describe 3D roadside high buildings and tunnel scenarios in V2V channels [17].

1.3.2 Requirements for MIMO V2V Channel Modeling In conventional F2M channel models, as shown in Fig. 1.2b, the departure direction of the radio waves only produces a slight phase variation of the signal. Here, the differences in the propagation characteristics between the transmit antenna and receive antenna arrays are insignificant. In light of this, the point source assumption is applied to both ends of the antenna array in F2M channels [19]. However, when applying MIMO technology in V2V channels, it is possible to improve the communication reliability, spectral efficiency, and energy efficiency. In light of this, each antenna has a different perspective according to its position. Therefore, the angular parameters of the propagated signals from the p-th transmit antenna to the q-th receive antenna are non-linear along the array. Indeed, the AAoD αTp,l,n (t), EAoD βTp,l,n (t), AAoA αRq,l,n (t), and EAoA βRq,l,n (t) should be computed according to the geometry of the antenna array, as shown in Fig. 1.2d. In addition, MIMO antenna arrays, which have the advantage of high spatial resolution, are able to distinguish closely receivers in 3D space. In light of this, it is important to provide 3D angular information of the propagations to describe V2V communication environments. Hence, it is required to precisely model the path lengths Dl,pq,n (t) and angular parameters (αTp,l,n (t), βTp,l,n (t), αRq,l,n (t), and βRq,l,n (t)) in horizontal and perpendicular directions for each scattered multipath.

1.3.3 Narrowband and Wideband V2V Channel Models In multipath channels, the ellipse model (representing roadside environments) is usually the physical basis for modeling both narrowband and wideband channels. However, conventional narrowband channel models assume that waves experience similar propagation delays, which can be described by an ellipse model with an MT and an MR located at the foci, as shown in Fig. 1.3a. Then, considering moving vehicles in V2V communications, narrowband V2V channel models can be used to describe narrowband communications between MT and MR, as illustrated in Fig. 1.3b. In practice, however, different propagation delays should be considered in V2V channels. Thus, it is essential to model wideband V2V channel models based on

12

1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

the tapped-delay-line structure with different ellipses, as shown in Fig. 1.3c. For the ellipse model with foci at the MT and MR, every wave in the scattering region characterized by the -th ellipse undergoes the same discrete propagation delay τ = τ0 + τ , = 0, 1, 2, . . . , L − 1, where τ0 denotes the delay of direct LoS propagations, τ is an infinitesimal propagation delay, and L is the number of paths with different propagation delays. Here, the number of paths with different propagation delays exactly corresponds to the number of delay elements. Moreover, as the rays of single and double interactions contribute differently to the channel characteristics, complex fading envelope hl,pq (t) should be separated into different channel taps for wideband V2V communications. Under these considerations, statistical properties of V2V channels for different propagation delays can be derived and thoroughly investigated [18]. For instance, we have provided a 3D geometry-based wideband channel model for MIMO V2V communication environments in [17], whose top view is shown in Fig. 1.3c. In the model, the link for the first channel tap is composed of the direct LoS propagations, rays of single interactions caused by the scatterers located on either of two cylinders presented by the moving vehicles (i.e., MT → A → MR and MT → B → MR) or on the first semi-ellipsoid presented by the roadside environments (i.e., MT → C → MR), and rays of double interactions generated from the scatterers located on both cylinders (i.e., MT → E → F → MR), depicted as green lines in Fig. 1.3c. For other channel taps, the link is considered as a superposition of the rays of single interactions that are produced only from the scatterers located on the semi-ellipsoid (i.e., MT → G → MR), as well as the rays of double interactions caused by the scatterers from the combined single cylinder and the corresponding semi-ellipsoid (i.e., MT → H → K → MR and MT → U → W → MR), depicted as blue lines in Fig. 1.3c. In reality, the combinations of propagation components should be considered in advance to investigate the propagation characteristics for different propagation delays. Here, it is worth mentioning that the model introduces scatterers with identical delays on the same semi-ellipsoid, whereas different semi-ellipsoids represent the wideband channel characteristics.

1.4 Conclusions In this chapter, we have provided a systematic survey of V2V channel models, especially directed towards 5G communication systems. Some important V2V channel properties have been presented, as well as modeling differences between V2V and conventional F2M channels, including non-stationarity, moving vehicles, Doppler PSDs, channel frequency, and range of angular parameters. Then, we have described recent developments on 5G V2V channel models. Finally, we have discussed the future challenges for V2V channel modeling from different points of view, such as the impacts of moving scatterers, mixed V2V channel propagation scenarios, and intelligent V2V communications. We expect that this

References

13

paper will provide guidelines for designing V2V channel models and carrying out measurements for the upcoming deployment of the corresponding wireless networks.

References 1. N. Zhang, S. Zhang, J. Zheng, X. Fang, J.W. Mark, X. Shen, QoE driven decentralized spectrum sharing in 5G networks: potential game approach. IEEE Trans. Veh. Technol. 66(9), 7797–7808 (2017) 2. M. Shafi, J. Zhang, H. Tataria, A.F. Molisch, S. Sun, T.S. Rappaport, F. Tufvesson, S. Wu, K. Kitao, Microwave vs. millimeter-wave propagation channels: key differences and impact on 5G cellular systems. IEEE Commun. Magazine 56(12), 14–20 (2018) 3. J. Zhang, L. Dai, Z. He, S. Jin, X. Li, Performance analysis of mixed ADC massive MIMO systems over Rician fading channels. IEEE J. Sel. Areas Commun. 35(6), 1327–1338 (2017) 4. S. Sun, T.S. Rappaport, M. Shafi, P. Tang, J. Zhang, P.J. Smith, Propagation models and performance evaluation for 5G millimeter-wave bands. IEEE Trans. Veh. Technol. 67(9), 8422– 8439 (2018) 5. A.F. Molisch, F. Tufvesson, J. Karedal, C.F. Mecklenbrauker, A survey on vehicle-to-vehicle propagation channels. IEEE Wirel. Commun. 16(6), 12–22 (2009) 6. V. Kristem, C.U. Bas, R. Wang, A.F. Molisch, Outdoor wideband channel measurements and modeling in the 3–18 GHz band. IEEE Trans. Wirel. Commun. 17(7), 4620–4633 (2018) 7. J. Zhang, Y. Zhang, Y. Yu, R. Xu, Q. Zheng, P. Zhang, 3-D MIMO: how much does it meet our expectations observed from channel measurements? IEEE J. Sel. Areas Commun. 35(8), 1887–1903 (2017) 8. J. Zhang, Z. Zheng, Y. Zhang, J. Xi, X. Zhao, G. Gui, 3D MIMO for 5G NR: several observations from 32 to massive 256 antennas based on channel measurement. IEEE Commun. Magazine 56(3), 62–70 (2018) 9. C.X. Wang, X. Cheng, D.I. Laurenson, Vehicle-to-vehicle channel modeling and measurements: recent advances and future challenges. IEEE Commun. Magazine 47(11), 96–103 (2009) 10. S. Wu, C. Wang, E.M. Aggoune, M.M. Alwakeel, X. You, A general 3-D non-stationary 5G wireless channel model. IEEE Trans. Commun. 66(7), 3065–3078 (2018) 11. H. Jiang, Z. Zhang, J. Dang, L. Wu, A novel 3-D massive MIMO channel model for vehicleto-vehicle communication environments. IEEE Trans. on Commun. 66(1), 79–90 (2018) 12. H. Jiang, Z. Zhang, L. Wu, J. Dang, A non-stationary geometry-based scattering vehicle-tovehicle MIMO channel model. IEEE Commun. Lett. 22(7), 1510–1513 (2018) 13. J. Zhang, C. Pan, F. Pei, G. Liu, X. Cheng, Three-dimensional fading channel models: a survey of elevation angle research. IEEE Commun. Mag. 52(6), 218–226 (2014) 14. Y. Yuan, C.X. Wang, X. Cheng, Novel 3D geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channels. IEEE Trans. Veh. Technol. 13(1), 298–309 (2014) 15. X. Cheng, C.X. 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) 16. M. Patzold, Mobile Radio Channels, 2nd edn. (Wiley, Hoboken, 2012) 17. H. Jiang, Z.C. Zhang, L. Wu, J. Dang, G. Gui, A 3D non-stationary wideband geometry-based channel model for MIMO vehicle-to-vehicle communications in tunnel environments. IEEE Trans. Veh. Technol. 68(7), 6257–6271 (2019) 18. X. Cheng, Q. Yao, M. Wen, C. Wang, L. Song, B. Jiao, Wideband channel modeling and intercarrier interference cancellation for vehicle-to-vehicle communication systems. IEEE J. Sel. Areas Commun. 31(9), 434–448 (2013)

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1 Overview of Vehicle-to-Vehicle Channel Modeling in 5G Mobile Systems

19. R. Janaswamy, Angle and time of arrival statistics for the Gaussian scatter density model. IEEE Trans. Wirel. Commun. 1(3), 488–497 (2002) 20. A.G. Zajic, Impact of moving scatterers on vehicle-to-vehicle narrow-band channel characteristics. IEEE Trans. Veh. Technol. 63(7), 3094–3106 (2014) 21. Y. Yuan, C.X. Wang, Y. He, M.M. Alwakeel, E.H.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) 22. H. Jiang, Z.C. Zhang, J. Dang, L. Wu, Analysis of geometric multi-bounced virtual scattering channel model for dense urban street environments. IEEE Trans. Veh. Technol. 66(3), 1903– 1912 (2017)

Chapter 2

Geometry-Based Statistical MIMO Channel Modeling

2.1 Introduction With the considerably increasing requirements of the wireless frequency spectrum, MIMO wireless communication systems have attracted significant attention because of their potentially increased channel capacities. A practical MIMO channel model is imperative for system performance evaluations. The 3rd Generation Partnership Project (3GPP) standardization body developed a spatial channel model. Because the model was dedicated to outdoor-to-outdoor propagation, it defined three scenarios, including suburban macro, urban macro, and urban micro, in which 3GPP 25.996 [1, 2] provides the power azimuth profile (PAP), AoA, AoD, angle spread (AS), and other factors. In this standard, the AoA of the multipath components has an independent identically distributed (i.i.d.) Gaussian distribution for the macro and micro scenarios, and the AoD has a uniform distribution in the micro scenario. Because of the complicated radio environments and insufficient empirical data, these channel parameters for the outdoor-to-indoor scenario model need to be determined to design a MIMO antenna system and radio location system. A number of fundamental geometry-based statistical channel models have been proposed in many studies [3–15] to present the properties of different multipath environments such as outdoor and indoor propagation channels. References [3–10] present several analytical geometrical channel models in which different geometrical shapes are assumed in these models, including the following currently available models: circular scattering model, ellipse scattering model [3], and hollow-disc scattering model [6]. In these models, distributions of the scatterers are assumed to be uniform or non-uniform spatial distributions. The non-uniform spatial distributions are the Gaussian [9], inverted parabolic, conical, exponential, Rayleigh [5], and hyperbolic [8] distributions. The authors in [3–9] present empirical models that are derived from measurements for outdoor and indoor environments. It was found that a Gaussian spatial distribution matches an outdoor environment [4, 5, 9], and an exponential spatial distribution is the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_2

15

16

2 Geometry-Based Statistical MIMO Channel Modeling

best fit for an indoor environment [5, 8]. However, empirical models are only efficient and accurate for environments with the same specific characteristics as those where the measurements were made. They cannot be used for different environments without modifications, and they are even useless when applied to quite different environments. Janaswamy [9] showed that a geometrical channel model is applicable to both outdoor and indoor environments by changing the value of the standard deviation (σG ). However, a comparison with measurements showed that the performance of the model for indoor environments is worse than that for outdoor environments. A single spatial scatterer distribution is insufficient to model various propagation environments. Generally, a transmitter faces a large number of scatterers in the immediate neighborhood, where the scatterer density tapers off with distance from the transmitter, and the nearer scatterers may obscure the farther scatterers. There may also be physical phenomena in which scatterers have differing reflection probability (RP). The nearer scatterers can reflect signal waves, whereas further scatterers cannot. The reflecting scatterers are called effective scatterers (ES). Therefore, ES and RP principles are proposed in our geometrical channel model, making the model more applicable to physical environments. The aims of this paper are threefold. First, to introduce the concepts of ES and RP, a new scatterer distribution is proposed as a trade-off between the outdoor and the indoor propagation environments. The statistics of the AoA and time of arrival (ToA) of the multipath components are investigated in detail. Second, to study the effects of the directional antenna at the BS, we derive the closed-form expression for the joint and marginal probability density functions of the angle of arrival and time of arrival, the Doppler spectrum due to the motion of the MS can be determined. Third, we investigate the spatial fading correlation (SFC) for a MIMO uniform linear array (ULA) and uniform circular array (UCA) over a channel model that can be applied to determine the covariance matrix in MIMO systems for channel capacity evaluation. The channel capacity of different multiple MIMO antennas under the same channel conditions are elaborated and discussed. Then, our results of the AoA/ToA of the channel model can be used instead of deterministic considerations in 3GPP 25.996, to simulate the realistic correlated channels and to predict the performance of real MIMO antenna systems.

2.2 System Model and Joint PDFs of AoA/ToA 2.2.1 Geometrical Model and PDFs of AoA/ToA In outdoor and indoor environments, multipath components may originate near both the MS and BS. Hence, it is reasonable to include both inside the region of scatterers. We assume that the scatterers are distributed within a circular area around the mobile

2.2 System Model and Joint PDFs of AoA/ToA

17

receiver and that the BS is also located inside this area. To simplify the derivation, the following assumptions are made: 1. Each scatterer is an omnidirectional reradiating element that is independent of other scatterers. Between the BS and MS, each propagation path reflects off one scatterer. 2. Interference from neighboring cells is neglected, i.e., the MS is assumed to be located away from the border of the scatterer area. 3. The antennas of BS and MS are assumed to be omnidirectional to derive the geometrical channel model. To derive the Doppler spectrum at the MS, the MS transceiver antenna is assumed to be omnidirectional, and the BS antenna is directional with a main-lobe width of 2α. However, all the antenna polarization influences are overlooked. Figure 2.1 shows the geometry of the scatterers around the MS. The BS is marked as B (without loss of generality, the BS is located on the x-axis), and the radius of the cell is R. D denotes the distance between the BS and MS. A scattering point (SP) in this circle is denoted by polar coordinates (rs , φ) or rectangular coordinates (x, y) with the origin at the MS. (x  , y  ) denote the coordinates of the SP with the BS as the origin, as shown in Fig. 2.1. The corresponding polar coordinates of the rectangular coordinates (x  , y  ) are (rb , θb ). The PDFs of the scatterers around the

Fig. 2.1 Geometrical channel model

18

2 Geometry-Based Statistical MIMO Channel Modeling

MS are given by frs ,φ (rs , φ), fxy (x, y), fx  y  (x  , y  ), or frb ,θb (rb , θb ), according to the different coordinates. With a given frs ,φ (rs , φ), the joint PDFs of the AoA/ToA can be derived as follows. On the basis of the geometrical considerations in Fig. 2.1, the coordinates (x, y) are given by [10, 12] x = rb cos θb − D = y = rb sin θb =

(c2 τ 2 − D 2 ) cos θb −D 2cτ − 2D cos θb

(2.1)

(c2 τ 2 − D 2 ) sin θb , 2cτ − 2D cos θb

(2.2)

where θb symbolizes the multipath azimuth arrival angle. We firstly derive the general joint PDFs of the AoA/ToA and then show how these PDFs are simplified for a particular scatterer spatial distribution. It will be useful to express the scatterer distribution function with respect to the polar coordinates (rb , θb ) originating at the BS, which is an intermediate step in deriving the joint PDFs of the AoA/ToA. It is difficult to perform the transformation directly from the polar coordinates (rs , φ) to (rb , θb ) because the radial and angular coordinates will both change. By transforming from (x, y) to (x  , y  ), the joint PDF fxy (x, y) is found as   fxy (x, y) = |J (x, y)|frs ,φ (rs , φ) √ 2 2 rs = x +y φ=arctan(y/x)

=

1 x2 + y2

 2 2 frs φ x + y , arctan(y/x) ,

(2.3)

where |J (x, y)| is the Jacobian of the inverse transformation. Then, the joint PDF frb ,θb (rb , θb ) is written as   frb ,θb (rb , θb ) = rb fx  y  (x , y )  x =rb cos θb −D 



y  =rb sin θb

= rb fxy (rb cos θb − D, rb sin θb ).

(2.4)

The total path propagation delay time τ is obtained by τ=



 1 rb + rs = rb + D 2 + rb2 − 2rb D cos θb . c c

(2.5)

Squaring both sides of Eq. (2.4) to solve the eqnarray, we can obtain rb as rb =

D 2 − τ 2 c2 . 2(D cos θb − τ c)

(2.6)

2.2 System Model and Joint PDFs of AoA/ToA

19

The Jacobian transformation J (rb , θb ) is    ∂rb −1 2(D cos θb − τ c)2  =  J (rb , θb ) =  . ∂τ  D 2 c + τ 2 c3 − 2τ c2 D cos θb

(2.7)

Therefore, by interchanging the variables in Eq. (2.3) and performing some operations, we can obtain the joint PDF of the AoA/ToA as follows: fτ,θb (τ, θb ) = =

frb ,θb (rb , θb ) | J (rb , θb ) | D 2 c + τ 2 c3 − 2τ c2 D cos θb · frb ,θb (rb , θb ). 2(D cos θb − τ c)2

(2.8)

Now, we have to find the range of τ and θb for fτ,θb (τ, θb ). From the geometrical model of scatterers in Fig. 2.1 by the distributed area, the following eqnarray is obtained: 0 ≤ (x − D)2 + y 2 ≤ R 2 .

(2.9)

By transforming Eq. (2.8) in polar coordinates and performing some operations, we obtain the following expressions:   2 2   2 

2 1 − τDc 1 − τDc cos θb R 0≤  +1≤ .  − τc τc 2 cos θ − D b 4 cos θb − D D

(2.10)

2.2.2 Scatterer Distributed Density and ES/RP In wireless environments, the scatterer density tapers off with distance from the BS and MS. In a city where the MS is in the middle of a commercial area, the density of the buildings and scatterers will be reduced with distance from the town center as we move into the suburbs. This is an example of a scene where the scatterer density decreases with distance from the BS to the MS. A zero-mean Gaussian distribution with a standard deviation σG of the scatterers around the MS is the assumption that is most often used for an outdoor environment. An exponential distribution is applicable to an indoor environment. In order to compare them, the models we specifically address in the paper are those where the scatterer density around the MS is assumed to follow Gaussian frGs ,φ (rs , φ) and exponential frEs ,φ (rs , φ) distributions as [5, 9]

20

2 Geometry-Based Statistical MIMO Channel Modeling

frGs ,φ (rs , φ) =

1 2 2 e−rs /2σG , 0 ≤ rs ≤ R 2π NR σG2

(2.11)

a −ars e , 0 ≤ rs ≤ R, NE

(2.12)

frEs ,φ (rs , φ) =

where rs2 = x 2 + y 2 , NR = 1 − e−R /2σG , and NE = 1 − e−aR are the normalization constant because we restrict rs to [0, R ], and a is a constant. As described before, the MS will likely face a large number of scatterers in the immediate neighborhood, and farther buildings can also act as scatterers for waves. However, the farther we travel from the MS, there are fewer buildings that may act as scatterers, as they will be obscured by nearer buildings. By the definitions of ES and RP, each ES cannot reflect the signal if its RP is equal to zero. For an ES, the RP is utilized to express its ability to reflect a signal and its probability to reach the receivers. The RP decay parameter is defined as D = eλ , where λ is the RP decay exponent. Then, the RP is 2

1 P (rs ) = D

2

τ0 rs

= e−λτ0 rs = e−Lrs ,

(2.13)

where L = λτ0 , and τ0 = D/c is the minimum time delay of the LoS between the MS and BS. Then, the effective numbers of scatterers N (rs ) in radius rs is expressed as  rs P (rs )frs ,φ (rs , φ)drs . (2.14) N(rs ) = 2π 0

Therefore, the cumulative density function (CDF) is calculated as Frs (rs ) =

N(rs ) . N(R)

(2.15)

The closed-form expressions of the PDF of the modified scatterer distribution can be obtained directly by taking the derivative of Eq. (2.14) with respect to rs as follows: A. Uniform scatterer distribution e−Lrs e−Lrs frUs (rs ) =  R = . −Lr dr NU 0 e

(2.16)

2.2 System Model and Joint PDFs of AoA/ToA

21

B. Gaussian scatterer distribution     2 2 − Lrs +rs2 /2σG − Lrs +rs2 /2σG e e   = frGs (rs ) =  . 2 NG R − Lr+r 2 /2σG dr e 0

(2.17)

C. Exponential scatterer distribution e−(L+a)rs e−(L+a)rs = . frEs (rs ) =  R −(L+a)r dr NE 0 e

(2.18)

By introducing the ES and RP, the closed-form expressions of the original scatterer distributions in Eqs. (2.11) and (2.12) are modified to Eqs. (2.16)–(2.18). Please note that Eq. (2.16) represents a new scatterer distribution. When L → 0, it becomes a Gaussian distribution and becomes an exponential distribution as σG → ∞. The three types of distributions can be modified as a unified expression, which can be described as a trade-off between the outdoor and the indoor environments because a Gaussian distribution matches the outdoor environment, and an exponential distribution is the best fit for the indoor environments, as shown in Fig. 2.2, that have been discussed in Ref. [5].

1.0 L=0.001,σG=120 L=0,σG=150 0.8

L=0.01,σG=inf

f(rs)

0.6

0.4

0.2

0 −600

−400

−200

Fig. 2.2 Scatterer spatial distribution

0 rs

200

400

600

22

2 Geometry-Based Statistical MIMO Channel Modeling

2.3 Marginal PDFs of AoA and ToA 2.3.1 Marginal PDFs of AoA at BS The marginal PDFs of the uplink azimuth AoA can be found by integrating the joint PDFs of the AoA/ToA over the whole delay time τ . Another more straightforward method is to integrate the polar coordinate system representation of the scatterer density function by the distances. Then the marginal PDF of the AoA at the BS is given by [3, 4] fθb (θb ) =

1 NG



rb,m (θb )

  2 2 e− Lrs +rs /2σG rb /rs drb .

(2.19)

0

From Eq. (2.4), the quadratic formula is used, which obtains the maximum solutions rb,m (θb ) as the scatterers are at the circular boundary of radius R as follows:  rb,m (θb ) = D cos θb + R 2 − D 2 sin2 θb (2.20) By substituting Eq. (2.19) into Eq. (2.18), the marginal PDF of the AoA can be numerically calculated, where θb ∈ [−α, α]. The standard deviation of the ensemble average AS is found by  σθb =

π −π

θb2 fθb (θb )dθb .

(2.21)

2.3.2 Marginal PDFs of AoA at MS The proposed geometrical model shown in Fig. 2.3 can be used to analyze the effects of a directional antenna at the BS and derive the Doppler spectrum and investigate the MIMO system performance at the MS due to its motion. It is necessary to characterize the AoA of multipath components at the MS. Figure 2.3 shows the model when the BS has a directional antenna with unity gain and a main-lobe width of 2α. When α = π , the directional antenna at the BS is an omnidirectional antenna. When α < π , the directional antenna at the BS will partially illuminate the scatterers. Firstly, we calculate the weighted areas covered by the antenna mainlobe. It is obvious that the distribution of the arrival angle is symmetrical with respect to 0◦ . To simplify the derivation, we start with the distribution for positive angles by replacing | φ | with φ; hence, the following relationships can be obtained:

2.3 Marginal PDFs of AoA and ToA

23

Fig. 2.3 Model of directional antenna with main-lobe width of 2α at the BS

ρ=

D tan α . sin φ + tan α cos φ

(2.22)

If we let BMG be γ , then

γ = arccos

D2 + R2 − d 2 2RD

,

(2.23)

where d is the length of the line BG and is equal to d = D cos α+ R 2 − D 2 sin2 α. Then, the total area A2α illustrated by the directional antenna is expressed as A2α

2 = G N



γ





ρ

e 0

2 + G N

0



π



2 − Lrs +rs2 /2σG



R

e γ

0

 rs drs dθs

2 − Lrs +rs2 /2σG

 rs drs dθs .

(2.24)

24

2 Geometry-Based Statistical MIMO Channel Modeling

Case 1 0 ≤ φ ≤ γ Aarea (φ) =

1 NG



φ



0

ρ

  2 2 e− Lrs +rs /2σG rs drs dθs .

(2.25)

0

The CDF F (φ) is Eq. (2.24) divided by Eq. (2.23). Then, the marginal PDF of the AoA at the MS is derived by taking the derivative of F (φ) with respect to φ as 

2 G N A2α

f1 (φ) =

ρ

  2 2 e− Lrs +rs /2σG rs drs .

(2.26)

0

Case 2 γ < φ ≤ π Aarea (φ) =

1 NG +



γ





φ

0

1 NG

ρ

  2 2 e− Lrs +rs /2σG rs drs dθs

0

γ



R

  2 2 e− Lrs +rs /2σG rs drs dθs

(2.27)

0

then ∂Aarea (φ)/A2α ∂φ  R   2 2 2 = G e− Lrs +rs /2σG rs drs . N A2α 0

f2 (φ) =

(2.28)

2.3.3 Marginal PDFs of ToA Deriving a general marginal PDF of the ToA is more difficult, as integrating the joint PDF of the AoA/ToA over the AoA even for a simple case may be nearly intractable and does not yield manageable results. A more promising approach is to first derive the CDF of the ToA and then take the derivative with respect to τ to obtain the marginal PDF of the ToA. The CDF of the ToA is calculated as the probability of a scatterer being placed inside the ellipse corresponding to a delay equal to τ . The area of the overlap of the ellipse with the scatterer region is illustrated in Fig. 2.3, with three cases listed as: (1) τ ∈ [D/c, (D + 2R)/c]: when arc GC lies entirely outside the elliptical rim, and the ellipse and the boundary lines BG and BC cross each other at two points. (2) τ ∈ [(D + 2R)/c, (d + R)/c]: arc GC partly lies inside the elliptical rim. (3) τ ∈ [(d + R)/c, +∞): the pie-cut region illustrated by the directional antenna lies completely inside the ellipse. From Fig. 2.3, the following relationships are obtained as [12–14] (Fig. 2.4).

2.3 Marginal PDFs of AoA and ToA

25

Fig. 2.4 Illustration of the scatterer region that causes positive and negative frequency components when the MS is moving [14, 16] (a) toward the BS and (b) perpendicular to the BS. The scatterers in the regions A1 and A2 results in the positive and negative Doppler frequency components, respectively

Case 1 D/c ≤ τ ≤ (D + 2R)/c  α  ρ1

0 f (rb , θb )rb drb dθb F1 (τ ) =  α −α ,  rb,m (θb ) f (rb , θb )rb drb dθb −α 0

(2.29)

where ρ1 =

c2 τ 2 − D 2 . 2cτ − 2D cos θb

(2.30)

Taking the derivative with respect to τ gives the desired PDF of the ToA as follows:  α c2 τ 2 − D 2 1 f1 (τ ) =  α  r (θ ) × b,m b −α 2cτ − 2D cos θb f (rb , θb )rb drb dθb −α 0  

  c2 τ 2 − D 2 c2 τ c(c2 τ 2 − D 2 ) f × − , θb dθb . cτ − D cos θb 2cτ − 2D cos θb 2(cτ − D cos θb )2 (2.31)

26

2 Geometry-Based Statistical MIMO Channel Modeling

Case 2 (D + 2R)/c ≤ τ ≤ (D + R)/c F2 (τ ) =  α  r (θ ) b,m b −α 0



+ δ

α



ρ1



1 f (rb , θb )rb drb dθb

× 

f (rb , θb )rb drb dθb +

0

−δ −α

δ

−δ





ρ1

f (rb , θb )rb drb dθb 0



rb,m (θb )

f (rb , θb )rb drb dθb , 0

(2.32) where

c2 τ 2 − 2cτ R + D 2 δ = arccos . 2(cτ − R)D

(2.33)

The PDF of the ToA is the derivative of the CDF with respect to τ as 1

f2 (τ ) =   r (θ ) × α b,m b f (rb , θb )rb drb dθb 0 0

×

 α

c2 τ 2 − D 2 δ 2cτ − 2D cos θb

    c2 τ 2 − D 2 c(c2 τ 2 − D 2 ) c2 τ f dθ − , θ b b . (2.34) cτ − D cos θb 2cτ − 2D cos θb 2(cτ − D cos θb )2

Case 3 (d + R)/c ≤ τ < +∞ f3 (τ ) = 0

(2.35)

2.4 Doppler Spectrum The received signal at the MS experiences Doppler spread because of the moving MS. The multipath components at the MS experience Doppler shift depending on the direction of the motion. The i-th multipath component experiences a Doppler shift vi of vi = fm cos(φi − φv ), where fm is the maximum possible Doppler shift of v/λ, v is the velocity of the MS, and λ is the wavelength. φi is the angle between the i-th multipath component and the direct path, and φv is the angle between the direction of the motion and the direct path. Let the baseband complex representation L−1 of the received signal be r(t) = E0 i=1 χi ej 2π vi t , where χi is the complex multipath amplitude for the i-th multipath component. The Doppler spectrum L−1 S(f2) was shown in Ref. [14] to be S(f ) = A20 fv (f ), where A20 = E02 /4 i=1 |χi | , and fv (f ) is the probability density function of the distribution of the Doppler frequency. Assuming an omnidirectional antenna at the MS, it was shown in Ref. [14] that fv (f ) is given by

2.5 MIMO Performance Over the Channel Model

    fφ φv + | cos−1 (f/fm )| fφ φv − | cos−1 (f/fm )| fv (f ) = + , fm 1 − (f/fm )2 fm 1 − (f/fm )2

27

(2.36)

where |f | < fm , and fφ (φ) is the PDF of the AoA of the multipath components at the MS derived in Eqs. (2.25) and (2.27). Therefore, the Doppler spectrum S(f ) is given by S(f ) =



A20

fm 1 − (f/fm )2      × fφ φv + | cos−1 (f/fm )| + fφ φv + | cos−1 (f/fm )| .

(2.37)

Substituting the marginal PDF of the AoA into Eq. (2.36), the Doppler spectrum can be derived. If the AoA of the receiving signal at the MS has a uniform distribution, S(f ) becomes Clarke’s model [14]: S(f ) =

A2 0 . πfm 1 − (f/fm )2

(2.38)

2.5 MIMO Performance Over the Channel Model Information-theoretic results have shown a potentially large capacity gain for MIMO wireless systems that use multielement antenna arrays at both the BS and MS. One method to achieve sufficiently high performance is to separate the antenna elements at both the BS and MS such that a large diversity order can be obtained. Employing multiple antennas at the BS is not a significant problem, but accommodating more compact antennas on the MS introduces several constraints for practical implementation. However, in the antenna arrays, the SFC among the antenna elements due to the spatial proximity may limit the increase in capacity [10–15, 17].

2.5.1 SFC of MIMO Multiple Antennas In wireless systems, an antenna array is used to receive information from the users of wireless networking operating under the same or different multiple access schemes such as frequency division multiple access (FDMA), time division multiple access (TDMA), and code division multiple access (CDMA). The used MIMO antenna arrays may assume different geometries, such as ULAs, UCAs, and URAs. In the ULA shown in Fig. 2.5a, the locations of the antennas form a straight line, whereas

28

2 Geometry-Based Statistical MIMO Channel Modeling

Fig. 2.5 MIMO (a) ULA and (b) UCA multiple antenna arrays (Nr = L)

in a planar array, such as the UCA shown in Fig. 2.5b, the positions of the antenna elements are specified by two variables representing polar or Cartesian coordinates. Although the propagation delay between the antenna elements encountered as the signal travels across a linear array is only a function of the elevation angle, both the elevation and the azimuth AoAs define the propagation delay in the case of planar arrays. The SFC between antenna elements at positions m and n can be expressed as [18–20]   E (hm −  hm )(hn −  hn ) ρ(n, m) =      E (hm −  hm )2 E (hn −  h n )2 

Ψm (φ)Ψn∗ (φ)fφ (φ)dφ  . (2.39) 2 | Ψm (φ) |2 fφ (φ)dφ × φ | Ψn (φ) | fφ (φ)dφ φ

=  φ

In above eqnarray, E[·] denotes the expectation, the superscript(·)∗ is the complex conjugate, hm is the mean value of the channel response at antenna element m, and Ψm (φ) is the m-th entry signal at antenna element m. fφ (φ) is the PDF of the AoA of the multipath channel presented in Eqs. (2.25) and (2.27). Figure 2.5a and b only consider the azimuth plane, and the scalars 0 ≤ φ ≤ 2π is the azimuth angles defined with respect to x. For a frequency nonselective Rayleigh fading channel model, the complex amplitude of the j -th multiple wave h(t) can be presented as [17–20] h(t) =

J MP C j =1

aj (t) · Ψ (φ),

(2.40)

2.5 MIMO Performance Over the Channel Model

29

where aj (t) is one of a set of zero-mean complex i.i.d. random variables, Ψ (φ) is the steering vector of the compact antenna arrays, and JMP C is the total number of multipath components. On considering the antenna array in Fig. 2.5, the steering vectors of the ULA and UCA are given by [18, 19] T  ΨU LA (φ) = 1, ej kw d cos φ , . . . , ej kw d(L−1) cos φ

(2.41)

 T ΨU CA (φ) = 1, ej ζ cos(φ−ψ0 ) , ej ζ cos(φ−ψ1 ) , . . . . . . ,

(2.42)

and

where T denotes the transpose operation. For a MIMO ULA, L is the number of antennas, d is the antenna spacing, kw = 2π/λ, λ is the wavelength, and [·]T denotes the transpose. For a MIMO UCA, ψl = 2π l/L, l = 1, 2, . . . , L − 1, and ζ = kw r, in which r is the radius of the UCA.

2.5.2 Capacity Performance Depending on the elements of interest, the SFC can differ significantly from one element pair to another even under the same channel conditions. Hence, the general characteristics of a MIMO antenna array shown in Fig. 2.5 that is sensitive to the channel parameters cannot be fully represented. To examine the antenna array sensitivity to the channel parameters on the system performance, the system capacity is considered in estimation. The instantaneous channel capacity in bits per second per hertz of a stochastic MIMO channel under an average transmitting power constraint and where the transmitter has no channel knowledge is given as [20, 21]



 SN R T , C = log2 det INr + HH Nt

(2.43)

where INr is an Nr × Nr identity matrix. SN R is the average signal-to-noise ratio, H is the Nr ×Nt complex fading envelope and (·)T denotes the transpose conjugate, det(·) denotes the matrix determinant. The entries of channel matrix H are generated by making use of the correlation information of the transmitting and receiving fading signals. H is given by 1/2

1/2

H = Rr Hw Rt ,

(2.44)

where (·)1/2 denotes the matrix square root operation. Rr is the Nr × Nr receiving correlation matrix, and Rt is the Nt × Nt transmitting correlation matrix. Nr and Nt are the numbers of antenna elements at the receiver and the transmitter, respectively. Hw is the Nr × Nr stochastic matrix with complex Gaussian i.i.d. entries.

30

2 Geometry-Based Statistical MIMO Channel Modeling

2.6 Results and Discussions In order to demonstrate the efficiency and accuracy of the proposed geometry-based statistical channel model that introduced ES and RP, a series of illustrative examples for the downlink and uplink PDFs of AoA, PDFs of ToA, Doppler spectrum and the performance of a MIMO ULA and UCA in the wireless channel are provided. A geometric radius R of 100 m, a distance between the BS and MS D of 50 m, and BS directional antenna main-lobes of α = 40◦ and 60◦ are assumed. For MIMO ULA and UCA antenna arrays, it is assumed that Nr = 4, and SN R = 20 dB.

2.6.1 Joint and Marginal PDFs of AoA/ToA The joint PDFs of the ToA/AoA observed at the MS corresponding to the beamwidth α are plotted in Fig. 2.6. It is observed that the receiving signals mostly have small AoAs and short time delays. Note that the statistical channel model is symmetric °

°

α=60

65 60 55 50 80

0 AOA (deg)

1 0.5 TOA (μs)

−80 0

Probability Density

Probability Density

α=30

80 60 40 80

0 AOA (deg)

°

60 40 80

1 0.5 TOA (μs)

Probability Density

Probability Density

α=180

80

−80 0

−80 0 °

α=90

0 AOA (deg)

1 0.5 TOA (μs)

95 70 45 80

0 AOA (deg)

1 −80 0

0.5 TOA (μs)

Fig. 2.6 Joint PDFs of the geometrical channel model shown in Fig. 2.1 when R = 100 m, D = 50 m, L = 0.01, and σG = 120

2.6 Results and Discussions

31

1.0 L=0,σG=120 0.9

L=0.01,σ =120

0.8

L=0.01,σ =inf

0.7

Janaswamy (Outdoor) [9] Janaswamy (Indoor) [9]

Probability Density

G G

0.6 0.5 0.4 0.3 0.2 0.1 0 −60

−40

−20

0 20 Angle of Arrival (deg)

40

60

Fig. 2.7 Marginal PDFs of uplink AoA for geometrical channel model when R = 100 m, D = 50 m, and α = 40◦

with respect to the x-axis, and the missing section of graphs enlarges slowly, accompanied by the increase in the beamwidth α, and the results are symmetric at −φm and +φm . Note that the joint probability density near the LoS enlarges p greatly. Moreover, the joint ToA/AoA PDFs have two peaks at the points (τ p , ±φm ), p + p + where τ = τLoS and φm = 0 , and fall sharply from the peak points to the LoS point (τLoS , 0). Because of the geometric symmetry, the missing section of the graphs enlarges slowly, accompanied by a decreasing value of θb , and the results are symmetric at +θb and −θb . The marginal PDFs of the AoA observed at the BS in Eq. (2.18) are shown in Fig. 2.7. It is obvious that the AoA significantly decreases with an increase in θb . Moreover, it is observed that the PDFs of the receiving signals with smaller AoAs are relatively large; however, a larger AoA has a comparatively lower PDF, an observation that is in agreement with the results reported by Janaswamy [9]. In addition, it is obvious that the AoA of the proposed scatterer distribution is larger than that of the exponential distribution and lower than that of the Gaussian distribution. For a more realistic consideration of the positions of the MS, the effects of different positions of the MS with a beamwidth α spanning the azimuth angle range of [−ψ2 , ψ1 ] (i.e., different ψ1 and ψ2 ) on the PDFs of the downlink AoA are shown in Fig. 2.8. It is clearly observed that the curves of the PDFs are asymmetric and have two “corners" each in the right and left parts. When the MS’s position is close to the BS, the PDFs with smaller AoAs sharply decrease to a local minimum

32

2 Geometry-Based Statistical MIMO Channel Modeling ψ =60°, ψ =30°, D=40m 1

ψ =60°, ψ =30°, D=80m

2

1

2

1.0 Probability Density

Probability Density

1.0 0.8 0.6 0.4 0.2

0.8 0.6 0.4 0.2

0 0 −240−180−120−60 0 60 120 180 240 L=0,σ =120 −240−180−120−60 0 60 120 180 240 G Angle of Arrival (deg) Angle of Arrival (deg) L=0.01,σG=120 °

°

ψ1=80 , ψ2=20 , D=40m

L=0.01,σ =inf G

0.8 0.6 0.4 0.2 0 −240−180−120−60 0 60 120 180 240 Angle of Arrival (deg)

1.0 Probability Density

Probability Density

1.0

ψ1=80°, ψ2=20°, D=80m

0.8 0.6 0.4 0.2 0 −240−180−120−60 0 60 120 180 240 Angle of Arrival (deg)

Fig. 2.8 Marginal PDFs of downlink AoA for geometrical channel model when R = 100 m and D = 50 m

with lower AoAs and then increase slowly. In addition, it is obvious that the AoA of the proposed scatterer distribution is larger than that of the exponential distribution, and lower than that of the Gaussian distribution. The closed-form expressions of the ToA are plotted by the derived marginal PDFs of the ToA in Eqs. (2.30) and (2.33) versus the non-uniform distribution parameters σG and α to simulate radio propagation environments. Figure 2.9 shows the various situations of the ToA, also comparing the results with different values of L and σG . Each curve in Fig. 2.9 takes on a nonzero amplitude only for τ ≥ D/c, as τ = D/c corresponds to the LoS, which represents the shortest propagation path between the BS and MS. It is obvious that the ToA decreases with a decrease in α. That is because the decrease in α with fewer scatterers could reduce the multipath components. Additionally, the ToA of the proposed scatterer distribution is larger than the exponential distribution but smaller than the Gaussian distribution at the same AoA. Janaswamy [9] proposed a geometry-based channel model with a Gaussian density of scatterers. The results predicted by the model were compared with experimental data available both for outdoor and indoor environments. For comparison

2.6 Results and Discussions

33

9

2.5

9

x 10

2.5

°

x 10

L=0.01,σ =120,α=60°

L=0,σ =120,α=60

G

G

°

L=0.01,σ =120,α=60 L=0,σ =120,α=40° G

L=0.01,σ =120,α=40°

1.5

G

Janaswamy Model(Outdoor)[9] 1

0.5

0 0.1

L=0.01,σ =inf,α=60°

2

G

Probability Density

Probability Density

2

G

L=0.01,σ =120,α=40° G

L=0.01,σ =inf,α=40°

1.5

G

Janaswamy Model(Indoor)[9] 1

0.5

0.2

0.3 0.4 Time of Arrival(μsec)

0.5

0.6

0 0.1

0.2

0.3 0.4 Time of Arrival(μsec)

0.5

0.6

Fig. 2.9 Marginal PDFs of ToA versus to L, σG and α when R = 100 m and D = 50 m

of our results with the existing channel models in Ref. [9], we introduced the results in Figs. 2.7 and 2.9. By varying the parameters L and σG in Eq. (2.16), good agreement between our results and those shown in Ref. [9] was found. As expected, the channel model should find use both in indoor as well as outdoor environments with a unified expression, which can be described as a trade-off among different propagation environments. The PDFs of the AoA/ToA of the geometrical-statistical channel model are investigated carefully. There are other statistical models based on deterministic considerations in Refs. [19, 20, 22], which assumed the incident multipath signals to be deterministic Gaussian or Laplacian distributions. For example, Ref. [22] deals with the implementation and the assessment of a wideband directional channel model, makes use of a Gaussian generator to generate the incident signal PAP in the simulation, and implements a link level simulator. Then, the numerical results of the joint and marginal PDFs of the AoA/ToA are related with the PAP described in Ref. [22]. The results may be used to simulate a MIMO system and design a link level simulator or diversity algorithms.

2.6.2 Doppler Spectrum We assume for a test case that the velocity v of the MS is 54 km/h and the carrier frequency fc is 2 GHz. The radius of the cell and the distance between the BS and MS are chosen to be 100 m and 50 m, respectively. The effects of σG and φv on the Doppler spectral density are shown in Fig. 2.10. It is observed that the Doppler spectrum decreases with an increase in σG , which is because the increase in σG could diminish the number of scatterers and reduce the multipath components. Moreover, it can be found that the Doppler spectrum is smaller than that of the Gaussian distribution but larger than that of the exponential distribution at the same AoA.

34

2 Geometry-Based Statistical MIMO Channel Modeling −5

°

L=0,σ =120,φ =0 G

v

°

L=0.01,σG=120,φv=0 power spectral density (dB)

−10

°

L=0,σG=120,φv=90

°

L=0.01,σG=120,φv=90 Clarke Model

−15

−20

−25

−30 −150

−100

−50

0 frequency (Hz)

50

−5

100

150

100

150

°

L=0.01,σ =inf,φ =0 G

v

L=0.01,σ =120,φ =0° G

power spectral density (dB)

−10

v

°

L=0.01,σG=inf,φv=90

L=0.01,σ =120,φ =90° G

v

Clarke Model

−15

−20

−25

−30 −150

−100

−50

0 frequency (Hz)

50

Fig. 2.10 Doppler spectrum in correspondence with the direction of the motion of the MS’ φv and σG when R = 100 m, D = 50 m, α = 40◦ , v = 54 km/s, fc = 2 GHz, and fm = 100 Hz

We quantify the effects of using a directional antenna at the BS on the fading envelopes in terms of the movement direction φv on the Doppler spectrum, shown in Fig. 2.10. For φv = 0◦ , the spectrum is skewed to the right, and there are more negative Doppler frequency components than positive frequency components.

