Wireless Communications and Mobile Computing

Volume 2017 (2017), Article ID 6394653, 15 pages

https://doi.org/10.1155/2017/6394653

## Modeling of Non-WSSUS Double-Rayleigh Fading Channels for Vehicular Communications

^{1}Faculty of Science, Universidad Autónoma de San Luis Potosí, Av. Salvador Nava Martinez s/n, 78290 San Luis Potosí, SLP, Mexico^{2}Department of Engineering, Universidad de Quintana Roo, Blvd. Bahía Esq. Ignacio Comonfort s/n, 77019 Chetumal, QR, Mexico

Correspondence should be addressed to Carlos A. Gutiérrez

Received 29 April 2017; Revised 14 July 2017; Accepted 1 August 2017; Published 3 October 2017

Academic Editor: Xianfu Lei

Copyright © 2017 Carlos A. Gutiérrez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the modeling of nonstationary time-frequency (TF) dispersive multipath fading channels for vehicle-to-vehicle (V2V) communication systems. As a main contribution, the paper presents a novel geometry-based statistical channel model that facilitates the analysis of the nonstationarities of V2V fading channels arising at a small-scale level due to the time-varying nature of the propagation delays. This new geometrical channel model has been formulated following the principles of plane wave propagation (PWP) and assuming that the transmitted signal reaches the receiver antenna through double interactions with multiple interfering objects (IOs) randomly located in the propagation area. As a consequence of such interactions, the first-order statistics of the channel model’s envelope are shown to follow a worse-than-Rayleigh distribution; specifically, they follow a double-Rayleigh distribution. General expressions are derived for the envelope and phase distributions, four-dimensional (4D) TF correlation function