2.6 Results and Discussions

35

This is because the MS is moving in the direction of the LoS toward the BS (see Fig. 2.4a). Because the area of region A2 is larger than that of region A1 , there are more negative Doppler frequency components than positive frequency components in the Doppler spectrum. However, for φv = 90◦ , the spectrum is symmetrical about the zero Doppler frequency component. When the MS moves perpendicular to the LoS (see Fig. 2.4b), the area of scattering region A1 is equal to that of region A2 ; therefore, the Doppler spectrum is symmetrical about a Doppler frequency of zero [12]. As the directional antenna is able to mitigate multipath components with large AoAs, region A1 approaches region A2 gradually as the beamwidth of the antenna decreases, meaning that the curves in Fig. 2.10 at φv = 0◦ are becoming more symmetrical gradually.

2.6.3 Performance of MIMO Multiple Antennas On the basis of the marginal PDFs of the AoA at the MS, the SFCs between antenna elements 1 and 2 for MIMO transceivers in Fig. 2.5 can be numerically calculated in Figs. 2.11 and 2.12, which are fundamental results for investigating the performance of MIMO wireless systems. Figure 2.11 shows the SFCs between ULA elements 1 and 2 versus d/λ and α in the proposed channel model. It can be observed that as

1 L=0,σ =120,α=60° G

0.9

L=0.01,σ =120,α=60°

0.8

L=0,σ =120,α=40°

G

G

L=0.01,σ =120,α=40° G

correlation |ρ(1,2)|

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8 1 1.2 1.4 Antenna spacing (d/λ)

1.6

1.8

2

Fig. 2.11 Spatial fading correlations of MIMO ULA (Fig. 2.5a) versus d/λ for R = 100 m, D = 50 m, and Nr = 4

36

2 Geometry-Based Statistical MIMO Channel Modeling 1 L=0.01,σ =120,α=60°

correlation |ρ(1,2)|

G

0.9

L=0.01,σG=inf,α=60°

0.8

L=0.01,σ =120,α=40°

0.7

L=0.01,σG=inf,α=40°

G

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1 Antenna spacing (r/λ)

1.5

2

Fig. 2.12 Spatial fading correlations of MIMO UCA (Fig. 2.5b) versus to r/λ for R = 100 m, D = 50 m, and Nr = 4

d/λ increases, the SFCs decrease rapidly, which agrees with the results of the UCA as r/λ increases, as shown in Fig. 2.12. If α increases with larger spacing between two antenna elements in MIMO ULA, the antenna array becomes weakly compact and its SFCs decrease monotonously. However, with increasing α in MIMO UCA and with smaller spacing between two elements, the antenna array becomes strongly compact, increasing the SFCs. Figure 2.13 shows that increasing d/λ leads to higher capacity for a MIMO ULA. The capacity at lower d/λ ≤ 0.5 increases rapidly because of the freely decreasing SFCs between any two elements at the same value of d/λ. However, the capacity tends to saturate the upper bound at d/λ ≥ 0.5 as α increases. It is observed that the capacity of the Gaussian distribution is smaller than that of the proposed distribution. When α is increased to 60◦ , the SFCs tend to be weaker, which increases the system capacity. Figure 2.14 presents the capacities of the MIMO UCA versus r/λ in the proposed channel model. The results are familiar with those of the exponential distribution at the same α. It can be observed that increase in r/λ leads to higher capacity. Unlike the increasing of d/λ in ULA, the capacity increases rapidly with no obvious effects. It is observed that the capacity of the exponential distribution is larger than that of the proposed distribution. However, by increasing α to 60◦ , the SFCs reach the same values, meaning that the capacities will tend to have identical upper bounds.

2.7 Conclusion

37

15 14

Capacity(bits/s/Hz)

13 12 11 10 9

°

L=0,σG=120,α=60

°

8

L=0.01,σG=120,α=60 L=0,σ =120,α=40° G

7

°

L=0.01,σG=120,α=40 6

0

0.5

1 Antenna spacing(d/λ)

1.5

2

Fig. 2.13 Capacities of MIMO ULA versus α and σG when R = 100 m, D = 50 m, and Nr = 4

In order to obtain a statistical channel model that can represent outdoor and indoor environments, a series of comparisons have been made with reference to α and Gaussian, Exponential, and our proposed scatterer distributions. Because it is known that α and σG have significant impacts on the performance of MIMO, the parameter σG is determined by the radio environment. By increasing σG , the density of the scatterers tapers off quickly with the distance from the MS as we move into the suburbs, which diminishes the number of scatterers and multipath components in our geometrical model. From Figs. 2.13 and 2.14, we observe that with increasing α, the capacity of a ULA tends to increase, whereas it decreases in a UCA. It is obvious that increasing d/λ and r/λ reduces the capacities of the proposed scatterer distribution more than with an exponential distribution but they are larger than those of the Gaussian distribution. From the above analysis, we observe that the statistical channel model introduces ES and RP is suitable for outdoor and indoor environments.

2.7 Conclusion In this chapter, we develop a geometry-based statistical channel model in which we introduced ES and RP to form a new scatterer distribution as a trade-off between outdoor and indoor propagation environments. General formulations of the joint

38

2 Geometry-Based Statistical MIMO Channel Modeling 15 14

Capacity(bits/s/Hz)

13 12 11 10 9

°

L=0.01,σ =120,α=60 G

8

L=0.01,σG=inf,α=60°

7

L=0.01,σG=120,α=40

°

L=0.01,σ =inf,α=40° 6

G

0

0.5

1 Antenna spacing(r/λ)

1.5

2

Fig. 2.14 Capacities of MIMO UCA versus α and σG when R = 100 m, D = 50 m, and Nr = 4

PDF of the AoA/ToA and marginal PDFs of the AoA and ToA are outlined by employing different scatterer distributions. We also analyzed the effects of the directional antenna at the BS, simply parameterized by the main-lobe width of 2α on the AoA, ToA, and the Doppler spectrum due to the motion of the MS. Furthermore, we investigated the SFCs for MIMO ULA and UCA over the proposed channel model, which can be applied to determine the covariance matrix in MIMO systems. Then, the results of the AoA/ToA can be used instead of deterministic considerations in 3GPP to simulate the realistic correlated channels and to predict the real MIMO performance.

References 1. 3GPP, Spatial channel model for MIMO simulations, TR 25.996. V7.0.0. (2007). Available: http://www.3gpp.org 2. H. Xiao, A.G. Burr, R.C. Lamare, Reduced complexity cluster modelling for the 3GPP Channel Model, in 2007 IEEE International Conference on Communications (Glasgow, 2007), pp. 4622–4627 3. R.B. Ertel, J.H. Reed, Angle and time of arrival statistics for circular and elliptical scattering model. IEEE J. Sel. Areas Commun. 17(11), 1829–1840 (1999) 4. L. Xin, Gaussian angular distributed MIMO channel model, in 2011 IEEE Vehicular Technology Conference (VTC Fall) (2011), pp.1–5

References

39

5. L. Jiang, S.Y. Tan, Geometrically based statistical channel models for outdoor and indoor propagation environments. IEEE Trans. Veh. Technol. 56(6), 3587–3593 (2007) 6. J. Zhou, L. Qiu, K. Hisakazu, Analyses and comparisons of geometrical-based channel model arisen from scatterers on a hollow-disc for outdoor and indoor wireless environments. IET Commun. 6, 2775–2786 (2012) 7. T.L. Ngoc Nguyen, Y. Shin, A new approach for positioning based on AOA measurements, in International Conference on Computing, Management and Telecommunications (ComManTel) (Ho Chi Minh City, 2013), pp. 208–211 8. A. Mittal, R. Bhattacharjee, B.S. Paul, Angle and time of arrival statistics for a far circular scattering model, in Proceedings of NCC2009 (2009), pp. 141–145 9. R. Janaswamy, Angle and time of arrival statistics for the Gaussian scatter density model. IEEE Trans. Wirel. Commun. 1(3), 488–497 (2002) 10. E. Tsalolihin, I. Bilik, Analysis of AOA-TOA signal distribution in indoor environments, in Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP) (Rome, 2011), pp. 1646–1650 11. S.J. Nawaz, M.N. Patwary, On the performance of AoA estimation algorithms in cognitive radio networks, in International Conference on Communication, Information & Computing Technology (ICCICT) (Mumbai, 2012), pp.1–5 12. S.H. Kong, TOA and AOD statistics for down link Gaussian scatterer distribution model. IEEE Trans. Wirel. Commun. 8(5), 2609–2617 (2009) 13. C.L.J. Lam, K.T. Wong, Y.I. Wu, The TOA-distribution of multipaths between an omnidirectional transceiver and a mis-oriented directional transceiver. IEEE Trans. Commun. 58(4), 1042–1047 (2010) 14. P. Petrus, H. Jeffrey, Geometrically-based statistical macrocell channel model for mobile environments. IEEE Trans. Commun. 50(3), 495–502 (2002) 15. S.J. Nawaz, B.H. Qureshi, N.M. Khan, A generalized 3-D scattering model for a macrocell environment with a directional antenna at the BS. IEEE Trans. Veh. Technol. 59(7), 3193– 3204 (2010) 16. J. Zhou, H. Jiang, K. Hisakazu, Geometrical statistical channel model and performance investigation for multi-antenna systems in wireless communications. Acta Phys. Sin. 63(12), 123201 (2014) 17. S. Buyukcorak, G.K. Kurt, Simulation and measurement of spatial correlation in MIMO systems with ray tracing, in 5th International Conference on Signal Processing and Communication Systems (ICSPCS) (2011), pp. 1–5 18. J.A. Tsai, R.M. Buehrer, B.D. Woerner, Spatial fading correlation function of circular antenna arrays with Laplacian energy distribution. IEEE Commun. Lett. 6(5), 178–180 (2002) 19. J. Zhou, S. Sasaki, S. Muramatsu, Spatial correlation for a circular antenna array and its applications in wireless communication, in Proceedings of IEEE Global telecommunications Conference (GLOBECOM), vol. 2 (2003), pp.1108–1113 20. A. Qahtani, Z.M. Hussain, Spatial correlation in wireless space-time MIMO channels, in Australasian Telecommunication Networks and Applications Conference (2007), pp. 358–363 21. S.H. Jin, Online social relations and country reputation. Int. J. Commun. Syst. 7, 1–20 (2013) 22. S. Morosi, M. Tosi, R.E. Del, Implementation of a wideband directional channel model for a UMTS link level simulator, in Proceedings of IEEE Global Telecommunications Conference (GLOBECOM), (San Francisco, USA, 2003)

Chapter 3

3D Scattering Channel Modeling for Microcell Communication Environments

3.1 Introduction The MIMO antenna system [1–5] has been accepted as a promising technology, given its potential to achieve a low bit error ratio (BER) by spatial temporal coding and high capacity by multiplexing. MIMO technologies such as the VSFOFCDM system and the European MATRICE project are integrated into 4G mobile networks. The study of the spatial and temporal characteristics of multipath channels has proven to be useful for improving the performance of MIMO antenna systems. Detailed knowledge about a mobile communication propagation channel is essential for designing wireless MIMO communication systems. In particular, a clear understanding of the spatial properties of the channel is a prerequisite in order to design and evaluate these systems [6–10]. The wireless channel model generally assumes a statistical distribution of scatterers in open environments. The properties of the multipath channel must be derived from the positions of the scatterers distributed in outdoor or indoor environments by applying the fundamental laws of the propagation of electromagnetic waves. Geometrical modeling of the propagation channel has always been attractive for researchers owing to its many advantages. Determination of the AoA, the ToA, and the probability density functions (PDFs) of the received signals is important and difficult. A number of fundamental geometrybased statistical channel models have been proposed in the literature [11–27]. In these models, different scatterer distributions have been considered, and 2D/3D geometrical models have been developed to present the properties of multiple paths in outdoor and indoor environments. In general, single-bounce scattering geometrical models are most widely used. Almost all of the geometrical models belong to the classes of circular models and elliptical models. The simplest and most common class among them is the geometrically based single-bounce CM, which is suitable for outdoor environments. For an indoor environment with low-height antennas, the geometrically based single-bounce EM is more © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_3

41

42

3 3D Scattering Channel Modeling for Microcell Communication Environments

attractive, and this model has been widely reported in the literature [11–15]. Refs. [11–15] focused on 2D models, and they assume that the propagation takes place in the plane joining the tips of the transmitting and receiving antennas. A detailed comparison of various 2D models, together with the measurements, has been presented in Ref. [15]. The 2D models are adequate for describing the angular spreading of the incoming multipath waves in the azimuth plane. Many researchers [16–27] have also observed the angular spreading of the incident waves in the elevation plane due to the interaction of the waves with building rooftops, ground, or other vertically disposed objects. The typical spreading of the waves in the elevation plane takes place at angles up to 20◦ . The description of the vertical spreading of the waves is particularly important for vertical or planar antenna arrays. It is well known that the spatial correlation properties of the received signal depend directly on the angular spreading of the waves in MIMO antenna arrays. In previous works [11–16], several 3D geometrical scattering channel models have been developed for various types of wireless communication environments. Several decades ago, an experiment conducted by Lee [16] confirmed that the scattered signals arriving under NLoS conditions are spread over a wide elevation angle, e.g., with a large portion of the signals outside 16◦ , and the elevation angle for scattered signals received in urban areas is greater than that in suburban areas. Although the azimuthal angle had been assumed to be uniformly distributed in Clarke’s model, a particular form of the PDF for the elevation angle was proposed to obtain a closed-form expression for the Doppler PSDs when only the elevation angle is very small, as it has an unrealistic flat section near the maximum frequency end. To overcome this problem and obtain the analytical solution, Ref. [17] considered an asymmetrical distribution of the elevation angle, treated the symmetrical distribution as a special case, and investigated the auto-correlation function and Doppler shift. Recently, more 3D models can be found in Refs. [10, 11]. Sahalos [10] proposed a simple 3D geometry-based channel model for outdoor wireless environments and extended the 2D geometrically based single-bounce model (GBSBM). Sebak [19– 21] presented a 3D ellipsoidal model in which both the BS and MS are considered to be inside the ellipsoid for simulating indoor environments. A simple semispheroid model for the signal that is received by the MS in 3D environments has been proposed by Janaswamy [15]. Given these numerous models, there is a strong need for a generalized 3D geometrical scattering model from which any 3D or 2D scattering model can be deduced with an appropriate combination of channel parameters. In addition, there is a need for more in-depth research on the application of the generalized model for MIMO antenna receivers. In this paper, a generalized 3D SS channel model is proposed in which both the MS and BS are considered to be inside the SS-shaped scattering region. The proposed model is a special case of the 3D ellipsoidal model [19–21]. In Ref. [12], the presented circular model, in which both the BS and MS are considered inside the circular scattering region to simulate indoor and outdoor environments, is a special case of our proposed model in the azimuth plane. The proposed model may be used for practical applications, e.g., to simulate special microcell environments such as Wi-

3.1 Introduction

43

Fig. 3.1 (a) Generalized 3D scattering channel model. (b) Threshold angles in the azimuth and elevation planes

Fi or ad hoc/mesh networks, when the antennas are located at low altitudes close to the ground. Refs. [26] and [27] presented a generalized 3D scattering channel model for land mobile radio cellular systems (outdoor environments). Refs. [28] and [29] mainly derived the closed-form expressions for the spatial fading correlation functions of MIMO antenna arrays in a 3D multiple channel, assuming the incident signal as some special statistical distributions which may be not good agreed to the practical wireless environments. As shown in Fig. 3.1, a generalized 3D scattering channel model applicable to different environments is proposed, in which a BS is located at the center of a 3D SS scattering region, and an MS is located within the region. In this work, we first derive the closed-form expression for the joint and marginal PDFs of the AoA and ToA measured at the MS corresponding to the azimuth and elevation angles. Then, we proceed to describe the Doppler spectra due to the motion of the MSs. Generally, the BS is equipped with a directional antenna. The effects of the directional antenna on the spatial and temporal parameters of the proposed model are analyzed, and the expressions for the PDFs of the AoA and ToA are derived. From numerical results and observations, the main advantages are: (1) the only one model proposed in the paper can be used to simulate the indoor and outdoor environments, instead of the 3D spheroid model and 3D ellipsoidal model [10–21], (2) it is possible to deduce all of the 2D models [11, 12] previously proposed, from our proposed 3D channel model, (3) firstly, the effects of the directional antenna employed at BS in indoor environments are investigated, otherwise Ref. [24] only analyzed the outdoor environment, (4) the Doppler spectra and performance analysis of MIMO antenna system in 3D channel models are first derived and discussed carefully, which works have not been regular found in the previous literature. Moreover, the results are compared with those of the previously proposed channel models and measurement results to validate the generalization of our model. Then we considered the proposed

44

3 3D Scattering Channel Modeling for Microcell Communication Environments

model may be used to the practical justification. Furthermore, our proposed model may be used to analyze the performance of massive MIMO related to 5G antenna receiving systems [30, 31].

3.2 System Channel Model This section describes the proposed 3D propagation channel model for land mobile radio cellular systems. For this purpose, uniformly distributed scatterers are assumed to be present in the 3D space with their mean at the BS and limited by an SS-shaped virtual boundary. A directional antenna with a controllable beamwidth α in the azimuth plane is deployed at the BS located at the center of the scattering region. The proposed 3D scattering channel model is depicted in Fig. 3.1a, where the major and minor dimensions of the scattering SS are a and b, respectively. The angles created in the azimuth and elevation planes with the direction of the incident signal at the BS are denoted by φb and βb , and at the MS, they are denoted by φm and βm , respectively. The distance between the BS and the MS is D. Here, we assume that D ≤ a and b ≤ a, and the distances of the scattering object from the MS and BS are rm and rb , respectively. Further, we assume that the BS is equipped with a directional antenna with a beamwidth α, the same assumption made in Ref. [24]. This means that the beamwidth is α in the horizontal plane and π/2 in the vertical plane. If we consider the other values of the beamwidth in the vertical plane, the analyses will become more complicated; therefore, this will be considered in our future work. Because of the directional antenna at the BS, clipping occurs in the scattering region. The scattering region illuminated by the antenna beam of width α is indicated by IRegion and can be expressed as [10–19] V =

Vspheroid 2a 2 bα − V1 = , 2 3

(3.1)

where Vspheroid = 2π a 2 b/3 is the volume of the entire spheroid, and V1 is the volume of the non-illuminated region of the SS. At α = π , the volume V is determined by the relation V = 2π a 2 b/3, which means with the omnidirectional antenna at the BS. If the Cartesian coordinates of a point with respect to the MS are (xm , ym , zm ), and those with respect to the BS are (xb , yb , zb ), then the transformations between the various coordinates are given as follows: xm = xb + D,

xb = rb cos βb cos φb ,

xm = rm cos βm cos φm

ym = yb ,

yb = rb cos βb sin φb ,

ym = rm cos βm sin φm

zm = z b ,

zb = rb sin βb ,

zm = rm sin βm .

(3.2)

3.2 System Channel Model

45

Now, the SS can be defined by the following equations or combinations thereof as [15] xb2 + yb2 zb2 + =1 a2 b2 rb2 sin2 βb rb2 cos2 βb + =1 a2 b2

(3.3)

2 cos2 β − 2Dr cos β cos φ 2 sin2 β D 2 + rm rm m m m m m + = 1. a2 b2

The scattering region illuminated by the beam from the directional antenna is shown in Fig. 3.1b. The dihedral angle βM can be defined as the angle between the plane P MQ and the plane xoy. Thus, the azimuth threshold angles φt1 and φt2 can be expressed as φt1 = 0, 0 ≤ βM ≤ π/2 ⎧   ⎨arccos P M 2 +QM 2 −P Q2 , 0 ≤ β ≤ arctan  b  M 2P M×QM D sin α φt2 =  b  ⎩0, arctan D sin α ≤ βM ≤ π/2,

(3.4)

(3.5)

where  PQ =

a2 −

a2 2 2 D sin α tan2 βM b2

 QM = d 2 + D 2 sin2 α tan2 βM  P M = D 2 + D 2 sin2 α tan2 βM + P Q2 − 2D × P Q cos α.

(3.6)

If βM = 0, the angles φ1 and φ2 are calculated to separate the aforementioned two partitions in the zero-elevation plane. The angle βM,t is defined to separate IRegion in the elevation plane and to be a function of φm and α of the directional antenna. Making methodical simplifications, the equation can be written in closed form as ⎧   ⎨ cot−1 √ aD csc(α+φm ) sin α , φ1 ≤ |φm | ≤ φ2 b a 2 −D 2 csc2 (α+φm ) sin2 α βM,t = (3.7) ⎩0, otherwise.

46

3 3D Scattering Channel Modeling for Microcell Communication Environments

As discussed earlier, we can separate the illustrated region into the partitions P1 and P2 as follows: ⎧ ⎪ ⎪0 ≤ βM ≤ βM,t ⎨ P1 → or   ⎪ ⎪ ⎩φt1 ≤ φ  ≤ φt2 m ⎧ ⎪ ⎪ ⎨βM,t ≤ βM ≤ π/2 P2 → or   ⎪ ⎪ ⎩φt2 ≤ φ  ≤ 2π − φt2 . m

(3.8)

(3.9)

The distances from the boundary of the scattering SS to the MS and BS, i.e., rm and rb , respectively, can be derived as functions of the angles measured at the BS and written as  (ab)2 rb (φb , βb ) = , −α ≤ φb ≤ α and 0 ≤ βb ≤ π/2 b2 cos2 βb + a 2 sin2 βb (3.10)

rm (φb , βb ) =

 D 2 + rb2 − 2rb D cos βb cos φb .

(3.11)

3.3 Spatial Characteristics of the 3D Scattering Channel Model In radio channel, Spatial Characteristics of channel models that describe the AoA and ToA statistics of the multipath components are very useful in the performance evaluation of wireless communication systems employing MIMO antenna arrays at the BS and MS. Here, focusing our on the downlink AoA and ToA to the MS. The reasons are (1) Because the proposed 3D scattering channel model is depicted as Fig. 3.1a, all scatterers are located within the 3D space with its center at BS, the geometrical model is a realistic situation to analyze the statistics of the channel observed at the MS side (see Ref. [12], only analyzed the channel of observed at the MS side). Similar methods are used to analyze the channel observed at the BS side. (2) Employing multiple antennas at BS does not present a significant problem because of having enough space to design antenna arrays to achieve high performance, but accommodating more antennas at the MS introduces several constraints for practical implementation. Therefore, it seems more important to also investigate the MIMO performance at the MS side.

3.3 Spatial Characteristics of the 3D Scattering Channel Model

47

3.3.1 Probability Density Functions of the AoA Here, the PDFs of the AoA observed at the MS are derived for the 3D scattering channel model. The joint density function corresponding to the azimuth and elevation angles observed at the MS can be expressed as [15, 23]  f (xm , ym , zm )  p(rm , φm , βm ) = , |J (xm , ym , zm )| P1 ,P2

(3.12)

where J (xm , ym , zm ) is the Jacobian transformation. When scatterers are uniformly distributed in the illuminated scattering region of volume V , the scatterer density can then be expressed as 1/V . Therefore, the joint density function can be obtained as p(rm , φm , βm ) =

2 cos β rm m . V

(3.13)

The distance rm was determined earlier in (3.11), which is the distance of the MS from the scattering objects located at the boundary of the scattering region, and can be rewritten as follows: rm (φm , βm ) ⎧ ⎪ P1 D sin α csc(α + φm ) sec βm , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ b2 cos2 βm +a 2 sin2 βm  = ⎪ 2 ⎪ × ⎪ Db cos βm cos φm ⎪ ⎪  ⎪  ⎪ 2    ⎪ ⎪ 2 2 2 2 2 2 2 2 2 ⎩+ Db cos βm cos φm − b cos βm + a sin βm b D − a b , P2 .

(3.14) As V and rm have been derived earlier in (3.1) and (3.14), respectively, we can obtain the simplified solution for the joint PDFs of the AoA as p(φm , βm ) ⎧  3 cos βm ⎪ D sin α csc(α + φm ) sec βm , P1 ⎪ ⎪ 3V ⎪ ⎪  ⎪ ⎪ ⎪ cos βm ⎪ 1 ⎨ 3V b2 cos2 βm +a 2 sin2 βm =  ⎪ ⎪ × Db2 cos βm cos φm ⎪ ⎪ ⎪  ⎪  ⎪    3  ⎪ ⎪ ⎩+ Db2 cos βm cos φm 2 − b2 cos2 βm + a 2 sin2 βm b2 D 2 − a 2 b2 , P2 .

(3.15)

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3 3D Scattering Channel Modeling for Microcell Communication Environments

As in earlier derivations, the marginal PDFs of the azimuth AoA can be obtained by integrating (3.15) over βm with the appropriate limits as follows: 

βM,t

p(φm ) = 0

  p(φm , βm ) dβm P1



π/2

+

βM,t

  p(φm , βm ) dβm , 0 ≤ φm ≤ 2π.

(3.16)

P2

Similarly, the marginal PDFs of the AoA in elevation can be obtained by integrating (3.15) over φm with the appropriate limits as  p(βm ) =

φt2 −φt2



  p(φm , βm ) dφm P1

2π −φt2

+

φt2

  p(φm , βm ) dφm , 0 ≤ βm ≤ π/2.

(3.17)

P2

3.3.2 Probability Density Functions of the TOA In this section, the temporal statistics for the 3D channel model are derived and discussed, and the joint and marginal PDFs of the ToA/AoA are derived. Here, τ is the propagation delay of a multipath signal reflected from any one scatterer, which is assumed to be located in the illuminated 3D scattering region. τ can be then generally expressed as [23–25] τ=

r m + rb . c

(3.18)

If the delay in the LoS link is τ0 , and τmax is the longest propagation delay, they can be expressed as τ0 = D/c and τmax = D/c · ρ12 , respectively, where c is the velocity of light. Now, ρ12 is given by ρ12

a + = D



 a a 2 cos α + 1. −2 D D

(3.19)

In previous sections, rb and rm were given by (3.10) and (3.14), respectively. rm may also be expressed in another simplified form in (3.11). Substituting (3.11) into (3.18) and solving for rb , we obtain rb (τ, φb , βb ) =

c2 τ 2 − D 2 . 2(cτ − D cos βb cos φb )

(3.20)

3.3 Spatial Characteristics of the 3D Scattering Channel Model

49

Similarly, rm of the scatterer from the MS can be rewritten as rm (τ, φm , βm ) =

c2 τ 2 − D 2 . 2(cτ − D cos βm cos φm )

(3.21)

Now, the joint density function for the AoA/ToA can be expressed as p(τ, φm , βm ) =

 p(rm , φm , βm )  |J (rm , φm , βm )| P1 ,P2

(3.22)

and the Jacobian transformation |J (rm , φm , βm )| can be expressed as [24]  ∂r −1 2(D cos βm cos φm − cτ )2  m . (3.23) J (rm , φm , βm ) =   = ∂τ c(D 2 + c2 τ 2 − 2cτ D cos βm cos φm ) In substituting (3.21) and (3.23) into (3.22), the joint density function for the ToA/AoA can be rewritten in a simplified form as p(τ, φm , βm ) =

c(c2 τ 2 − D 2 )2 (D 2 + c2 τ 2 − 2cτ D cos βm cos φm ) cos βm , 8V (D cos βm cos φm − cτ )4 where 0 ≤ φm ≤ 2π, 0 ≤ βm ≤ π/2.

(3.24)

The joint PDFs of the TOA in the azimuth and elevation planes observed at the MS can then be obtained by integrating (3.24) over the elevation and azimuth angles, respectively. Using Mathematica, the closed-form expressions can be obtained as 

π/2

p(τ, φm ) =

p(τ, φm , βm )dβm 0

k1 = 2 2 6k5 (k4 − k52 )7/2



  2 2 k4 − k5 k3 k5 2k43 + 13k4 k52

  + k2 k44 − 10k42 k52 − 6k54



  k4 + k5 − 6k52 k3 k5 (4k42 + k52 ) − k2 k43 + 4k4 k52 atanh  k42 − k52 (3.25)

50

3 3D Scattering Channel Modeling for Microcell Communication Environments

and  p(τ, βm ) =

2π

p(τ, φm , βm )dφm 0

   2 − 2k k 3 k6 k8 k93 − 3k7 k92 k10 + 4k8 k9 k10 2k9 7 10 = × 1+ π, 3 4 −k9 + k10 (k9 − k10 ) (k9 + k10 ) (3.26) where ⎧ ⎪ k1 = c(c2 τ 2 − D 2 )2 /8V ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎨k2 = c τ + D k3 = 2cτ D cos φm ⎪ ⎪ ⎪ ⎪ ⎪k4 = D cos φm ⎪ ⎪ ⎩k = cτ 5

⎧ ⎪ k6 = c(c2 τ 2 − D 2 )2 cos βm /8V ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎨ k7 = c τ + D and

k8 = 2cτ D cos βm ⎪ ⎪ ⎪ ⎪ ⎪k9 = D cos βm ⎪ ⎪ ⎩k = cτ. 10

Thus, the marginal PDFs of the TOA can be found by integrating (3.24) over the corresponding angles as follows: 

π/2  π

p(τ ) =

p(τ, φm , βm )dφm dβm . 0

(3.27)

0

Similar to the derivations of (3.25) and (3.26), the closed-form expression for p(τ ) can be obtained by Mathematica. In all of these expressions, we can deduce the 2D channel model from the proposed 3D channel model, not just by substituting βm = 0 in (3.26) but on the basis of the extensive analyses in [1–12]. Further, the joint function for the ToA/AoA can be obtained as p2D (τ, φm ) =

c(c2 τ 2 − D 2 )(D 2 + c2 τ 2 − 2cτ D cos φm ) , 0 ≤ φm ≤ 2π. 4V (cτ − D cos φm )3 (3.28)

By integrating (3.28) over the azimuth angle with the appropriate limits, the marginal PDF p2D (τ ) of the TOA can be obtained in a closed form, as derived in [12] for the 2D scattering channel model.

3.4 Performance of the MIMO Antenna Receiving Systems MIMO antenna receiving systems can provide a potentially high gain in capacity by using multiple antennas at both the MS and BS ends. Because of the motion of the MSs, it has been shown that the Doppler spectra can be related to the

3.4 Performance of the MIMO Antenna Receiving Systems

51

distribution of power over the AoA at the MS, which can be classically calculated by assuming that the received signal power and antenna gains are functions of the AoA. We begin this section with a discussion of the Doppler spectra distribution of a signal that is received by the MS in the 3D scattering channel model. We then analyze the performance of MIMO ULA and UCA receiving systems and their numerical results and present our observations in the following subsections.

3.4.1 Doppler Distribution Here, we use the PDFs of the AoA derived in Sect. 3.3 to derive the Doppler spectrum. The received signal at the MS experiences Doppler spread owing to its motion. Figure 3.1a illustrates the condition when the mobile is moving at an angle of with respect to the direct LoS component. The characteristics of the 3D channel model are analyzed and decomposed in azimuth and elevation planes [27]. Then, we adopted the same method used in Ref. [11], which can be used to analyze the Doppler shift in a plane, such as −α ≤ ϕm ≤ α, a specific value of βm . From the results of the Doppler shift for each plane in the 3D model, we can analyze and discuss the properties of the proposed channel model for any AoA. Here, we considered the situation when the MS is moving at an angle of φv with  respect to the direct LoS in Fig. 3.1a. We let the received signal be L−1 r(t) = E0 i=0 χi ej 2π vi t , where vi is the Doppler shift that the i-th multipath component experiences. Then, the Doppler spectrum S(f ) is A20 F (f ) [32], where L−1 |χi |2 , and F (f ) is the PDF of the distribution of the Doppler A20 = E02 /4 i=1 frequency. If an omnidirectional antenna is assumed at the MS, F (f ) can be given by [11] F (f ) =

p(φv + | cos−1 (f/fm )|) p(φv − | cos−1 (f/fm )|)   + , fm 1 − (f/fm )2 fm 1 − (f/fm )2

(3.29)

where fm is the maximum possible Doppler shift given by fm = v/λ, where v is the velocity of the MS, and λ is the wavelength of the carrier signal. When |f | < fm , the Doppler spectra is given by S(f ) =



A20

fm 1 − (f/fm )2      × fφ φv + | cos−1 (f/fm )| + fφ φv + | cos−1 (f/fm )| .

(3.30)

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3 3D Scattering Channel Modeling for Microcell Communication Environments

If the AoA of the signal at the mobile is uniform, the Doppler spectra is given by Clarke’s model [8] as SClarke (f ) =

A2 0 , when |f | < fm . πfm 1 − (f/fm )2

(3.31)

In (3.29) and (3.30), p(φ) is the PDF of the AoA of the multipath components at the MS, which have been derived in (3.15) and (3.16). Here, we discuss their physical characteristics. If we substitute (3.15) into (3.30), the Doppler spectra can be obtained in each azimuth plane (at various fixed values of βm , as in Ref. [27]). The marginal PDF of (3.30) is the cumulative probability of the AoA at the azimuth plane. When substituting (3.16) (the marginal PDF of the AoA) into (3.30), the results of the Doppler spectra are the cumulative spectra without the parameter βm . From all of the analytical results at each azimuth plane and the cumulative spectra, the Doppler characteristics of the proposed 3D channel model can be presented and discussed.

3.4.2 MIMO ULA and UCA Antenna Receiving Systems In wireless systems, a compact antenna array is used at the MS to receive information from users of wireless networks operating with the same or different multiple access schemes such as FDMA, TDMA, and CDMA. The compact MIMO antenna arrays used may have different geometries such as ULA, UCA, and Y-shaped. The antenna elements form a straight line, as shown in Fig. 3.2a, whereas the positions of the antenna elements in a planar array such as the UCA in Fig. 3.2b are specified by two variables representing polar or Cartesian coordinates. The propagation delay encountered between antenna elements as the signal travels across a linear array is a function of only the elevation angle. On the other hand, both the elevation and azimuth AoAs define the propagation delay in the case of planar arrays.

Fig. 3.2 MIMO ULA and UCA antenna arrays in the 3D scattering channel model. (a) ULA antenna array. (b) UCA antenna array

3.4 Performance of the MIMO Antenna Receiving Systems

3.4.2.1

53

Spatial Fading Correlations

In the 3D scattering channel model, we develop SFCs for various compact antenna arrays that can be applied to determine the covariance matrix at the transmitter and receiver in MIMO systems to estimate the channel capacity. Here, we study the importance and dependency of the angular parameters in detail according to the azimuth of arrival and the elevation of arrival in the proposed 3D scattering channel model. First, the SFCs between any two antenna elements such as p-th and q-th elements can be expressed as [28, 29]   hp )(hq −  hq ) E (hp −  ρ(p, q) =      E (hp −  hp )2 E (hq −  hq )2  

=   φ β

Ψp (φ, β)Ψq∗ (φ, β) sin(β)p(φ, β)dβdφ   , | Ψp (φ, β) |2 sin(β)p(φ, β)dβdφ φ β | Ψq (φ, β) |2 sin(β)p(φ, β)dβdφ φ β

(3.32) where E{·} denotes the expectation, the superscript (·)∗ is the complex conjugate, the scalar  hp is the mean value of the channel response at the p-th antenna element, and Ψp (φ, β) is the p-th entry signal of Ψ (φ, β), which can be obtained from (3.33) or (3.34) for the MIMO ULA and UCA antenna arrays, respectively. p(φ, β) is the joint PDF of the AoA in the 3D multipath channel. For the downlink at the MS, it is given by (3.15), and the SFCs can be estimated by numerical integration. Considering the spatial fading due to the antenna arrays in Fig. 3.2, Ψ (φ, β) of the MIMO ULA and UCA arrays are given by T  Ψ (φ, β)ULA = 1 , ej kw d cos φ sin β , . . . , ej kw d(L−1) cos φ sin β

(3.33)

 T Ψ (φ, β)UCA = ej ζ cos(φ−φ0 ) , ej ζ cos(φ−φ1 ) , . . . , ej ζ cos(φ−φl ) , . . . , ej ζ cos(φ−φL−1 ) ,

(3.34) where [·]T denotes the transpose operation, and L is the total number of antenna arrays. As shown in Fig. 3.2a for the ULA MIMO system, d is the antenna element spacing, kw = 2π/λ, and λ is the wavelength of the incident signals. For the MIMO UCA system, φl = 2π l/L, l = 1, 2, . . . , L − 1, and ζ = kw r sin β.

3.4.2.2

Capacity of MIMO ULA and UCA in Correlated Channels

Here, we analyze the performance of the compact antenna arrays in Fig. 3.2 to illustrate the efficacy of these compact arrays under various scenarios. Further, we focus on capacity analyses of the MIMO ULA and UCA arrays in the 3D scattering

54

3 3D Scattering Channel Modeling for Microcell Communication Environments

channel model. The instantaneous channel capacity, in bits per second per hertz, of a stochastic MIMO channel with an average transmitting power constraint and a transmitter having no channel knowledge has been given by Foschini and Gans [33, 34] as



 SN R C = log2 det I + H H T , bps/Hz, NT

(3.35)

where SN R is the average received signal-to-noise ratio assuming identically distributed noise at each receiver and transmit power equally distributed among the transmit antennas. H is the normalized channel transfer matrix, and (·)T corresponds to the conjugate transpose. The hij entry of the channel matrix represents the normalized channel transfer function evaluated at the frequency of operation between the transmit antenna j and the receive antenna i. In statistical modeling, these entries are usually chosen from a distribution of zeromean unity-variance complex Gaussian processes, which are independent and thus uncorrelated. When the correlation is introduced, one way to affect the antenna signal correlation is to post-multiply the channel transfer matrix H when modeling the correlation among the receiver and transmitter array elements independently from one another [28, 29, 33, 34]. For a MIMO system with NT transmit antennas and NR receive antennas, the well-known i.i.d. assumption for which the fades between pairs of transmit–receive antennas is independent and identical does not hold for many practical cases, particularly in compact antenna array MIMO systems, as shown in Fig. 3.2. The channel capacity is then given by



 SN R T , bps/Hz, C = log2 det I + ΦR H ΦT H NT

(3.36)

where ΦR and ΦT are the covariance matrices of the transmit and receive antenna arrays of sizes NR × NR and NT × NT , respectively. The entries of the covariance matrices can be given by the correlation coefficients derived in the previous subsection and can be easily incorporated into the MIMO channel models. Therefore, I is the identity matrix of size NR , and H is an i.i.d. NR × NT channel transfer matrix whose entries describe the channel response from the transmit antenna to the receive antenna. Here, we assume in the ideal case that the transmit antennas are sufficiently far apart so that ΦT becomes the identity matrix even under small-angle spread scenarios. Applying the entries ρ(p, q) of ΦR obtained from the previous subsection for the various antenna arrays, the capacity can be numerically estimated in our proposed 3D scattering channel model.

3.5 Numerical Results and Discussions

55

3.5 Numerical Results and Discussions In this section, we present the analytical and simulation results. To demonstrate the validity of the AoA PDFs in the azimuth and elevation planes, the proposed model has been compared with some available models and measured data. Comparisons of the results illustrate good agreement between the proposed model and the 2D Jiang model [12], the 3D Nawaz model [24], and the measured data.

3.5.1 PDFs of the AoA and ToA Figure 3.3 shows the derived marginal PDFs of the AoA in the azimuth plane. The effect of decreasing the beamwidth of the directional antenna on the PDFs of the azimuth AoA at the MS is shown in Fig. 3.3a. It is obvious that the PDFs apparently decrease with a decrease in the beamwidth of the directional antenna. Furthermore, it can be clearly observed that each curve of the AoA PDFs is symmetric because of the symmetry of the channel model in Fig. 3.1a. Moreover, the obtained theoretical results show good agreement between the proposed 3D model and the 2D Jiang model [12]. In addition, compared with the omnidirectional transceiver [12], we observe that the missing section of the graphs enlarges slowly with increasing α. Figure 3.3b shows the marginal PDFs of the AoA versus the ratio b/a. Note that the AoA significantly decreases with a decrease in b/a. This is due to the fact that decreasing b/a with fewer scatterers could reduce the multipath components. It is also observed that each curve of the AoA PDFs is symmetric and has a similar changing trend with the two “corners” in the right and left regions that occur at the azimuth angle φm = φ2 , implying that the PDFs of the AoA have break points due to the abrupt changes caused by the directional antenna. 5

5

14

x 10

7

°

x 10

α=30

°

12

φ =φ

°

α=90

5

°

α=180 (Omni) Jiang Model [12]

m

8

p(φ )

m

p(φ )

10

6

2

1

−60

0

60 °

φm [ ] (a)

120

180

2

3 2

−120

m

4

4

0 −180

b/a=0.25 b/a=0.50 b/a=0.75 b/a=1.00

6

α=60 (Proposed)

0 −180

−120

−60

0 φm [°]

90

120

180

(b)

Fig. 3.3 Marginal PDFs of the AoA in the azimuth plane at (a) a = 100 m, b = 50 m, and D = 50 m and (b) a = 100 m, D = 50 m, and α = 60◦

56

3 3D Scattering Channel Modeling for Microcell Communication Environments 6

4

6

x 10

4

°

x 10

b/a=0.25

b/a=0.5,α=60 (Proposed) °

3.5

b/a=0.5,α=180° (Omni)

3

m

2

p(βm)

b/a=1,α=180° (Omni) Nawaz Model [25]

2.5 p(β )

3.5

b/a=1,α=60 (Proposed)

βm=tan−1(b/d)

1.5

b/a=0.75

2.5

b/a=1.00

2

βm=tan (b/d)

−1

1.5

1

1

0.5

0.5

0

b/a=0.50

3

0

20

40

60

80

0

100

0

20

40

60

80

100

β [°]

βm [°]

m

(b)

(a)

Fig. 3.4 Marginal PDFs of the AoA in the elevation plane at (a) a = 100 m and D = 50 m and (b) a = 100 m, D = 50 m, and α = 60◦ Proposed Model Nawaz Model[25] 16

16

14

14

12

12

10

10

8

8

6 200

6 100 0.8 φm [°]

0.6

0

0.4 −200

0.2 0 (a)

τ (μ s)

0.8 βm [°]

0.6

50

0.4 0

0.2 0

τ (μ s)

(b)

Fig. 3.5 Joint PDFs of the ToA/AoA of the proposed model and the Nawaz model [24] for a = 100 m, b = 50 m, D = 50 m, and α = 60◦

The marginal PDFs of the AoA in the elevation plane are shown in Fig. 3.4. It is obvious from Fig. 3.4a that the PDFs of the AoA tend to be the same as βm increases from tan−1 (b/d) to π/2. Comparing the proposed 3D model with the Nawaz model [24], we can deduce that the PDFs of the AoA have similar changing trends for 0 ≤ βm ≤ π/2 in the vertical direction, irrespective of macrocell or microcell environments. Figure 3.4b shows the marginal PDFs of the AoA versus the ratio b/a. It is observed that the AoA significantly decreases with a decrease in b/a, and each curve of the AoA PDFs has one “corner,” which occurs at βm = tan−1 (b/d). Moreover, when βm is substituted with zero, the PDFs of the AoA agree with the 2D models proposed in [24], and the PDFs are all in agreement with each other. The joint PDFs of the ToA/AoA observed at the MS corresponding to the azimuth and elevation angles are plotted in Fig. 3.5, in which the effect of the directional antenna can be clearly observed. The closed-form expression of the ToA/AoA

3.5 Numerical Results and Discussions

57

in the azimuth plane is shown in Fig. 3.5a, and the statistical channel model is symmetric with respect to the xoz plane, on which the joint PDFs of the ToA/AoA in (3.25) are plotted. It can be seen that the missing section of graphs enlarges slowly, accompanied by an increase in α, and the results are symmetric for −φm and +φm . Note that the joint probability density near the LoS greatly enlarges. p Moreover, the joint ToA/AoA PDFs have two peaks at the points (τ p , ±φm ) p + p + where τ = τLoS and φm = 0 , and the joint PDFs decrease sharply from the peak points to the LoS point (τLoS , 0). Figure 3.5b shows that the joint PDFs of the ToA/AoA are in the elevation plane, and the receiving signals are mostly at small AoAs and short time delays. On the basis of the PDFs, we investigated the effect of α on the ToA/AoA. In addition, the scattering region increases with increasing α, implying the existence of scatterers farther away from the transmitter and receiver, thus resulting in longer propagation time delays. In addition, it is observed that the theoretical analysis above is in agreement with the Nawaz model [24], demonstrating that the results are accurate and applicable to microcell wireless environments. For a more realistic consideration of the position of the MSs, the effects of different positions of the MSs with α spanning the azimuth angle range of [−ψ2 , ψ1 ] (i.e., different ψ1 and ψ2 ) on the PDFs of the azimuth AoA are shown in Fig. 3.6. It is clearly observed that the curves of the PDFs are asymmetric and have two “corners” located in the right and left parts, seemingly only depending on the geometrical model, as in Fig. 3.1a. With increasing ψ1 + ψ2 and more scatterers in the areas

°

°

ψ =80 , ψ =40 , D/a=0.4

5

8

x 10

1

2

°

°

ψ1=80 , ψ2=40 , D/a=0.8 °

°

ψ =60 , ψ =30 , D/a=0.4

7

1

2

°

°

ψ =60 , ψ =30 , D/a=0.8 1

6

2

°

ψ =ψ =180 , D/a=0.5 1

Zhou Model [33]

5 m

p(φ )

2

4 3 2 1

0 −200

−150

−100

−50

0 φ [°]

50

100

150

200

m

Fig. 3.6 Marginal PDFs of the AoA in the azimuth plane for a = 100 m and b = 50 m

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3 3D Scattering Channel Modeling for Microcell Communication Environments

illuminated by the directional antenna, the PDFs have higher values on both sides of the curves, whereas the values at φm = 0 tend to be equal. In addition, when the position of an MS is close to the BS, the PDFs with smaller AoAs sharply increase to a local maximum with larger AoAs and then decrease slowly, an observation that is in agreement with the results reported by Zhou [15].

3.5.2 Doppler Distribution The effect of changing the geometry of the scattering region is very important because both high-rise and low-height buildings contribute to the Doppler shift. This effect is studied in Fig. 3.7 considering α and b/a. For an omnidirectional antenna, the PDFs of the Doppler shift are U shaped [11]. However, the PDFs of the Doppler shift increase for the directional antenna by increasing the minor axis b of the 3D scattering region, which means that the scatterers located vertically high correspond to a significant reduction in Doppler shift for a particular scattering environment for all beamwidths of a directional antenna (i.e., α = 60◦ ) [27]. Therefore, it is important to maintain the vertical beamwidth of the directional antenna such that it can illuminate the scattering objects that are located vertically high.

10

b/a=0.25, α=60° b/a=0.50, α=60° °

5

b/a=0.75, α=60

°

b/a=0.25, α=180 (Omni) °

b/a=0.50, α=180 (Omni)

0

°

10

log [p(γ)]

b/a=0.75, α=180 (Omni) −5

−10

−15

−20 −1

−0.5

0 γ = f /f

0.5

1

DS m

Fig. 3.7 Distribution of the Doppler shift with respect to b/a and α at a = 100 m, D = 50 m, and φv = 90◦

3.5 Numerical Results and Discussions

59

10

φv=0°,D/a=0.2 φ =0°,D/a=0.5 v

φ =0°,D/a=0.8

5

v

φ =180°,D/a=0.5

0

v

10

log [p(γ)]

φv=90°,D/a=0.5

−5

−10

−1

−0.5

0 γ=f

0.5

1

/f

DS m

Fig. 3.8 Distribution of the Doppler shift with respect to D/a and φv at a = 100 m, b = 50 m, and α = 60◦

The distribution of the Doppler shift versus the motion of the MS and the ratio D/a for the directional antenna is shown in Fig. 3.8 for the following values: φv = 0◦ , 90◦ , and 180◦ . However, the PDFs are skewed toward lower and higher frequencies when φv = 0◦ and 180◦ , respectively, resulting from the unbalanced number of illuminated scattering objects with respect to the axis of motion of the MS. In addition, it is obvious that the PDFs are skewed from lower to higher frequencies by increasing D/a at φv = 0◦ (i.e., the MS moves towards the BS). For physical consideration of the proposed model, it is noteworthy that the area of the tangent plane in the vertical direction decreases first and then increases at the azimuth angle φt1 ≤ φm ≤ φt2 in Fig. 3.3. It is obvious that the tangent plane monotonically decreases for φt2 ≤ φm ≤ π . It is also impressive that the AoA decreases with an increase in βm , as shown in Fig. 3.4, because the area of the tangent plane tends to be zero at βm = π/2 when βm increases with the lower area of the tangent plane in the horizontal direction. This implies that the receiving signals in the vertical direction tend to be zero. Furthermore, it is obvious from Fig. 3.8 that the PDFs are symmetrical about γ = 0 when φv = 90◦ . This is because the clipped scattering region is geometrically symmetric when the MS moves perpendicular to the LoS, and the numbers of multipath components corresponding to the positive and negative Doppler frequencies are equal.

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3 3D Scattering Channel Modeling for Microcell Communication Environments

3.5.3 Performance of MIMO Systems On the basis of the joint and marginal PDFs of the ToA/AoA, the SFCs between any two antenna elements for the MIMO transceivers can be numerically calculated and are presented in Figs. 3.9 and 3.10. These SFCs are the fundamental results for investigating the performance of MIMO systems. Figure 3.9a shows the SFCs between ULA elements 1 and 2 versus βM and α in the proposed 3D scattering channel model. By increasing d/λ, the correlation decreases rapidly. However, the correlation in the MIMO UCA system decreases gradually with increasing r/λ, as shown in Fig. 3.9b. Further, the correlation tends to be higher as the elevation angle increases. In addition, we can deduce that the correlation becomes higher as α increases, and the same conclusion can be obtained in Fig. 3.9b, which demonstrates the excellent performance of the directional antenna.

1

βM=0°,α=60° °

Spatial Fading Correlation|ρ(1,2)|

Spatial Fading Correlation|ρ(1,2)|

1

°

βM=30 ,α=60

0.8

°

°

βM=30 ,α=180 °

β =60 ,α=Any angle

0.6

M

0.4

0.2

0

0

1

2 3 Antenna spacing (d/λ) (a)

°

°

°

βM=30 ,α=60

0.8

°

βM=30 ,α=180° β =60°,α=Any angle

0.6

M

0.4

0.2

0

4

°

βM=0 ,α=60

0

1

2 3 Antenna spacing (r/λ) (b)

4

Fig. 3.9 SFCs in correspondence with βM and α for MIMO ULA and UCA systems at a = 100 m, b = 50 m, and D = 50 m 1

1

β =0°,D/a=0.5

β =0°,D/a=0.5 M

Spatial Fading Correlation|ρ(1,2)|

Spatial Fading Correlation|ρ

(1,2)

|

M

°

β =30 ,D/a=0.25

0.8

M

°

βM=30 ,D/a=0.5 °

βM=30 ,D/a=0.75

0.6

°

βM=60 ,D/a=0.5 0.4

0.2

0

0

1

2 3 Antenna spacing (d/λ) (a)

4

°

βM=30 ,D/a=0.25

0.8

°

βM=30 ,D/a=0.5 βM=30°,D/a=0.75

0.6

β =60°,D/a=0.5 M

0.4

0.2

0

0

1

2 3 Antenna spacing (r/λ) (b)

4

Fig. 3.10 SFCs in correspondence with βM and D/a for MIMO ULA and UCA systems at a = 100 m, b = 50 m, and α = 60◦

3.5 Numerical Results and Discussions

61

Figures 3.10a and b explore the effects of the elevation angle βM and distance D/a on the SFCs for the MIMO ULA and UCA systems, respectively. It can be observed that the correlation decreases and is reduced to zero gradually as βM decreases. Note that there are several zero-crossing points in the MIMO ULA system, where the first zero occurs appropriately when r/λ ≈ 0.5 in Fig. 3.10a. Figure 3.10b presents the SFCs for the MIMO UCA system versus D/a at various values of βM . When the elevation angle decreases, the correlation decreases rapidly with no obvious effects. In addition, it is notable that an increase in D/a leads to a higher correlation at the same angle βM . To gain more insight, several comparisons are made with reference to α and D/a in Figs. 3.11 and 3.12, respectively. The beamwidth at the BS and the distance between the BS and the MS thus has significant effects on the system performance. For a consistent and fair comparison of the MIMO ULA and UCA systems, the capacities are computed with the same size and with all the other parameters fixed at Nr = 4 elements, a = 100 m, b = 50 m, and SN R = 20 dB. We can deduce 15

15

14

14 β =30°

13

M

Capacity (bits/s/Hz)

Capacity (bits/s/Hz)

13 12 11 10

°

β =0 ,α=60 M

9

°

°

β =30 ,α=60 M

8

°

10

M

°

M

°

M

6

2

°

β =30 ,α=180 M

7

β =60 ,α=Any angle 1 1.5 Antenna spacing (d/λ) (a)

°

β =30 ,α=60

8

°

°

0.5

β =0°,α=60°

9

M

0

M

11

β =30 ,α=180

7 6

°

°

β =30

12

°

β =60 ,α=Any angle M

0

0.5

1 1.5 Antenna spacing (r/λ) (b)

2

15

15

14

14

13

13

12 11

°

β =0 ,D/a=0.5 M

10

°

β =30 ,D/a=0.25 M

9

°

β =30 ,D/a=0.5 M

8 °

β =30

7 6

M

0

0.5

12 11

M M

β =30°,D/a=0.5 M

°

7

2

°

β =30 ,D/a=0.25

9 8

βM=60 ,D/a=0.5

°

β =0 ,D/a=0.5

10

°

βM=30 ,D/a=0.75

1 1.5 Antenna spacing (d/λ) (a)

Capacity (bits/s/Hz)

Capacity (bits/s/Hz)

Fig. 3.11 Capacities with respect to βM and α for MIMO ULA and UCA systems for a = 100 m, b = 50 m, D = 50 m, Nr = 4, and SN R = 20 dB

°

β =30 M

β =30°,D/a=0.75 M

β =60°,D/a=0.5 M

6

0

0.5

1 1.5 Antenna spacing (r/λ) (b)

2

Fig. 3.12 Capacities with respect to βM and D/a for MIMO ULA and UCA systems for a = 100 m, b = 50 m, α = 60◦ , Nr = 4, and SN R = 20 dB

62

3 3D Scattering Channel Modeling for Microcell Communication Environments

that increases in d/λ and r/λ lead to larger capacities for the MIMO ULA and UCA systems, respectively. It is also noted that increasing βM will lead to lower capacities. In addition, the capacities for decreasing beamwidths are larger for an omnidirectional antenna with α = 180◦ and βM = 30◦ . The analysis agrees with the results in Figs. 3.11a and b, clearly demonstrating the MIMO system as a promising technology for wireless communications. Figures 3.12a and b present the capacities of the MIMO ULA and UCA systems versus D/a at various values of βM . At βM = 30◦ , the capacities tend to increase as D/a increases owing to the increase in r/λ. It is obvious that the capacities in Fig. 3.12b for the UCA MIMO system increase more rapidly than those in Fig. 3.12a for the ULA MIMO system.

3.6 Conclusions In this chapter, we have derived a generalized 3D scattering channel model assuming that the scatterers are located around the BS in a uniform distribution. General formulations of the joint and marginal PDFs of the ToA/AoA have been carefully derived corresponding to the azimuth and elevation planes. On the basis of the 3D channel model, we have analyzed the Doppler spectra due to the motion of the MS and the SFCs of the MIMO ULA and UCA antenna arrays and determined their channel capacities. Comparisons between our theoretical results and some customary 2D [11, 12] and 3D [27] scattering channel models show that the analysis is accurate and applicable to depict radio environments.

References 1. W. Jakes, Microwave Mobile Communications (IEEE, Piscataway, 1974) 2. J. Karedal, P. Almers, A.J. Johansson, F. Tufvesson, A.F. Molisch, A MIMO channel model for wireless personal area networks. IEEE Trans. Commun. 9(1), 245–255 (2010) 3. K. Deng, Frequency synchronization in MIMO systems, in 2nd International Conference on Consumer Electronics, Communications and Networks (CECNet) (Yichang, 2012), pp. 1832– 1835 4. L. Taponecco, Joint TOA and AOA estimation for UWB localization applications. IEEE Trans. Wirel. Commun. 10(7), 2207–2217 (2011) 5. A. Intarapanich, P.L. Kafle, R.J. Davies, A.B. Sesay, J.G. McRory, Geometrically based broadband MIMO model with tap-gain correlation. IEEE Trans. Veh. Technol. 56(6), pp. 3631– 3641 (2007) 6. C.H. Lim, J.T. Kim, D.S. Han, Channel selective diversity for DTV mobile reception with adaptive beamforming. IEEE Trans. Consum. Electron. 51(2), 357–364 (2005) 7. A. Kuchar, J.P. Rossi, E. Bonex, Directional macro-cell channel characterization from urban measurements. IEEE Trans. Antennas Propag. 48(2), 137–146 (2000) 8. A. Hernandez, R. Badorrey, J. Choliz, I. Alastruey, A. Valdovinos, Accurate indoor wireless location with IR UWB systems a performance evaluation of joint receiver structures and TOA based mechanism. IEEE Trans. Consum. Electron. 54(2), 381–389 (2008)

References

63

9. K.T. Wong, X. Yuan, Vector cross product direction finding with an electromagnetic vector sensor of six orthogonally oriented but spatially non-collocation diploes/loops. IEEE Trans. Signal Process. 59(1), 160–171 (2011) 10. K.B. Baltzis, On the geometric modeling of the uplink channel in a cellular system. J. Eng. Sci. Technol. Rev. 1(1), 75–82 (2008) 11. P. Petrus, J.H. Reed, T.S. Rappaport, Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans. Commun. 50(3), 495–502 (2002) 12. L. Jiang, S.Y. Tan, Geometrically based statistical channel models for outdoor and indoor propagation environments. IEEE Trans. Veh. Technol. 56(6), 3587–3593 (2007) 13. Y.I. Wu, K.T. Wong, A geometrical model for the TOA distribution of uplink/downlink multipaths assuming scatterers with a conical spatial density. IEEE Antennas Propag. Mag. 50(6), 196–205 (2008) 14. S.H. Kong, TOA and AOD statistics for down link Gaussian scatterer distribution model. IEEE Trans. Wirel. Commun. 8(5), 2609–2617 (2009) 15. R. Janaswamy, Angle and time of arrival statistics for the Gaussian scatter density model. IEEE Trans. Wirel. Commun. 1(3), 488–497 (2002) 16. S.J. Nawaz, M. Riaz, N.M. Khan, S. Wyne, Temporal analysis of a 3D ellipsoid channel model for the vehicle to vehicle communication environments. Wirel. Pers. Commun. 82(3), 1337– 1350 (2015) 17. S.X. Qu, T. Yeap, A three-dimensional scattering model for fading channels in land mobile environment. IEEE Trans. Veh. Technol. 48(3), 765–781 (1999) 18. K.B. Baltzis, J.N. Sahalos, A simple 3D geometric channel model for macrocell mobile communication. Wirel. Pers. Commun. 51(2), 329–347 (2009) 19. M. Alsehaili, A. Sebak, S. Noghanian, A 3D geometrically based ellipsoidal wireless channel model, in Proceedings 12th International Symposium on Antenna Technology and Applied Electromagnetics (2006), pp. 407–410 20. A. Mohammad, S. lsehaili, Generalized three dimensional geometrical scattering channel model for indoor and outdoor propagation environments. PHD dissertation, Department of electrical and computer engineering, University of Manitoba, Winnipeg Manitoba, Canada, 2010 21. M. Alsehaili, Angle and time of arrival statistics of a three dimensional geometrical scattering channel model for indoor and outdoor propagation environments. Prog. Electromagn. Res. 109, 191–209 (2010) 22. R. Janaswamy, Angle of arrival statistics for a 3-D spheroid model. IEEE Trans. Veh. Technol. 51(5), 1242–1247 (2002) 23. A.Y. Olenko, K.T. Wong, S.A. Qasmi, J. Ahmadi-Shokouh, Analytically derived uplink/downlink TOA and 2-D DOA distributions with scatterers in a 3-D hemispheroid surrounding the mobile. IEEE Trans. Antenna Propag. 54(9), 2446–2454 (2006) 24. S.J. Nawaz, B.H. Qureshi, N.M. Khan, A generalized 3-D scattering model for a macrocell environment with a directional antenna at the BS. IEEE Trans. Veh. Technol. 59(7), 3193– 3204 (2010) 25. P. Petrus, J.H. Reed, T.S. Rappaport, Geometrical based statistical macrocell channel model for mobile environments. IEEE Trans. Commun. 50(3), 495–502 (2002) 26. S.X. Qu, An analysis of probability distribution of Doppler shift in three dimensional mobile radio environments. IEEE Trans. Veh. Technol. 58(4), 1634–1639 (2009) 27. S.J. Nawaz, N.M. Khan, Effect of directional antenna on the Doppler spectrum in 3-D mobile radio propagation environment. IEEE Trans. Veh. Technol. 60(7), 2895–2903 (2011) 28. S.K. Yong, J.S. Thompson, Three dimensional spatial fading correlation models for compact MIMO receivers. IEEE Trans. Commun. 4(6), 2856–2869 (2005) 29. D. Zhong, Z.M. Li, The study of MIMO channel modeling and simulation based on 3GPP LTE, in IEEE Transactions on Consumer Electronics (ICCE) (Yichang, 2012), pp. 3646–3651 30. DOCOMO 5G White Paper, 5G Radio Access: Requirements, Concept and Technologies. NTT DOCOMO (2014)

64

3 3D Scattering Channel Modeling for Microcell Communication Environments

31. T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans. Wirel. Commun. 9(11), 3590–3600 (2010) 32. J.C. Liberti, T.S. Rappaport, A geometrically based model for line-of-sight multi-path radio channels, in Proceedings of IEEE Vehicular Technology Conference (1996), pp. 844–848 33. J. Zhou, H. Jiang, H. Kikuchi, Geometry-based statistical channel model and performance for MIMO antennas. Int. J. Commun. Syst. 29(3), 459–477 (2016) 34. J. Kim, J. Kim, J. Hwang, D. Shin, J. Ahn, Capacity of frequency selective fading channel in MIMO single frequency network for 3D-HDTV terrestrial broadcasting, in IEEE Transactions on Consumer Electronics (ICCE) (Las Vegas, NV, 2011), pp. 423–424

Chapter 4

Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street Environments

4.1 Introduction With the rapid development of mobile Internet and the Internet of things (IoT), there are still some challenges that cannot be accommodated by the 4G wireless communication networks, such as the spectrum crisis and high energy consumption [1]. Therefore, research on the 5G wireless communication networks has been started, which has attracted great attention around the world. Compared to 4G, 5G networks are supposed to provide greatly enhanced capacity, spectral efficiency, energy efficiency, cost efficiency, mobility, data rate, connection density, etc., with a much reduced end-to-end latency [2]. To meet the 5G requirements, it is necessary to design the radio propagation channel model for mobile and wireless systems, especially for car-to-car (C2C) mobile radio environments [3], which has been regarded as one of the main research trends in next generation of mobile communication technology. Furthermore, the spatial characteristics of the multipath channel are proven to be useful for high-performance MIMO antenna systems. To efficiently create these systems, it is essential to have a reliable understanding of the radio propagation characteristics of the transmission path between the transmitter and receiver, which leads to the design of effective signal processing techniques. Geometrical modeling of the propagation channel has always been attractive for researchers, due to its many advantages [4]. The spatial characteristics of the multipath channel, such as the PDFs of the AoD and AoA of the signals, can be utilized to analyze the performance of the MIMO and massive MIMO antenna systems. Therefore, a number of fundamental geometry-based statistical channel models have been proposed in the literature. References [5–9] focused on 2D scattering channel models. For example, circular scattering models, ellipse scattering models [5], and asymmetric scattering models [9], which indicate that propagation takes place within the plane joining the tips of the transmitting and receiving antennas. Subsequently, many researchers have observed an angular spreading of the waves in the elevation plane because of the interaction of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_4

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4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

waves with street buildings, ground, and other vertically disposed objects [10–13]. In 2010, Nawaz [10] presented a 3D scattering model for macrocell environments, with a BS employing a directional antenna located outside the hemispheroid. Furthermore, Jiang [11] derived a generalized 3D scattering channel model for microcell environments, in which the scatterers are assumed to be located around the BS with a uniform distribution. Recently, a 3D ellipsoidal model for the M2M radio propagation environments was proposed, in which the most frequent occurrence of the AoA of the received multipath waves is located around the relative direction at each MS [12]. In general, the geometric scattering channel models above only focus on singlebounce, while multi-bounced propagation paths are not taken into account. In particular, for the dense urban street environments, the single-bounced assumption is rather restrictive because the street width is not sufficient to match the maximum of the elliptical scattering region. Therefore, there is a strong need for a generalized scattering channel model, which can be used to describe multi-bounced propagation paths. To overcome this problem and obtain the analytical solution, the concept of effective street width was presented in [14]. Subsequently, Ref. [15] provided a pseudo-geometrical scattering channel model, which visualized the double- and tri-bounced paths as single-bounce. Ghoraish [16] presented the polar directional analysis of the urban NLoS propagation channel at 2.2 GHz, which is based on the measured data in Tokyo and Yokohama. The authors in [17] presented a reference model for a wideband MIMO channel based on the geometric elliptical scattering model. However, in [17], the mobile properties between the transmitter and receiver were not discussed in detail, and they are still restricted to the single-bounced propagation paths. In [18], a geometric street scattering channel model under LoS and NLoS propagation conditions for the outdoor communication environments was proposed, but they ignored multi-bounced propagation paths. MacCartney [19] conducted two measurement campaigns in urban microcellular environments, and path loss models suitable for the development of 5G standards were presented. The authors in [20] and [21] later provided a V2V channel model, which combines a two-ring model and a multiple confocal ellipses model, consisting of LoS, single-, and double-bounced waves. Rasekh [22] considered a street canyon approximation model, for the 60 GHz wireless channel in an urban environment, with rough surfaces. The authors in [23] proposed an adaptive geometry based on stochastic model for non-isotropic M2M communication environments, but they only focused on the single- and double-bounced conditions, and the condition of the multi-bounce has not been discussed in detail. However, when the signal endures unlimitedbounced propagation paths (i.e., the bounce number tends to be infinity), the previous scattering channel models are no longer applicable. Furthermore, the results and discussions above are restricted to the perspective of time domain, and the influence of the relative motion between the MSs on the distribution of the Doppler frequency, for the C2C mobile radio environments, has not been instigated in the previous literature.

4.1 Introduction

67

Fig. 4.1 Illustration of a typical C2C mobile radio environment and its corresponding visual scattering channel model for dense urban street environments

In this chapter, a generalized virtual scattering channel model for the microcell C2C street environments is proposed, in which multi-scatterings are taken into account and we visualize them as single-bounce scattering, as illustrated in Fig. 4.1. In this work, we first derive the closed-form expression for the marginal PDFs of the AoD and AoA for the odd- and even-numbered-bounced propagation paths. We then proceed to describe the Doppler shift due to the relative motion between the MT and MR. From numerical results and observations, the main advantages of this paper are given as follows: (1) the proposed visual scattering channel model can be used to simulate the dense urban street environments, (2) it is possible to deduce single-, double-, triple-, and other multi-bounced paths of the street channel model, from our proposed visual scattering channel model. To the best of the authors’ knowledge, this work has not been investigated before, (3) the effect of the asymmetric directional antenna employed at the MT in street environments is investigated, which analyzes more realistic positions of the MRs, (4) the total Doppler frequency due to the relative motion between the MT and MR is first taken into analysis in the proposed visual channel model, which broadens the research from the perspective of the frequency domain. Furthermore, the results are compared with those of the previous scattering channel models and measurement results, to validate the generalization of the proposed model. Accordingly, our proposed visual street channel model may be used to analyze the performance of M2M and IoT related to 5G wireless communication networks [2, 24].

68

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

4.2 Generalized Visual Street Channel Model 4.2.1 Virtual Model Description In order to sufficiently analyze and design the proposed visual street channel model, here we mainly concentrate on two important channel properties: scattering power distribution and Doppler power spectral density. In the proposed model, the scatterer density tapers off with the distance from the transmitter and receiver. Therefore, we adopt the scatterer non-uniform distribution came from the experimental measurements in [15] to describe the proposed mobile radio environments. On the other hand, the Doppler frequency distribution of the proposed street channel vary significantly with different moving properties in C2C mobile radio environments [24]. Furthermore, the Doppler distribution for the C2C channels is significantly different from the conventional Clarke distribution for scattering channels. To make the proposed street channel model more systematic, and in correspondence with the dense urban street environments, several common assumptions, as shown in [10] and [11], are used to design the proposed visual channel model. They are as follows: (1) either the MT or the MR is surrounded by non-uniformly distributed scatterers confined with a visual elliptical channel model, (2) the channel model is two-dimensional, which means that the MT, the MR, and the scatterers are all within the same plane, (3) all the scatterers have uniform random phases and the same scattering coefficient, (4) each scatterer is an omnidirectional reradiating element, independent of other scatterers. The schematic of the multi-bounced propagation paths and its corresponding geometric elliptical scattering channel model are depicted in Fig. 4.1. In general, the transmission signal from the MT to the MR under multi-bounced propagation paths. However, for the conventional scattering channel models in [5–13], they almost concentrated on the single- and double-bounced models. For this we should propose multi-bounced scattering channel model, which can be utilized to accurately describe mobile radio communication environments. However, it is difficult to analyze multi-bounced propagation links in the proposed street channel model, here we can visualize them as single-bounced scattering scenarios. It can be observed that the scattering point s2 is identified in the roadside environments, and path p2 might be multi-scatterings. Obviously the visual scattering point for path p2 is identified in s2 which located on the cross point of the ellipse, note that the geometric path length of p2 is equal to that of p1 . Also, we assume that both the visual path p2 and the multi-bounced path p1 take equal path losses. Furthermore, we can observe that the reflected number of the signal is related to the angle at the MT, which is represented as ϕ = arctan(2nW/D), where D denotes the distance between the MT and MR, and W is street width. When the reflected number n is an odd number, then the signal will endure odd numbered-bounced propagation paths. Similarly, the signal will endure even-numbered-bounced propagation paths when n is an even number. Additionally, for the dense urban street environments, the propagation paths are not so long since the small distance between the MT and

4.2 Generalized Visual Street Channel Model

69

Fig. 4.2 The proposed visual street channel model for different positions of the MRs. (a) The receiver is located at the MR1 . (b) The receiver is located at the MR2 . (c) The receiver is located at the MR3

MR, which means that the channel always endures multi-scattering. Therefore, the dominant loss in the channel is the scattering loss. Figure 4.2 illustrates the proposed visual street channel model for different positions of the MRs and their corresponding visual single-bounced scattering channel models. It can be observed that the visual elliptical model is gradually skewed from the left to right when the receiver moves from the MR1 to MR3 . From this we should consider the rotation of the elliptical channel model. Let us define θ as the tilt angle of the elliptical model. Moreover, the major and minor dimensions of the proposed visual scattering channel model are defined as a and b correspondingly, here we assume b ≤ a. For this rotatable ellipse, the generalized equation can be expressed as [10] (Fig. 4.3)  2  2 x cos θ + y sin θ − x cos θ + y sin θ + = 1. a2 b2

(4.1)

From Fig. 4.2 we can note that as the MR moves on the connection between the MR1 and MR3 , the proposed visual street channel model corresponds to different multi-bounced propagation conditions, including even- and odd-numbered-bounced paths. It is clearly observed that when the receiver is located at the MR1 , the proposed visual elliptical channel model can be utilized to describe even-numberedbounced propagation paths, as illustrated in Fig. 4.4. However, the proposed model presented the odd-numbered-bounced propagation paths as the receiver is identified at the other positions on the connection between the MR1 and MR3 . Based on the aforementioned analysis, it is necessary to separate the multi-bounced propagation paths into odd- and even-numbered-bounced propagation conditions, which will be analyzed in detail in the following subsections.

70

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

Fig. 4.3 The schematic of the odd-numbered-bounced propagation paths and its corresponding visual single-bounced geometric scattering channel model. (a) The geometrical description of the odd-numbered-bounced paths. (b) Geometric angles and path lengths of the proposed model for the odd-numbered-bounced conditions.

4.2 Generalized Visual Street Channel Model

71

Fig. 4.4 The schematic of the even-numbered-bounced propagation paths and its corresponding visual single-bounced geometric scattering channel model. (a) The geometrical description of the even-numbered-bounced paths. (b) Geometric angles and path lengths of the proposed model for the even-numbered-bounced conditions

4.2.2 Odd-Numbered-Bounced Propagation Paths Figure 4.3 illustrates the proposed visual street channel model for the oddnumbered-bounced propagation paths. For simplicity, the model assumes that the receiver is located at the MR2 , i.e., θ = 0, and its connection with the MT parallels the street walls for all odd-numbered-bounced propagation conditions. Therefore, let the connection of the MT and MR be x-axis, as shown in Fig. 4.3. Note that the focal length of the proposed visual elliptical channel model is larger than the distance between the MT and MR, and the focus are both assumed located on the x-axis. The multipath azimuth departure angle at the MT is denoted by θb , and the distance of the scattering object from the MT is defined as rb . In order to take more realistic positions of the MRs into account, the visual channel model assumes that an asymmetric directional antenna is deployed at the MT, spanning the azimuth range of [−ψ1 , ψ2 ]. The same assumption is made in [9]. Therefore, the generalized equation of the visual channel model for odd-numbered-bounced propagation paths can be expressed as [10] D 2 − 4Drb cos θb + 4rb2 cos2 θb rb2 sin2 θb + = 1. 4a 2 b2

(4.2)

72

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

In this case, the major and minor dimensions of the visual elliptical channel model can be expressed as   a = W0 + (n − 1)W csc ϕ

(4.3)

  b = n − 1 W + W0

(4.4)

and   2c = 2 W0 + (n − 1)W cot ϕ,

n = 1, 2, 3, . . . .

(4.5)

In Eqs. (4.3)–(4.5), W0 denotes the distance of the upper building to the MT, and 2c is the focal length of the visual elliptical channel model, as illustrated in Fig. 4.3. By inserting (4.3)–(4.4) into (4.2), the standard equation of the proposed visual street channel model for odd-numbered-bounced paths can be obtained. Further, the distances from the visual boundary of the elliptical scattering channel model to the MT and MR are denoted by rb and rm , respectively. They are derived as functions of the angles measured at the MT and written as rb (θb ) =

1 2b2 cos2 θb + 2a 2 sin2 θb +



 × Db2 cos θb

    D 2 b4 cos2 θb − b2 cos2 θb + a 2 sin2 θb D 2 b2 − 4a 2 b2

rm (θb ) =



D 2 + rb2 − 2Drb cos θb , −ψ1 ≤ θb ≤ ψ2 .

(4.6)

(4.7)

Based on the proposed visual street model in Fig. 4.3, the lengths of the boundary of the scattering region, illustrated by the beam from the asymmetric directional antenna at the MT, are given by   ρ1 = rb (θb ) =

θb =ψ1

   1 × Db2 cos ψ1     2 2 2 2 2b cos ψ1 + 2a sin ψ1      + D 2 b4 cos2 ψ1 − b2 cos2 ψ1 + a 2 sin2 ψ1 D 2 b2 − 4a 2 b2

(4.8)

4.2 Generalized Visual Street Channel Model

  ρ2 = rb (θb ) =

73

θb =ψ2

    × Db2 cos ψ2

1

   2b2 cos2 ψ2 + 2a 2 sin2 ψ2      2 2 4 2 2 2 2 2 2 2 2 + D b cos ψ2 − b cos ψ2 + a sin ψ2 D b − 4a b .

(4.9)

4.2.3 Even-Numbered-Bounced Propagation Paths The proposed visual street channel model for the even-numbered-bounced paths is shown in Fig. 4.4. Note that defining the connection of the MT and MR as the x-axis of the visual elliptical model is quite unreasonable [15]. Let us assume that the major dimension of the visual elliptical channel model is located on the x  -axis. While it is difficult to obtain the generalized equation of the visual scattering channel model directly from the rectangular coordinate (x, y) since the number of the paths and the number of reflections are not equal. By transforming (x, y) into (x  , y  ), the generalized equation can be derived as   2   2 x y + 2 =1 2 a b 

2x − W0 cot ϕ0

2 

cos ϕ0 − sin ϕ0

(4.10)

2

4a 2 2  2  2y − W0 cos ϕ0 + sin ϕ0 + = 1. 4b2

(4.11)

In Eq. (4.11), the transformation angle from the rectangular coordinate (x, y) to (x  , y  ) is given by  W0 . ϕ0 = arctan D − W0 cot ϕ 

(4.12)

Similar as before, the major and minor dimensions of the visual elliptical channel model for even-numbered-bounced paths can be expressed as   a = 2nW − W0 csc ϕ

(4.13)

b = nW − W0

(4.14)

74

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

and   c = 2nW − W0 cot ϕ,

n = 1, 2, 3, . . . .

(4.15)

As described before, substituting (4.13)–(4.14) into (4.11), the standard equation of the visual street channel model for even-numbered-bounced paths can be obtained.

4.3 Spatial Characteristics of the Street Channel Model In wireless environments, the spatial characteristics that describe the AoD and AoA statistics of the multi-bounced components are quite useful. This can be used in the performance evaluation of wireless communication systems employing MIMO antenna arrays, and broaden the research from the perspective of frequency domain. We begin this section with a discussion of the scattering distribution for dense urban street environments, which is measured in [15]. We then analyze the marginal PDFs of the AoD and AoA statistics, viewed at the MT and MR, respectively.

4.3.1 Scattering Distribution From the illustration of the C2C mobile radio environments in Fig. 4.1, it can be found that the scattering region gradually increases with an increase in the number of reflections. In general, the scatterer density tapers off with the distance from the transmitter and receiver. Therefore, it is necessary to present scatterer non-uniform distribution in the proposed C2C street channel model. However, we can observe if we perform experimental measurements for the proposed C2C dense urban street environments, the analysis will be very complex. Here, we adopt the experimental measurements in [15] of the scattering distribution to describe the proposed mobile radio environments. Furthermore, it is observed that the scattering distribution has an elliptical shape, but it is different from the conventional scatterer Gaussian and Exponential distributions in that most scatterers are distributed along the street. From this we can clearly describe the spatial characteristics of the proposed visual street channel model. Therefore, the scattering power distribution in [15] can be expressed as  p(x, y) = exp

 −  −

 2 Ax x − c + Ay · y 2 

Ax x + c

2

+ Ay

· y2

 + Cxy ,

(4.16)

4.3 Spatial Characteristics of the Street Channel Model

75

where Ax and Ay are loss coefficients along x and y axes, respectively, Cxy is a constant. After Jacobian of the inverse transformation, the joint PDF of the AoD can be written as      p x, y     p(rb , θb ) =   |J x, y |  x=rb cos θb −D/2 y=rb sin θb

⎧ 

2 ⎨ D = rb exp − Ax rb cos θb − − c + rb2 Ay sin2 θb ⎩ 2 ⎫

2 ⎬ D + c + rb2 Ay sin2 θb + Cxy . − Ax rb cos θb − ⎭ 2 

(4.17)

Similarly, at the MR, the joint PDF of the AoA can be given by ⎧ 

2 ⎨ D 2 A sin2 θ − rm cos θm − c + rm p(rm , θm ) = rm exp − Ax y m ⎩ 2  − Ax



D − rm cos θm + c 2

2 2 A sin2 θ + C + rm y m xy

⎫ ⎬ ⎭

.

(4.18)

In the following, we performed experimental measurements in [15] for three streets, with the different widths of 26, 18, and 10 m. The distances of the MT and MR from the walls of the same side are set as 5m, 5m, and 3m, respectively. The MT to the MR distance is fixed at 60m, and the asymmetric directional antennas at the MT having main-lobes of [−40◦ , 80◦ ] and [−20◦ , 40◦ ] are assumed. From the experimental measurements, for the dense urban areas [15], the loss coefficients (Ax , Ay ) for the above three cases are derived as (1.0 × 10−4 , 1.0 × 10−3 ), (2.0 × 10−4 , 2.0 × 10−3 ), and (2.1 × 10−4 , 3.0 × 10−3 ), respectively.

4.3.2 Marginal PDF of the AoD Here, the marginal PDF of the AoD can be derived by integrating the joint PDF in (4.17) over the geometric path length rb . Therefore, the AoD PDF is given by [6] p(θb ) =

1 A



rb (θb )

p(rb , θb ) drb . 0

(4.19)

76

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

For the proposed visual street channel model, the total scattering region created by the directional antenna at the MT can be calculated as  A=

ψ2 −ψ1

p(rb , θb ) dθb

⎧ 

2 ⎨ D = rb exp − Ax rb cos θb − − c + rb2 Ay sin2 θb ⎩ 2 −ψ1 

ψ2

 − Ax



D rb cos θb − +c 2

2 + rb2 Ay sin2 θb + Cxy

⎫ ⎬ ⎭

dθb .

(4.20)

In previous sections, rb (θb ) and A were given by (4.6) and (4.20), respectively. Substituting (4.6) and (4.20) into (4.19), the marginal PDF of the AoD can be obtained.

4.3.3 Marginal PDF of the AoA Similarly, at the MR, the marginal PDF of the AoA can be obtained within the proposed visual street channel model. Note that the arriving angles θt1 and θt2 (i.e., θt1 ≤ θt2 ) are calculated to separate the above scattering region into three partitions within the azimuth plane. Here the angle θt1 and θt2 are defined as azimuthal threshold angles, and can be written as the functions of the number of reflections and beamwidth of the directional antenna. Therefore, the equations can be written in the closed form as   ρ1 sin ψ1 (4.21) θt1 = arctan D − ρ1 cos ψ1  ρ2 sin ψ2 θt2 = arctan . D − ρ2 cos ψ2 

(4.22)

Owing to the directional antenna at the MT (i.e., MT’), clipping occurs in the scattering region. Note that the distances from the MR to the visual boundary of the street channel model are obviously different among these three parts, they are: the areas of T RU (i.e., T  RU ), U V W R, and T RW (i.e., T  RW ), respectively, as shown in Figs. 4.3 and 4.4. Therefore, the joint PDFs of the AoA in 0 ≤ θm ≤ 2π are shown as follows:

4.3 Spatial Characteristics of the Street Channel Model

77

Case 1 0 ≤ θm ≤ θt2 p1 (θm ) =

1 A



rm1 (θm )

p(rm , θm ) drm .

(4.23)

0

It can be found in the scattering area of T RU (i.e., T  RU ) from Figs. 4.3 and 4.4. Let the function rm1 (θm ) be the scatterers at the boundary of the line MS −P , which can be expressed as   rm1 (θm ) = D sin ψ1 csc ψ1 + θm .

(4.24)

Substitute (4.20) into (4.23), the marginal PDF of the AoA for the Case 1 can be obtained by integrating the joint PDF of the AoA in (4.18) over the function in (4.24), as derived in [6]. Case 2 θt2 ≤ θm ≤ 2π − θt1 p2 (θm ) =

1 A



rm2 (θm )

p(rm , θm ) drm .

(4.25)

0

For the scattering area of U V W R, the function rm2 (θm ) denotes the distance of the MR from the scattering objects located at the visual boundary of the elliptical scattering region, which has been determined in (4.7), and can be further rewritten as  1 rm2 (θm ) = × Db2 cos θm 2b2 cos2 θm + 2a 2 sin2 θm      2 2 4 2 2 2 2 2 2 2 2 + D b cos θm − b cos θm + a sin θm D b − 4a b . (4.26) As in earlier derivations, the marginal PDF of the AoA statistics for the Case 2 can be obtained by integrating the joint PDF in (4.18) over the Eq. (4.26). Case 3 2π − θt1 ≤ θm ≤ 2π 1 p3 (θm ) = A



rm3 (θm )

p(rm , θm ) drm .

(4.27)

0

In this part (i.e., the area of T RW or T  RW ), we can define the function rm3 (θm ) as the scatterers at the boundary of the line MS − Q, which can be calculated as   rm3 (θm ) = D sin ψ2 csc ψ2 + θm .

(4.28)

78

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

Like Case 1 and 2 above, substituting (4.20) into (4.27), the marginal AoA PDF for Case 3 can be obtained by integrating the joint AoA PDF in (4.18) over the path length rm3 (θm ) in (4.28).

4.4 Doppler Frequencies In the proposed visual street channel model, the received signal at the MR experiences spread in the frequency spectrum caused by the relative motion between the MT and MR. For readability purposes, we assume an omnidirectional antenna at the MR, then the relationship between the arriving angles (i.e., θb and θm ) and the Doppler shift of a sinusoidal signal can be expressed as [13, 16]     vb fc1 cos φvb − θb = fm1 cos φvb − θb c

(4.29)

    vm fc2 cos φvm − θm = fm2 cos φvm − θm , c

(4.30)

fb = fm =

where vb denotes the moving velocity of the MT, φvb is the moving direction with respect to the direct LoS component, c is the velocity of light, fc1 and fc2 are the frequencies of the carrier signal and baseband signal, respectively. Similarly, at the MR, vm is the relative moving velocity of the MR, φvm denotes the angle between the direction of the MR with respect to the x-axis (i.e., x  -axis). While fb and fm stand for the maximum Doppler frequencies for the MT and MR, respectively (i.e., fb ≤ |fm1 | and fm ≤ |fm2 |). Based on the marginal PDF of the AoD in (4.19), the PDF of the Doppler frequency at the MT can be obtained by [16] 

p(fb ) = fm1

1  2 1 − fb /fm1

        × p φvb − arccos fb /fm1 + p φvb + arccos fb /fm1 ,     fm1 cos φvb − ψ2 ≤ fb ≤ fm1 cos φvb + ψ1 .

(4.31)

Note that the MT and MR are both in motion. For simplicity, we introduce the concept of the relatively moving velocity, which indicates that the MT is relatively static, while the MR is in relative motion. Here we can define MR moving in an arbitrary direction φv in the azimuth plane. Owing to the relative motion between the MT and MR, it has been shown that the PDF of the Doppler frequency is significantly related to the direction of relative motion. Moreover, on the basis of the marginal PDF of the AoA in (4.23), (4.25), and (4.27), when the MR moves towards the MT, i.e., φv = 0, the PDF of the Doppler shift can be expressed as

4.4 Doppler Frequencies

p(fm ) =

79

⎧     ⎪ 1 ⎪  − arccos f × p /f ⎪ 3 m m2  2 ⎪ ⎪ ⎪ ⎪ ⎪ fm2 1− fm /fm2  ⎪  ⎪   ⎪ ⎪ ⎪ arccos f , fm2 cos θt1 ≤ fm ≤ fm2 +p /f 1 m m2 ⎪ ⎪ ⎪ ⎪  ⎪  ⎪   ⎪ 1 ⎪  − arccos f × p /f ⎪ 2 m m2 ⎨   fm2 1− fm /fm2

2

  ⎪   ⎪ ⎪ ⎪ +p1 arccos fm /fm2 , fm2 cos θt2 ≤ fm ≤ fm2 cos θt1 ⎪ ⎪ ⎪   ⎪ ⎪   ⎪ ⎪ 1 ⎪  − arccos f × p /f 2 m m2 ⎪  2 ⎪ ⎪ ⎪ fm2 1− fm /fm2 ⎪ ⎪  ⎪  ⎪   ⎪ ⎪ , −fm2 ≤ fm ≤ fm2 cos θt2 . +p2 arccos fm /fm2 ⎩ (4.32)

On the other hand, when the receiver moves perpendicular to the line joining the MT and MR (i.e., φv = π/2), the PDF of the Doppler frequency at the MR can be written as   ⎧   ⎪ 1 ⎪  × p /f π/2 − arccos f 1 m m2 ⎪   ⎪ 2 ⎪ ⎪ ⎪ fm2 1− fm /fm2 ⎪  ⎪  ⎪   ⎪ ⎪ , fm2 sin θt2 ≤ fm ≤ fm2 /f π/2 + arccos f +p ⎪ 3 m m2 ⎪ ⎪ ⎪  ⎪  ⎪   ⎪ 1 ⎪ ⎪ × p2 π/2 − arccos fm /fm2 ⎪    2 ⎪ ⎪ ⎪ fm2 1− fm /fm2 ⎪  ⎪  ⎪   ⎪ ⎪ ⎪ +p3 π/2 + arccos fm /fm2 , 0 ≤ fm ≤ fm2 sin θt2 ⎨   p(fm ) =   ⎪ ⎪ ⎪  1 × p2 π/2 − arccos fm /fm2 ⎪  2 ⎪ ⎪ ⎪ ⎪ fm2 1− fm /fm2  ⎪  ⎪   ⎪ ⎪ ⎪ /f π/2 + arccos f +p , −fm2 sin θt1 ≤ fm ≤ 0 1 m m2 ⎪ ⎪ ⎪ ⎪   ⎪ ⎪   ⎪ 1 ⎪ ⎪ 2 × p3 π/2 − arccos fm /fm2 ⎪   ⎪ ⎪ ⎪ fm2 1− fm /fm2 ⎪  ⎪  ⎪   ⎪ ⎪ ⎩ +p1 π/2 + arccos fm /fm2 , −fm2 ≤ fm ≤ −fm2 sin θt1 .

(4.33) Next, the concept of the total Doppler frequency of the proposed visual street channel model is introduced, which is formed by the relative motion between the MT and MR. Moreover, the PDFs of the Doppler frequency at the MT and MR are assumed to be two independent random variables. To obtain the PDF of the total Doppler frequency, the characteristic functions are taken into account, which can be defined as

80

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

 pb (ω) =  pm (ω) =

∞

  p fb ej ωfb dfb

(4.34)

  p fm ej ωfm dfm .

(4.35)

−∞ ∞ −∞

By using (4.34) and (4.35), we can find the characteristic function p(ω) of the total Doppler frequency by the product of two characteristic functions as p(ω) = pb (ω) · pm (ω).

(4.36)

Consequently, by using the inversion Fourier transform formula for (4.36), the PDF can be derived as  ∞   1 p(f ) = p ω ej ωf dω. (4.37) 2π −∞ In substituting the (4.36) into (4.37), the PDF of the total Doppler frequency can be obtained. Further, we can analyze the proposed visual street model from the perspective of the frequency domain. Additionally, we can see if the AoA of the signal at the receiver is assumed to be uniform, the Doppler spectra is given by Clarke’s model [8].

4.5 Numerical Results and Discussions To establish the validity and generalizability of the proposed visual street channel model, several comparisons are made between the proposed model and some notable models and measured data. The results illustrate that the distributions of the AoD/AoA statistics and the Doppler shift of the proposed visual street model fit the Ghoraishi model [15], the Avazov model [18], the Zhou model [6], and the measured data [23] very well.

4.5.1 Distribution of the AoD and AoA PDFs The marginal PDF of the AoD statistics corresponding to the beamwidth of the directional antenna (i.e., ψ1 and ψ2 ) and the street width (i.e., W ), is shown in Fig. 4.5. Note that the values of the AoD PDF tend to be lower with decreasing the width of the streets. It also can be observed that the PDF of the AoD significantly decreases in 0 ≤ θb ≤ ψ2 , and similar behavior can be seen in −ψ1 ≤ θb ≤ 0, which agrees with the results in the Petrus model [7]. Meanwhile, the PDF of the submitted signals with smaller values of the AoD is relatively large. Furthermore, when the MT

4.5 Numerical Results and Discussions

81 ψ =40°, ψ =80°, W=26, W =5

1.0

1

2

0

°

°

ψ1=40 , ψ2=80 , W=18, W0=5 ψ =40°, ψ =80°, W=10, W =3 1

0.8

2

0

°

ψ1=ψ2=180 , W=26, W0=26 Petrus Model [7] Simulation result [25]

p(θ ) b

0.6

0.4

0.2

0 −200

−150

−100

−50

0 50 θ (degree)

100

150

200

b

Fig. 4.5 Marginal PDF of the AoD in terms of the different beamwidths and street widths when D = 60 m

is equipped with the omnidirectional antenna, the distribution of the AoD PDF fits very well with the simulation results in [25], demonstrating that the results above are accurate and applicable to describe the street wireless environments. The impacts of changing the beamwidths ψ1 and ψ2 on the PDFs of the AoA statistics are shown in Fig. 4.6. It can be found that the missing sections of curves enlarge slowly accompanied by an increase in beamwidths, resulting from the irregular scattering region. Furthermore, the curves of the PDFs are asymmetric and have two “corners” located in the left and right parts, which occur at the azimuth angle θm = −θt1 and θm = θt2 , respectively, only depending upon the visual asymmetric geometrical channel model, as seen in Figs. 4.3 and 4.4. With decreasing ψ1 + ψ2 with less scatterers in the areas illuminated by the directional antenna, the AoA PDFs have lower values on both sides of the curves in −θt1 ≤ θm ≤ θt2 , whereas the values at φm = 0 tend to be equal. The analysis above agrees with the results in [15], clearly demonstrating the MIMO and massive MIMO compact antenna systems as promising technologies for wireless street environments. The effects of changing the width of the visual street and the number of reflections (i.e., n) of the received signal for the AoA PDFs are illustrated in Fig. 4.7. It is observed that the AoA statistics decrease first and increase to a local maximum value with higher AoAs, and then decreases gradually in θt2 ≤ θm ≤ π , a similar behavior can be seen in −π ≤ θm ≤ 0. Moreover, it is clear that when the street width is fixed (i.e., W = 60 m), the PDFs of the AoA tend to be equal for

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4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

ψ =2π/9, ψ =4π/9, n=3

1.0

1

2

ψ =π/9, ψ =2π/9, n=3 1

2

n=7

ψ =π/9, ψ =2π/9, n=5 1

0.8

2

ψ1=π/9, ψ2=2π/9, n=7 ψ1=ψ2=π, n=3

p(θm)

0.6

0.4

n=5 0.2 n=3 0 −200

−150

−100

−50

0 50 θ (degree)

100

150

200

m

Fig. 4.6 Marginal PDF of the AoA in terms of the different beamwidths and the number of reflections when W = 18 m, W0 = 5 m, and D = 60 m

1.0

W=10, W =3, n=2 0

W=26, W =5, n=2 0

0.8

W=26, W =5, n=4 0

W=26, W =5, n=6 0

Zhou Model [6]

p(θm)

0.6 n=6 0.4 n=4 0.2

0 −200

n=2 −150

−100

−50

0 50 θ (degree)

100

150

200

m

Fig. 4.7 Marginal PDF of the AoA in terms of the different street widths and the number of reflections when ψ1 = 40◦ , ψ2 = 80◦ , and D = 60 m

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83

0 ≤ θm ≤ θt2 . Further, the AoA statistics are increasing gradually with increasing the number of reflections when the values are in θt2 ≤ θm ≤ π . Comparing the proposed visual street channel model with the Zhou Model [6], we can see that the distribution of the AoA PDFs has similar trends for the omnidirectional antenna (i.e., ψ1 = ψ2 = 180◦ ) at the MT, irrespective of macrocell and microcell environments. From the numerical results and discussions above, we can draw a conclusion that the distributions of the odd- and even-numbered-bounced propagation paths would tend to be the same when the number of reflections of the received signal tends to infinity (i.e., n → ∞ ).

4.5.2 Distribution of the Doppler Frequency The distribution of the total Doppler shift versus the direction of relative motion of the MR (i.e., φv ) and the beamwidths of the directional antenna at the MT is shown in Fig. 4.8. It can be noted that when the MR moves towards the MT, the values of the Doppler shift significantly increase, accompanied by an increase in the beamwidth ψ1 and ψ2 , which is due to the fact that a decrease in the beamwidth could diminish the amount of scatterers and then reduce the multipath components. On the other hand, for an omnidirectional antenna, the Doppler PDFs

12 10

°

10

log [p(f)]

4

2

v

ψ =40°, ψ =80°, φ =0° 1

6

°

ψ =20°, ψ =40°, φ =90° 1

8

°

ψ1=20 , ψ2=40 , φv=0

2

°

v °

ψ1=ψ2=180 , φv=0 Avazov Model [18] Measured data [26]

2 0 −2 −4 −6 −8 −100

−50

0 f [Hz]

50

100

Fig. 4.8 Distribution of the total Doppler shift with respect to the different relative moving direction, the beamwidth, and the number of reflections when D = 60 m, W = 26 m, W0 = 5 m, fm1 = fm2 = 100 Hz, Ax = 1.0 × 10−4 , and Ay = 1.0 × 10−3

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4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

of the proposed visual street channel models are significantly different from the traditional U -shaped Doppler distribution [8, 18]. It also can be noted that the PDFs are skewed toward higher frequencies when the omnidirectional antenna is deployed at the MT, resulting from the unbalanced number of illuminated scattering objects with respect to the x-axis of motion of the MR. The analysis in Fig. 4.8 fits very well with the results in [23], demonstrating the accuracy of the total Doppler shift in the proposed model. Furthermore, the curves of the Doppler PDFs are asymmetric and have two “corners” located in the left and right parts, which is in agreement with the discussion in Fig. 4.5. This clearly indicates that the Doppler frequency is significantly related to the distribution of power over the AoD and AoA statistics. The influence of the distance between the MT and MR (i.e., D) and the number of reflections on the total Doppler distribution is shown in Fig. 4.9. When the receiver moves perpendicular to the direct LoS component, most of the spectral components remain near the central frequencies (i.e., |f | = 0), an observation that is in agreement with the results reported in [9]. Meanwhile, the curves of all PDFs are asymmetric. This is because the clipped scattering region is geometrically asymmetric, as illustrated in Figs. 4.3 and 4.4, and the number of multipath components corresponding to the positive and negative Doppler frequencies is unequal [12]. Moreover, it is observed that the PDFs of Doppler frequency gradually increase with an increase in the distance between the MT and MR or the number of reflections. Besides that, comparisons between the above theoretical discussions with the Wang model [20] show that the distribution trends are both in agreement

4

ψ =ψ =π/9, D=20, n=3 1

2

2

ψ =ψ =π/9, D=40, n=3

0

ψ1=ψ2=π/9, D=40, n=4

1

2

Zhou Model [9] Conventional Clarke Model [8]

10

log [p(f)]

−2 −4 −6 −8 −10 −12 −100

−50

0 f [ Hz ]

50

100

Fig. 4.9 Distribution of the total Doppler shift with respect to the distance between the MT and MR, and the number of reflections when φv = π/2, W = 26 m, W0 = 5 m, fm1 = fm2 = 100 Hz, Ax = 1.0 × 10−4 , and Ay = 1.0 × 10−3

References

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with each other, which validate the generalization of the proposed virtual street channel model.

4.6 Conclusion In this chapter, a generalized visual scattering channel model for multi-bounced propagation paths in the dense urban street environments has been observed. The proposed model first has the ability to describe the multi-reflecting propagation paths for odd- and even-numbered-bounced propagation paths. On the basis of the marginal PDFs of the AoD and AoA statistics, the distributions of the total Doppler frequency due to the relative motion between the MT and MR have been analyzed, which broaden the analysis of the visual street channel model from the perspective of the frequency domain. Furthermore, we have performed the spatial characteristics for different beamwidths, street widths, distance between the MT and MR, the number of reflection, and the relative moving direction. Comparisons between our theoretical results and several previous scattering channel models show that the proposed visual scattering channel model is accurate and applicable to depict dense urban street scenarios.

References 1. P.P. Tayade, V.M. Rohokale, Enhancement of spectral efficiency, coverage and channel capacity for wireless communication towards 5G, in International Conference on Pervasive Computing (ICPC) (Pune, India, 2015), pp. 1–5 2. DOCOMO 5G White Paper, 5G radio access: requirements, concept and technologies, NTT DOCOMO (2014) 3. J. Karedal et al., A geometry-based stochastic MIMO model for vehicle-to-vehicle communications. IEEE Trans. Wirel. Commun. 8(7), 3646–3657 (2009) 4. Y.F. Chen, V.K. Dubey, Accuracy of geometric channel-modeling methods. IEEE Trans. Veh. Technol. 53(1), 82–93 (2004) 5. R.B. Ertel, J.H. Reed, Angle and time of arrival statistics for circular and elliptical scattering model. IEEE J. Sel. Areas Commun. 17(11), 1829–1840 (1999) 6. J. Zhou, H. Jiang, H. Kikuchi, Geometry-based statistical channel model and performance for MIMO antennas. Int. J. Commun. Syst. 29(3), 459–477 (2016) 7. P. Petrus, T.S. Rappaport, Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans. Commun. 50(3), 495–502 (2002) 8. L. Jiang, S.Y. Tan, Geometrically based statistical channel models for outdoor and indoor propagation environments. IEEE Trans. Veh. Technol. 56(6), 3587–3593 (2007) 9. J. Zhou, Z.G. Cao, H. Kikuchi, Asymmetric geometrical-based statistical channel model and its multiple-input and multiple-output capacity. IET Commun. 8(1), 1–10 (2014) 10. S.J. Nawaz, B.H. Qureshi, N.M. Khan, A generalized 3-D scattering model for a macrocell environment with a directional antenna at the BS. IEEE Trans. Veh. Technol. 59(7), 3193– 3204 (2010) 11. H. Jiang, J. Zhou, H. Kikuchi, Angle and time of arrival statistics for a 3-D pie-cellular-cut scattering channel model. Springer Wirel. Pers. Commun. 78(2), 851–865 (2014)

86

4 Multi-Bounced Virtual Scattering Channel Model for Dense Urban Street. . .

12. M. Riaz, N.M. Khan, S.J. Nawaz, A generalized 3-D scattering channel model for spatiotemporal statistics in mobile-to-mobile communication environment. IEEE Trans. Veh. Technol. 64(10), 4399–4410 (2015) 13. S.X. Qu, An analysis of probability distribution of Doppler shift in three dimensional mobile radio environments. IEEE Trans. Veh. Technol. 58(4), 1634–1639 (2009) 14. M. Marques, M. Correia, A wideband directional channel model for UMTS micro-cells, in Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’01), vol. 1 (San Diego, CA, 2001), pp. B-122–B-126 15. M. Ghoraishi, J.I. Takada, T. Imai, A pseudo-geometrical channel model for dense urban line-of-sight street microcell, in IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communication (Helsinki, Finland, 2006), pp. 1–5 16. M. Ghoraish, G. Ching, N. Lertsirisopon, Polar directional characteristics of the urban mobile propagation channel at 2.2 GHz. 3rd European Conference on Antennas and Propagation (Berlin, Germany, 2009), pp. 892–896 17. M. Patzold, B.O. Hogstad, A wideband MIMO channel model derived from the geometrical elliptical scattering model, in International Symposium on Wireless Communication and Mobile Computing (Valencia, Spain, 2008), pp. 597–605 18. N. Avazov, M. Patzold, A geometric street scattering channel model for car-to-car communication systems, in IEEE International Conference on Advanced Technol. for Communication (Da Nang, Vietnam, 2011), pp. 224–230 19. G.R. MacCartney, J.H. Zhang, S. Nie, Path loss models for 5G millimeter wave propagation channels in urban microcells, in IEEE Global Communications Conference (GLOBECOM), (Atlanta, GA, 2013), pp. 3948–3953 20. C.X. Wang, X. Cheng, D.I. Laurenson, Vehicle-to-vehicle channel modeling and measurements: recent advances and future challenges. IEEE Commun. Magazine 47(11), 96–103 (2009) 21. X. Cheng, C.X. Wang, B. Ai, Envelope level crossing rate and average fade duration of nonisotropic vehicle-to-vehicle Ricean fading channels. IEEE Trans. Intell. Transp. Syst. 15(1), 62–72 (2013) 22. M.E. Rasekh, F. Farzaneh, A.A. Shishegar, A street canyon approximation model for the 60 GHz propagation channel in an urban environment with Rough Surfaces, in International Symposium on Telecommunication (IST) (Tehran, Iran, 2010), pp.132–137 23. X. Cheng, C.X. 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) 24. S.B. Wu, C.X. Wang, E.-H.M. Aggoune, A non-stationary 3-D wideband twin-cluster model for 5G massive MIMO channels. IEEE J. Sel. Areas Commun. 32(6), 1207–1218 (2014). 25. Y. Yuan, C.X. Wang, X. Cheng, Novel 3D geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channels. IEEE Trans. Veh. Technol. 13(1), 298–309 (2014)

Chapter 5

A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication Environments

5.1 Introduction MIMO technologies have received considerable attention for modern wireless communication systems because they can substantially increase spectral efficiency [1, 2]. Recently, massive MIMO systems, which are equipped with hundreds or even thousands of antennas, have emerged as enhanced MIMO systems to meet the increasing traffic demands of 5G wireless communication networks [3]. In general, massive MIMO has been regarded as an efficient means of improving both spectrum efficiency and energy efficiency for wireless communication systems. They are thus significantly better than the conventional MIMO systems [4]. More specifically, it has been suggested that massive MIMO V2V communication environments comprise a promising technology for 5G wireless communication networks. To create the above communication systems, a solid understanding of the radio propagation characteristics of the transmission paths between the mobile transmitter and mobile receiver is required [5]. This understanding can result in the design of effective signal processing techniques. Additionally, the authors in [6] demonstrated that correlation-based conventional MIMO channel models, such as the respective Kronecker and Weichselberger models, can be used to analyze the performance of compact MIMO systems. The authors of [7–10] presented a series of geometric configurations of compact MIMO antenna arrays in 3D multipath channels, including ULA, UCA, uniform rectangular array (URA), L-shaped array, and uniform concentric circular array (UCCA). However, they were restricted to the planar antenna array. In 2009, Mammasis [10] provided the concept of the Von Mises Fisher Distribution and derived the SFCs for signals at the UCA antenna array. For the sake of the multipath richness of the wireless channel, it is necessary to design and evaluate MIMO systems with an accurate small-scale multipath fading MIMO channel model. Thus far, the respective spatial channel, Okumura/Hata, IEEE 802.16d,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_5

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Winner II, and Clarke’s models have been widely utilized [11]. References [12– 16] presented many two-dimensional (2D) geometry-based channel models for conventional MIMO channels, such as the respective circular, ellipse [13–15], and hollow-disc models [16]. Subsequently, many researchers have observed an angular spreading of the waves in the elevation plane because of the interaction of the waves with street buildings, the ground, and other vertically disposed objects [17–22]. In 2010, Nawaz [17] presented a 3D scattering model for macrocell communication environments. Additionally, in [18] and [19], a generalized 3D scattering channel model for microcell environments was proposed. In that model, the scatterers are assumed to be located around the BS in Gaussian distribution. Furthermore, the authors of [20] proposed a 3D theoretical regular-shaped geometry-based stochastic model (RSGBSM) for non-isotropic MIMO V2V Ricean fading channels. However, that model cannot describe the channel statistics for different time delays, which are important for V2V channels. The authors in [21] and [22] presented a 3D geometry-based channel model for cross-polarization discrimination in narrowband fixed-to-fixed or F2M wireless channels. However, most previous channel models rely on the WSS assumption, which assumes that channel statistics are unchanged with respect to time. According to the measurement results in [23] and [24], the WSS assumption is valid only for very short time intervals, i.e., in the order of milliseconds. Furthermore, the above models focus on the far-field assumption, which may not be appropriate for large-scale MIMO channel models [25]. According to the measurement observations in [26] and [27], when the number of antennas is large, the plane wavefront assumption is not fulfilled for massive MIMO channels. Instead, a spherical wavefront channel model should be considered. The authors in [28, 29], and [30] employed only non-stationarities at time-variant properties; however, they ignored the antenna array properties. Furthermore, plane wavefront models and spherical wavefront models were compared in [31] and [32]. The results demonstrated that the plane wavefront assumption underestimates the rank of the channel matrix. Nevertheless, they only focused on short range or constant distance communications, which are not applicable for general massive MIMO systems. Recently, many researchers have focused on the study of massive MIMO channel modeling by the METIS project. For example, the authors in [33] considered near-field effects and non-stationarities in the array properties. However, in [33], the impact of the spherical wavefront for NLoS components was not studied in detail; moreover, the time-variant properties of the geometric relationships were not included. Furthermore, the authors in [34] presented a 2D wideband ellipse channel model with non-stationarities at both time-variant and array-variant properties for massive MIMO communication systems. They analyzed the time and frequency cross-correlation functions for two different propagation paths; however, they ignored angular spreading of the incident waves in the elevation plane in a 3D multiple channel. The authors of [35] proposed a theoretical nonstationary 3D wideband twin-cluster channel model for massive MIMO systems with carrier frequencies in the order of GHz. Recently, Cheng et al. [36, 37] presented a geometry-based stochastic model for non-isotropic scattering V2V Ricean fading channels. Moreover, in [36] and [37], the impact of vehicular traffic

5.1 Introduction

89

density on channel characteristics was considered. However, the above-mentioned massive MIMO channel models do not describe V2V communication scenarios. Additionally, the mobile properties between the transmitter and the receiver are not considered, and the time and frequency cross-correlation functions were not studied in detail. The authors in [38] presented an approach for mobile V2V scenarios by introducing highly directional sectored antennas into V2V communication environments. In [38], the knowledge of the maximum number of vehicles and the minimum data rate support are very meaningful for V2V scenarios. However, the channel model in [38] remained a conventional channel model (i.e., up to four or eight antennas), which did not consider the spherical wavefront assumption and massive MIMO antennas. For V2V mobile radio environments, large-scale omnidirectional antennas are employed at different surfaces of vehicles. To this end, the propagation links collecting different transmitting and receiving antennas should be discussed in detail. To the best of our knowledge, this endeavor has not been previously undertaken. For 3D massive MIMO V2V channel modeling, the conventional MIMO antenna array model cannot be directly taken into account owing to the changeable vehicle shape in 3D space. Furthermore, it is obvious that the previous work in [14–19] is improper because it neglects the impact of the non-stationarity or spherical assumptions on the channel statistics. To address this issue, we present a 3D vehicle massive MIMO antenna array model for non-stationary V2V communication scenarios. By adjusting the proposed channel parameters, our model can describe a variety of communication environments, such as macro-, micro-, and picocell scenarios. This channel model was developed to capture the spherical wavefront effect, spatial statistics, and mobile properties. Moreover, it assumes that the AoD and AoA are independent of each other. The main contributions of this paper are summarized as follows: (1) A 3D vehicle massive MIMO antenna array model that can be fully utilized for future V2V communication environments. The performance of the proposed antenna array model is analyzed in detail, including the impact of the antenna spacing on the spatial correlation between different antenna elements. (2) In the proposed model, we present a spherical wavefront assumption, instead of the plane wavefront assumption characterizing the conventional MIMO channel model. The impact of the spherical wavefront assumption on both LoS and NLoS components in time and frequency domains is studied. (3) A non-stationary V2V channel model is proposed. The important channel statistical properties are derived and thoroughly investigated, including AoD and AoA statistics, time and frequency cross-correlations, and the Doppler spectrum.

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

5.2 Proposed 3D Vehicle Massive MIMO Array Model As introduced in [32], the far-field assumption for the conventional MIMO channel models, which is equivalent to the plane wavefront assumption shown in Fig. 5.1a, is not fulfilled for the proposed 3D vehicle massive MIMO. This is because the dimension of the antenna array cannot be ignored. Additionally, the proposed channel statistics vary significantly over the large array. Accordingly, we herein assume that the wavefront emitted from the scatterer to the receiving array in the proposed model is spherical, as illustrated in Fig. 5.1b. In this case, the AoD and AoA are no longer linear along the array; they must be computed based on the geometric relationships. To analyze the performance of MIMO systems, it is necessary to adopt a frequency nonselective directional Rayleigh fading channel model [7]. As mentioned in [39], the channel impulse response is related to both the complex amplitude and the steering vector components of the antenna arrays. Therefore, the channel impulse response of the MIMO antenna array model can be expressed as h(t) =

N 

αj (t) · a(αA , βE ),

(5.1)

j =1

where αj (t) denotes the complex amplitude, a(αA , βE ) is the steering vector of the compact antenna array, and N is the total number of large-scale antennas. The scalars αA and βE denote the azimuth and elevation angles with respect to the positive x- and y-axes, respectively. Therefore, the steering vector in Fig. 5.1b can be expressed as

Fig. 5.1 Geometric uniform rectangular array model with (a) plane wavefront and (b) spherical wavefront

5.2 Proposed 3D Vehicle Massive MIMO Array Model

⎛⎡ ⎜⎢ ⎜⎢ ⎜⎢ a(αA , βE ) = vec ⎜⎢ ⎜⎢ ⎝⎣

1 ej m ej 2m ... ej (W −1)m

⎤

91

⎞

⎟ ⎥ ⎥ ⎟ ⎥ jp j (L−1)p ⎟ ⎟, ⎥ 1, e , . . . , e ⎟ ⎥ ⎠ ⎦

(5.2)

where m = kw dx cos αA cos βE , p = kw dy cos αA cos βE , kw = 2π/λ, and λ denotes the wavelength. The scalars dx and dy denote the respective spacing between the array elements parallel to the x- and y-axes. Additionally, the operator vec(·) maps a W × L matrix to a W L × 1 vector by stacking the columns of the matrix. However, when the transmission distance gradually increases, the properties of the singular values of the MIMO channel matrix of the spherical wave model slowly approach the properties of the plane wave model. Accordingly, the steering vector given by the plane wavefront can be obtained when the elevation angle (i.e., βE = 0) in 3D space is neglected, as illustrated in Fig. 5.1a. For V2V mobile radio environments, we first introduce the proposed 3D vehicle massive MIMO antenna array model, including the spherical wavefront assumption and geometric properties. Here, large-scale omnidirectional antennas are employed at different surfaces of vehicles. The bottom side of the vehicle is close to the ground. Thus, the signals cannot be effectively transmitted by multipath links. We therefore present an approach for vehicles to form a 3D ULA in the shape of a box, wherein large-scale antennas are equally situated on the surfaces of U V KT , T KGS, U V EF , U T SF , and KV EG, as illustrated in Fig. 5.2. We can extend this approach to any number of surfaces, depending on the vehicle shape. To perform the proposed massive MIMO system, we provide a serial number to each antenna

Fig. 5.2 Proposed 3D vehicle massive MIMO antenna array model

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element of the proposed 3D array model from the surface of U V KT to the surfaces of T KGS, U V EF , U T SF , and KV EG, respectively. Therefore, the steering vector for the proposed model can be expressed as  T a(αA , βE ) = a1 (αA , βE ), a2 (αA , βE ), . . . , ampq (αA , βE ), . . . , aN (αA , βE ) , (5.3) where ampq (αA , βE ) denotes the phase steering of the mpq-th element for the proposed 3D array model, which can be defined as the m-, p-, and q-th element along with the positive x-, y-, and z-axes, i.e., m = 1, 2, . . . , W , p = 1, 2, . . . , L, and q = 1, 2, . . . , H , as shown in Fig. 5.2. Thus, the total number of the proposed 3D antenna elements can be derived as N = W L + 2LH + 2W H . In this case, the proposed channel impulse response can be obtained by submitting (5.3) into (5.1). When modeling the slowly varying and frequency-flat fading channel systems, it is common to assume that the received wavefront is plane [32], as shown in Fig. 5.1a. However, the plane wave model is accurate for many communication scenarios, it almost relies on the far-field assumption, it underestimates the MIMO gain in some situations, such as in short-range communications. In this case, the true spherical nature of the wave propagation should be considered. Nonetheless, in the proposed 3D vehicle massive MIMO antenna array model, large-scale antennas are situated on each vehicular surface, and incident signals are transmitted from various directions. Therefore, the closed-form expressions of the phase delay ampq (αA , βE ) of the above five surfaces are significantly different as mpq changes, i.e., mpq = 1, 2, . . . , N, which can be shown as ⎧  ⎪ exp j kw mdx cos αA cos βE + j kw pdy sin αA cos βE ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ , 0 ≤ mpq ≤ W L +j k H sin β ⎪ w E ⎪ ⎪  ⎪   ⎪ ⎪ exp j kw W − 1 dx cos αA cos βE + j kw pdy sin αA cos βE ⎪ ⎪ ⎪  ⎪   ⎪ ⎪ ⎪ , W L ≤ mpq ≤ W L + LH q − W L d +j k sin β ⎪ w z E ⎪ ⎪   ⎪   ⎪ ⎪ exp j kw pdy sin αA cos βE + j kw q − W L − LH dz sin βE , ⎪ ⎪ ⎨ ampq (αA , βE ) = W L + LH ≤ mpq ≤ W L + 2LH ⎪ ⎪   ⎪   ⎪ ⎪ exp j k , q −W L−2LH d md cos α cos β + j k sin β ⎪ w x A E w z E ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ W L + 2LH ≤ mpq ≤ W L + 2LH + W H ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ exp j kw mdx cos αA cos βE + j kw Ldy sin αA cos βE ⎪ ⎪ ⎪  ⎪   ⎪ ⎪ ⎪ +j kw q − W L − 2LH − W H dz sin βE , ⎪ ⎪ ⎪ ⎪ ⎩ W L + 2LH + W H ≤ mpq ≤ N. (5.4)

5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model

93

For the proposed 3D array model, we assume that the incident signals are uniformly distributed in the azimuth and elevation planes [7], i.e., 0 ≤ αA ≤ 2π and 0 ≤ βE ≤ π/2. Therefore, the SFC between the mpq- and MP Q-th antenna elements is defined as   1 ρ mpq, MP Q = 2π

 0

2π



π/2

ej kw dT x +j kw dT y +j kw dT z sin βE dαA dβE , (5.5)

0

where dT x = (m−M)dx cos αA cos βE , dT y = (p −P )dy sin αA cos βE , and dT z = (q − Q)dz sin βE .

5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model In this section, we first discuss the time-varying geometric path lengths of the proposed non-stationary V2V channel model. Then, we analyze the space-time and frequency cross-correlations for the LoS and NLoS components.

5.3.1 Geometric Properties of V2V Communication Environments The authors of [40] introduced a 3D geometric channel model for mobile-to-mobile (M2M) communication environments. However, the above channel model solely assumes that the distance between a scatterer and an antenna array is far beyond the Rayleigh distance [41], i.e., 2M/λ, where M denotes the dimension of the antenna array and λ is the carrier wavelength. Namely, the plane wavefront and far-field assumptions are applied to simplify the channel models. However, as the number of antennas is enormous, the plane wavefront assumption is not fulfilled for massive MIMO channels[27]. Instead, a spherical wavefront channel model should be considered. Therefore, we present a 3D geometric channel model, where the MT and MR are equipped with the proposed 3D vehicle massive MIMO antenna arrays. Additionally, spherical wavefront is assumed in the proposed channel model, instead of the plane wavefront that is assumed in conventional MIMO channel models. However, there are three key differences between the proposed model and that of [40]. For one, the geometry of the proposed model is significantly different from the geometry of the channel model in [40]. The authors in [40] proposed a geometrically based 3D channel model for M2M communication environments, where both MRs are located at two different semi-ellipsoids. On the other hand, we present a 3D semi-ellipsoidal channel model for V2V scenarios, where the MT and MR are located in the focal points of the ellipse in the azimuth plane. Second, the authors of [40] relied on the WSS assumption which means, in domain, that the channel statistics remain invariant over a short period of time [37]. In the proposed model,

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

on the other hand, we present a 3D semi-ellipsoidal non-stationary channel model that analyzes the proposed channel statistics for different relative moving directions and different moving time instants. Additionally, the proposed channel statistics for different path delays can be investigated. Third, the spherical wavefront assumption is herein introduced in the proposed massive MIMO channel model. Conversely, in [40], the plane wavefront assumption was introduced for a conventional MIMO channel model. In a typical V2V communication environment, the roadside environment significantly affects the channel characteristics. For example, the authors in [40] and [42], respectively, presented the semi-ellipsoid and cylinder models to describe the scatterers around the MT and MR; however, in [40] and [42], the effect of the roadside environment on the channel statistics was not discussed. In general, most structures in V2V environments (e.g., buildings, highways, urban spaces) have many kinds of vertical surfaces. Thus, the channel statistics for different path delays should be considered. For example, in [18] and [20], the authors proposed semisphere and elliptic-cylinder models to depict the roadside environments. However, the channel statistics for different path delays were not discussed. Furthermore, the authors in [14] and [15] stated that 2D elliptical channel models with MT and MR located at the foci can depict realistic V2V communication scenarios. Therefore, we adopt a 3D non-stationary semi-ellipsoid channel model to describe the real V2V communication environments, as illustrated in Fig. 5.3.

Fig. 5.3 model

Typical 3D V2V mobile radio environment and its corresponding scattering channel

5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model

95

In the proposed model, the scalars dx , dy , and dz represent the antenna spacing between the transmit antenna parallel to the x-, y-, and z-axes, respectively. Similarly, at the receiver, they are denoted as dx  , dy  , and dz , respectively. Let us define a, b, and c as the half-lengths of the three dimensions of the semiellipsoid, i.e., a ≥ b ≥ c. The distance between the MT and MR is denoted by D. Furthermore, the angles created in the azimuth and elevation planes with the direction of the incident signal at the MT are denoted by αT and βT ; at the MR, they are denoted by αR and βR , respectively. For V2V scenarios, MT and MR are both moving, which can be equivalent to a static MT situation with the principles of relative motion. Here, we define MR moving in an arbitrary direction φv with a speed of vR in the azimuth plane. Figure 5.3 shows the geometric properties and moving statistics (i.e., relative moving time and moving directions) of the proposed model in the azimuth plane. For V2V scenarios, the geometric path lengths are time-variant because of the relative movement between the MT and MR. In general, the proposed channel model can be more effectively utilized compared to describe many existing 2D and 3D scattering channel models of V2V and F2M communication environments. For example, the proposed model tends to be WSS channels as the relative time t = 0, which describes those of F2M channels [12–14]. However, when the variable t is not equal to zero, the proposed model denotes the non-stationary channels, as presented in [20] and [43]. Therefore, the direct distance between the mpq−th transmit antenna and m p q  −th receive antenna at time instant t can be computed as  LoS ξmpq,m  p  q  (t) =

 2  2 m dx  − mdx + vR t sin φv + p dy  − pdy + vR t cos φv . (5.6)

For the NLoS components, the closed-form expression of the distance from the mpq-th transmit antenna element to the scatterer at time t can be computed as a function of the angles at the MT and written as T ξmpq (t) =

 1 × − k2 + 2k1



 k22 − 4k1 k3 ,

(5.7)

where k1 = b2 c2 cos2 βT cos2 αT + a 2 c2 cos2 βT sin2 αT + a 2 b2 sin2 βT k2 = 2b2 c2 pdy cos βT cos αT − Db2 c2 cos βT cos αT +2a 2 c2 mdx cos βT sin αT 2  2  k3 = D/2 − pdy b2 c2 + a 2 c2 mdx − a 2 b2 c2 . It is correct that the distance from the m p q  −th receiving antenna to the scatterer can be expressed as a function of the receiving angles at the MR, which is

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

different from the independent variables (i.e., αR and βR ) in (5.7). In this case, the discrete expressions of the receiving angles αR and βR can be replaced by the continuous departure angles αT and βT , respectively [15]. Therefore, it can be derived as ξmR p q  (t)   2   2 T (t) T ξmpq + ξmpq,m =  p  q  (t) − 2ξmpq (t)ξmpq,m p  q  (t) cos βT cos αT − α0 ,

(5.8) where the azimuth angle α0 can be approximately derived as α0 ≈ vR t/D · sin φv . Based on the above time-varying geometric properties, we further analyze the proposed model in the following subsections. As mentioned in Sect. 5.1, the proposed 3D channel model can depict a wide variety of communication environments by adjusting the geometric model parameters. For example, when we define the relative moving time t as zero, the proposed model can describe the previous 3D semi-ellipsoid channels, as shown in [44] and [45]. However, the proposed channel model tends to be a 2D model, i.e., c = 0. In this case, when we set t = 0, our model can be transformed into the non-stationary V2V channels, such as the Yuan model [20] and Ghazal model [43]. Likewise, the proposed model describes other models in previous work; we omit them for brevity.

5.3.2 Spatial Characteristics of the Radio Channel Based on the above-mentioned geometric angles and path lengths for the LoS and NLoS components, the PDF of the AoD at the transmitter measured in the azimuth and elevation planes is derived for the proposed 3D V2V channel model. Here, we adopt the same method used in [12] and [13]. It focuses only on the first and last bounces of the transmitted signal. In general, the scatterer density tapers in accordance with the increasing the distance from the transmitter to receiver [16]. Therefore, it is necessary to present a scatterer non-uniform distribution in the proposed V2V channel model. In the literature, many different scatterer distributions have been proposed to characterize the geometric path lengths and geometric angles, such as Gaussian, Laplacian, and von Mises. We herein adopt the Gaussian PDF to characterize the distribution of scatterers in the proposed model, because it approximates many of the previously mentioned distributions and leads to closedform solutions for many useful situations [13]. The Gaussian scatterer density function can be expressed as   f ξ, α, β =

  1 2 2 , × exp − ξ /2σ 2π σ 2

(5.9)

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97

where σ denotes the standard deviation. As shown in Fig. 5.3, the proposed 3D semi-ellipsoid is centered at the origin of the global Cartesian coordinate system O. The scatterers are non-uniformly distributed within the semi-ellipsoid’s volume. However, for graphical simplicity, the figure shows only one scatterer denoted by s. The scatterer’s spherical and Cartesian coordinates with respect to the original point are denoted as (ξ, α, β) and (X, Y, Z), respectively. Additionally, T , α , β ) denotes the spherical coordinates of the MT, with the MT as the (ξmpq T T origin. However, it is difficult to directly transform from the polar coordinates T , α , β ) since the radial and angular coordinates will both (ξ, α, β) to (ξmpq T T change. Therefore, it is necessary to firstly transform (ξ, α, β) to (X, Y, Z), and T , α , β ). Here, the joint PDF of f (X, Y, Z) can then transform (X, Y, Z) to (ξmpq T T be written as        f X, Y, Z = J X, Y, Z f ξ, α, β ,

(5.10)

   where J X, Y, Z  is the Jacobian of the inverse transformation. Therefore, the joint PDF of the AoD can be written as 

T f ξmpq (t), αT , βT



=

 2 T (t) ξmpq cos βT 2π σ 2  ×e

−

1 × 2σ 2

−

1 × 2σ 2

 ×e

−

1 × 2σ 2

2

T (t) cos β cos α −d/2 ξmpq T T

 ×e

2

T (t) sin β ξmpq T

T (t) cos β sin α ξmpq T T

2 .

(5.11)

As mentioned in [19], spatial characteristics of channel models that describe the AoA statistics of the multipath components are very useful in the performance evaluation of wireless communication systems employing massive MIMO antenna arrays at the transmitter and receiver. For example, the time and frequency crosscorrelations, Doppler shift, and channel capacity.

5.3.3 Time Cross-Correlation Function Analysis Based on the above geometrical relationships, the massive MIMO channel for V2V communication scenarios is presented in Fig. 5.3. The transmitter and receiver are equipped with the proposed vehicle massive MIMO antenna model with NT and NR omnidirectional antennas, respectively [34]. Here, our convention for numbering the antenna elements is such that 1 ≤ mpq ≤ MP Q ≤ NT and 1 ≤ m p q  ≤ M  P  Q ≤ NR . Assume that the waves are spherical, unlike in the conventional MIMO channel models, as shown in [35]. Let us define l1 as the propagation link

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connecting the mpq-th transmitting antenna and m p q  -th receiving antenna. As observed from the proposed V2V channel model, the multipath channel impulse response for the link l1 at time t and delay τ can be expressed as hl1 (t, τ ) =

∞ 

  hl1 ,i (t) · δ τ − τi (t) .

(5.12)

i=1

In general, the multipath channel impulse response for link l1 contains LoS and N LoS (t). As NLoS components, which can be expressed as hl1 (t) = hLoS l1 (t) + hl1 introduced in [34], the spherical waves emitted from the mpq-th transmit antenna array element travel over paths of different lengths. After being scattered by the local scatterers around the receiver, they show that the array element comes from different directions, i.e.,    K j ϕ + j 2π ×ξ LoS (t) LoS × e o λ mpq,m p q  hl1 ,i (t) = δ n − 1 × K +1 ×ej 2πfmax t cos αR cos βR  LoS hN l1 ,i (t)

=

(5.13)

S Pn 1  × lim √ gi K + 1 S→∞ S i=1



×e





T (t)+ξ R j ϕo +j 2πfmax t cos αR −φv cos βR + j 2π ξmpq (t) λ m p  q 

 ,

(5.14)

where fmax represents the maximum Doppler shift, K denotes the Rice factor, and Pn is the mean power transmitted through the link collecting mpq-th transmit antenna and m p q  -th receive antenna. Here, power Pn can be normalized to 1 so that the Rice factor K is effective. Moreover, ϕo denotes the initial phase of the transmit signal. Assume that ϕo is an independent and identically distributed random variable with Gaussian distribution over [0, 2π ) in the azimuth plane. Furthermore, S denotes the number of independent scatterers in the proposed 3D space, and gi represents  the amplitude of the wave scattered by the scatterer toward the receiver; thus, Si=1 E[ gi2 ]/S = 1 as S → ∞. Subsequently, let l2 denote the propagation link collecting the MP Q-th transmitting antenna and M  P  Q -th receiving antenna. Here, we assume that l1 and l2 are independent of each other. Therefore, the normalized time cross-correlation functions for the two different NLoS components, link l1 and link l2 , at different time instants of t are given by 

hl1 (t)h∗l2 (t − τ )   ∗  RhNLoS ,hNLoS (t, τ ) = E  hl (t) · h (t − τ ) l1 ,i l2 ,i 1

l2

S 1 1  2 = × lim E gi K + 1 S→∞ S i=1

5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model

×ej 2πfmax ·τ cos  ×e

j 2π λ ×



99



αR −φv cos βR



R R T (t)−ξ T ξmpq MP Q (t)+ξm p q  (t)−ξM  P  Q (t)

. (5.15)

Because the transmitter and receiver are equipped with large scales of NT and NR omnidirectional antennas, respectively, the total power of the NLoS components Pn is proportional to E[ gi2 ]/S. This is equal to the infinitesimal power from the differential of the 3D angles, dαT dβT , i.e., E[ gi2 ]/S = f (αR , βR )dαT dβT , where f (αR , βR ) denotes the joint PDF of the AoA observed at the MR [13]. Therefore, (5.15) can be rewritten as follows: RhNLoS ,hNLoS (t, τ ) = l1 ,i

l2 ,i

1 × K +1  ×e

j 2π λ



π/2  π −π

0

  f αR , βR 

R R T (t)−ξ T ξmpq MP Q (t)+ξm p q  (t)−ξM  P  Q (t)

×ej 2πfmax ·τ cos





αR −φv cos βR

dαT dβT .

(5.16)

Obviously, the time cross-correlation between the two different NLoS components link l1 and link l2 is not only related to the geometric path lengths, but also to the moving properties. Moreover, the joint PDF of the AoA statistics f (αR , βR ) is observed at the receiver.

5.3.4 Frequency Cross-Correlation Function Analysis To analyze the frequency cross-correlation between the two different propagation paths, along with the auto-correlation for a single path, we define f as the frequency variable. Therefore, by taking the Fourier transform of hl1 (t, τ ), the time-variant transfer function Hl1 (t, f ) can be defined as  Hl1 ,i (t, f ) =

∞

−∞

hl1 ,i (t, τ ) · e−j 2πf τ dτ.

(5.17)

If hl1 (t, τ ) is modeled as a complex-valued zero-mean Gaussian random process in the time variable, it follows that Hl1 (t, f ) has the same statistics. Under the assumption that the channel is wide-sense stationary, we can obtain the crosscorrelation function between link l1 and link l2 as ⎡     ⎤ Hl1 ,i t, f Hl∗2 ,i t, f + Δf   (5.18) Rhl1 ,i ,hl2 ,i t, f, Δf = E ⎣     ∗   ⎦ ,  · t, f t, f + Δf  Hl2 ,i  Hl1 ,i

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

where ∗ denotes the complex conjugate. Substitute (5.17) into (5.18), the frequency cross-correlation function can be rewritten as ⎤ ⎡  ∞ hl1 ,i (t) h∗l2 ,i (t)ej 2π Δf τ dτ   −∞ (5.19) Rhl1 ,i ,hl2 ,i t, Δf = E ⎣       ⎦ ,  Hl1 ,i t, f  · Hl∗2 ,i t, f + Δf  where τ is the propagation delay of a multipath signal reflected from any one scatterer, which is assumed to be located in the illuminated 3D scattering region. In the proposed 3D channel model, let us define τ0 and τmax as the shortest and longest propagation delays, i.e., τ0 = D/c and τmax = (D/2 + a)/c, as shown in [17] and [19]. Therefore, the normalized frequency cross-correlation function for two different NLoS propagation paths (i.e., link l1 and link l2 ) can be rewritten as  τmax   1 × RhNLoS ,hNLoS (t) RhNLoS ,hNLoS t, Δf = l1 ,i l2 ,i l1 ,i l2 ,i K +1 τ0   ×ej 2π Δf τ f αR , βR dτ.

(5.20)

If we set t = 0, the frequency cross-correlation function between link l1 and link l2 at the initial time can be obtained. However, because the aforementioned analysis focuses on the different NLoS propagation paths, let us set m = M, p = P , q = Q, m = M  , p = P  , and q  = Q . Accordingly, the frequency time auto-correlation function for the NLoS component link l1 can be derived as  τmax   1 LoS N LoS × hN RhNLoS t, Δf = l1 ,i (t, τ ) hl1 ,i (t, τ ) l1 ,i K +1 τ0   ×ej 2π Δf τ f αR , βR dτ,

(5.21)

LoS (t) denotes the channel impulse response of link l for the NLoS where hN 1 l1 components, which has been given in (5.14) and can be rewritten as

 hl1 ,i (t, τ ) =

π/2  π −π

0

×e

j 2π λ

ej 2πfmax τ cos







T (t)+ξ R ξmpq (t) m p  q 



αR −φv cos βR

  f αR , βR dαT dβT .

(5.22)

Furthermore, the frequency auto-correlation function for the LoS component can be expressed as   RhLoS t, Δf = l1 ,i

K K +1 ×e

j 2π λ





τmax

e



j 2πfmax ·τ cos αR −φv cos βR

τ0

LoS ×ξmpq,m  p q  (t)+j 2π Δf τ

  f αR , βR dτ.

(5.23)

5.3 Proposed 3D Vehicle Massive MIMO Antenna Array Model

101

In a typical V2V environment, the received waves arrive at the MR from various directions (i.e., NLoS components) with different time delays through multiple paths. In addition to the fluctuations in the signal envelope and phase, the received signal frequency constantly varies as a result of the relative motion between the MT and MR. Therefore, in the proposed V2V channel model, the received signal at the MR incurs a spread in the frequency spectrum caused by the relative motion between the MT and MR. Let us define Stotal (γ ) as the total Doppler spectrum of the proposed 3D V2V time-varying channel model. Moreover, the PDFs of the Doppler frequency at the MT and MR are assumed to be two independent random variables. To obtain the PDF of the total Doppler frequency, the characteristic functions are considered. These can be defined as  ∞       c c RLoS t, ω = t, Δf ej ωΔf f αR , βR dΔf (5.24) RLoS −∞

c RN LoS

  t, ω =



∞ −∞

  j ωΔf   c RN f αR , βR dΔf. LoS t, Δf e

(5.25)

If we take (5.24) and (5.25) into the inverse Fourier transform formula, the PDF of the total Doppler frequency can be derived as   1 c t, Δf = Rtotal 2π



∞ −∞

  c   c t, ω RN RLoS LoS t, ω

  ×ej ωΔf f αR , βR dω.

(5.26)

c Subsequently, we define the Fourier transform of Rtotal (t, Δf ) with respect to the variable t to be the function Stotal (γ , Δf ), i.e.,

  Stotal γ , Δf =



∞

−∞

    c t, Δf e−j 2π γ t f αR , βR dt. Rtotal

(5.27)

c (t, 0) = If we set Δf = 0, we can then obtain Stotal (γ , 0) = Stotal (γ ) and Rtotal Therefore, the equation in (5.27) can be rewritten as

c Rtotal (t).

  Stotal γ =



∞

−∞

  −j 2π γ t   c t e Rtotal f αR , βR dt.

(5.28)

Thus far, the total Doppler spectrum Stotal (γ ) can be obtained. Additionally, c we can observe that if the proposed channel is time-invariant, Rtotal (t) = 1 and Stotal (γ ) becomes equal to δ(γ ). Therefore, when there are no time variations in the channel, no spectral broadening is observed in the transmission of a pure frequency tone.

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

5.4 Numerical Results and Discussions In this section, the numerical results of the proposed vehicle massive MIMO model for the V2V communication environments are analyzed. We present the proposed vehicle massive MIMO antenna elements with NT = NR = 500 (i.e., a 500×500 massive MIMO channel). The number of each antenna element is shown in Fig. 5.4.

Fig. 5.4 Layout of the proposed vehicle massive MIMO antenna elements for different vehicle surfaces. (a) For UVKT, 0 ≤ mpq ≤ 100. (b) For TKGS, 100 ≤ mpq ≤ 200. (c) For UVEF, 200 ≤ mpq ≤ 300. (d) For UTFS, 300 ≤ mpq ≤ 400. (e) For VKGE, 400 ≤ mpq ≤ 500

5.4 Numerical Results and Discussions

103

Assume dx = dy = d and L = W = H = 10. Comparisons of the results illustrate good agreement between the proposed 3D model and the conventional MIMO models, the massive MIMO models [34], and the measured data [24].

5.4.1 Performance Analysis The SFCFs between different antenna elements for the proposed 3D vehicle massive MIMO antenna array model can be numerically calculated and are presented in Fig. 5.5. It is obvious that the spatial corrections rapidly decrease with an increase in the normalized antenna spacing d · λ−1 . This conforms with the results in the massive MIMO array model in [35] and the conventional MIMO array model in [7]. Moreover, for the area of the U V KT (see Fig. 5.4a), the PDFs of the spatial correlation gradually decrease by changing the m p q  -th element from 12 to 45, 50, and 100 owing to the geometric distance between them, which continuously increases. For (mpq, m p q  ) = (1, 200), the geometric distance between any two elements of the proposed model tend to be a local maximum value. The spatial correlation between two elements is relatively lower than any other conditions. It can thus be inferred that the proposed array model can significantly decrease the spatial correlation of the compact antenna system. This point clearly demonstrates

Spatial Fading Correlation | ρ(mpq,MPQ) |

1.0 (mpq, m’p’q’) = (1, 12) (mpq, m’p’q’) = (1, 45) (mpq, m’p’q’) = (1, 50) (mpq, m’p’q’) = (1, 100) (mpq, m’p’q’) = (1, 200) Yong Model [7]

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

Normalized antenna spacing d ⋅ λ−1

Fig. 5.5 Spatial fading correlations with respect to different antenna elements for the proposed 3D vehicle massive MIMO antenna array model

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

the excellent performance of the proposed 3D vehicle massive MIMO antenna array model. Thus, it can be fully utilized to design future 5G communication systems.

5.4.2 AoD and AoA Statistical Analysis The influence of distance d and channel parameter b on the distribution of the marginal PDF of the AoD statistics in the azimuth plane is illustrated in Fig. 5.6. As mentioned in [19], the marginal PDFs of the AoD statistics at the MT are strongly related to the first geometric path length in multipath channels. In [40], we note that the MT and MR are located at two different semi-ellipsoid models. However, when the waves from the MT antenna elements impinge on the scatterers around the transmitter, the marginal AoD statistics in the azimuth plane are similar to that in the proposed model. It also can be noted that each curve of the AoD PDFs is symmetric owing to the symmetry of the geometric channel model in Fig. 5.3, as presented in [40]. Therefore, we can conclude that the proposed 3D semi-ellipsoid channel model is able to characterize real roadside environments. Moreover, the AoD significantly decreases with an increase in the azimuth angle αT for D ≥ (a 3 − ab2 )/(a 2 + b2 ). On the other hand, for D ≤ (a 3 − ab2 )/(a 2 + b2 ), the AoD gradually decreases to a local minimum value in 0 ≤ αT ≤ π/2; it then gradually increases in π/2 ≤ αT ≤ π . A similar behavior is evident in −π ≤ αT ≤ π . Furthermore, 1.0

b=80m, D=100m b=100m, D=100m b=80m, D=5m b=80m, D=195m Riaz Model [36] Simulation Results in [20]

Marginal PDF of AoD Statistics p(α T)

Riaz Model 0.8

0.6 Simulation Results

0.4

0.2

0 −200

−150

−100

−50 0 50 AoD, α (degree)

100

150

200

T

Fig. 5.6 Marginal PDF of the AoD statistics in the azimuth plane when a = 100 m and c = 60 m

5.4 Numerical Results and Discussions

105

the obtained theoretical results show good agreement between the proposed model with the simulation results in [20] for V2V communication scenarios. The distribution of the marginal PDF of the AoD in the elevation plane is shown in Fig. 5.7. From the figure, we note that each curve of the AoA PDFs in the vertical plane has a similar changing trend with an increase in βT from 0 to π/2, which is in agreement with the results in [40]. It is also shown that the AoD PDF in the vertical plane is not related to the distance between the MT and MR; the curves of D = 50 m and that of D = 100 m tend to be completely coincident. Moreover, when βT is substituted with zero, the values of the AoD in the elevation plane significantly increase. This is accompanied by an increase in parameter b, depending on the proposed V2V channel model, as shown in Fig. 5.3. Additionally, it is observed that the AoD significantly decreases when the height of the channel model (i.e., c) decreases. In comparing the results of the proposed 3D model of the macrocell environments with the results in [18] and [19], it is evident that the PDFs have similar changing trends at 0 ≤ βR ≤ π/2 in the vertical direction. This is because the first path lengths in [18] and [19] are similar to that in the proposed model. Meanwhile, beamforming technology has received considerable attention in massive MIMO systems. The transmitter emits the signal to the receiver in significantly small beamwidths, spanning the azimuth range of [−α1 , α2 ]. To this end, we can only analyze the LoS propagation components. However, in V2V mobile radio environments, the beamwidth at the MT is relatively large and the propagation paths include LoS and NLoS components. For a more realistic consideration of the

Marginal PDF of AoD Statistics p(βT)

1.0 D=100m, b=80m, c=60m D=50m, b=80m, c=60m D=100m, b=100m, c=60m D=100m, b=80m, c=100m D=100m, b=80m, c=20m Simulation Results in [36]

0.8

0.6

0.4

0.2

0

0

10

20

30

40 AoD, β

T

50 60 (degree)

70

80

Fig. 5.7 Marginal PDF of the AoD statistics in the elevation plane when a = 100 m

90

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5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

Fig. 5.8 Marginal PDF of the AoA statistics in the azimuth plane when a = 100 m, b = 80 m, and c = 60 m

proposed massive MIMO system, we assume that the massive antennas at the MT are directional. However, the MT and MR located at the major axis (i.e., x-axis) of the ellipse in the azimuth plane is a special situation. Thus, it is necessary to analyze the proposed channel statistics for more realistic MRs’ positions. For example, the MT is located at the x-axis, while the MR is not located at the x-axis, which makes the proposed geometric channel model asymmetric [16]. Figure 5.8 shows the marginal PDF of the AoA statistics in the azimuth plane. It is obvious that the AoA PDFs in 0 ≤ αR ≤ π firstly decrease to a local value of AoA and then increase to a local maximum with a “corner,” the AoA PDFs finally sharply decrease. A similar behavior is evident in −π ≤ αR ≤ 0. By increasing the beamwidths α1 and α2 with more scatterers in the scattering region illuminated by the directional antenna, the PDFs firstly have higher values on both sides of the curves, and then gradually tend to be equal. The effects of the beamwidth and distance d on the marginal PDF of the AoA statistics in the azimuth plane are shown in Fig. 5.9. From the figure, we note that the curves of the AoA PDFs have two “corners,” which are located in the right and left parts. In general, the marginal AoA statistics at the MR are related to the last receive path lengths in multipath channels. However, the proposed channel model is different from the channel model in [34]. Here, if we set c → ∞, the last geometric path length arrives at the MR is similar to that in the proposed model. Thus, the marginal PDF of the AoA statistics in the case of the waves scattered from the

5.4 Numerical Results and Discussions

107

1.0

α = 60°, D=120m α = 60°, D=80m

Marginal PDF of AoA P(βR)

0.8

α = 30°, D=80m Yuan Model [20] Zhou Model [19]

Yuan Model

0.6 Zhou Model 0.4

0.2

0 −200

−150

−100

−50 0 50 AoA, β [degree]

100

150

200

R

Fig. 5.9 Marginal PDF of the AoA statistics in the azimuth plane when a = 100 m, b = 80 m, and c = 60 m

roadside environments can be obtained from [20]. Additionally, the authors in [19] presented a semi-spheroid model to describe a wireless propagation channel. Note that the last geometric path length is also similar to that in the channel model in Fig. 5.3. Thus, when the position of the receiver is close to the transmitter, the PDFs with smaller AoAs increase sharply to a local maximum with larger AoAs and then slowly decrease. This observation is in agreement with the results in [19].

5.4.3 Time and Frequency Cross-Correlation Analysis The distribution of the time cross-correlation in terms of different normalized antenna spacing d · λ−1 for two different NLoS components is shown in Fig. 5.10. It is clearly observed that the spatial correlation gradually decreases when d · λ−1 increases, which confirms the analysis in Fig. 5.5. Comparing the numerical results of the antenna correlation curves of the proposed V2V model with measurements on the antenna spatial correlation of massive MIMO in [27], it is evident that they share a similar trend. Furthermore, as the MP Q-th element changes from 12 to 200, the value of the time cross-correlation gradually decreases. In addition, it is observed that the above-mentioned theoretical analysis is in agreement with the results in [34], demonstrating that they are accurate and applicable to describing the time

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(mpq=1, m’p’q’=1) −−− (MPQ=12, M’P’Q’=1) (mpq=1, m’p’q’=1) −−− (MPQ=45, M’P’Q’=1) (mpq=1, m’p’q’=1) −−− (MPQ=100, M’P’Q’=1) (mpq=1, m’p’q’=1) −−− (MPQ=200, M’P’Q’=1) Measurements in [24] Wu Model [34]

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cross-correlation between any two NLoS components of the proposed V2V radio channel. In our 3D non-stationary V2V channel model, it is important to analyze the impact of non-stationarity, including of the relative moving times t and moving directions φv , on the proposed channel characteristics. Accordingly, we herein analyze the time cross-correlation and Doppler frequency with respect to the different relative moving time and different moving directions in Figs. 5.11, 5.12, and 5.13. Meanwhile, Fig. 5.11 plots the distribution of the time cross-correlation with respect to different relative moving directions φv for two different NLoS components. We herein present four cases of the moving directions: φv = 0, φv = π/3, φv = 2π/3, and φv = π . It is observed that, for the fixed moving direction and moving instant, when the receiver moves away from the MT (i.e., φv = 0), the value of the time cross-correlation is relatively lower than that at φv = π/3, φv = 2π/3, and φv = π . This is because higher geometric path lengths result in lower correlations, and the path length at φv = 0 is obviously longer than in the other cases [20]. Meanwhile, when the MR moves towards the MT (i.e., φv = π ), the time cross-correlation is significantly large. Furthermore, the value of the time cross-correlation of two different NLoS components at φv = 2π/3 is relative lower than that at φv = π/3, which only depends on the proposed geometric V2V channel model, as shown in Fig. 5.3.

5.4 Numerical Results and Discussions

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Fig. 5.11 Distribution of the time cross-correlation in terms of different relative moving directions for NLoS-to-NLoS components when a = 100 m, b = 80 m, c = 60 m, D = 100 m, mpq = 1, m p  q  = 1, MP Q = 2, M  P  Q = 2, vR = 5 m/s, and t = 2 s

The distribution of the total Doppler frequency versus the different relative moving direction is depicted in Fig. 5.12. For the direction of motion φv = π and φv = 0, the Doppler spectrum is, respectively, positively and negatively shifted, which is in agreement with the results in [13] and [19]. It is stated in [16] that the Doppler frequency in the geometric channel model is related not only to the geometric shape, but also to the last path length at the receiver. Thus, in the proposed model, when the receiver moves perpendicular to the direct LoS component (i.e., φv = π/2), each curve of the Doppler frequency of the conventional V2V MIMO channel models in [46] is symmetrical with respect to the central frequencies (i.e., |γ | = 0). However, this is not necessary for the proposed vehicle massive MIMO channel model [35]. Additionally, for the moving direction π/3, the curve of the Doppler frequency has one “corner” in the left part, which occurs at γ = fm cos φv . This clearly indicates that the Doppler frequency is strongly related to the distribution of power over the AoD and AoA statistics. The influence of distance D and geometric parameter b at different time instants t on the total Doppler distribution is illustrated in Fig. 5.13. The figure shows that the PDFs of the Doppler frequency gradually increase with a decrease in the distance between the MT and MR. Moreover, as channel parameter b increases, the values of the Doppler frequency slowly increase. Furthermore, the PDFs of the Doppler frequency at different time instants vary because of the geometric properties

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Normalized Power Spectral Density log10 [S(λ )]

16

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Wu model 12

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Normalized Power Spectral Density log [S(λ)] 10

21 18

Wu Model

15 12

t = 2s, D = 100m, b = 100m t = 2s, D = 100m, b = 50m t = 4s, D = 100m, b = 50m t = 4s, D = 50m, b = 50m a = b, c = 0, t = 0 Clarke Model [13] Wu Model [35]

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Fig. 5.13 Distribution of the Doppler frequency in terms of different relative moving time instants and channel parameters when a = 100 m, c = 60 m, fm = 100 Hz, φv = π/2, and vR = 2 m/s

References

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changing as t continuously increases. The curves of the Doppler distribution constantly shift to the left region by increasing t at φv = π/2. We can conclude that, for the moving direction 3π/2, the Doppler constantly shifts to the right part as the time instant continuously increases. Additionally, the proposed model gradually tends toward the Clarke’s model when a = b, c = 0, and t = 0, as shown in [13]. Comparisons between the theoretical discussions above with the Wu model [35] show that the distribution trends are in agreement with each other, which validates the generalization of the proposed V2V channel model.

5.5 Conclusion In this chapter, we have presented a 3D vehicle massive MIMO antenna array model for V2V communication environments. In this model, spherical wavefronts are assumed to characterize near-field effects, resulting in AoD and AoA statistics, time and frequency cross-correlation functions, and Doppler shift variations on the antenna array. It has also demonstrated that the non-stationarity, including that of the relative moving time t and moving directions φv , impact the proposed channel characteristics. It is evident that, when the receiver gradually moves from φv = 0 to φv = π , resulting in an increase of distance between the MT and MR, the time cross-correlation slowly decreases. Additionally, the Doppler frequency positively and negatively shift when the relative moving direction is φv = π and φv = 0, respectively. Furthermore, the curves of the Doppler distribution constantly shifts to the left region by increasing time t, which significantly differs from the conventional MIMO channels. The proposed 3D model results align with those of the previous scattering channel models and measurement results very well, which validate the generalization of the proposed 3D V2V channel model. For future work, we intend to obtain the proposed channel parameters by performing experiments. Moreover, because conventional MIMO models do not depend on the configurations of antenna arrays, polarized antenna arrays and antenna lobe pattern analysis can be employed in future extensions of the channel model.

References 1. G.J. Foschini, M.J. Gans, On limits of wireless communications in a fading environment when using multiple antennas. Wirel. Pers. Commun. 6(3), 311–335 (1998) 2. P.P. Tayade, V.M. Rohokale, Enhancement of spectral efficiency, coverage and channel capacity for wireless communication towards 5G, in International Conference on Pervasive Computing (ICPC), Pune (2015), pp. 1–5 3. E.G. Larsson, O. Edfors, F. Tufvesson, T.L. Marzetta, Massive MIMO for next generation wireless systems. IEEE Commun. Mag. 52(2), 186–195 (2014) 4. P. Patcharamaneepakorn, et al., Spectral, energy, and economic efficiency of 5G multicell massive MIMO systems with generalized spatial modulation. IEEE Trans. Veh. Technol. 65(12), 9715–9731 (2016)

112

5 A 3D Massive MIMO Channel Model for Vehicle-to-Vehicle Communication. . .

5. 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) 6. L. Wood, W.S. Hodgkiss, Understanding the Weichselberger model: a detailed investigation, in IEEE Military Communications Conference, San Diego, CA (2008), pp. 1–7 7. S.K. Yong, J.S. Thompson, Three-dimensional spatial fading correlation models for compact MIMO receivers. IEEE Trans. Commun. 4(6), 2856–2869 (2005) 8. F. Harabi, A. Gharsallah, S. Marcos, Three-dimensional antennas array for the estimation of direction of arrival. IET Microw. Antennas Propag. 3(5), 843–849 (2009) 9. J. Zhou, H. Jiang, H. Kikuchi, Performance of uniform concentric circular arrays in a threedimensional spatial fading channel model. Wirel. Pers. Commun. 83(4), 2949–2963 (2015) 10. K. Mammasis, R.W. Stewart, J.S. Thompson, Spatial fading correlation model using mixtures of Von Mises Fisher distributions. IEEE Trans. Wirel. Commun. 8(4), 2046–2055 (2009) 11. S.Y. Cho, J. Kim, W.Y. Yang, MIMO-OFDM Wireless Communications with MATLAB (WileyIEEE Press, Singapore, 2010) 12. R.B. Ertel, J.H. Reed, Angle and time of arrival statistics for circular and elliptical scattering models. IEEE J. Sel. Areas Commun. 17(11), 1829–1840 (1999) 13. P. Petrus, J.H. Reed, T.S. Rappaport, Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans. Commun. 50(3), 495–502 (2002) 14. A. Abdi, M. Kaveh, A space-time correlation model for multielement antenna systems in mobile fading channels. IEEE J. Sel. Areas Commun. 20(3), 550–560 (2002) 15. X. Cheng, C.X. 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) 16. H. Jiang, Z.C. Zhang, J. Dang, L. Wu, Analysis of geometric multi-bounced virtual scattering channel model for dense urban street environments. IEEE Trans. Veh. Technol. 66(3), 1903– 1912 (2017) 17. S.J. Nawaz, B.H. Qureshi, N.M. Khan, A generalized 3-D scattering model for a macrocell environment with a directional antenna at the BS. IEEE Trans. Veh. Technol. 59(7), 3193– 3204 (2010) 18. H. Jiang, J. Zhou, H. Kikuchi, Angle and time of arrival statistics for a 3-D pie-cellular-cut scattering channel model. Wirel. Pers. Commun. 78(2), 851–865 (2014) 19. J. Zhou, H. Jiang, H. Kikuchi, Generalised three-dimensional scattering channel model and its effects on compact multiple-input and multiple-output antenna receiving systems. IET Commun. 9(18), 2177–2187 (2015) 20. Y. Yuan, C.X. Wang, Y. He, M.M. Alwakeel, e.H.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) 21. S.C. Kwon, G.L. Stuber, A.V. Lopez, J. Papapolymerou, Geometrically based statistical model for polarized body-area-network channels. IEEE Trans. Veh. Technol. 62(8), 3518–3530 (2013) 22. S.C. Kwon, G.L. Stuber, Cross-polarization discrimination in vehicle-to-vehicle channels: geometry-based statistical modeling, in IEEE Global Telecommunications Conference GLOBECOM, Miami, FL, December (2010), pp. 1–5 23. M. Boban, T.T.V. Vinhoza, M. Ferreira, J. Barros, O.K. Tonguz, Impact of vehicles as obstacles in vehicular ad hoc networks. IEEE J. Sel. Areas Commun. 29(1), 15–28 (2011) 24. A. Paier, et al., Non-WSSUS vehicular channel characterization in highway and urban scenarios at 5.2 GHz using the local scattering function, in International ITG Workshop on Smart Antennas, Vienna, Austria, February (2008), pp. 9–15 25. T. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans. Wirel. Commun. 9(1), 3590–3600 (2010) 26. X. Gao, O. Edfors, F. Rusek, F. Tufvesson, Massive MIMO performance evaluation based on measured propagation data. IEEE Trans. Wirel. Commun. 14(7), 3899–3911 (2009)

References

113

27. S. Payami, F. Tufvesson, Channel measurements and analysis for very large array systems at 2.6 GHz, in 6th European Conference on Antennas and Propagation (EUCAP), Prague, Czech Republic, March (2012), pp. 433–437 28. T. Zwick, C. Fischer, W. Wiesbeck, A stochastic multipath channel model including path directions for indoor environments. IEEE J. Sel. Areas Commun. 20(6), 1178–1192 (2002) 29. H. Xiao, A.G. Burr, L. Song, A time-variant wideband spatial channel model based on the 3GPP model, in IEEE Vehicular Technology Conference (VTC), Montreal, QC, September (2006), pp. 1–5 30. D.S. Baum, J. Hansen, J. Salo, An interim channel model for beyond-3G systems: extending the 3GPP spatial channel model (SCM), in 61st Vehicular Technology Conference, Stockholm, Sweden, May (2005), pp. 3132–3136 31. F. Bohagen, P. Orten, G.E. Oien, Design of optimal high-rank line-of-sight MIMO channels. IEEE Trans. Wirel. Commun. 6(4), 1420–1425 (2007) 32. F. Bohagen, P. Orten, G.E. Oien, On spherical vs. plane wave modeling of line-of-sight MIMO channels. IEEE Trans. Commun. 57(3), 841–849 (2009) 33. S.B. Wu, C.X. Wang, H. Aggoune, Non-stationary wideband channel models for massive MIMO systems, in Proceedings of WSCN, Jeddah, Saudi Arabia, December (2013), pp. 1–8 34. S. Wu, C. Wang, H. Haas, e.M. Aggoune, M.M. Alwakeel, B. Ai, A non-stationary wideband channel model for massive MIMO communication systems. IEEE Trans. Wirel. Commun. 14(3), 1434–1446 (2015) 35. S.B. Wu, C.X. Wang, E.-H.M. Aggoune, A non-stationary 3-d wideband twin-cluster model for 5G massive MIMO channels. IEEE J. Sel. Areas Commun. 32(6), 1207–1218 (2014) 36. X. Cheng, C.X. Wang, B. Ai, H. Aggoune, Envelope level crossing rate and average fade duration of nonisotropic vehicle-to-vehicle Ricean fading channels. IEEE Trans. Intell. Trans. Sys. 15(1), 62–72 (2014) 37. X. Cheng, C.X. Wang, B. Ai, Envelope level crossing rate and average fade duration of nonisotropic vehicle-to-vehicle Ricean fading channels. IEEE Trans. Intell. Transp. Syst. 15(1), 62–72 (2013) 38. Q. Zhan, V.K.V. Gottumukkala, A. Yokoyama, H. Minn, A V2V communication system with enhanced multiplicity gain, in Proceedings of the IEEE GLOBECOM Workshop Vehicular Network Evolution, Atlanta, GA, December (2013), pp. 1326–1332 39. S.K. Yong, J.S. Thompson, The effect of various channel conditions on the performance of different antenna array architectures, in IEEE 58th Vehicular Technology Conference, VTC 2003-Fall, Orlando, FL, October (2003), pp. 198–202 40. M. Riaz, N.M. Khan, S.J. Nawaz, A generalized 3-D scattering channel model for spatiotemporal statistics in mobile-to-mobile communication environment. IEEE Trans. Veh. Technol. 64(10), 4399–4410 (2015) 41. S.R. Saunders, A. Aragon-Zavala, Antennas and propagation for wireless communication systems, 2nd edn. (Wiley, West Sussex, 2007) 42. 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) 43. A. Ghazal, C.X. Wang, B. Ai, D. Yuan, H. Haas, A non-stationary wideband MIMO channel model for high-mobility intelligent transportation systems. IEEE Trans. Intell. Transp. Syst. 16(2), 885–897 (2015) 44. L. Bai, C.X. Wang, S. Wu, H. Wang, Y. Yang, A 3-D wideband multi-confocal ellipsoid model for wireless MIMO communication channels, in IEEE International Conference on Communications (ICC), Kuala Lumpur (2016), pp. 1–6 45. J. Chen, S. Wu, S. Liu, C. Wang, W. Wang, On the 3-D MIMO channel model based on regular-shaped geometry-based stochastic model, in International Symposium on Antennas and Propagation (ISAP), Hobart, TAS, November (2015), pp. 1–4 46. N. Avazov, M. Patzold, A geometric street scattering channel model for car-to-car communication systems, in International Conference on Advanced Technologies for Communications (ATC 2011), Da Nang, Vietnam, August (2011), pp. 224–230

Chapter 6

A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel Communications

6.1 Introduction Recently, V2V communications have received widespread applications on account of the rapid development of 5G wireless communication networks [1–3]. Unlike conventional F2M cellular systems, V2V systems are employed with low-elevation multiple antennas, and the MT and MR are both in motion. For facilitating the design and analysis of V2V communication systems, the radio propagation characteristics must be designed between the MT and MR [4, 5]. Reliable knowledge of the realistic propagation channel models, which provide effective and simple means to approximately express the statistical properties of V2V channels [6]. To evaluate the performance of V2V communication systems, accurate channel models are indispensable. Regarding the approach of V2V channel modeling, the models can be categorized as deterministic models (mainly indicates the raytracing method) and stochastic models. In particular, stochastic models can be roughly divided into several categories, such as nonregular-shaped geometry-based stochastic models (NGBSMs) and RS-GBSMs [7–15]. The former are also known as parametric models, which are constructed based on the channel measurements, while the latter are based on the regular geometric shape of interfering objects. In [7], the authors demonstrated that the LoS is likely to be obstructed by buildings and obstacles between the MT and MR. Thus, it is necessary to adopt Ricean channels to describe V2V communication environments. The authors in [8] proposed a generalized visual scattering channel model for multi-bounced propagation paths in dense urban communication environments; however, the impacts of moving vehicles around the MT and MR on the V2V channel characteristics were not discussed. Cheng et al. [9] introduced an RS-GBSM for V2V scenarios. The authors presented a two-ring model to depict moving vehicles and an ellipse model to mimic roadside environments. Yuan et al. [10] adopted a two-sphere model to describe moving vehicles as well as multiple confocal elliptic-cylinder models to depict roadside scenarios. The authors in [11] proposed a semi-ellipsoidal © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_6

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non-stationary channel model for V2V scenarios. In 2014, Zajic et al. [12] proposed a two-cylinder model to depict moving and stationary scatterers in the vicinity of the transmitter and receiver. In [12], the authors stated that the mobility of interfering objects significantly affect the Doppler spectrum. Furthermore, 3D RS-GBSMs for macrocell and microcell communication environments were presented in [13] and [14], respectively. However, most of the above RS-GBSMs focus on narrowband channel models, wherein all rays experience a similar propagation delay [15]. This is not a realistic description of wireless V2V communication environments. According to the channel measurements between the narrowband and wideband V2V channels in [16], Sen et al. concluded that the channel characteristics for different propagation delays in wideband channels should be addressed. In 2009, the authors in [17] first proposed a wideband RS-GBSM for MIMO V2V Ricean fading channels. However, in [17], the model was shown to be unable to reflect the channel statistics for different propagation delays, which are significant for wideband channels. Based on two measured scenarios in [18], Cheng et al. [19] introduced the concept of vehicle traffic density (VTD) to represent moving vehicles. Those authors presented the channel statistics for different propagation delays (i.e., per-tap channel statistics); however, the angular spreading of incident waves in an elevation plane was ignored. The authors in [20, 21] proposed a wideband MIMO V2V channel model based on a geometrical semi-circular tunnel scattering model, which assumed that an infinite number of scatterers are randomly distributed on the tunnel wall. In [22], the authors first provide an analytical channel model for the tunnel with a deterministic vehicular traffic flow, which accurately predict the signal propagation and field distribution in the tunnel. The authors in [23] investigated the spatio-temporal radio propagation channel inside an arched highway tunnel employing a wideband directional channel sounding data for future cellular systems in terms of coverage, delay spread, and dominant scatterers. Most previous channel models rely on the wide-sense stationary uncorrelated scattering (WSSUS) assumption, which is valid only for very short time intervals (in the order of milliseconds [18]). Measurements in [24] have shown that in V2V scenarios, the stationary interval, during which the WSS assumption is valid, is much shorter than the observation time interval. This means that the WSS condition is no longer fulfilled in V2V scenarios. To fill the above gaps, models that consider the non-stationarities of the channel are needed. In [25] and [26], the authors proposed 2D geometry-based non-WSS narrowband channel models for T-junction and straight road environments, respectively. The authors in [27] and [28] presented 2D non-stationary theoretical wideband MIMO Ricean channel models for V2V scenarios. Furthermore, Yuan et al. [29] presented a 3D wideband MIMO V2V channel. Nonetheless, the authors only focused on two relative special movement directions: the same direction and opposite direction. In [30], the authors investigated the algebraic structure of the poles in the Doppler spectrum for arbitrary delays and velocity configurations, in which speed vectors are introduced to describe the arbitrary movement directions in V2V communication environments. The authors in [31] proposed a novel algorithm based on the Hough transform, which can be used to identify multiple clusters observed in the measured

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Fig. 6.1 Proposed 3D wideband MIMO V2V channel model combining the two-cylinder model and multiple confocal semi-ellipsoid models for the LoS propagation rays

Fig. 6.2 Geometric angles and path lengths of the proposed V2V channel model for the rays with single and double interactions

channels. The authors of [32] presented a wideband MIMO model for V2V channels based on extensive measurements taken in highway and rural environments. In this section, we present a 3D non-stationary wideband channel model for MIMO V2V tunnel communication environments, as illustrated in Figs. 6.1 and 6.2. The model is operated at 5.9 GHz, with a bandwidth of 50 MHz, which is dedicated

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for vehicular communications. The major contributions of this paper are outlined as follows: (1) Based on the two measured scenarios mentioned above in [32], we propose a 3D non-stationary wideband geometric channel model for two different V2V communication environments, i.e., highway scenarios and urban scenarios. (2) We, to the best of our knowledge, for the first time introduce a 3D wideband semi-ellipsoid model to describe the curved walls of tunnels in V2V communication environments, which is able to model interfering objects with identical delays on the same semi-ellipsoid, while different semi-ellipsoids represent the different propagation delays. In light of this, important time-varying channel characteristics for different taps can be derived and thoroughly investigated. (3) In the proposed model, the time-varying propagation path lengths are first taken into account to capture the V2V channel non-stationarity. To be specific, the impacts of movement velocities and directions on the time-varying V2V channel characteristics are investigated in comparison with those of the corresponding WSS model and measured results. The results show that the model is an excellent approximation of realistic V2V scenarios. (4) In the model, the propagation conditions of the MT and MR located in different azimuth planes (i.e., slope V2V scenarios) are firstly studied. In this case, the NLoS rays with ground reflection on the proposed V2V channel characteristics are investigated.

6.2 System Channel Model for Flat Communications In general, the geometric model specifies the mathematical model and the algorithms used for channel modeling that applied to all scenarios. It is based on the WINNER II channel model. This geometric model employs a GBSM approach to represent the multipath propagation channel between a transmitter and receivers. In multipath channels, the path length of each wave determines the propagation delay and essentially also the average power of the wave at the MR. In [6], the authors state that the ellipse model forms to a certain extent the physical basis for the modeling of frequency-selective channels. Therefore, when the MT and MR are located in the focus of the ellipse, every wave in the scattering region characterized by the l-th ellipses undergoes the same discrete propagation delay τ = τ0 + τ , = 0, 1, 2, . . . , L − 1, where τ0 denotes the propagation delay of the LoS component, τ is an infinitesimal propagation delay, and L is the number of paths with different propagation delays. In particular, the number of paths with different propagation delays exactly corresponds to the number of delay elements required for the tappeddelay-line (TDL) structure of modeling frequency-selective channels. It is worth mentioning that in V2V communication scenarios with different contributions of rays with single and double interactions to the V2V channel statistics; thus, it is necessary to design different taps of the proposed wideband V2V channel model.

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Fig. 6.3 The ellipse model describing the path geometry (a) first tap; (b) other taps

As mentioned in [33], the tap is strongly related to the delay resolution in V2V channels. Here, let us define al as the semimajor of the l-th ellipse in the azimuth plane. Then, for the next propagation delay, the semimajor of the (l + 1)-th ellipse in the azimuth plane can be derived as al+1 = al + cτ/2 with c = 3 × 108 m/s. Modeling V2V channels by using a TDL structure with time-varying coefficients gives a deep insight into the channel statistics in the proposed model. In Fig. 6.3a, we notice that the received signal for the first tap is composed of an infinite number of delayed and weighted replicas of the transmitted signal in a multipath channel, including direct LoS propagations (i.e., MT → MR), rays with single interactions caused by the interfering objects located on either of two cylinders presented by the moving vehicles (i.e., MT → A → MR and MT → B → MR) or on the first semiellipsoid (i.e., MT → C → MR), and rays with double interactions generated from the interfering objects located on both cylinders (i.e., MT → U → V → MR). Here, let us define the combination of the above cases as the first tap. In light of this, we can analyze the proposed channel characteristics for different propagation

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delays, i.e., per-tap channel statistics. However, for other taps (l ≥ 1), the link is a superposition of the rays with single interactions that are produced only from the interfering objects located on the semi-ellipsoid (i.e., MT → G → MR), as well as the rays with double interactions caused by the interfering objects from the combined single cylinder and the corresponding semi-ellipsoid (i.e., MT → E → F → MR, MT → M → N → MR), as shown in Fig. 6.3b.

6.2.1 Descriptions of the Proposed V2V Channel Model Figures 6.1 and 6.2 illustrate the geometry of the proposed V2V channel model, which is a combination of direct LoS propagations, rays with single and double interactions. It is assumed that the propagation components are independent of each other. Furthermore, we assume that the MT and MR are located in the same azimuth plane. Thus, the model is mainly applicable for flat road communications. A similar assumption can be seen in [12] and [29]. In the proposed model, we use a two-cylinder model to depict moving vehicles (i.e., around the MT or MR), as well as multiple confocal semi-ellipsoid models to mimic curved walls of tunnels. It is worth mentioning that the ellipse model is usually the physical basis for modeling wideband channels. Conventional channel models assume that waves experience similar propagation delays, which can be described by an ellipse model with MT and MR located at the foci. Measurements in [34] demonstrate that there is a wealth of angular information in the vertical plane. This indicates that the 3D channel model is more accurate than the 2D channel model when calculating the channel capacity, i.e., the assumption of 2D propagation may lead to an inaccurate estimation of the system performance. In light of this, many kinds of 3D geometric channel models (including azimuth and elevation angles) are introduced to describe V2V environments. To be specific, the authors in [10] and [29] used the cylinder model to describe roadside environments. However, the interfering objects located on the cylinder did not have identical delays and the complexity of the channel model increased, which negated the advantage of elliptic models for wideband channels. For V2V tunnel scenarios, the interfering objects are randomly distributed on the curved walls of tunnels [35]. Therefore, it is reasonable to adopt semi-ellipsoid models to describe the wireless communications in roadside environments [36, 37]. To the best of our knowledge, this paper is the first to propose a 3D wideband semi-ellipsoid model to describe V2V environments, which is able to efficiently describe the distributions of interfering objects on the interval surface of tunnel walls. The model introduces interfering objects with identical delays on the same semi-ellipsoid, whereas different semi-ellipsoids represent the different propagation delays. In light of this, we are able to investigate the V2V channel characteristics for different propagation delays, i.e., per-tap channel statistics. On the other hand, it is worth mentioning that the MT and MR have similar heights as the surrounding interfering objects (such as vehicles and pedestrians). Note that the vehicles and pedestrians are hardly above the transmitter and receiver.

6.2 System Channel Model for Flat Communications

121

Measurements in [17] showed that the differences in the power levels of the Doppler spectrum when the interfering objects are modeled with the 2D (circle) and the 3D (cylinder) models are insignificant. Accordingly, we introduce the two-cylinder model to describe the interfering objects around the MT and MR. As shown in Figs. 6.1 and 6.2, suppose that the MT and MR are equipped with ULA MT and MR omnidirectional antenna elements. The proposed model is also capable of introducing other MIMO geometric antenna systems, such as UCA, URA, and L-shaped array [38]. The distance between the centers of the MT and MR cylinders is denoted as D = 2f0 , where f0 designates the half-length of the distance between the two focal points of the ellipse in the azimuth plane. Let us define al and bl as the semi-ellipsoid’s semi-lengths on the major axis (i.e., x-axis) and minor axis (i.e., y-axis), respectively. Here, we assume that the  l-th semi-ellipsoid’s semi-

length on the z-axis is also denoted as bl , i.e., bl = al2 − f02 . It is assumed that the radius of the cylindrical surface around the MT is denoted as Rt1 ≤ Rt ≤ Rt2 . Note that Rt1 and Rt2 correspond with the respective urban and highway scenarios in [32]. Similarly, at the MR, the radius of the cylindrical surface is denoted as Rr1 ≤ Rr ≤ Rr2 . Let AntTp represent the p-th (p = 1, 2, . . . , MT ) antenna of the transmit array, and let AntR q represent the q-th (q = 1, 2, . . . , MR ) antenna of the receive array. The spaces between the two adjacent antenna elements at the MT and MR are denoted as δT and δR , respectively. The orientations of the transmit antenna array in the azimuth plane (relative to the x-axis) and elevation plane (relative to the x − y plane) are denoted as ψT and θT , respectively. Similarly, the orientations at the receiver are denoted as ψR and θR , respectively. Here, we assume that there are N1,1 interfering objects (vehicles) existing on the cylindrical surface around the MT, and the n1,1 -th (n1,1 = 1, . . . , N1,1 ) object is defined as s (n1,1 ) . N1,2 interfering objects likewise exist around the MR lying on the cylinder model, and the n1,2 -th (n1,2 = 1, . . . , N1,2 ) object is defined as s (n1,2 ) . For the multiple confocal semiellipsoid models, Nl,3 interfering objects lie on a multiple confocal semi-ellipsoid with the MT and MR located at the foci. The nl,3 -th (nl,3 = 1, . . . , Nl,3 ) object is designated as s (nl,3 ) .

6.2.2 Complex Impulse Response In general, MIMO V2V channel model can be described by matrix  the proposed  H(t) = hpq (t, τ ) M ×M of size MR × MT , where hpq (t, τ ) denotes the complex R T impulse response (CIR) between the p-th transmit antenna and q-th receive antenna L(t) in our model, i.e., hpq (t, τ ) = l=1 ωl hl,pq (t)δ(τ − τl ), with l represents the tap number, hl,pq (t) denotes the complex tap coefficient of the AntTp → AntR q link, L(t) is the total number of taps, ωl is the gain of the l-th tap, and τl is the propagation delays [27]. For a unit transmit power, suppose the power transferred through the 2 AntTp → AntR q link is Ωl,pq , i.e., Ωl,pq = E[|hl,pq (t)| ] ≤ 1. Then, the complex tap coefficient for the first tap at the carrier frequency fc can be expressed as [19]

122

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

h1,pq (t) = hLoS 1,pq (t) +

3 

SB

1,i h1,pq (t) + hDB 1,pq (t)

(6.1)

i=1

with  hLoS 1,pq (t) =

  KΩ1,pq j ϕ0 −2πfc ξpq /c e K +1    2π

×e

j

λ

 SB1,i (t) h1,pq

Ω1,pq ηSB1,i K +1

=

2π λ vT t

× ej  hDB 1,pq (t) =

N1,i 

lim

N1,i →∞



(n1,i )

cos αT



×e

ϕ0 −2πfc ξpq,n1,1 ,n1,2 /c



(n1,1 )

1 N1,i

(n1,i )

−γT cos βT



j 2π λ vT t cos αT



n1,i =1

Ω1,pq ηDB × lim N1,1 ,N1,2 →∞ K +1

× ej







LoS −γ LoS vT t cos αTLoS −γT cos βTLoS +vR t cos αR R cos βR





e

j ϕ0 −2πfc ξpq,n1,i /c



(n

)

(6.2)





(n

1,i +j 2π −γR cos βR 1,i λ vR t cos αR

N1,1 ,N1,2



n1,1 ,n1,2 =1

)

(6.3)

 1 N1,1 N1,2

 (n1,1 )

−γT cos βT



(n

)



(n

1,2 +j 2π −γR cos βR 1,2 λ vR t cos αR

)

, (6.4)

where ξpq,n1,i = ξpn1,i + ξqn1,i and ξpq,n1,1 ,n1,2 = ξpn1,1 + ξn1,1 n1,2 + ξqn1,2 denote the travel distance of the waves through the link AntTp → s (n1,i ) → AntR q and , respectively. Here, K denotes the Rice factor, AntTp → s (n1,1 ) → s (n1,2 ) → AntR q λ is the wavelength. The parameters vT and vR are the movement velocities of the MT and MR, respectively. The γT and γR are the movement directions of the MT and MR, respectively. αRLoS and βRLoS denote the AAoA and EAoA of the LoS (n

)

path, respectively. For the NLoS rays, the symbol αR 1,1 represents the AAoA of (n

)

the wave scattered from the interfering object s (n1,1 ) around the MT, whereas αR 1,2 represents the AAoA of the wave scattered from the interfering object s (n1,2 ) around (n ) (n ) the MR. Similarly, βR 1,1 and βR 1,2 denote the EAoAs of the waves scattered from (n

)

the interfering object s (n1,1 ) and s (n1,2 ) , respectively. On the other hand, αR 1,3 and (n

)

βR 1,3 denote the AAoA and EAoA of the waves scattered from the interfering object s (n1,3 ) in the semi-ellipsoid model for the first tap. It is assumed that the phase ϕ0 is an independent random variable, which has a uniform distribution in the interval from −π to π , i.e., ϕ0 ∼ [−π, π ). Furthermore, energy-related parameters

6.2 System Channel Model for Flat Communications

123

ηSB1,i and ηDB specify the numbers of the rays with single and double interactions, respectively,  contribute to the total scattered power, which can be normalized to satisfy 3i=1 ηSB1,i + ηDB = 1 for brevity [10]. However, for other taps (l ≥ 1), the complex tap coefficient of the AntTp → AntR q link can be derived as SB

DB

DB

hl,pq (t) = hl,pql,3 (t) + hl,pql,1 (t) + hl,pql,2 (t)

(6.5)

with SB

hl,pql,3 (t) =

 Ωl,pq ηSBl,3

×e

DB hl,pql,1 (t)

=



(nl,3 )

j 2π λ vT t cos αT

Ωl,pq ηDBl,1





nl,3 =1



1 Nl,3

(nl,3 )

−γT cos βT



lim

N1,1 ,Nl,3 →∞



ϕ0 −2πfc ξpq,n1,1 ,nl,3 /c

2π λ vT t



(n1,1 )

cos αT

ej

× ej × ej

2π λ vT t



(n1,1 )

lim



(n

)

(6.6)

1 N1,1 Nl,3



(n

)



(n

l,3 +j 2π −γR cos βR l,3 λ vR t cos αR



Nl,3 ,N1,2 →∞

(nl,3 )

cos αT



)





ϕ0 −2πfc ξpq,nl,3 ,n1,2 /c

(n





n1,1 ,nl,3 =1

−γT cos βT



ϕ0 −2πfc ξpq,nl,3 /c



Nl,3 ,N1,2

Ωl,pq ηDBl,2 ×



l,3 +j 2π −γR cos βR l,3 λ vR t cos αR

N1,1 ,Nl,3

× ej

=

Nl,3 →∞



× ej

DB hl,pql,2 (t)

Nl,3 

lim

)

(6.7)



nl,3 ,n1,2 =1

1 Nl,3 N1,2

 (nl,3 )

−γT cos βT



(n

)



(n

1,2 +j 2π −γR cos βR 1,2 λ vR t cos αR

)

, (6.8)

where ξpq,nl,3 = ξpnl,3 + ξqnl,3 , ξpq,n1,1 ,nl,3 = ξpn1,1 + ξn1,1 nl,3 + ξqnl,3 , and ξpq,nl,3 ,n1,2 = ξpnl,3 + ξnl,3 n1,2 + ξqn1,2 denote the travel distance of the waves T (n1,1 ) → s (nl,3 ) → AntR , and through the link AntTp → s (nl,3 ) → AntR q , Antp → s q (n )

(n )

l,3 and βR l,3 denote the AAoA AntTp → s (nl,3 ) → s (n1,2 ) → AntR q , respectively. αR and EAoA of the waves scattered from the interfering object s (nl,3 ) in the l-th semiellipsoid model. Similar to the above case, energy-related parameters ηSBl,3 and ηDBl,1 (ηDBl,2 ) specify the amount that the rays with single and double interactions, respectively, contribute to the total scattered power, which can be normalized to satisfy ηSBl,3 + ηDBl,1 + ηDBl,2 = 1 for brevity. In addition, because the derivations

124

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

of the condition that guarantees the fulfillment of the TDL structure are the same, we only detail the derivation of the condition for the second tap. As introduced in [29], we note that the impulse response of the proposed model is related to the scattered power in V2V channels. Therefore, it is important to define the received scattered power in different taps and different V2V scenarios (i.e., highway and urban scenarios) in the proposed model. In short, for the first tap, the rays with single interactions are caused by the interfering objects located on either of the two cylinders or the first semi-ellipsoid, while the rays with double interactions are generated from the interfering objects located on the both cylinders, as shown in Fig. 6.2. For highway scenarios (i.e., Rt = Rt2 and Rr = Rr2 ), there exist few moving vehicles around the MT and MR. Therefore, the received scattered power is mainly from waves reflected by the curved walls of tunnels described by the interfering objects located on the first semi-ellipsoid. The moving vehicles represented by the interfering objects located on the two cylinders are more likely to be single-bounced, rather than double-bounced. This indicates that the power of the rays with single interactions scattered from the semi-ellipsoid model is larger than the others; while the power of the rays with double interactions scattered from the two-cylinder model is the lowest, i.e., ηSB1,3 > max{ηSB1,1 , ηSB1,2 } > ηDB . For urban scenarios (i.e., Rt = Rt1 and Rr = Rr1 ), the moving vehicles are relatively dense around the MT and MR. In this case, the rays with double interactions of the two-cylinder model can bear more energy than the rays with single interactions of the two-cylinder and semi-ellipsoid models. This means that the power of the rays with double interactions of the two-cylinder model is larger than the others, i.e., ηDB > max{ηSB1,1 , ηSB1,2 , ηSB1,3 }. However, for the second tap, it is assumed that the rays with single interactions are produced only from the interfering objects located on the corresponding semiellipsoid, while the rays with double interactions are caused by the interfering objects from the combined one cylinder (either of the two cylinders) and the corresponding semi-ellipsoid [19]. It is worth mentioning that the rays with double interactions in the first tap must be smaller in distance than the rays with single interactions on the second semi-ellipsoid, i.e., max{Rt , Rr } < min{a2 − a1 }. It is stated in [39] that the delay resolution is approximately the inverse of bandwidth and therefore, we assume that the delay resolution in the proposed model is 20 ns for 50 MHz. In this paper, we define different propagation delays with the different ellipses. Thus, the second ellipse should produce at least 6 m excess path length than the first ellipse, i.e., 2a2 − 2a1 = cτ with τ = 20 ns. In this case, the proposed channel statistics for different propagation delays can be investigated. For highway scenarios, the received scattered power is mainly from waves reflected by the curved walls of tunnels described by the interfering objects located on the semiellipsoid. This indicates that the power of the rays with single interactions of the l-th semi-ellipsoid model is larger than the others, i.e., ηSBl,3 > max{ηDBl,1 , ηDBl,2 }. For urban scenarios, the rays with double interactions from the combined single cylinder and semi-ellipsoid models can bear more energy than the rays with single

6.2 System Channel Model for Flat Communications

125

interactions of the semi-ellipsoid model. This means that the power of the rays with single interactions of the l-th semi-ellipsoid model is smaller than the others, i.e., min{ηDBl,1 , ηDBl,2 } > ηSBl,3 .

6.2.3 Non-stationary Time-Varying Parameters In [29], the authors introduced the time-varying AAoD, EAoD, AAoA, and EAoA to describe the movement properties of V2V channels. Nevertheless, the effects of arbitrary movement directions on the channel statistics, which are meaningful for V2V channels, cannot be investigated in this study. To this aim, the original fixed geometric path lengths are replaced by the time-varying parameters. It is worth mentioning that the MR is relatively far from the MT in the proposed V2V tunnel communication environments; thus, we can make the following assumptions: min{R t , Rr , u − f }  max{δT , δR }, D  max{δT , δR }, and the approximation √ x + 1 ≈ 1 + x/2 is used for small x. Accordingly, based on the law of cosines in appropriate triangles and small angle approximations (i.e., sin x ≈ x and cos x ≈ 1 for small x), the corresponding time-varying geometric path lengths can be approximated as  ξpq (t) ≈

 2  2   D − δRx + vR t − 2 D − δRx vR t cos γR

  −δT cos θT sin ϕT + cos ϕT

(6.9)

 (n ) (n ) ξpn1,1 (t) ≈ Rt − δT cos θT cos ψT cos βT 1,1 cos αT 1,1 (n1,1 )

+ cos θT sin ψT cos βT  ξpn1,2 (t) ≈

(n1,1 )

sin αT

(n1,1 )

+ sin θT sin βT

ξpnl,3 (t) ≈

(6.10)

 2  (n )  ξT2,n1,2 + vT t − 2ξT ,n1,2 vT t cos αT 1,2 − γT

  (n ) −δT cos θT Rr /ξT ,n1,2 sin ϕT sin αR 1,2 + cos ϕT 



(6.11)

 2  (n )  ξT2,nl,3 + vT t − 2ξT ,nl,3 vT t cos αT l,3 − γT

  (n ) −δT cos θT ξR,nl,3 /ξT ,nl,3 sin ϕT sin αR l,3 + cos ϕT

(6.12)

126

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

 (n ) (n ) ξqn1,2 (t) ≈ Rr − δR cos θR cos ψR cos βR 1,2 cos αR 1,2 (n

)

(n

)

(n

+ cos θR sin ψR cos βR 1,2 sin αR 1,2 + sin θR sin βR 1,2  ξqn1,1 (t) ≈

)

ξqnl,3 (t) ≈

(6.13)

  2  (n ) 2 ξR,n + vR t − 2ξR,n1,1 vR t cos αR 1,1 − γR 1,1

  (n ) −δR cos θR Rt /ξR,n1,1 sin ϕR sin αT 1,1 + cos ϕR 



(6.14)

  2  (n ) 2 ξR,n + vR t − 2ξR,nl,3 vR t cos αR l,3 − γR l,3

  (n ) −δR cos θR ξT ,nl,3 /ξR,nl,3 sin ψR sin αT l,3 + cos ψR ,

(6.15)

bl2 f0 +al bl2

(n )

(nl,3 ) (n ) = bl2 cos2 βT l,3 (n ) , QT QT l,3 (n ) (n ) (n ) (n ) (n ) cos2 αT l,3 + al2 cos2 βT l,3 sin2 αT l,3 + al2 sin2 βT l,3 , ξR,n1,1 = D + Rt cos αT 1,1 ,

where ξT ,n1,2 = D + Rr cos αR l,2 , ξT ,nl,3 = and ξR,nl,3 = arctan



(nl,3 )

ξT ,nl,3 cos βT

(nl,3 )

D − ξT ,nl,3 cos βT (nl,i )

Note that the AAoD/EAoD, (i.e., αT

(n )

(nl,i )

, βT

(nl,3 )

sin αT

(nl,3

cos αT

 . ) (n )

), and AAoA/EAoA, (i.e., αR l,i ,

βR l,i ), are independent for double-bounced rays, while they are correlated for the rays with single interactions. According to the proposed geometric model in Figs. 6.1 and 6.2, the relationships between the AAoDs/EAoDs and AAoAs/EAoAs are derived as (n )

(nl,i )

αT

= arctan

(n )

ξqnl,i cos βR l,i sin αR l,i (n )

(n )

D − ξqnl,i cos βR l,i cos αR l,i

(6.16)

(n )

(nl,i )

βT

= arctan 

ξqnl,i sin βR l,i ,  (nl,i ) 2 2 (nl,i ) 2 F0 sin αR + D − F0 cos αR

(6.17)

(n )

where F0 = ξqnl,i cos βR l,i , and ξqnl,i is the distance from the MR to the interfering (nl,1 )

object s (nl,i ) . Substituting (6.13), (6.14), and (6.15) into (6.16), the AAoDs αT (nl,2 )

αT

(nl,3 )

, and αT

,

related to the AAoA/EAoA can be derived, respectively. However,

6.3 Proposed Theoretical Channel Characteristics (n )

(n )

127

(n )

the EAoDs βT l,1 , βT l,2 , and βT l,3 related to the AAoA/EAoA can be derived by substituting (6.13), (6.14), and (6.15) into (6.17), respectively. To jointly characterize the azimuth and elevation angular parameters on the channel characteristics, we use the von Mises PDF in [40] because it approximates many of the aforementioned distributions and admits closed-form solutions for many useful situations. The von Mises PDF is derived as (n )

l,i (nl,i )  (nl,i ) (nl,i )  k cos βT (R) × ek sin β0 sin βT (R) p αT (R) , βT (R) = 4π sinh k  (nl,i )  (nl,i ) × ek cos β0 cos βT (R) cos αT (R) −α0

(n )

(6.18)

(n )

l,i l,i with αT (R) and βT (R) ∈ [−π, π ), α0 ∈ [−π, π ). In addition, β0 ∈ [−π, π ) denotes

(n )

(n )

l,i l,i the mean values of the azimuth angle αT (R) and elevation angle βT (R) , respectively. In addition, k (k ≥ 0) is a real-valued parameter that controls the angles spread of α0 and β0 .

6.3 Proposed Theoretical Channel Characteristics 6.3.1 Spatial Correlation Functions For V2V wireless communications, the channel effect is completely characterized by the CIRs [6]. Once calculated, the CIR can be used to analyze or simulate the effect of the V2V wireless channel on the performance of vehicular communication systems. As shown in [39], the spatial correlation properties of two arbitrary CIRs hpq (t, τ ) and hp q  (t, τ ) of a MIMO V2V channel are completely determined by the correlation properties of hl,pq (t) and hl,p q  (t) in each tap, so that no correlations exist between the underlying processes in different taps. In previous studies, the spatial correlation function (CF) is an important statistic in designing a communication link that characterizes how fast a wireless channel changes with respect to time, movement, or frequency. The normalized time-varying spatial CF can be expressed as [33]   E hl,pq (t)h∗l,p q  (t − τ  ) ρhl,pq ,hl,p q  (t, τ  ) =   , 2   2        E hl,pq (t) E hl,p q  (t − τ )

(6.19)

where E[·] denotes the expectation operation and (·)∗ is the complex conjugate operation. It is assumed that the direct LoS propagations, rays with single and double interactions are independent of each other, the spatial CF for the first tap can be expressed as

128

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

ρh1,pq ,h1,p q  (t, τ  ) = ρhLoS 1,pq ,h

1,p q

(t, τ  ) + 

3 

SB

1,i ρh1,pq ,h

1,p q 

(t, τ  )

i=1

+ρhDB (t, τ  ). 1,pq ,h1,p q 

(6.20)

However, for other taps, we have ρhl,pq ,hl,p q  (t, τ  ) = ρhl,pql,3,h SB

(t, τ  ) + ρhl,pql,1,h DB

l,p  q 

DB

+ρhl,pql,2,h

l,p  q 

l,p  q 

(t, τ  )

(t, τ  ).

(6.21)

By applying the corresponding scatterer non-uniform distribution, and by following similar reasoning in [40], we can obtain the time-varying spatial CFs of the direct LoS propagations, rays with single and double interactions, as outlined below. Specifically, by submitting (6.2) into (6.19), the time-varying spatial CF in the case of the LoS rays can be expressed as ρhLoS 1,pq ,h

1,p q

2π λ

(t, τ  ) = ej 









LoS −γ τ  vT cos αTLoS −γT −vR cos αR R

×Ke







j 2πfc ξpq (t)−ξp q  (t) /c

(6.22)

.

In submitting (6.3) into (6.19), the time-varying spatial CF in the case of the rays with single interactions SB1,i can be derived as SB

1,i ρh1,pq ,h

1,p q

(t, τ  ) = ηSB1,i × 

N1,i 

lim

N1,i →∞

n1,i =1

1 N1,i



×e

j 2πfc ξpq,n1,i (t)−ξp q  ,n

× ej

2π λ





(n1,i )

τ  vT cos αT



1,i

(t) /c





(n

)

−γT −vR cos αR 1,i −γR

 .

(6.23)

N1,i It is stated in [40] that n1,i =1 1/N1,i = 1 as N1,i → ∞. Thus, the total power of the SB1,i rays is proportional to 1/N1,i . This is equal to the (n ) (n ) infinitesimal power coming from the differential of the 3D angles, dαR 1,i dβR 1,i , (n

)

(n

)

(n

)

(n

)

(n

)

(n

)

i.e., 1/N1,i = p(αR 1,i , βR 1,i )dαR 1,i dβR 1,i , where p(αR 1,i , βR 1,i ) denotes the joint von Mises PDF in (6.18). Therefore, (6.23) can be rewritten as

6.3 Proposed Theoretical Channel Characteristics SB1,i  ρh1,pq ,h1,p q  (t, τ )

 = ηSB1,i

π



−π

129



π −π

e

j 2πfc ξpq,n1,i (t)−ξp q  ,n



1,i

(t) /c

  (n1,i )   (n1,i )  2π  × ej λ τ vT cos αT −γT −vR cos αR −γR   (n1,i ) (n1,i ) (n1,i ) (n1,i ) dαT (R) × p αT (R) , βT (R) dβT (R) .

(6.24)

Similarly, submitting (6.4) into (6.19), the time-varying spatial CF in the case of the rays with double interactions DB can be expressed as ρhDB (t, τ  ) 1,pq ,h1,p q 

 = ηDB



π

−π



π −π

e

j 2πfc ξpq,n1,1 n1,2 (t)−ξp q  ,n



1,1 n1,2

(t) /c

  (n1,1 )   (n1,2 )  2π  × ej λ τ vT cos αT −γT −vR cos αR −γR   (n1,2 ) (n1,2 ) (n1,2 ) (n1,2 ) dαT (R) × p αT (R) , βT (R) dβT (R) .

(6.25)

Submitting (6.6) into (6.19), the time-varying spatial CF in the case of the rays with single interactions SBl,3 can be expressed as SB

ρhl,pql,3,h

l,p  q

(t, τ  ) = ηSBl,3 



π



−π



π −π

e

j 2πfc ξpq,nl,3 (t)−ξp q  ,n



l,3

(t) /c



 (nl,3 )   (nl,3 )  2π  × ej λ τ vT cos αT −γT −vR cos αR −γR   (nl,3 ) (nl,3 ) (nl,3 ) (nl,3 ) dαT (R) , βT (R) dβT (R) . × p αT (R)

(6.26)

Submitting (6.7) into (6.19), the time-varying spatial CF in the case of the rays with double interactions DBl,1 can be derived as DB ρhl,pql,1,h   (t, τ  ) l,p q

 = ηDBl,1

π

−π





π −π

e

j 2πfc ξpq,n1,1 nl,3 (t)−ξp q  ,n



1,1 nl,3

  (n1,1 )   (nl,3 )  2π  × ej λ τ vT cos αT −γT −vR cos αR −γR   (nl,3 ) (nl,3 ) (nl,3 ) (nl,3 ) dαT (R) × p αT (R) , βT (R) dβT (R) .

(t) /c

(6.27)

130

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

In submitting (6.8) into (6.19), the time-varying spatial CF in the case of the rays with double interactions DBl,2 can be derived as DB

ρhl,pql,2,h

l,p  q

(t, τ  ) = ηDBl,2 



π

−π





π −π

e

j 2πfc ξpq,nl,3 n1,2 (t)−ξp q  ,n



l,3 n1,2

 (nl,3 )   (n1,2 )  2π  × ej λ τ vT cos αT −γT −vR cos αR −γR   (n1,2 ) (n1,2 ) (n1,2 ) (n1,2 ) dαT (R) × p αT (R) , βT (R) dβT (R) .

(t) /c



(6.28)

From (6.22)–(6.28), we notice that the spatial CFs of the proposed model are related to the propagation path lengths; hence, the spatial correlations have different properties for different taps. In the following, by setting p = p and q = q  , the temporal auto-correlation function (ACF) can be expressed as [41]   E hl,pq (t)h∗l,pq (t + Δt) rhl,pq (t, Δt) =   . 2   2       E hl,pq (t) E hl,pq (t + Δt)

(6.29)

It is worth mentioning that, the temporal ACF is time-varying because of the motion of the MT and MR.

6.3.2 Frequency CFs The frequency CFs, which measures the frequency selectivity of the proposed channel model [42], can be expressed as   E hpq (t, f ) h∗p q  (t, f − Δf ) ρhpq ,hp q  (t, Δf ) =   , 2   2   E hpq (t, f ) E hp q  (t, f − Δf )

(6.30)

where hpq (t, f ) can be derived by the Fourier transform of the CIR hpq (t, τ ),  −j 2πf τ . Here, substituting the CIRs of the LoS, i.e., hpq (t, f ) = L(t) l=1 ωl hl,pq (t)e single-, and double-bounced propagation rays into (6.30), the corresponding timevarying frequency CFs can be derived.

6.3 Proposed Theoretical Channel Characteristics

131

6.3.3 Doppler PSD In the proposed model, the signals propagate from the MT to the MR via different propagation paths. In addition to the fluctuations in the signal envelope and phase, the received signal frequency constantly varies because of the motion between the MT and MR. Note that the received signal is a superposition of the rays with single and double interactions, which jointly influence the distributions of the Doppler PSDs. Here, by applying the Fourier transform of the temporal ACF rhl,pq (t, Δt) with respect to the time interval Δt, the Doppler PSD of the proposed model can be derived as  ∞ S(f, t) = rhl,pq (t, Δt)e−j 2πf Δt dΔt. (6.31) −∞

It is worth mentioning that the Doppler PSDs of the proposed model are impacted by the movement directions (i.e., γT and γR ), velocities (i.e., vT and vR ), and time t of the MT and MR. Furthermore, we notice that the propagation paths for the first tap are different from those for other taps; therefore, the Doppler PSDs of the first tap are different from those of other taps.

6.3.4 Power Delay Profile In wireless communications, the power delay profile (PDP) gives the intensity of a signal received through a multipath channel as a function of the propagation delay, which is easily measured empirically and can be used to extract certain parameters of the channel such as the delay spread. Therefore, the PDP in the proposed model can be derived by taking the spatial average of the CIRs, i.e., L(t)  2   Phl,pq (t, τ ) =  ωl hl,pq (t)δ(τ − τl ) ,

(6.32)

l=1

where ωl denotes the gain of the l-th tap. It is worth mentioning that the proposed V2V channel model is non-stationary because of the motion of the MT and MR. Therefore, when we aim to reproduce the non-exponential power decay behaviors of proposed model, it is important to derive the time-varying propagation path lengths for the first tap and other taps, which have been discussed in Section II.C.

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

6.3.5 Stationary Interval The stationary interval is the maximum time duration over which the WSS assumption is valid, and can be calculated as [41] P hl,pq (ts , τ ) =

1

s+N P DP −1

NP DP

s

ωl2 h2l,pq (ts )δ(τ − τl ),

(6.33)

where NP DP denotes the number of averaged PDPs, ts is the time of the s-th drop (snapshot). The correlation coefficient between two averaged PDPs can be expressed as  Ph (ts , τ ) · Ph (ts + Δt, τ )dτ ρ(ts , Δt) = (6.34)  / 0. max Ph (ts , τ )2 dτ, Ph (ts + Δt, τ )2 dτ The stationary interval can be then written as    T0 (ts ) = max Δt ρ(ts , Δt ≥ ρ0 ) ,

(6.35)

where ρ0 is a given threshold of the correlation coefficient. It is worth mentioning that the above analysis is mainly for flat communication environments, where the MT and MR are located in the same plane. However, in reality, the moving vehicles can be anywhere above, below, or on the actual slope, requiring a more careful analysis to accurately model this V2V propagation condition [43]. To overcome the above problem, it is important to investigate the V2V channel characteristics where the MT and MR are located in different azimuth planes. Detailed discussions can be seen in the following section.

6.4 System Channel Model for Slope Communications In this section, we investigate the V2V channel characteristics for the NLoS propagation rays with ground reflection. It is assumed that there are Ng interfering objects uniformly existing on the ground in different azimuth plane. The heights from the center points of the MT and MR antenna elements to the ground are denoted as Ht and Hr , respectively. Then, the complex coefficient for the NLoS rays with ground reflection can be expressed as (n ) hpqg (t)

=



Ωg × lim

Ng →∞

× ej

2π λ vT t



Ng  ng

1 ej Ng =1

(ng )

cos αT





(ng )

−γT cos βT



 

ϕ0 −2πfc ξpng +ξqng /c



(ng )

+j 2π λ vR t cos αR



(ng )

−γR cos βR

,

(6.36)

6.5 Moving Vehicles on the Channel Characteristics

133 (n )

(n )

where Ωg denotes the energy-related parameter, αT g and βT g denote the AAoD and EAoD of the waves that impinge on the interfering objects on the ground, (n ) (n ) respectively, αR g and βR g are the AAoA and EAoA of the waves scattered from the interfering objects on the ground, respectively. The ξpng and ξqng are the distances from the MT and MR to the interfering objects on the ground, respectively, which can be expressed as  ξpng (t) ≈

  2  (n ) ξT2,ng + vT t − 2ξT ,ng vT t cos αT g − γT

  (n ) −δT cos θT ξR,ng /ξT ,ng sin ϕT sin αR g + cos ϕT  ξqng (t) ≈

(6.37)

  2  (n ) 2 ξR,n + vR t − 2ξR,ng vR t cos αR g − γR g

  (n ) −δR cos θR ξT ,ng /ξR,ng sin ϕR sin αT g + cos ϕR , (n )

(6.38)

(n )

where ξT ,ng = Ht csc βT g and ξR,ng = Hr csc βR g . In the model, we assume that the received signals scattered from the ground are single-bounced, rather than double-bounced. Therefore, the spatial CF for the NLoS rays with ground reflection can be expressed as   (SB ) ρhl,pqg,hl,pa´˛rq a´˛r t, τ

 =



π

−π

×e



π −π

e 



j 2πfc ξpq,ng −ξp q  ,ng /c



(ng )

 j 2π λ τ vT cos αT





(ng )

−γT −vR cos αR

−γR

  (ng ) (ng ) (ng ) (ng ) dαT (R) × p αT (R) , βT (R) dβT (R) .



(6.39)

It is worth mentioning that, the spatial CFs of the proposed model for the groundbounced rays are related to the heights of the slopes on which the MT and MR are located.

6.5 Moving Vehicles on the Channel Characteristics As shown in Fig. 6.4a, when the MT moves from position P1 to P2 , the MR moves from position P3 to P4 , and the vehicles move from position P5 to P6 , the channel model varies over time. Note that the MT and MR are symmetric about the y − z plane in the proposed model; therefore, we consider the movement directions and

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

Fig. 6.4 Diagrams of geometric channel models in different propagation environments: (a) geometric relations in V2V channels; (b) geometric update of V2V channels

velocities of the vehicles around the MT in this section. It is assumed that the interfering objects represented by the moving vehicles around the MT are fixed as a reference point. Then, the velocities of the MT and MR relative to the moving vehicles are denoted as vT /s and vR/s , respectively. Note that the MT, MR, and moving vehicles are located in the same plane; therefore, the movement directions of the MT and MR relative to the moving vehicles are denoted as γT /s and γR/s , respectively. Here, the complex fading envelope between the p-th transmit antenna and q-th receive antenna can be expressed as

6.5 Moving Vehicles on the Channel Characteristics

m) h(n pq (t) =



Nm 

Ωm × lim

Nm →∞

×e



√

nm =1

135

1 Nm



 

j ϕ0 −2πfc Dpnm (t)+Dqnm (t) /c

× ej

2π λ vT /s t



(nm ) −γT /s

cos αT



(nm ) +j 2π λ vR/s t

cos βT



(n )



(n )

cos αR m −γR/s cos βR m

,

(6.40) where Ωm denotes the energy-related parameter, Dpnm (t) and Dqnm (t) are the distances from the p-th transmit and q-th receive antenna to the moving vehicles around the MT, respectively, which can be expressed as Dpnm (t) ≈

Dqnm (t) ≈ with DR,0 =





  DT2 (t) + (kp δT )2 − 2DT (t)kp δT cos αT(nm ) − ψT



  (n ) DR2 (t) + (kq δR )2 − 2DR (t)kq δR cos αR m − ψR (nm )

Rt2 + D 2 − 2Rt D cos αT

DR (t) =



DT (t) =

(6.41)

(6.42)

,

  (nm ) 2 + (v 2 DR,0 − γR/s , R/s t) − 2vR/s tDR,0 cos αR 

  (n ) Rt2 + (vT /s t)2 − 2Rt vT /s t cos αT m − γT /s .

Based on the principle of the relative motion, the velocities of the MT and MR relative to the moving vehicles can be derived as vT /s =

vR/s =



  vT2 + vs2 − 2vT vs cos γT − γs

(6.43)

   vR2 + vs2 − 2vR vs cos γR − γs ,

(6.44)

where vs and γs denote the movement velocity and direction of the moving vehicles, respectively. Then, the relative motion angles ϕT /s and γR/s can be, respectively, derived as ϕT /s = arctan

vT sin γT − vs sin γs vT cos γT − vs cos γs

(6.45)

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

ϕR/s = arctan

vR sin γR − vs sin γs . vR cos γR − vs cos γs

(6.46)

Here, when the waves from the MT impinge on moving vehicles (around the MT) before reaching the MR, the spatial CF can be expressed as (n )

ρhpqm,h

p q

  t, τ = 



π

−π

× ej



π

−π

e−j 2πfc

2π  λ vT /s τ cos

× ej 2πfc









Dpnm (t)+Dqnm (t) /c

(nm ) −γT /s

αT



(nm )  +j 2π λ vR/s τ cos

cos βT



Dp nm (t−τ  )+Dq  nm (t−τ  ) /c



(n )



(n )

αR m −γR/s cos βR m

  (nm ) (nm ) (nm ) (nm ) dαT (R) p αT (R) , βT (R) dβT (R) .

(6.47) It is worth mentioning that, the spatial CFs for the rays with single interaction of moving vehicles around the MT are related to the movement directions and velocities of the vehicles.

6.6 Numerical Results and Discussions In this section, the statistical properties of the proposed 3D non-stationary wideband V2V channel model are evaluated and analyzed. The time slots for the stationary and non-stationary conditions are set t = 0 and t = 2 s, respectively. To investigate the proposed channel statistics for different propagation delays, i.e., per-tap statistics, we define the semimajor dimensions for the first tap and second tap are, respectively, a1 = 180 m and a2 = 200 m, i.e., τ = 2(a2 − a1 )/c ≈ 133 ns > 20 ns. Unless otherwise specified, all the channel related parameters used in this section are using D = 200 m, fc = 5.9 GHz, Ωl,pq = 1, MT = MR = 8, Ht = 10 m, ψT = θT = π/3, and ψR = θR = π/3. Considering the constraints of the Ricean factor and energy-related parameters, we have Rt = Rr = 40 m, vT = vR = 25 m/s, K = 3.942, ηSB1,1 = 0.371, ηSB1,2 = 0.212, ηSB1,3 = 0.402, ηDB = 0.015, k (1,1) = 8.9, k (1,2) = 2.7, and k (1,3) = 12.3 for tap one highway scenarios. For tap one urban scenarios, we have Rt = Rr = 20 m, vT = vR = 8.3 m/s, K = 1.062, ηSB1,1 = ηSB1,2 = 0.142, ηSB1,3 = 0.085, ηDB = 0.631, k (1,1) = 0.55, k (1,2) = 1.21, and k (1,3) = 12.3. For tap two highway scenarios, we have Rt = Rr = 40 m, vT = vR = 25 m/s, K = 3.942, ηSB2,3 = 0.724, ηDB2,1 = ηDB2,2 = 0.138, k (2,1) = 8.9, k (2,2) = 2.7, and k (2,3) = 12.3. For tap two urban scenarios, we have Rt = Rr = 20 m, vT = vR = 8.3 m/s, K = 1.062, ηSB2,3 = 0.056, ηDB2,1 = ηDB2,2 = 0.472, k (2,1) = 0.55, k (2,2) = 1.21, and k (2,3) = 12.3.

6.6 Numerical Results and Discussions

137

1. Ricean factor: In Rice channels, the Ricean factor is related to the real communication environments. Specifically, • For urban scenarios, the waves from the MT impinge on interfering objects before reaching the MR (i.e., NLoS rays). Thus, the Ricean factor K is small (i.e., normally smaller than 1.5) as the LoS component does not have dominant power. • For highway scenarios, K is large (i.e., normally larger than 3.5) because fewer moving vehicles/obstacles (between the MT and MR) on the road result in the strong LoS propagation component. 2. Energy-related parameters: As mentioned in [44], the energy-related parameters 3 in V2V channels for tap one and other taps should be equal to unity, i.e., i=1 ηSB1,i + ηDB = 1 and ηSBl,3 + ηDBl,1 + ηDBl,2 = 1. Note that the energyrelated parameters ηSBl,1 , ηSBl,2 , ηSBl,3 , ηDB , ηDBl,1 , and ηDBl,2 are related to the scattered cases of NLoS rays. Specifically, • For tap one highway scenarios, the received scattered power is mainly from waves reflected by the stationary roadside environments. The moving vehicles represented by the interfering objects located on the two cylinders are more likely to be single-bounced, rather than double-bounced. This indicates that ηSB1,3 > max{ηSB1,1 , ηSB1,2 } > ηDB , i.e., ηSB1,3 is normally larger than 0.4, ηSB1,1 and ηSB1,2 are normally both larger than 0.2 but smaller than 0.4, while ηDB is normally smaller than 0.1. • For tap one urban scenarios, the received scattered power is mainly from the waves scattered from the two-cylinder model, i.e., ηDB > max{ηSB1,1 , ηSB1,2 , ηSB1,3 } (normally, ηDB is larger than 0.6, while ηSB1,1 , ηSB1,2 , and ηSB1,3 are all smaller than 0.15). • For tap two highway scenarios, the received scattered power is mainly from waves reflected by the stationary roadside environments described by the interfering objects located on the semi-ellipsoid. Thus, ηSB2,3 > max{ηDB2,1 , ηDB2,2 }, i.e., ηSB2,3 is normally larger than 0.7, while ηDB2,1 and ηDB2,2 are both smaller than 0.15. • For tap two urban scenarios, the received scattered power is mainly from the rays with double interactions from the combined single cylinder and semi-ellipsoid models, i.e., min{ηDB2,1 , ηDB2,2 } > ηSB2,3 (normally, ηSB2,3 is smaller than 0.1, while ηDB2,1 and ηDB2,2 are both larger than 0.4). 3. Environment-related parameters: In V2V channels, the environment-related parameters k (l,1) , k (l,2) , and k (l,3) are related to the distribution of interfering objects. Specifically, • For highway scenarios, the values of k (l,1) and k (l,2) (i.e., normally both smaller than 10) are high, because there exist few moving vehicles in V2V channels. • In both the highway and urban scenarios, k (l,3) is large (i.e., normally larger than 10) as the interfering objects reflected from roadside environments are normally concentrated.

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

6.6.1 AAoA Statistics In general, the PDFs of the AAoA statistics at the MR are strongly related to the last geometric path length in multipath channels. Although the channel model in [11] is different from the proposed model, when the waves scattered from the roadside environments (semi-ellipsoid model), the AAoA statistics are similar to those in the proposed model. Here, we assume that the transmitter emits the signal to the receiver in significantly small beamwidth, spanning the azimuth range of [−α, α]. As shown in Fig. 6.5, when the MT is employed with the directional antenna elements, the (n ) AAoA PDFs in 0 ≤ αR l,3 ≤ π firstly decrease to a local value of AAoA and then increase to a local maximum with a “corner”, the AAoA PDFs finally decrease (n ) sharply. A similar behavior can be seen in −π ≤ αR l,3 ≤ 0. By increasing the beamwidth α with more interfering objects in the scattering region, the PDFs firstly have higher values on both sides of the curves, and then gradually tend to be equal. It can also be noted that when the road width b1 increases from 120 m to 160 m, the values of the AAoA PDFs increase sharply.

Fig. 6.5 Proposed AAoA statistics in the azimuth plane for the different road width b1 and different beamwidth α of the directional antenna at the MT

6.6 Numerical Results and Discussions

139

6.6.2 Spatial and Frequency CFs By adopting an MT antenna element spacing δT = λ, the absolute values of the time-varying spatial CFs of the proposed V2V channel model are illustrated in Figs. 6.6, 6.7, and 6.8. By imposing i = 1 and 3 in (6.24), Fig. 6.6 shows that the spatial correlation decreases slowly as the transmit antenna angles ψT and θT decrease. By using (6.26), the time-varying spatial CFs of the first and second taps of the single-bounced semi-ellipsoid model (i.e., SBl,3 ) for different movement properties (i.e., t and γR ) are shown in Fig. 6.7. Measurement results in [45] have shown that the spatial CFs gradually decrease when the normalized antenna spacing δR /λ increases, which confirms the results in Fig. 6.7. In this figure, the correlation in the first tap is higher than that in the second tap because of the dominant LoS rays, which is in correspondence with the results in [29]. By using (6.25) and imposing i = 1 and 3 in (6.24), Fig. 6.8 illustrates the time-varying spatial CFs of the rays with single (i.e., SB1,1 and SB1,3 ) and double (i.e., DB) interactions of the first tap in the WSS condition (i.e., t = 0). The figure shows that the movement directions (i.e., γR ) have no impact on the distribution of the time-varying spatial CFs when the proposed channel model is under WSS assumption. It can be observed that the

Fig. 6.6 Proposed time-varying spatial CFs for the rays with single interactions (i.e., SB1,1 and SB1,3 ) for different transmit antenna angles in tap one highway scenarios

140

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

Fig. 6.7 Proposed time-varying spatial CFs of the single-bounced semi-ellipsoid model for different taps of the proposed model in highway scenarios

Fig. 6.8 Proposed time-varying spatial CFs of the single-bounced semi-ellipsoid model for different taps of the proposed model in highway scenarios

6.6 Numerical Results and Discussions

141

Fig. 6.9 Proposed time-varying spatial CFs for the rays with single interactions of ground reflection with respect to different antenna heights (i.e., Ht and Hr ) and different distance D between the MT and MR

spatial correlation of the single-bounced rays SB1,3 is lower than that of the singlebounced rays SB1,1 . Figure 6.9 shows the time-varying spatial CFs for the rays with ground interactions. From the figure, we can easily notice that when the heights from the center points of the MT and MR antenna elements to the ground increase from 20 m (Hr = 2Ht ) to 50 m (Hr = 5Ht ), the spatial CFs decrease slowly. It can also be noted that the spatial correlation decreases gradually as the MR moves away from the MT. By using (6.26) and (6.39), Fig. 6.10 illustrates the proposed spatial CFs for the single-bounced rays scattered by ground surface and roadside environments. It can be observed that the spatial correlation gradually decreases when the propagation delay τ  increases, which is in agreement with the results in [29]. Furthermore, the correlations scattered by roadside environments are clearly higher than those scattered by the ground surface. Moreover, when the road width b1 decreases from 160 m to 120 m, the values of the correlation increase slowly.

142

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

Fig. 6.10 Proposed spatial CFs for the rays with single interactions scattered by ground surface and roadside environments with respect to different road width b1

By using (6.25) and (6.28), Fig. 6.11 shows the time-varying frequency CFs of the double-bounced models (i.e., DB and DB2,2 ) corresponding to the different movement directions and different movement time instants. It is clearly observed that, for the double-bounced DB WSS model, regardless of what the movement directions are (i.e., γR = π/3 or 2π/3), the curves of the frequency CFs between them tend to be the same, which confirms the analysis in Fig. 6.6. Furthermore, it is evident that when the receiver’s movement direction γR is π/3, the value of the frequency CFs is relatively higher than that at γR = π/6. Then, we observe that the frequency correlations of the double-bounced DB2,2 are lower than those of the double-bounced DB in the proposed non-stationary V2V channel model. In Fig. 6.12, the movement velocities and directions of the MT/MR are fixed at vT = vR = 8.3 m/s and γT = γR = −π/2, respectively, to highlight the impacts of the movement velocities/directions of the moving vehicles on the V2V channel characteristics [13]. It is obvious that the spatial correlation decreases more rapidly as the velocity vs increases from 5 m/s to 20 m/s. It also can be noted that when the vehicles moves towards the x axis, i.e., γs = −π/2, the spatial correlation is higher than that when the vehicles moves away the x axis, i.e., γs = π/2.

6.6 Numerical Results and Discussions

143

Fig. 6.11 Proposed time-varying frequency CFs for the rays with double interactions with respect to different movement directions and different time instants in highway scenarios

Fig. 6.12 Proposed spatial CFs for the rays with single interactions scattered by the moving vehicles around the MT

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

6.6.3 Doppler PSDs By using (6.31), Figs. 6.13 and 6.14 show the Doppler PSDs of the proposed V2V channel model for different movement directions and different taps, respectively. It is observed from Fig. 6.13 that the Doppler PSD is positively shifted when the MT and MR move toward each other, i.e., γT = 0 and γR = π . In contrast, the Doppler spectrum is negatively shifted when the MT and MR move in opposite directions, i.e., γT = π and γR = 0. It is also noted that the Doppler spectrum decreases gradually as the road width b1 decreases from 160 m to 120 m. In Fig. 6.14, we notice that the Doppler spectrum gradually decreases with an increase in the taps of the proposed channel model. For the MR movement perpendicular to the direct LoS rays (i.e., γT = γR = π/2), the Doppler spectrum in the stationary channel model has a peak at zero, which is in agreement with the measurements in [46]. However, this is not necessary for the proposed non-stationary V2V channel model. Note that the Doppler spectrum in non-stationary V2V channels changes continually at different time instants when γT and γR are set to π/2, as reported in [27].

Fig. 6.13 Doppler PSDs of the proposed channel model with respect to different movement directions and different road width b1 in highway scenarios

6.6 Numerical Results and Discussions

145

Fig. 6.14 Doppler PSDs of the proposed model with respect to different taps and different movement directions in highway scenarios

6.6.4 PDPs and Stationary Intervals By using (6.32), Fig. 6.15 illustrates the values of the PDPs of the proposed 3D model for different propagation delays. The number of averaged PDP is set 15. It is worth mentioning that the propagation delay τ can be defined as the ratio of the geometric path length to the light velocity c; thus, the shortest and longest propagation delays of the proposed WSS model are, respectively, obtained as τmin = D/c and τmax ≈ 2al /c [47]. From the figure, we can notice that the PDP gradually decreases with an increase in the propagation delay τ , which agrees with the results of [29]. Furthermore, the values of the PDPs decrease as the MT and MR are surrounded by more vehicles (i.e., Rt = Rr = 10 m). Overall, the proposed model can be easily fitted by the measured PDP at different time slots. In this way, we can efficiently design and analyze the vehicular communication systems. Figure 6.16 shows the stationary intervals of the proposed model for V2V communication environments using the following simulation parameters obtained from [28]: fc = 5.9 GHz, ωl = −3.2, NP N P = 15, and ρ0 = 0.8. It is observed that the stationary intervals for the V2V highway scenarios are obviously lower than

146

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

Fig. 6.15 PDPs of the proposed V2V channel model. (a) Rt = Rr = 20 m; (b) Rt = Rr = 10 m

those for the V2V urban scenarios. This is because the movement velocities of the MT and MR in V2V highway scenario are both 25 m/s, which are larger than those in the V2V urban scenarios (8.3 m/s). However, [41] demonstrates that the stationary intervals gradually decrease as the movement velocity increases. The excellent agreement between the theoretical results and measured data in [47] confirms the utility of the proposed model.

6.7 Conclusion

147

Fig. 6.16 Stationary intervals of the proposed 3D non-stationary channel model for the different V2V scenarios

6.7 Conclusion In this chapter, we have proposed a 3D wideband geometry-based channel model for MIMO V2V communications in tunnel environments. By adjusting some model parameters, the model is adaptable to a wide variety of communication environments. In addition, the 3D multiple confocal semi-ellipsoid models are for the first time taken into account in the GBSMs for modeling wideband channels; therefore, we are able to practically investigate the channel characteristics for different propagation delays. Based on the proposed models, comprehensive statistical properties have been derived and thoroughly investigated. It has been demonstrated that the proposed channel characteristics show different behaviors at different propagation delays. It is additionally shown that the dominance of the LoS component results in a higher correlation in the first tap of the proposed channel model than in the second one. From the numerical results, we conclude that the time-varying spatial CFs and frequency CFs are significantly affected by the different taps of the proposed timevarying channel model, the movement times, and the directions between the MT and MR. Finally, the proposed results closely agree with the measured data, which validates the utility of our model. Our research work therefore as a theoretical guidance for providing designing V2V communication systems in future wireless networks.

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6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

References 1. J.J. Liu et al., Device-to-device communication for mobile multimedia in emerging 5G networks. ACM Trans. Multimed. Comput. Commun. Appl. 12(5), 75:1–75:20 (2016) 2. H. Nishiyama, M. Ito, N. Kato, Relay-by-smartphone: realizing multihop device-to-device communications. IEEE Commun. Mag. 52(4), 56–65 (2014) 3. M. Liu, J. Yang, G. Gui, DSF-NOMA: UAV-assisted emergency communication technology in a heterogeneous internet of things. IEEE Internet Things J. 6(3), 5508–5519 (2019) 4. N. Avazov, M. Patzold, A geometric street scattering channel model for car-to-car communication systems, in International Conference on Advanced Technologies for Commun. (ATC 2011), Da Nang, Vietnam (Aug 2011), pp. 224–230 5. F. Tang et al., AC-POCA: anticoordination game based partially overlapping channels assignment in combined UAV and D2D-based networks. IEEE Trans. Veh. Technol. 67(2), 1672–1683 (2018) 6. 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) 7. A.S. Akki, F. Haber, A statistical model for mobile-to-mobile land communication channel. IEEE Trans. Veh. Technol. 35(1), 2–7 (1986) 8. H. Jiang, Z. Zhang, J. Dang, L. Wu, Analysis of geometric multi-bounced virtual scattering channel model for dense urban street environments. IEEE Trans. Veh. Technol. 66(3), 1903– 1912 (2017) 9. 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. Wireless Commun. 8(9), 4824–4835 (2009) 10. Y. Yuan, C. Wang, X. Cheng, B. Ai, D.I. Laurenson, Novel 3D geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channels. IEEE Trans. Wireless Commun. 13(1), 298–309 (2014) 11. H. Jiang, Z. Zhang, J. Dang, L. Wu, A novel 3D massive MIMO channel model for vehicle-tovehicle communication environments. IEEE Trans. Commun. 66(1), 79–90 (2018) 12. A.G. Zajic, Impact of moving scatterers on vehicle-to-vehicle narrow-band channel characteristics. IEEE Trans. Veh. Technol. 63(7), 3094–3106 (2014) 13. M. Riaz, N.M. Khan, S.J. Nawaz, A generalized 3-D scattering channel model for spatiotemporal statistics in mobile-to-mobile communication environment. IEEE Trans. Veh. Technol. 64(10), 4399–4410 (2015) 14. J. Zhou, H. Jiang, H. Kikuchi, Generalised three-dimensional scattering channel model and its effects on compact multiple-input and multiple-output antenna receiving systems. IET Commun. 9(18), 2177–2187 (2015) 15. H. Xiao, A.G. Burr, L.T. Song, A time-varying wideband spatial channel model based on the 3GPP model, in IEEE Vehicular Technology Conference, Montreal, QC (Sept. 2006), pp. 1–5 16. I. Sen, D.W. Matolak, Vehicle-vehicle channel models for the 5-GHz band. IEEE Trans. Intell. Transp. Syst. 9(2), 235–245 (2008) 17. 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) 18. A. Paier et al., Non-WSSUS vehicular channel characterization in highway and urban scenarios at 5.2GHz using the local scattering function, in International ITG Workshop on Smart Antennas, Vienna, Austria (Feb 2008), pp. 9–15 19. X. Cheng, Q. Yao, M. Wen, C. Wang, L. Song, B. Jiao, Wideband channel modeling and intercarrier interference cancellation for vehicle-to-vehicle communication systems. IEEE J. Sel. Areas Commun. 31(9), 434–448 (2013) 20. N. Avazov, M. Patzold, A novel wideband MIMO car-to-car channel model based on a geometrical semi-circular tunnel scattering model. IEEE Trans. Veh. Technol. 65(3), 1070– 1082 (2016)

References

149

21. N. Avazov, S.M.R. Islam, D. Park, K.S. Kwak, Statistical characterization of a 3-D propagation model for V2V channels in rectangular tunnels. IEEE Antennas Wirel. Propag. Lett. 16, 2392– 2395 (2017) 22. Z. Sun, I.F. Akyildiz, A mode-based approach for channel modeling in underground tunnels under the impact of vehicular traffic flow. IEEE Trans. Wirel. Commun. 10(10), 3222–3231 (2011) 23. G.S. Ching et al., Wideband polarimetric directional propagation channel analysis inside an arched tunnel. IEEE Trans. Antennas Propag. 57(3), 760–767 (2009) 24. C. Wang, A. Ghazal, B. Ai, Y. Liu, P. Fan, Channel measurements and models for high-speed train communication systems: a survey. IEEE Commun. Surveys Tuts. 18(2), 974–987 (2016); 2nd Quart. 25. A. Chelli, M. Patzold, A non-stationary MIMO vehicle-to-vehicle channel model based on the geometrical T-junction model, in International Conference on Wireless Communication and Signal Processing (WCSP), Nanjing, China (Nov 2009), pp. 1–5 26. A. Chelli, M. Patzold, A non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical street model. IEEE Veh. Technol. Conf. (VTC Fall), San Francisco, CA, USA (Sept 2011), pp. 1–6 27. S. Wu, C. Wang, H. Haas, E.M. Aggoune, M.M. Alwakeel, B. Ai, A non-stationary wideband channel model for massive MIMO communication systems. IEEE Trans. Wirel. Commun. 14(3), 1434–1446 (2015) 28. A. Ghazal, C.X. Wang, B. Ai, D. Yuan, H. Haas, A non-stationary wideband MIMO channel model for high-mobility intelligent transportation systems. IEEE Trans. Intell. Transp. Syst. 16(2), 885–897 (2015) 29. Y. Yuan, C.X. Wang, Y. He, M.M. Alwakeel, e.H.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) 30. M. Walter, D. Shutin, A. Dammann, Algebraic analysis of the poles in the Doppler spectrum for vehicle-to-vehicle channels. IEEE Wirel. Commun. Lett. 99, 1–1 (2017) 31. X. Cai, B. Peng, X. Yin, A.P. Yuste, Hough-transform-based cluster identification and modeling for V2V channels based on measurements. IEEE Trans. Veh. Technol. 99, 1–1 (2017) 32. J. Karedal et al., A geometry-based stochastic MIMO model for vehicle-to-vehicle communications. IEEE Trans. Wirel. Commun. 8(7), 3646–3657 (2009) 33. M. Patzold, Mobile Radio Channels, 2nd edn. (Wiley, West Sussex, 2012) 34. J. Zhang, C. Pan, F. Pei, G. Liu, X. Cheng, Three-dimensional fading channel models: a survey of elevation angle research. IEEE Commun. Mag. 52(6), 218–226 (2014) 35. Y. Liu et al., 3D non-stationary wideband circular tunnel channel models for high-speed train wireless communication systems. Sci. China Inf. Sci. 60(8), 082304 (2017) 36. L. Bai, C. Wang, S. Wu, H. Wang, Y. Yang, A 3-D wideband multi-confocal ellipsoid model for wireless MIMO communication channels, in IEEE International Conference on Communications (ICC), Kuala Lumpur, Malaysia (May 2016), pp. 1–6 37. J. Chen, S. Wu, S. Liu, C. Wang, W. Wang, On the 3-D MIMO channel model based on regular-shaped geometry-based stochastic model, in International Symposium Antennas and Propagation (ISAP), Hobart, TAS (Nov. 2015), pp. 1–5 38. H. Huang, Y. Song, J. Yang, G. Gui, F. Adachi, Deep-learning-based millimeter-wave massive MIMO for hybrid precoding. IEEE Trans. Veh. Technol. 68(3), 3027–3032 (2019) 39. R. Vaughan, J.B. Andersen, Channels, Propagation and Antennas for Mobile Communications (Institution of Electrical Engineers, London, 2003) 40. A. Abdi, J.A. Barger, M. Kaveh, A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station. IEEE Trans. Veh. Technol. 51(3), 425–434 (2002) 41. J. Bian et al., A WINNER+ based 3-D non-stationary wideband MIMO channel model. IEEE Trans. Wirel. Commun. 17(3), 1755–1767 (2018)

150

6 A 3D Non-stationary Wideband Channel Model for MIMO V2V Tunnel. . .

42. D. Takaishi, Y. Kawamoto, H. Nishiyama, N. Kato, F. Ono, R. Miura, Virtual cell based resource allocation for efficient frequency utilization in unmanned aircraft systems. IEEE Trans. Veh. Technol. 67(4), 3495–3504 (2018) 43. P. Liu, D.W. Matolak, B. Ai, R. Sun, Path loss modeling for vehicle-to-vehicle communication on a slope. IEEE Trans. Veh. Technol. 63(6), 2954–2958 (2014) 44. X.F. Yin, X. Cheng, Propagation Channel Characterization, Parameter Estimation, and Modeling for Wireless Communications (Wiley-IEEE Press, Hoboken, Oct 2016) 45. S. Payami, F. Tufvesson, Channel measurements and analysis for very large array systems at 2.6 GHz, in 6th European Conference on Antennas and Propagation (EUCAP), Prague, Czech Republic (Mar 2012), pp. 433–437 46. N. Naz, D.D. Falconer, Temporal variations characterization for fixed wireless at 29.5 GHz, in Proceedings of IEEE Vehicular Technology Conference (VTC), Tokyo, Japan (May 2000), pp. 2178–2182 47. B. Chen, Z. Zhong, B. Ai, Stationarity intervals of time-varying channel in high speed railway scenario. J. China Commun. 9(8), 64–70 (2012)

Chapter 7

An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation Algorithm

7.1 Introduction To satisfy the efficient requirements of 5G wireless communication networks, a variety of advanced technologies have been introduced in the existing literature [1]. To be specific, V2V communications, which constitute one of the most potentially useful 5G research topics, have received great attention in recent years. The successful design and analysis of V2V communication systems requires investigation on the propagation characteristics between an MT and an MR in V2V channels. In general, realistic channel models provide effective means to approximate the propagation characteristics and serve as the basis for performance evaluation in general communication systems. Thus far, there have been a variety of studies of the V2V propagation characteristics between a transmitter and a receiver. GBSMs, which assume that all the interfering objects between a transmitter and receivers are located on a geometric shape, are widely used to describe V2V communication environments [2], mainly because the geometric model, which has the advantage of low complexity, specifies the mathematical model and the algorithms used for channel modeling that are applied to all communication environments. In 1986, the authors of [3] were the first to develop a geometric Rayleigh fading model for singleinput single-output (SISO) V2V communication environments. As demonstrated in [4], the ellipse model with an MT and MR located at the foci can be used to determine the waves of the same propagation path lengths in multipath channels. Therefore, it is reasonable to use an ellipse to describe the distribution of the interfering objects in roadside communication environments in V2V channels. In previous studies, the authors of [5] proposed a generic Ricean fading channel model with propagation rays with single and double interactions for MIMO V2V scenarios. The authors of [6] developed a geometric Ricean channel model for MIMO M2M environments. Furthermore, in [7] and [8], we proposed ellipse channel models for V2V communication environments, which investigated the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_7

151

152

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

channel characteristics for different model parameters and moving velocities and directions of the MT and MR. For the above channel models, the ellipse scattering region of the V2V communication environments, which can be determined by the major and minor axes of the ellipse models, is assumed to be known. However, because the practical vehicular scattering environments are complex and variable, it is impossible to accurately determine the values of the major and minor axes of the ellipse models. Therefore, it is important to propose an angular estimation algorithm to estimate the major and minor axes of ellipse models, which can be further used to determine the ellipse scattering region in V2V channels. The authors [5–8] focus on narrowband channel models, i.e., frequency nonselective channels, wherein all rays are subjected to a similar propagation delay. However, this scenario is not a realistic description of V2V communication environments. In [9], the authors conducted channel measurements in wideband and narrowband V2V channels; the results demonstrated that the propagation rays scattered by the roadside environments make different contributions to the V2V channel characteristics. Accordingly, it is important to analyze different taps of the wideband V2V channel model, i.e., frequency-selective channels. Cheng et al. [10] proposed multiple confocal ellipses to reflect vehicular roadside environments. In [10], the authors investigated the propagation characteristics for different time delays, i.e., per-tap channel statistics, which provide efficient solutions to investigate the propagation characteristics in wideband V2V channels. Furthermore, the authors in [11] developed deterministic and stochastic simulation models for wideband V2V channels. For the above-mentioned wideband V2V channels, the ellipse models, which correspond to the scattering regions for the different propagation delays, are assumed to be known in advance. In reality, this condition is too idealistic to reflect the real V2V communication environments. To overcome this challenge, it is important to estimate ellipse channel models for different taps. Consequently, an efficient angular estimation algorithm is indispensable. Thus far, a variety of angular estimation algorithms have been developed to analyze the performance of wireless communication systems [12–16]. Specifically, in [12], Xiao et al. proposed a novel triply selective Rayleigh channel model that incorporated a discrete-time MIMO communication system over space-, time-, and frequency-selective Rayleigh channels. To reduce the variation of time-averaged correlations of a fading realization, the authors of [13] improved Clarke’s model. In [14], Zheng et al. proposed a hardware implementation scheme for discretetime MIMO triply selective fading emulators, which uses a mixed parallel-serial structure to achieve the best trade-off of hardware usage and output speed. Gui et al. [15] proposed an algorithm with pre-estimation to reduce the dimensionality of the measurement matrix for direction-of-arrival (DoA) estimation. Furthermore, in [16], the authors proposed deep-learning-based schemes for achieving super-resolution DoA estimation and channel estimation in massive MIMO communication systems. To address the above issues, we present an estimated wideband geometry-based V2V channel model based on the AoD and AoA estimation algorithm, as illustrated in Fig. 7.1. The major contributions of this paper are outlined as follows:

7.2 System Model

153

Fig. 7.1 Geometric model parameters in the proposed estimated V2V channel model

(1) We propose a computationally efficient approach to estimate the ellipse scattering region in V2V channels. Then, we estimate the wideband ellipse channel models for different taps. (2) In the proposed estimated wideband geometry-based channel model, we estimate the V2V channel characteristics for different propagation delays, which have not been studied previously. (3) In the proposed model, the spatial cross-correlation functions (CCFs) for different taps are thoroughly investigated and compared to prior results. The results demonstrate that the proposed model has the ability to efficiently approximate the performances of realistic V2V communication systems.

7.2 System Model 7.2.1 Wideband Channel Model In multipath channels, an ellipse model typically forms the physical basis for modeling wideband channel models, i.e., frequency-selective channels. Here, the model introduces scatterers with identical delays on the same ellipse, whereas different ellipses represent the wideband channel characteristics. To be specific, as shown in Fig. 7.2, when the MT and MR are located in the focus of the ellipse and the communications between the MT and MR are confined to the l-th ellipse scattering region, every wave is exposed at the same discrete propagation delay τl = τ0 + lτ , l = 0, 1, 2, . . . , L − 1, where τ0 denotes the delay of direct LoS propagation, τ is an infinitesimal propagation delay, and L is the number of paths with different propagation delays. The number of paths l with different propagation delays in the ellipse models exactly corresponds to the number of delay elements.

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7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

Fig. 7.2 Illustration of the wideband ellipse channel models

For V2V scenarios, it is important to design different taps of wideband V2V channel models. As demonstrated in [10], the tap is strongly related to the delay resolution in V2V channels. Here, when we define al as the semimajor of the l-th ellipse, the semimajor of the (l + 1)-th ellipse can be derived as al+1 = al + cτ/2 with the speed of light c = 3 × 108 m/s.

7.2.2 Description of the Proposed Channel Model In this subsection, we provide a wideband geometry-based MIMO V2V channel model, as shown in Fig. 7.1, where the MT and MR are equipped with MT and MR ULA omnidirectional receiving antennas. In the model, we define the major of the l-th ellipse, on which the center points of the MT and MR are located, as the x-axis; we define the semiminor of the l-th ellipse as the y-axis. In poor V2V communication environments, the LoS components are typically blocked by interfering objects; therefore, the LoS components are very weak, while the NLoS components are dominant in the signal received at the MR. To sum up, the signal from the MT in the model impinges on the interfering obstacles (cluster) before reaching the MR. In the proposed model, we introduce multiple confocal ellipses to depict the roadside environments in V2V channels. For the transmitting and receiving antennas, let us define δT and δR as the antenna spacing elements at the transmitter and receiver, respectively, and ψT and ψR as the orientations of the transmitting and receiving antenna arrays relative to the x-axis, respectively. The semimajor and semiminor axes of the l-th ellipse are designated al and bl , respectively. The distance from the center point of the MT to that of the MR is designated D0 . Here, we suppose that Nl effective scatterers lie on the l-th confocal ellipse with the MT and MR located at the foci, and the nl -th (nl = 1, . . . , Nl ) scatterer is designated s (nl ) . The distances from the p-th (p = 1, 2, . . . , MT ) transmitting and q-th (q = 1, 2, . . . , MR ) receiving antenna to the scatterer s (nl ) are designated Dpnl and Dqnl , respectively.

7.3 AoD and AoA Estimation for the First Tap

155

In V2V environments, the MT and MR are both in motion, and we define vT and vR as the movement velocities of the MT and MR, respectively, and ϕT and ϕR as the movement directions of the MT and MR, respectively. The AoA of the wave (n ) traveling from an effective scatterer s (nl ) toward the MR is designated αR l . The AoD of the wave that impinges on the effective scatterer s (nl ) is designated αT(nl ) . For the proposed MIMO V2V  channel, the physical properties can be described by a matrix H(t) = hpq (t, τ ) M ×M of size MR × MT , where hpq (t, τ ) represents R T the complex channel impulse response (CIR) between the p-th antenna of the transmitting array and the q-th antenna of the receiving array, i.e., L 

hpq (t, τ ) =

  ωl hl,pq (t)δ τ − τl ,

(7.1)

l=1

where l denotes the tap number, L is the total number of the taps, and ωl and τl are the complex amplitude and the delay time of the l-th tap, respectively. In reality, the delay time τl in the proposed channel model is discrete, which corresponds to the time delay for different taps in wideband ellipse models [10]. Furthermore, hl,pq (t) denotes the complex fading envelope of the l-th tap, which can be expressed as hl,pq (t) = lim

Nl →∞

× ej



Nl  nl =1

2π λ vT t

1 j e Nl 



ϕ0 −2πfc τl,pq,n

cos αl,T −ϕT



× ej



2π λ vR t



cos αl,R −ϕR

 ,

(7.2)

where τl,pq,n denotes the travel times of the waves from the p-th antenna of the transmitting array and the q-th antenna of the receiving array. Here, fc denotes the carrier frequency and λ is the carrier wavelength. To proceed further with a stochastic description of the channel model in (7.2), we assume that the phase ϕ0 is an independent random variable, which has a uniform distribution in the interval from −π to π , i.e., ϕ0 ∼ [−π, π ).

7.3 AoD and AoA Estimation for the First Tap In general, when we aim to estimate the wideband ellipse scattering channel models for different taps, it is important to determine the major and minor axes of the ellipse models in advance. However, it seems impossible to estimate the ellipse models for every tap because of the high computational complexity of the estimation process. Note that the ellipse models for different taps can be derived based on the model parameters and the major and minor axes for the first tap, i.e., l = 1. In light of this condition, we propose a three-step approach to estimate the ellipse channel models for different taps. To be specific, we propose an approach that estimates the AoD and

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7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

AoA for the first tap. Then, based on the known distance between the center points of the MT and MR, we can estimate the major and minor axes of the ellipse model for the first tap. In this case, the ellipse scattering region for the first tap has been determined. Finally, we estimate the major and minor axes of the ellipse models for other taps (l ≥ 1) based on the estimated model parameters. For the first tap, let us define α1,T and α1,R as the AoD and AoA of the propagation path, respectively, D1,T and D1,R as the distances from the center points of the MT and MR to the scatterers, respectively, and Dpn1 and Dqn1 as the distances from the p-th transmitting and q-th receiving antenna to the scatterers, respectively. Here, the complex fading envelope can be calculated as N1 

h1,pq (t) = lim

N1 →∞

× ej

 1 j e N1

n1 =1

2π λ vT t





ϕ0 −2πfc τ1,pq,n

cos α1,T −ϕT



× ej

2π λ vR t





cos α1,R −ϕR

 ,

(7.3)

where τ1,pq,n denotes the delay time of the wave from the p-th antenna of the transmitting array and the q-th antenna of the receiving array for the first tap, which can be derived as   τ1,pq,n = Dpn1 + Dqn1 /c   1  2 × D1,T + kp2 − 2D1,T kp cos α1,T − ψT c    2 + k 2 − 2D + D1,R , 1,R kq cos α1,R − ψR q

=

(7.4)

where kp = (MT − 2p + 1)δT /2, kq = (MR − 2q + 1)δR /2, and D0 sin α1,R cot α1,T + cos α1,R

(7.5)

D0 sin α1,R . sin α1,R cos α1,T + cos α1,R sin α1,T

(7.6)

D1,R = D1,T =

The distance D0 is assumed to be known in advance; thus, the parameters D1,R and D1,T can be determined when the angular parameters α1,T and α1,R are estimated. In the proposed communication system, the received signal of the q-th antenna of the receiving array can be calculated as [12] yq (τ ) =

MT  p=1

xp (τ ) ∗ hpq (t, τ ),

(7.7)

7.3 AoD and AoA Estimation for the First Tap

157

where xp (τ ) denotes the p-th transmitted signal and ∗ is the convolution operation. In substituting (7.3) into (7.7), the received signal of the q-th receiving antenna for the first tap can be expressed as N1 MT  

yq (τ ) = lim

N1 →∞

×e



p=1 n1 =1

1 ω1 ej N1









j 2π λ vT t cos α1,T −ϕT

× ej

2π λ vR t

cos α1,R −ϕR



ϕ0 −2πfc τ1,pq,n



  xp τ − τ1 + nq (τ ),

(7.8)

where nq (τ ) denotes the complex noise of the q-th receiving antenna. As previously mentioned, there are MR receiving antennas in the proposed communication system; therefore, the received signal can be expressed as an MR -dimensional vector, i.e., [17]  T y(τ ) = y1 (τ ), y2 (τ ), . . . , yq (τ ), . . . , yMR (τ ) .

(7.9)

Therefore, y(τ ) = lim

N1 →∞

× ej ⎡

N1 MT   p=1 n1 =1

2π λ vT t





  1 ω1 xp τ − τ1 N1 



cos α1,T −ϕT +j 2π λ vR t cos α1,R −ϕR

  ⎤ ⎡ ⎤ ej ϕ0 −2πfc τ1,p1,n  n1 (τ ) ⎢ ⎥ ⎢ ej ϕ0 −2πfc τ1,p2,n ⎥ ⎢ n (τ ) ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ . ⎥ .. ⎢ ⎥ ⎢ . ⎥ . ⎢ . ⎥ ⎢ ⎥ ×⎢  ⎥+⎢ ⎥. ⎢ ej ϕ0 −2πfc τ1,pq,n ⎥ ⎢ nq (τ ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ .. ⎢ ⎥ ⎣ ⎦ . . ⎣  ⎦ (τ ) n MR ej ϕ0 −2πfc τ1,pMR ,n



(7.10)

The continuous received signal yq (τ ) can be converted to a discrete sequence of samples {yq (1T ), . . . , yq (kT ), . . . , yq (KT )} with no loss of information. Therefore, the received signal of the k-th (k = 1, 2, . . . , K) sequence of the q-th receiving antenna element can be expressed as

158

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . . MT  N 

yq (kT ) = lim

N →∞



p=1 n=1

1 ω1 ej N









× ej

2π λ vT t

cos α1,T −ϕT

× ej

2π λ vR t

cos α1,R −ϕR



ϕ0 −2πfc τ1,pq,n



  xp kT − τ1 + nq (kT ),

(7.11)

where nq (kT ) denotes the complex white Gaussian noise of the k-th sequence of the q-th antenna of the receiving array, i.e., nq (kT ) ∼ C N (0, σ 2 ). Therefore, the received signal in the proposed communication system can be expressed as T  y = y1 , y2 , . . . , yq , . . . , yMR .

(7.12)

Assume that y ∼ C N (μ, σ 2 IMR ×K ), where μ denotes the MR × K mean element. It is assumed that the elements in y are independent and identically distributed (i.i.d.). Then, according to Euler’s theorem, the PDF for the complex normal distribution can be expressed as   1 f y = MR ×K × exp 2 πσ ×

MR  K



 −

1 σ2

  yq (kT ) yq (kT ) − yq (kT ) + yq (kT )

q=1 k=1

×

MT 

    ω1 xp kT − τ1 cos ϕ0 − 2πfc τ1,pq,n

p=1

+

 MT

    ω1 xp kT − τ1 cos ϕ0 − 2πfc τ1,pq,n

2

p=1

+

 MT

    ω1 xp kT − τ1 sin ϕ0 − 2πfc τ1,pq,n

2  .

(7.13)

p=1

To estimate the angular parameters αˆ 1,T and αˆ 1,R for the first tap in the proposed V2V communication system, we can adopt a series of classic estimation solutions, i.e., the Newton–Raphson method and grid search method [18]. Therefore, the semimajor and semiminor axes of the ellipse model for the first tap can be estimated as

7.3 AoD and AoA Estimation for the First Tap

aˆ 1 =

 1 D0 × 2 sin αˆ 1,R cot αˆ 1,T + cos αˆ 1,R +

bˆ1 =

159



 D0 sin αˆ 1,R sin αˆ 1,R cos αˆ 1,T + cos αˆ 1,R sin αˆ 1,T

aˆ 12 − (D0 /2)2 .

(7.14)

(7.15)

Thus far, the ellipse scattering region for the first tap has been estimated. As demonstrated in [10], the delay resolution is approximately the inverse of the bandwidth; therefore, we assume that the delay resolution in the proposed model is 20 ns for 50 MHz. Based on the above estimated parameters aˆ 1 and bˆ1 , the semimajor and semiminor axes of the ellipse models for other taps can be estimated as 1 aˆ l = aˆ 1 + (l − 1)cτ 2  bˆl = aˆ l2 − (Dl /2)2 .

(7.16) (7.17)

Then, we can estimate the distances from the centers of the MT and MR to the scatterer s (nl ) as Dˆ l,T =

Dˆ l,R =

4aˆ l2 − D02

(7.18)

4aˆ l − 2D0 cos αT(nl ) 4aˆ l2 − D02 4aˆ l − 2D0 cos αR(nl )

.

(7.19)

Consequently, the distances from the p-th transmitting and q-th receiving antenna to the scatterer s (nl ) can be expressed as Dˆ pnl =

   2 + k 2 − 2D ˆ l,T kp cos α (nl ) − ψT Dˆ l,T p T

(7.20)

Dˆ qnl =

   2 + k 2 − 2D ˆ l,R kq cos α (nl ) − ψR . Dˆ l,R q R

(7.21)

(n ) (n ) The parameters Dˆ l,T , Dˆ l,R , Dˆ pnl , Dˆ qnl , αT l , and αR l are set to be variables in the proposed communication system. To be specific, the angular parameters (n ) (n ) αT l and αR l have non-uniform distributions in the interval from −π to π , i.e., (nl ) (nl ) αT , αR ∼ [−π, π ). Thus far, the ellipse models and propagation path lengths for other taps have been estimated. Then, we can investigate the propagation characteristics in the proposed model.

160

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

7.4 Channel Characteristics of the Proposed Model In wireless channels, the spatial CCFs can be used to measure the spatial correlation between two different propagation links. It was reported in [7] that the spatial CCF between the two propagation links from p-th transmitting antenna to q-th receiving antenna and from the p -th (p = 1, 2, . . . , MT ) transmitting antenna to the q  -th (q  = 1, 2, . . . , MR ) receiving antenna is defined as the correlation between the complex fading envelope. Therefore, the spatial CCF of the proposed model can be expressed as   ρhl,pq ;hl,p q  (τ  ) = E hl,pq (t) h∗l,p q  (t − τ  ) ,

(7.22)

where E[·] denotes the expectation operation and (·)∗ is the complex conjugate operation. Here, we assume that the E[·] applies only to the random phases ϕ0 . In substituting (7.2) into (7.22), the spatial CCF of the proposed model can be expressed as ρˆhl,pq ;hl,p q  (τ  ) = lim

Nl →∞

×e ×e

Nl 



nl =1

1 j 2π vT t cos e λ Nl 

(n )

l j 2π λ vR t cos αR −ϕR



(nl ) −ϕ

αT

 T





−j 2πfc Dˆ pnl +Dˆ qnl −Dˆ p n −Dˆ q  n l

 l

/c

.

(7.23)

 Nl

nl =1 1/Nl = 1 as Nl → ∞. Consequently, the discrete angular (nl ) parameters αT and αR(nl ) can be replaced by the continuous random variables (l) (l) (l) (l) αT and αR , respectively. Then, we can obtain 1/Nl = f (αT (R) )dαT (R) , where (l) f (αT (R) ) denotes the PDF of the AoD and AoA distributions. Here, we adopt von

Note that

Mises PDF to characterize the angular distribution, which can be expressed as f (α) =

1 ek cos(α−μ) , 2π I0 (k)

(7.24)

where I0 (·) denotes the zeroth-order modified Bessel function of the first kind, μ ∈ [−π, π ] is the mean value of the angular parameter α, and k(k ≥ 0) is an environment-related parameter. The von Mises PDF approximates a variety of scattering distributions in V2V communication environments. For example, we obtain f (α) = 1/(2π ) (isotropic scattering) as k = 0, while k = ∞ yields f (α) = δ(α − μ) (extremely non-isotropic scattering). For small k, this function approximates the cardioid PDF, while for large k, it resembles a Gaussian PDF with mean μ and standard derivation [19]. As a result, (7.23) can be rewritten as

7.5 Numerical Results and Discussion

161

1 ρˆ SBl SBl (τ ) = hl,pq ;hl,p q  2π I0 (k (l) ) 

× ej ×e

2π λ vR t





π −π



e

(l)

j 2π λ vT t cos αT −ϕT

(l)

cos αR −ϕR





−j 2πfc Dˆ pnl +Dˆ qnl −Dˆ p n −Dˆ q  n

× ek

l

(l) cos



(l)

(l)



αT (R) −μT (R)



d αT(l)(R) .

 l

/c

(7.25)

In substituting (7.24) into (7.25), the spatial CCF of the proposed model can be derived. Note that the expression above can be obtained by averaging over the random phases ϕ0 . Furthermore, the proposed spatial correlation is related to the transmitting/receiving antenna element spacings and the estimated parameters for the first tap.

7.5 Numerical Results and Discussion In this section, we first discuss how to estimate AoD and AoA for the first tap; then, we investigate the spatial CCFs of the proposed model.

7.5.1 Estimated Angular Parameters In the proposed V2V communication   system,  we define  the p-th transmitted signal as a cosine signal, i.e., xp kT − τ1 = cos kT − τ1 . Unless otherwise specified, the essential parameters in this section are obtained using fc = 5.9 GHz, ω1 = 10, σ 2 = 5, ϕT = ϕR = π/12, λ = 1, and MT = MR = 2. Furthermore, we set the true angular values as α1,T = π/4 and α1,R = 5π/6. In the proposed communication system, the mean squared error (MSE) performances of the AoD and AoA estimations for the first tap (l = 1) and other taps (l > 1) are provided in Figs.  7.3 and 7.4,respectively. The MSE of angular estimation is defined as E = |vtrue − vest |2 , where vtrue represents the exact angle in the proposed communication system and vest is its estimated counterpart. Figure 7.3 shows that the MSEs of the AoD and AoA estimations gradually decrease as K increases. Furthermore, the MSEs of the angular estimations increase as the distance D0 increases from 100 m to 200 m. In Fig. 7.4, we observe that the MSE performance has different behaviors at different taps. The MSE gradually increases as the tap number increases.

162

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

Fig. 7.3 Mean squared error performance of the AoD and AoA estimations for the first tap. (a) AoD estimation; (b) AoA estimation

7.5 Numerical Results and Discussion

163

Fig. 7.4 Mean squared error performance of the AoD and AoA estimations for other taps. (a) AoD estimation; (b) AoA estimation

164

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

7.5.2 Spatial CCFs By using (7.25), the spatial CCFs of the proposed model for different antenna spacings can be determined as shown in Figs. 7.5, 7.6, and 7.7. The figures demonstrate that the spatial correlation gradually decreases as the MR antenna spacing δR increases. Figure 7.5 shows that when the distance from the center point of the MT to that of the MR increases from D0 = 50 m to D0 = 500 m, the spatial correlation decreases gradually. Note also that when the distance D0 is less than 100 m, the changing curves of the spatial correlations are insignificant. Figure 7.6 illustrates the spatial CCFs of the proposed model for different movement directions (i.e., ϕT and ϕR ) of the MT and MR. When the MT and MR move toward each other, i.e., ϕT = 0 and ϕR = π , the spatial correlations are different from those of the MT and MR moving in opposite directions, i.e., ϕT = π and ϕR = 0. Furthermore, the spatial correlations increase gradually as the moving time t increases from 2 s to 4 s, which is consistent with the results in [6]. Figure 7.7 shows that the channel characteristics of the proposed model have different properties for different taps. To be specific, when the tap number increases from 2 to 10, the spatial correlation decreases gradually.

Fig. 7.5 Spatial CCFs of the proposed model for different distances between the center points of the MT and MR when vT = vR = 10 m/s and t = 2 s

7.5 Numerical Results and Discussion

165

Fig. 7.6 Spatial CCFs of the proposed model for different movement velocities and directions of the MT and MR when D0 = 200 m, vT = vR = 10 m/s, and t = 2 s

Fig. 7.7 Spatial CCFs of the proposed model for different movement velocities and directions of the MT and MR when D0 = 200 m and vT = vR = 10 m/s

166

7 An Estimated Wideband V2V Channel Model Using an AoD/AoA Estimation. . .

7.6 Conclusion In this chapter, we have provided an estimated wideband geometry-based channel model for vehicle-to-vehicle communication environments. Based on the known distance between the center points of the MT and MR, the AoD and AoA for the first tap are estimated. Then, we estimate the ellipse channel models for other taps based on the above estimated angular parameters and the geometric properties of the channel model. It has been demonstrated that the MSE performances of the AoD and AoA perform satisfactorily as the parameter K gradually increases. The proposed V2V channel characteristics have different properties for different propagation delays. The results demonstrate that the spatial CCFs of the proposed model are impacted by the movement directions of the MT and MR. Our research work therefore provides a new and efficient guidance for providing designing V2V communication systems in future wireless networks.

References 1. J.G. Andrews, S. Buzzi, W. Choi, S.V. Hanly, A. Lozano, A.C.K. Soong, J.C. Zhang, What will 5G be? IEEE J. Sel. Areas Commun. 32(6), 1065–1082 (2014) 2. M. Patzold, C. Wang, B.O. Hogstad, Two new sum-of-sinusoids based methods for the efficient generation of multiple uncorrelated Rayleigh fading waveforms. IEEE Trans. Wirel. Commun. 8(6), 3122–3131 (2009) 3. A.S. Akki, F. Haber, A statistical model for mobile-to-mobile land communication channel. IEEE Trans. Veh. Technol. 35(1), 2–7 (1986) 4. C. 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) 5. A.G. Zajic, G.L. Stuber, Space-time correlated mobile-to-mobile channels: modelling and simulation. IEEE Trans. Veh. Technol. 57(2), 715–726 (2008) 6. 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) 7. H. Jiang, Z. Zhang, L. Wu, J. Dang, A non-stationary geometry-based scattering vehicle-tovehicle MIMO channel model. IEEE Commun. Lett. 22(7), 1510–1513 (2018) 8. H. Jiang, Z. Zhang, J. Dang, L. Wu, Analysis of geometric multibounced virtual scattering channel model for dense urban street environments. IEEE Trans. Veh. Technol. 66(3), 1903– 1912 (2017) 9. I. Sen, D.W. Matolak, Vehicle-vehicle channel models for the 5-GHz band. IEEE Trans. Intell. Transp. Syst. 9(2), 235–245 (2008)

References

167

10. X. Cheng, Q. Yao, M. Wen, C. Wang, L. Song, B. Jiao, Wideband channel modeling and intercarrier interference cancellation for vehicle-to-vehicle communication systems. IEEE J. Sel. Areas Commun. 31(9), 434–448 (2013) 11. Y. Li, X. Cheng, N. Zhang, Deterministic and stochastic simulators for non-isotropic V2VMIMO wideband channels. China Commun. 15(7), 18–29 (2018) 12. C.S. Xiao, J.X. Wu, S.Y. Leong, Y. Zheng, K.B. Letaief, A discrete-time model for triply selective MIMO Rayleigh fading channels. IEEE Trans. Wirel. Commun. 3(5), 1678–1688 (2004) 13. C.S. Xiao, Y.R. Zheng, N.C. Beaulieu, Novel sum-of-sinusoids simulation models for Rayleigh and Rician fading channels. IEEE Trans. Wirel. Commun. 5(12), 3667–3679 (2006) 14. F. Ren, Y.R. Zheng, A novel emulator for discrete-time MIMO triply selective fading channels. IEEE Trans. Circuits Syst. I Regul. Pap. 57(9), 2542–2551 (2010) 15. B. Liu, G. Gui, S. Matsushita, L. Xu, Dimension-reduced direction-of-arrival estimation based on 2,1 -norm penalty. IEEE Access 6, 44433–44444 (2018) 16. H. Huang, J. Yang, H. Huang, Y. Song, G. Gui, Deep learning for super-resolution channel estimation and DOA estimation based massive MIMO system. IEEE Trans. Veh. Technol. 67(9), 8549–8560 (2018) 17. L. Wan, G. Han, J. Jiang, J.J.P.C. Rodrigues, N. Feng, T. Zhu, DOA estimation for coherently distributed sources considering circular and noncircular signals in massive MIMO systems. IEEE Syst. J. 11(1), 41–49 (2017) 18. L. Wan, G. Han, L. Shu, S. Chan, T. Zhu, The application of DOA estimation approach in patient tracking systems with high patient density. IEEE Trans. Ind. Inf. 12(6), 2353–2364 (2016) 19. A. Abdi, M. Kaveh, A space-time correlation model for multielement antenna systems in mobile fading channels. IEEE J. Sel. Areas Commun. 20(3), 550–560 (2002)

Chapter 8

3D Non-stationary Wideband UAV Channel Model for A2G Communications

8.1 Introduction Unmanned aerial vehicles (UAVs) are an emerging technology that have been widely used in the public and civil domains [1, 2]. In recent years, a variety of advanced technologies, e.g., massive MIMO and cooperative spectrum sensing [3, 4], have been introduced together with 5G wireless communication systems. To efficiently design communication systems and signal processing techniques for airto-ground (A2G) communications, it is important to have reliable statistical channel model to express the propagation properties between a transmitter and a receiver. In previous studies, GBSMs, which have the advantages of low computational complexity and high accuracy, are widely utilized to reflect the propagation properties of UAV communication environments. In reality, UAV transmitters are located at a relatively high altitude compared with the ground receiver [5, 6]. Therefore, it is important to introduce a 3D A2G channel model to reflect the propagation properties of UAV communications. To be specific, the authors in [7] introduced a 3D MIMO channel model to describe A2G communication scenarios, which assumed that the propagation paths are single interaction and the transmitter follows a cylindrical shape. The authors in [8] and [9] applied the 3D geometry-based MIMO channel modeling to reflect the UAV communications in A2G environments, which assumed that all effective scatterers around the ground receiver are distributed on the surface of a cylinder and a sphere, respectively. In [10], we developed a 3D geometric MIMO channel model in A2G communication environments, which adopted the elliptic-cylinder model to describe the distribution of effective scatterers around the UAV and MR. Furthermore, the authors in [11] proposed a 3D geometric channel model for A2G communication scenarios, which considered the air transmitter and ground receiver located at foci points of a virtual bounding ellipsoid. As demonstrated in [12], one of the main characteristics of V2V channels is the non-stationarity, which is caused by the motion of the MT and MR. The authors in [13] introduced the time-varying angular parameters of AAoD, EAoD, AAoA, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_8

169

170

8 3D Non-stationary Wideband UAV Channel Model for A2G Communications

and EAoA to describe the V2V channel non-stationarity. In [14], we introduced the time-varying parameters of propagation paths to reflect the channel non-stationarity. Furthermore, the authors in [15] proposed a 3D half-sphere model to describe the distribution of scatterers in UAV MIMO channels, which introduced the update and computation methods of time-varying model parameters to describe the channel non-stationarity. Accordingly, in A2G scenarios, it is important to investigate the impacts of the movements of UAV transmitter, mobile objects, and MR on the channel characteristics. To the best of authors’ knowledge, this work has not been endowed before. In real A2G scenarios, the channel non-stationarity because of the appearance and disappearance of effective scatterers, which is caused by the fast movement of the UAV [16]. In light of this, Cheng et al. [17] developed deterministic and stochastic simulation models to reflect the wideband wireless channels. The authors in [18] and [19] proposed 3D wideband stochastic multi-ellipsoid models for wireless MIMO communication channels, which assumed that interfering objects were distributed on the surface of the ellipsoids whose foci are at the center of transmitter and receiver ends. However, the models in [18] and [19] cannot be used to investigate the statistical properties for different propagation delays, which are meaningful for wideband channels, i.e., frequency-selective channels. Furthermore, Yuan et al. [13] developed a 3D wideband MIMO V2V channel, which introduced different cylinders represent the propagation properties for different time delays. However, the effective scatterers located on the cylinder did not have identical propagation delays, and meanwhile, the complexity of the channel model increased, which negated the advantage of ellipsoid models for frequency-selective channels. However, it is worth mentioning that the interesting obstacles are randomly distributed between the UAV transmitter and ground MR in A2G scenarios, which means that the waves from the UAV experience different propagation delays to the receiver. Furthermore, the authors in [20] proposed wideband twin-cluster channel models, which defined a large number of clusters with stochastic model parameters based on the measurements and simulations, to describe A2G scenarios. He et al. [21] presented a 3D cluster-based channel model to reflect V2V communication scenarios, where the azimuth and elevation angles of the cluster are assumed to follow the von Mises distribution, while the distances from the clusters to the transceivers are exponential distribution. In this section, we present a novel 3D MIMO channel model to reflect A2G communication scenarios, as illustrated in Fig. 8.1. The proposed model introduces the UAV transmitter and ground MR located at the foci points of the bounding ellipsoid; therefore, the propagation of waves from the UAV to MR experiences similar delays. The major contributions of this paper are outlined as follows: 1. To the best of our knowledge, we first introduce a 3D wideband ellipsoid channel model to describe A2G communication environments, which can be used to model effective scatterers with identical delays on the same ellipsoid, while different ellipsoids represent the statistical properties for different propagation delays.

8.2 System Model

171

Fig. 8.1 A 3D wideband ellipsoid channel model for A2G communication environments

2. The time-varying angular parameters of AAoD/EAoD and AAoA/EAoA are derived to properly describe the channel non-stationarity, which is caused by the motion of the UAV transmitter, cluster, and ground MR. 3. In the proposed model, the impacts of the moving directions and velocities of the UAV, cluster, and MR on the channel characteristics are firstly and jointly considered, which have not been studied in existing channel models. Furthermore, the key time-varying spatial cross-correlation functions (CCFs), temporal ACFs, Doppler PSDs, and PDPs for different time instants are thoroughly investigated.

8.2 System Model 8.2.1 Model Description In real A2G communication environments, the NLoS components are usually blocked by mobile birds, airship, etc.; therefore, the LoS components are very weak, while the NLoS components are dominant in the signal received at the MR. Note that the MR is on the earth surface, it is improper to consider the ground reflection in the proposed model. It is worth mentioning that the UAV transmitter

172

8 3D Non-stationary Wideband UAV Channel Model for A2G Communications

Fig. 8.2 Proposed 3D MIMO channel model for A2G communication scenarios

mainly flies at low altitude; therefore, the signals received at the MR are reflected by the interfering obstacles (clusters) in the air. However, if we consider the high altitude A2G communication environments, e.g., multiple kilometers above ground, it is important to investigate the radio wave refracts in the atmosphere since the air density changes over altitude, which will be our future work. In this section, let us consider a novel 3D UAV MIMO geometric model to reflect A2G channels, where the UAV transmitter and ground MR are equipped with MT and MR ULA omnidirectional receive antennas, respectively, as illustrated in Fig. 8.2. In the preliminary stage, we assume that the UAV and MR are static, the line connecting the center point of the MR and the projection point of the center point of the UAV in the horizontal plane is defined as the x-axis. For the transmit and receive antenna elements, we define δT and δR as the antenna spacing elements at the UAV transmitter and ground receiver, respectively, ψT (ψR ) and θT (θR ) as the orientations of the transmit (receive) antenna array relative to the x-axis and to the ground plane, respectively. Note that when the UAV transmitter moves from position P1 to P2 , cluster moves from position P3 to P4 , and the MR moves from position P5 to P6 , the channel model varies over time, and therefore the time-varying model

8.2 System Model

173

Fig. 8.3 Geometric angles and propagation path lengths of the UAV transmitter and ground MR relative to the mobile cluster in the proposed model

parameters should be derived to describe the channel non-stationarity. It is assumed that the mobile cluster is fixed as a reference point and therefore, the equivalent channel model is shown in Fig. 8.3. Then, we define vT /c and vR/c as the moving velocities of the UAV and MR relative to the cluster, respectively, ϕT /c (ϕR/c ) and γT /c (γR/c ) as the moving directions of the UAV relative to the cluster in the azimuth and elevation planes, respectively. Furthermore, we define αT , (t) and βT , (t) as the time-varying angular parameters of AAoD and EAoD of the -th propagation path that impinge on the cluster, αR, (t) and βR, (t) as the time-varying angular parameters of AAoA and EAoA of the -th propagation path traveling from the cluster.

8.2.2 Channel Impulse Response In the proposed model, the waves from the UAV impinge on the large numbers of clusters (obstacles) before reaching the MR. For the proposed MIMO

174

8 3D Non-stationary Wideband UAV Channel Model for A2G Communications

the physical properties can be represented by a matrix H (t) = channel model,  hpq (t, τ ) M ×M of size MR × MT , where hpq (t, τ ) denotes the complex impulse R T response (CIR) between the p-th (p = 1, 2, . . . , MT ) transmit antenna and q-th (q = 1, 2, . . . , MR ) receive antenna, i.e., hpq (t, τ ) =

N (t) 

  hpq,n (t)δ τ − τn (t) .

(8.1)

n=1

In (8.1), hpq,n (t) represents a narrowband process where all the L(t) sub-paths within each of the N(t) clusters are irresolvable rays and have the identical propagation delay τn (t), Pn is the power of the n-th cluster associated with the delay τn (t). Here, the complex fading envelope hpq,n (t) can be expressed as  L(t)      Pn j ϕ0 −2πfc DTp, (t)+DRq, (t) /c e hpq,n (t) = L(t)

=1

















× ej

2π λ vT /c t

cos αT , (t)−ϕT /c cos βT , (t)−γT /c

× ej

2π λ vR/c t

cos αR, (t)−ϕR/c cos βR, (t)−γR/c ,

(8.2)

where fc represents the carrier frequency, λ is the carrier wavelength, and c is the speed of light. Here, we assume that the phase ϕ0 is an independent random variable, which has a uniform distribution in the interval from −π to π , i.e., ϕ0 ∼ [−π, π ). It is worth mentioning that the second and third terms in formula (8.2) are Doppler frequency components. Furthermore, the DTp, (t) and DRq, (t) are the distances from the p-th transmit and q-th receive antenna to the cluster, respectively, which can be expressed as  2 DTp, (t) = DT , (t) sin βT , (t) − kp sin θT 2  + D T , (t) cos βT , (t) cos αT , (t) − kp cos θT cos ψT 2 1/2  + D T , (t) cos βT , (t) sin αT , (t) − kp cos θT sin ψT DRq, (t) =

(8.3)

 2 DR, (t) sin βR, (t) − kq sin θR 2  + D R, (t) cos βR, (t) cos αR, (t) − kq cos θR cos ψR 2 1/2  + D R, (t) cos βR, (t) sin αR, (t) − kq cos θR sin ψR , (8.4)

where kp = (MT − 2p + 1)δT /2 and kq = (MR − 2q + 1)δR /2. Parameters DT , (t) and DR, (t) denote the distances from the center points of the UAV and MR to the

8.2 System Model

175

cluster, respectively, which close-form expressions can be seen in our previous work in [14], we omit them here for brevity.

8.2.3 Time-Varying Angular Parameters and Propagation Paths In the proposed model, we define vT as the moving velocity of the UAV transmitter in the 3D space, ϕT and γT as the moving directions of the UAV in the azimuth and elevation planes, respectively. For the mobile cluster, we define vu as the moving velocity, while ϕu and γu as the moving directions in the horizontal and vertical planes, respectively. However, at the ground receiver, we define vR and ϕR as the moving velocity and direction, respectively. Based on the principle of relative motion, the velocity of the UAV transmitter relative to the cluster can be derived as  2 vT /c = vT cos γT cos ϕT − vc cos γc cos ϕc 2  + vT cos γT sin ϕT − vc cos γc sin ϕc 2 1/2  + vT sin γT − vc sin γc .

(8.5)

While the azimuth and elevation moving angles ϕT /u and γT /u can be, respectively, derived as ϕT /c = arctan γT /c = arccot ×



vT cos γT sin ϕT − vc cos γc sin ϕc vR cos γT cos ϕT − vc cos γc cos ϕc

(8.6)

1 vT sin γT − vc sin γc

 2 vT cos γT sin ϕT − vc cos γc sin ϕc

2 1/2   . + vR cos γT cos ϕT − vc cos γc cos ϕc

(8.7)

However, at the MR, the velocity relative to the cluster can be expressed as vR/c =

 2  2 vR cos ϕR − vc cos γc cos ϕc + vc sin γc 2 1/2  + vR sin ϕR − vc cos γc sin ϕc .

(8.8)

176

8 3D Non-stationary Wideband UAV Channel Model for A2G Communications

Then, the relative moving angles ϕR/c and γR/c can be, respectively, derived as ϕR/c = arctan

γR/c = arccot ×



vR sin ϕR − vc cos γc sin ϕc vR cos ϕR − vc cos γc cos ϕc

(8.9)

1 −vc sin γc

 2 vR sin ϕR − vc cos γc sin ϕc

2 1/2   . + vR cos ϕR − vc cos γc cos ϕc

(8.10)

To capture the non-stationarity of wireless channels, many solutions have been introduced in previous literatures. To be specific, the authors in [13] and [14], respectively, derive the time-varying angular parameters and propagation paths to describe the V2V channel non-stationarity. Here, let us define α ,R (t0 ), β ,R (t0 ), α ,T (t0 ), and β ,T (t0 ) as the angular parameters of AAoA, EAoA, AAoD, and EAoD at the beginning time of the movement (t = t0 ), respectively. Then, the time-varying angular parameters αR, (t) and βR, (t) can be expressed as αR, (t) = arctan

βR, (t) = arctan 

F1 sin αR, (t0 ) − F2 sin ϕR/c F1 cos αR, (t0 ) − F2 cos ϕR/c

DR, (t0 ) sin βR, (t0 ) − vR/c t sin γR/c  , F12 + F22 − 2F1 F2 cos αR, (t0 ) − ϕR/c

(8.11)

(8.12)

where F1 = DR, (t0 ) cos βR, (t0 ) and F2 = vR/c t cos γR/c . Similarly, the time-varying angular parameters αT , (t) and βT , (t) can be derived as αT , (t) = arctan

F3 sin αT , (t0 ) − F4 sin ϕT /c F3 cos αT , (t0 ) − F4 cos ϕT /c

DT , (t0 ) sin βT , (t0 ) − vc t sin γT /c βT , (t) = arctan  ,  F32 + F42 − 2F3 F4 cos αT , (t0 ) − ϕT /c where F3 = DT , (t0 ) cos βT , (t0 ) and F4 = vT /u t cos γT /u .

(8.13)

(8.14)

8.3 Channel Statistics of the Proposed Model

177

Subsequently, the time-varying distances from the p-th transmit and q-th receive antenna to the cluster are, respectively, expressed as D ,T ,p (t) =

 2 DT , (t0 ) sin βT , (t0 ) − vT /u t sin γT /u  1/2  + F32 + F42 − 2F3 F4 cos αT , (t0 ) − ϕT /u

D ,R,q (t) =

(8.15)

 2 DR, (t0 ) sin βR, (t0 ) − vR/u t sin γR/u  1/2  + F12 + F22 − 2F1 F2 cos αR, (t0 ) − ϕR/u .

(8.16)

Thus far, the time-varying angular parameters and propagation paths of the proposed model have been derived.

8.3 Channel Statistics of the Proposed Model For the proposed communication system, the channel properties are completely characterized by the CIRs in (8.1). It is worth mentioning that the spatial CCF between the two propagation links from p-th transmit antenna to q-th receive antenna and from p -th (p = 1, 2, . . . , MT ) transmit antenna to q  -th (q  = 1, 2, . . . , MR ) receive antenna is defined as the correlation between the complex fading envelope, i.e.,   ρpq,p q  ,n (t, τ  ) = E hpq,n (t) h∗p q  ,n (t − τ  ) ,

(8.17)

where E[·] denotes the expectation operation and (·)∗ is the complex conjugate operation. In substituting (8.2) into (8.17), the spatial CCFs in the proposed model can be derived. Furthermore, the temporal ACFs, which can be used to investigate the correlation properties of hpq,n (t) at two different time instants, can be derived by   rpq,n (t, Δt) = E hpq,n (t) h∗pq,n (t + Δt)

(8.18)

In substituting (8.11)–(8.14) into (8.18), the proposed temporal ACF with respect to the different moving features of the cluster is derived. Furthermore, the Doppler PSD of the proposed channel model can be obtained by the Fourier transform of the temporal ACF rpq,n (t, Δt) with respect to the time interval Δt.

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The PDP gives the intensity of a signal received through a wireless channel as a function of propagation delay, which can be measured by the spatial average of the CIR. In the proposed model, the PDP can be obtained as  2   Phpq (t, τ ) = hpq (t, τ ) .

(8.19)

Obviously, the PDP of the proposed model is time-variant on account of the motion of the UAV transmitter, cluster, and ground MR.

8.4 Numerical Results and Discussions Figure 8.4 shows the spatial CCFs of the proposed model for different receive antenna angles. It is obvious that the spatial correlation gradually decreases as the spacing between two adjacent receive antenna elements increases, which is in agreement with the measurements in [22]. Furthermore, when the receive antenna angles ψR and θR increase, the spatial correlation decreases slowly.

Fig. 8.4 Spatial CCFs of the proposed model with respect to the different receive antenna angles when ψT = θT = π/6 and MT = MR = 8

8.4 Numerical Results and Discussions

179

Fig. 8.5 Temporal ACFs of the proposed model with respect to the different heights of the UAV when δT = δR = 0.5λ, ψT = ψR = π/3, θT = θR = π/4, and MT = MR = 8

It was reported in [23] that the heights of the UAV transmitter have great influences on the A2G channel characteristics. In Fig. 8.5, we notice that the temporal correlation gradually decreases as the time delay Δt increases, which is in agreement with the results in [14]. Furthermore, when the distance from the UAV to the ground increases, the temporal correlation decreases slowly. The impacts of the moving velocities of the UAV transmitter, cluster, and ground MR on the proposed temporal ACFs are investigated in Fig. 8.6. It is obvious that the temporal correlation gradually decreases when the velocities of the UAV and MR (vT and vR ) increase. Furthermore, when the velocity of the mobile cluster vu increases from 5 m/s to 10 m/s, the temporal correlation decreases rapidly [24]. Figure 8.7 illustrates the Doppler PSDs of the proposed model at different moving time instants. It can be observed that the proposed Doppler spectrum fits very well with the prior results in [25], verifying the correctness of the above derivations and simulations. In Fig. 8.8, we notice that when the MR moves away the origin point of the coordinate, i.e., ϕR = 0, the values of the proposed PDPs gradually decrease as the time delay τ increases. Furthermore, the PDPs of the proposed model vary more heavily as the velocity of the MR increases from 5 m/s to 25 m/s. The analysis above

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Fig. 8.6 Temporal ACFs of the proposed model with respect to the different moving velocities of the UAV and MR when δT = δR = 0.5λ, ψT = ψR = π/3, θT = θR = π/4, MT = MR = 8, ϕT = γT = π/6, ϕu = γu = π/6, ϕR = π/6, and vu = 5 m/s

fits very well with the results in [26], demonstrating the accuracy of the derivations of the proposed PDPs.

8.5 Conclusion In this chapter, we have provided a novel 3D MIMO channel model to reflect A2G communication scenarios. The time-varying angular parameters of AAoD/EAoD and AAoA/EAoA are derived to properly describe the channel non-stationarity. Numerical results have demonstrated that the receive antenna angles have great influences on the theoretical propagation properties of the A2G channel model. The temporal ACFs are impacted by the moving directions and velocities of the UAV transmitter, cluster, and ground MR. Furthermore, the Doppler PSDs have different properties at different moving time instants. The above observations, overall, are able to efficiently design the system of A2G communications.

8.5 Conclusion

181

Fig. 8.7 Doppler PSDs of the proposed model with respect to the moving time instants of the UAV and MR when δT = δR = 0.5λ, ψT = ψR = π/3, θT = θR = π/4, and MT = MR = 8

Fig. 8.8 PDPs of the proposed model with respect to the different moving velocities of the MR when vT = 0, vc = 0, and ϕR = 0

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References 1. L. Gupta, R. Jain, G. Vaszkun, Survey of important issues in UAV communication networks. IEEE Commun. Surv. Tutor. 18(2), 1123–1152 (2016) 2. A.A. Khuwaja, Y. Chen, N. Zhao, M. Alouini, P. Dobbins, A survey of channel modeling for UAV communications. IEEE Commun. Surv. Tutor. 20(4), 2804–2821 (2018); Fourthquarter 3. J. Zhang, L. Dai, X. Li, Y. Liu, L. Hanzo, On low-resolution ADCs in practical 5G millimeterwave massive MIMO systems. IEEE Commun. Mag. 56(7), 205–211 (2018) 4. X. Liu, M. Jia, X. Zhang, W. Lu, A novel multi-channel internet of things based on dynamic spectrum sharing in 5G communication. IEEE Internet Things J. https://doi.org/10.1109/JIOT. 2018.2847731 5. Y. Zhou, N. Cheng, N. Lu, X.S. Shen, Multi-UAV-aided networks: aerial-ground cooperative vehicular networking architecture. IEEE Veh. Technol. Mag. 10(4), 36–44 (2015) 6. J. Zhang, C. Pan, F. Pei, G. Liu, X. Cheng, Three-dimensional fading channel models: a survey of elevation angle research. IEEE Commun. Mag. 52(6), 218–226 (2014) 7. X. Gao, Y. Hu, Analysis of unmanned aerial vehicle MIMO channel capacity based on aircraft attitude. WSEAS Trans. Inform. Sci. Appl. 10(2), 58–67 (2013) 8. L. Zeng, X. Cheng, C. Wang, X. Yin, A 3D geometry-based stochastic channel model for UAVMIMO channels, in IEEE Wireless Communications and Networking Conference (WCNC) (March 2017), pp. 1–5 9. K. Jin, X. Cheng, X. Ge, X. Yin, Three dimensional modeling and space-time correlation for UAV channels, in IEEE 85th Vehicular Technology Conference (VTC Spring) (June 2017), pp. 1–5 10. H. Jiang, Z. Zhang, L. Wu, J. Dang, Three-dimensional geometry-based UAV-MIMO channel modeling for A2G communication environments. IEEE Commun. Lett. 22(7), 1438–1441 (2018) 11. S.M. Gulfam, J. Syed, M.N. Patwary, M. Abdel-Maguid, On the spatial characterization of 3-D air-to-ground radio communication channels, in IEEE International Conference on Communications (ICC) (June 2015), pp. 2924–2930 12. C. 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) 13. Y. Yuan, C. 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) 14. H. Jiang, Z. Zhang, J. Dang, L. Wu, A novel 3-D massive MIMO channel model for vehicleto-vehicle communication environments. IEEE Trans. Commun. 66(1), 79–90 (2018) 15. K. Jiang, X. Chen, Q. Zhu, W. Zhong, Y. Wang, X. Yu, B. Chen, A geometry-based 3D nonstationary UAV-MIMO channel model allowing 3D arbitrary trajectories, in 10th International Conference on Wireless Communications and Signal Processing (WCSP) (Oct 2018), pp. 1–6 16. X. Cheng, Y. Li, A 3D geometry-based stochastic model for UAV-MIMO wideband nonstationary channels. IEEE Internet Things J. 6, 1–1 (2018) 17. Y. Li, X. Cheng, New deterministic and statistical simulation models for non-isotropic UAVMIMO channels, in 9th International Conference on Wireless Communications and Signal Processing (WCSP) (Oct 2017), pp. 1–6 18. J. Chen, S. Wu, S. Liu, C. Wang, W. Wang, On the 3-D MIMO channel model based on regularshaped geometry-based stochastic model, in 2015 International Symposium on Antennas and Propagation (ISAP) (Nov 2015), pp. 1–4 19. L. Bai, C. Wang, S. Wu, H. Wang, Y. Yang, A 3-D wideband multi-confocal ellipsoid model for wireless MIMO communication channels, in IEEE International Conference on Communications (ICC) (May 2016), pp. 1–6 20. S. Wu, C. Wang, e.M. Aggoune, M.M. Alwakeel, Y. He, A non-stationary 3-D wideband twincluster model for 5G massive MIMO channels. IEEE J. Sel. Areas Commun. 32(6), 1207–1218 (2014)

References

183

21. Y. Li, R. He, S. Lin, K. Guan, D. He, Q. Wang, Z. Zhong, Cluster-based nonstationary channel modeling for vehicle-to-vehicle communications. IEEE Antennas Wirel. Propag. Lett. 16, 1419–1422 (2017) 22. S. Payami, F. Tufvesson, Channel measurements and analysis for very large array systems at 2.6 GHz, in 6th European Conference on Antennas and Propagation (EUCAP) (March 2012), pp. 433–437 23. E. Yanmaz, R. Kuschnig, C. Bettstetter, Channel measurements over 802.11a-based UAV-toground links, in IEEE GLOBECOM Workshops (GC Wkshps) (Dec 2011), pp. 1280–1284 24. X. Liang, X. Zhao, S. Li, Q. Wang, J. Li, A non-stationary geometry-based scattering model for street vehicle-to-vehicle wideband MIMO channels, in IEEE 26th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC) (Aug 2015), pp. 2239–2243 25. J. Bian, J. Sun, C. Wang, R. Feng, J. Huang, Y. Yang, M. Zhang, A winner+ based 3-D nonstationary wideband MIMO channel model. IEEE Trans. Wirel. Commun. 17(3), 1755–1767 (2018) 26. S. Kim, A. Zajic, Statistical modeling of THz scatter channels, in 9th European Conference on Antennas and Propagation (EuCAP) (April 2015), pp. 1–5

Chapter 9

Summary

In this chapter, we present a summary of main ideas of this book and discuss the future research directions about 5G channel modeling [1].

9.1 Summary of the Book The study of the 5G and B5G wireless communications is carried out by setting up a practical system and analyzing its performance or by making simulations for the communication systems over wireless channels [2, 3]. The development of wireless channel modeling in 5G/B5G communications is of great significance for the diversity schemes, wideband wireless communication systems, and MIMO techniques. However, in reality, the design and evaluation of 5G/B5G communication system is a time-consuming and expensive task. In light of this, the researchers have proposed a variety of channel models to efficiently and effectively study the statistical propagation properties [4]. The following summarizes the main contributions of this book: • An overview of current practices in V2V channel modeling is given. First, we investigate the statistical propagation properties, i.e., spatial CFs and Doppler PSDs. Then, we compare V2V channel models with conventional F2M cellular channel models to determine their fundamental distinctions. Finally, we briefly discuss the recent developments on V2V channel models for 5G systems. • A geometry-based statistical channel model for radio propagation environments is proposed. In the model, we assume that each multipath component of the propagating signal undergoes only one bounce traveling from the transmitter to the receiver and that scattering objects are located according to Gaussian and exponential spatial distributions, and a new scatterer distribution is proposed as a trade-off between the outdoor and the indoor propagation environments. Using the channel model, we analyze the effects of directional antennas at the base © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6_9

185

186

•

•

•

•

•

•

9 Summary

station on the Doppler spectrum of a mobile station due to its motion and the performance of its MIMO systems. A 3D scattering channel model for land mobile systems is proposed. The model considers a base station located at the center of a 3D semi-spheroid-shaped scattering region and a MS located within the region. Using this channel model, we first derive the closed-form expression for the joint and marginal PDFs of the AoA and ToA measured at the MS corresponding to the azimuth and elevation angles. Then, we derive an expression for the Doppler spectra distribution due to the motion of the MSs. A visual scattering channel model for C2C mobile radio environments is proposed. The signals received at the MR from the MT are assumed to experience multi-bounced propagation paths. In the model, we separate the multi-bounced propagation paths into the odd- and even-numbered-bounced propagations. General formulations of the marginal PDFs of the AoD and AoA have been derived for the above two propagation conditions, respectively. From the proposed model, we derive an expression for the Doppler frequency due to the relative motion between the MT and MR, which broadens the research of the proposed visual street scattering channel model. A 3D vehicle massive MIMO antenna array model for V2V communication environments is proposed. A spherical wavefront is assumed in the proposed model instead of the plane wavefront assumption used in the conventional MIMO channel model. In the proposed model, we first derive the closed-form expressions for the joint and marginal PDFs of the AoD and AoA in the azimuth and elevation planes. We additionally analyze the time and frequency crosscorrelation functions for the different propagation paths. A 3D wideband geometry-based channel model is proposed for MIMO V2V communications in tunnel environments. In the model, we introduce a twocylinder model to describe moving vehicles, as well as multiple confocal semi-ellipsoid models to depict internal surfaces of tunnel walls. The received signal is constructed as a sum of direct LoS propagations, rays with single and double interactions. The movement between the MT and MR results in time-varying geometric statistics that make our channel model non-stationary. Using this channel model, we investigate the proposed channel characteristics for different propagation delays. An estimated wideband geometry-based channel model is proposed for V2V communication environments. First, we estimate the AoD and AoA for the first tap. In this case, the ellipse scattering region for the first tap can be determined. Then, we estimate the ellipse channel models for other taps based on the estimated model parameters for the first tap. Furthermore, the spatial CCFs are derived and thoroughly investigated for different propagation delays. A novel 3D cluster-based MIMO channel model is proposed to describe the A2G communication environments. The model introduces the UAV transmitter and ground MR located at the foci points of the boundary ellipsoid, while different ellipsoids represent the propagation properties for different time delays. In light of this, we are able to investigate the propagation properties of the A2G channel

9.2 Future Research Directions

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model for different time delays. Furthermore, the time-varying parameters of AAoD, EAoD, AAoA, and EAoA are derived to properly describe the channel non-stationarity, which is caused by the motion of the UAV transmitter, cluster, and MR. The impacts of the movement properties of the cluster in both the azimuth and elevation planes are investigated on the channel characteristics.

9.2 Future Research Directions Although research on V2V channel modeling has been widely developed, there are still many research and implementation challenges, especially when considering 5G technologies [5, 6]. In this section, we discuss some of the most relevant challenges and outline potential solutions. • Impacts of Moving Scatterers on Channel Characteristics. Besides moving vehicles on the road, the propagation characteristics in V2V communications can be affected by moving obstacles in the air. In such situations, the impact of elevation angles and motion properties of these scatterers should be investigated on the V2V channel characteristics. Hence, when considering the motion among the MT, MR, and mobile scatterers in 3D space, it is essential to derive timevarying path and angular parameters of V2V channel models to accurately characterize the vehicular environments, which further extend the models that only consider the motion between the MT and MR. In addition, when incorporating aerial UAVs into V2V communications, the models could be extended to include vehicular networks aided by multiple UAVs. Such aiding system is composed of aerial and ground vehicular subnetworks. The resulting V2V channel models should then consider multi-dimensional channel characterizations in the time, frequency, and spatial domains. • Mixed V2V Channel Propagation Scenarios. Vehicles in motion encounter a plethora of complicated scenarios including a variety of obstacles such as city blocks, expressways, tunnels, and bridges. Vehicular communications can be described by combinations of one or more typical V2V channel models. Currently, however, V2V channel models fail to accurately reflect the characteristics of wireless channels in mixed vehicular environments. Hence, future developments must consider combined GBSMs for channels to support a smooth evolution across switching environments. • Estimation of V2V Scattering Region. For the geometry-based V2V channel models, the scattering region of communication environments can be determined by the semi-major and semi-minor axes of ellipse models, which are assumed to be known in previous studies. However, in reality, the practical vehicular scattering environments are complicated and changeable, it is improper to accurately determine the ellipse models in advance. In light of this, it is important to introduce the AoD and AoA estimation algorithm to estimate the values of

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semi-major and semi-minor axes of ellipse models, which can be further used to determine the ellipse scattering region in V2V channels. • Intelligent V2V Communications. As 5G technology will imply big data, the performance of V2V communication systems can be analyzed using channel measurements. Meanwhile, artificial intelligence technologies, e.g., deep learning (DL) and deep reinforcement learning (DRL), will apparently become the mainstream for developments in future wireless networks, essential steps to unveil V2V channel parameters will include data mining. Furthermore, prediction for intelligent V2V communication systems will likely be based on the machine learning algorithms. Therefore, several V2V channel measurements in different scenarios should be conducted.

References 1. C. Wang, J. Bian, J. Sun, W. Zhang, M. Zhang, A survey of 5G channel measurements and models. IEEE Commun. Surv. Tutorials 20(4), 3142–3168 (2018) 2. C. Wang, A. Ghazal, B. Ai, Y. Liu, P. Fan, Channel measurements and models for high-speed train communication systems: a survey. IEEE Commun. Surv. Tutorials 18(2), 974–987 (2016) 3. M. Yang, B. Ai, R. He, L. Chen, X. Li, J. Li, B. Zhang, C. Huang, Z. Zhong, A clusterbased three-dimensional channel model for vehicle-to-vehicle communications. IEEE Trans. Veh. Technol. 68(6), 5208–5220 (2019) 4. A. Bodi, J. Zhang, J. Wang, C. Gentile, Physical-layer analysis of IEEE 802.11ay based on a fading channel model from mobile measurements, in IEEE International Conference on Communications (ICC), Shanghai, China (2019), pp. 1–7 5. X. Zhao, F. Du, S. Geng, N. Sun, Y. Zhang, Z. Fu, G. Wang, Neural network and GBSM based time-varying and stochastic channel modeling for 5G millimeter wave communications. China Commun. 16(6), 80–90 (2019) 6. S.A. Busari, K.M. Saidul Huq, S. Mumtaz, J. Rodriguez, Impact of 3D channel modeling for Ultra-High speed beyond-5G networks, in IEEE GLOBECOM Workshops (GC Wkshps), Abu Dhabi, United Arab Emirates (2018), pp. 1–6

Index

A ACF, see Auto-correlation function (ACF) Air-to-ground (A2G) communication environments channel characteristics, 179 5G wireless communication systems, 169 NLoS components, 171 scenarios, 169, 170 3D MIMO channel model, 169, 170, 172 3D UAV MIMO geometric model, 172 3D wideband ellipsoid channel model, 170, 171 Angle of arrival (AoA) and AoD (see Angle of departure (AoD)) geometry-based statistical channel model, vi at MS, 22–24, 51 PDFs, vi, 51, 52 probability density functions, 47–48 and ToA, 22, 30–33, 38, 41, 43, 55–58 Angle of departure (AoD) and AoA estimation algorithms angular parameters, 158 ellipse scattering, 155, 156, 159 MT and MR, 156 propagation path, 156 received signal, 156–158 distribution, 80–83 geometrical modeling, 65 marginal PDF, 75–76 MIMO V2V channel modeling, 2 non-stationary V2V channel model, 89 PDF, 75–76 spatial characteristics, 74 statistical analysis, 104–107

3GPP, 15 time-varying angular parameters, 173 V2V communication environments, vii AoA/ToA PDFs azimuth plane, 55 closed-form expression, 56 directional antenna, 55 elevation plane, 56, 57 ES/RP, 19–21 joint and marginal PDFs, 30–33 Artificial intelligence technologies, 188 Auto-correlation function (ACF), 130 Azimuth angle of arrival (AAoA) interfering object, 123 statistics, 138–139 time-varying parameters, 125 Azimuth angle of departure (AAoD) MIMO V2V channel, 11 statistical distribution, 9 time-varying parameters, 125, 171, 173, 180

B Beamforming technology, 105

C Cartesian coordinate system, 97 Car-to-car (C2C) Doppler frequency, 66, 68 mobile radio environments, 186 next generation, mobile communication technology, 65 scattering region, 74

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 H. Jiang, G. Gui, Channel Modeling in 5G Wireless Communication Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-32869-6

189

190 Car-to-car (C2C) (cont.) urban street environments, 74 visual scattering channel model, vi, 67, 186 Channel characteristics CIRs, 177 moving vehicles, 133–136 PDP, 178 spatial CCFs, 160, 161, 177 statistics, 177–178 theoretical Doppler PSD, 131 frequency CFs, 130 PDP, 131 spatial CFs, 127–130 stationary interval, 132 von Mises PDF, 160 Channel non-stationarity A2G scenarios, 170 time-varying parameters, 170–172, 180 V2V, 166, 170, 176 wireless channels, 176 Clarke’s model, 110 Complex impulse response (CIR) energy-related parameters, 123 impulse response, 124 propagation delays, 124 semi-ellipsoid model, 122–125 unit transmit power, 121 Correlation based stochastic models (CBSMs), 9 Cross-correlation functions (CCFs), 153, 160, 163–166, 171, 177, 178, 186

D Deep learning (DL), 188 Deep reinforcement learning (DRL), 188 Development in V2V model categories, 9 GBSMs, 10–11 MIMO requirements, 10 narrowband and wideband, 11–12 stochastic models, 9 Directional antenna AoAs, 35 BS, 16, 22 main-lobes, 30 pie-cut region, 24 Direction-of-arrival (DoA), 152 Doppler frequencies AoD and AoA, 84 marginal PDF, AoA, 78, 80 MT and MR, 78–80, 85, 186

Index omnidirectional antenna, 84 relative motion, 78, 79 vs. relative moving direction, MR, 83 Doppler PSDs closed-form expression, 42 Fourier transform, 8 movement directions, 131, 144, 145 propagation paths, 131 temporal ACFs, 171, 180 V2V scenarios, 7 Doppler spectrum, 121, 144 directional antenna, 34 frequency components, 35 mobility of, 116 multipath components, 26–27 probability density function, 26 shift, 26 spectral density, 33 3D V2V time-varying channel model, 101 Double-bounced (DB) WSS model, 142

E Elevation angle of arrival (EAoA) interfering object, 122, 133 time-varying parameters, 125, 171, 173, 176, 180 Elevation angle of departure (EAoD) double-bounced rays, 126 energy-related parameter, 133 time-varying parameters, 125, 171, 173, 180 Energy-related parameters, 123, 136, 137 Environment-related parameters, 137–138, 160 Estimated angular parameters, 161–164, 166 Extended Saleh–Valenzuela stochastic models, 9

F 5G communication systems channel frequency, 9 V2V channel, 2 Fixed-to-mobile (F2M) BS and MS, 1 communication environments, 95 conventional, 6, 7, 11, 12, 115 narrowband, 88 wireless channel, 5 Frequency correlation functions (CFs), 130, 139–143 Frequency cross-correlation function analysis, 88, 99–101, 107–110

Index G Gaussian distribution, 15, 21, 32, 33, 36, 37, 88, 98 Gaussian scatterer density function, 96 Geometrically based single-bounce model (GBSBM), 42 Geometrical modeling, 65 assumptions, 17 considerations, 18 Jacobian transformation, 19 MS and BS, 16 scatterer spatial distribution, 18 SP, 17 Geometric scattering channel models, 66 Geometry-based channel model AoA/ToA, 16–21 Gaussian spatial distribution, 15 geometrical shapes, 15 marginal PDFs, 22–26 MIMO performance, 27–30 scatterers, 15 standard deviation, 15 Geometry-based stochastic models (GBSMs) advantages, 169 confocal semi-ellipsoid models, 147 geometric model, 118 interfering objects, 151 stochastic models, 9 V2V communication environments, 10–11 H Hough transform, 116 I Internet of things (IoT), 65, 67 L Line-of-sight (LoS) propagations, 12, 66, 105, 117, 119, 120, 127, 128, 137, 153, 186 M Macrocell communication environments, 88 Marginal PDFs AoA at BS, 22 at MS, 22–24 joint and BS, 31 closed-form expressions, 31 deterministic considerations, 32 Gaussian density, 32

191 probability density, 31 scatterer distribution, 31 ToA, 24–26 Massive MIMO communication systems, 88, 152 Mean squared error (MSE), 161–164, 166 METIS project, 88 MIMO antenna receiving systems Doppler distribution, 51–52 MS and BS, 50 ULA and UCA, 52–55 MIMO channel model, see 3D vehicle massive MIMO antenna array model MIMO performance capacities, 29, 61, 62 directional antenna, 60 elevation angle’s effect, 61 radio environment, 37 SFCs, 27–29, 35, 60 UCA, 36 ULA, 36 wireless systems, 35 Mobile-to-mobile (M2M), 93 Multi-bounced propagation paths even-numbered-bounced propagation paths, 69, 71, 73–74, 186 geometric elliptical scattering channel model, 68 odd-numbered-bounced propagation paths, 69–73, 186 spatial characteristics AoD PDF, 75–78 evaluation, wireless communication systems, 74 scattering distribution, 74–75 Multiple confocal semi-ellipsoid models, 117, 120, 121 Multiple-input and multiple-output (MIMO) cellular radio communications, 2 Doppler frequency component, 4 geometric update, 3 MT and MR, 3, 4 performance (see MIMO performance) technologies, 87 ULA and UCA capacity, 53–54 compact antenna array, 52 spatial fading correlations, 53

N Narrowband channel models, 116, 152 NGBSMs, see Nonregular-shaped geometrybased stochastic models (NGBSMs)

192 NGSMs, see Non-geometrical stochastic models (NGSMs) Non-geometrical stochastic models (NGSMs), 9 Non-line-of-sight (NLoS), 6, 7, 42, 66, 88, 95, 96, 98–101, 107, 108, 122, 132, 133, 137, 154, 171 Nonregular-shaped geometry-based stochastic models (NGBSMs), 115 Non-stationary time-varying parameters, 125–127 O Odd-numbered-bounced propagation paths, 69–72 Outdoor-to-indoor scenario model, 15 P PDP, see Power delay profile (PDP) Per-tap channel statistics, 116, 120, 152 Plane wavefront channel matrix, 88 conventional MIMO channel models, 93, 94 far-field assumptions, 93 steering vector, 91 Power azimuth profile (PAP), 15, 33 Power delay profile (PDP), 131, 145–146, 178–181 Pseudo-geometrical scattering channel model, 66 R Radio propagation environments, 32, 185 Rayleigh fading model, 28, 90, 151, 152 Rays with single and double interactions, 12, 117, 118, 123, 127–128, 131, 151, 186 Reflection probability (RP), 16 Regular-shape geometry-based stochastic models (RS-GBSMs), 9, 10, 88, 115, 116 Ricean factor, 136, 137 RS-GBSMs, see Regular-shape geometrybased stochastic models (RS-GBSMs) S Scatterer distributed density BS and MS, 19 CDF, 20 closed-form expressions, 20, 21

Index ES and RP, 20 normalization, 20 zero-mean Gaussian distribution, 19 Scattering point (SP), 17, 68 SFCs, see Spatial fading correlations (SFCs) Signal processing techniques, 65, 87, 169 Signal-to-noise ratio (SNR), 29, 54 Single-input single-output (SISO), 151 SISO, see Single-input single-output (SISO) SNR, see Signal-to-noise ratio (SNR) Spatial CCFs, 160, 161, 164–166, 177, 178, 186 Spatial characteristics of 3D scattering AoA PDFs, 47–48 ToA PDFs, 48–50 wireless communication systems, 46 Spatial correlation functions (CFs), 6, 7, 127–130, 136, 139–143, 147, 185 Spatial fading correlations (SFCs) antenna elements, 53, 60, 103 covariance matrix, 53 Doppler spectra, 62 MIMO systems, 60 of multiple antennas, 27–28 numerical integration, 53 3D vehicle massive MIMO antenna array model, 103 UCA antenna array, 87 ULA elements, 35 von Mises Fisher distribution, 87 Spherical wavefront channel statistics, 89 massive MIMO antennas, 89 MIMO channel matrix, 91 near-field effects, 110 NLoS components, 88 Stationary intervals, 1, 116, 132, 145–147 System channel model flat communications CIR, 121–125 propagation delays, 119–120 TDL, 118–119 time-varying parameters, 125–127 V2V channel model, 120–121 WINNER II channel model, 118 slope communications, 132–133 T Tapped-delay-line (TDL), 118, 119, 124 TDL, see Tapped-delay-line (TDL) 3rd Generation Partnership Project (3GPP), 15 3D cluster-based MIMO channel model, 186 3D ellipsoidal model, 66 3D MIMO channel model, 170, 172, 180

Index 3D multipath channels, 87 3D propagation model directional antenna, 45 land mobile radio cellular systems, 44 non-illuminated region, 44 scattering region, 44, 45 zero-elevation plane, 45 3D scattering channel model, 66, 186 advantages, 43 AoA PDFs, 55–58 Doppler distribution, 58–59 Fi/ad hoc/mesh networks, 43 GBSBM, 42 indoor and outdoor environments, 42 MIMO antenna receiving systems, 50–54 MIMO performance, 60–62 outdoor wireless environments, 42 scatterer distributions, 41 spatial characteristics, 46 3D propagation, 44–46 2D propagation, 42 wireless MIMO communication systems, 41 3D vehicle massive MIMO antenna array model AoD and AoA, 89, 104–107 channel statistics, 90 elements, vehicle surfaces, 102 energy efficiency, 87 frequency cross-correlation function analysis, 99–101, 107–110 geometric uniform rectangular, 90 impulse response, 90, 92 large-scale omnidirectional antennas, 91 microcell environments, 88 performance Analysis, 103–104 spatial characteristics, radio channel, 96–97 spectrum efficiency, 87 steering vector, 90, 92 time cross-correlation function analysis, 97–99, 107–110 time-variant and array-variant properties, 88 V2V, 186 communication environments, 93–96 mobile radio environments, 89, 91 3D wideband geometry-based channel model MIMO V2V communications, 186 propagation delays, 116, 147 system (see System channel model) theoretical channel characteristics (see Channel characteristics, theoretical) V2V communications, 115, 147, 186 vehicular communications, 117–118

193 Time cross-correlation function analysis, 97–99, 107–110 Time of arrival (ToA) AoA (see Angle of arrival (AoA)) marginal PDFs, 24–26, 33 ToA, see Time of arrival (ToA) Total Doppler frequency vs. relative moving direction, 109–111 Two-cylinder model, 11, 116, 117, 120, 124, 137 Two-dimensional (2D) geometry-based channel models, 88 2D scattering channel models, 65 Two-ring model, 6, 10, 66, 115 U Uniform circular array (UCA) antenna arrays, 30 antenna receiving systems, 52 MIMO, 53, 60–62 SFCs, 28, 36 ULA (see Uniform linear array (ULA)) Uniform linear array (ULA) antenna receiving systems, 52 capacities, 37, 53 MIMO, 28 omnidirectional receiving antennas, 154, 172 SFCs, 35 Unmanned aerial vehicle (UAVs) A2G (see Air-to-ground (A2G) communication environments) Doppler PSDs, 179, 181 GBSMs, 169 impulse response, 173–175 model description, 171–173 moving velocities, 179 PDPs, 179, 181 propagation paths, 175–177 public and civil domains, 169 spatial CCFs, 178 temporal ACFs, 179, 180 3D A2G channel model, 169 half-sphere model, 170 MIMO channel model, 169 time-varying angular parameters, 175–177 V Vehicle-to-vehicle (V2V) aim, 1 deterministic models, 115 development (see Development in V2V model)

194 Vehicle-to-vehicle (V2V) (cont.) energy-related parameters, 137 environment-related parameters, 137–138 vs. F2M, 4–9 MIMO, 2–4 multiple confocal ellipses, 66 propagation characteristics, 1 research, 2 stochastic models, 115 TDL structure, 119 two-ring model, 66 vehicular communications, 2 V2V communication environments angular estimation algorithms, 152 AoD and AoA estimation, 156–160 channel statistics, 94 CIR, 155 ellipse models, 152, 155 estimated angular parameters, 161–164 5G, 151, 185 F2M, 95 GBSMs, 10–11, 151 geometric model parameters, 152–154 LoS components, 154 M2M, 93 MT and MR, 95 narrowband channel models, 152 NLoS components, 95 propagation characteristics, 151–153 Rayleigh fading model, 151, 152 roadside environment, 94 SISO, 151 spatial CCFs, 153, 164–166 spherical and plane wavefront, 93–94 transmitting and receiving antennas, 154 wideband channel model, 153–154

Index V2V vs. F2M models angular parameter range, 9 channel frequency, 9 Doppler PSD, 7–8 moving vehicles, 5–7 non-stationarity, 5 Vehicle traffic density (VTD), 116 Vehicular communications, 2, 3, 118, 127, 145, 187 Visual scattering channel model, 115 advantages, 67 AoD PDF, 80–83 C2C (see Car-to-car (C2C)) description, 68 Doppler frequencies, 68, 78–80, 83–85 M2M and IoT, 67 multibounced propagation paths (see Multibounced propagation paths) von Mises Fisher distribution, 87 von Mises PDF, 127, 129, 160 VTD, see Vehicle traffic density (VTD)

W Wideband channel model, 11, 120, 147, 153–154 Wide-sense stationary uncorrelated scattering (WSSUS), 116 Wireless channel modeling 5G/B5G communications, 185 F2M, 5, 88 spatial CCFs, 160 V2V, 185, 187 WSSUS, see Wide-sense stationary uncorrelated scattering (WSSUS)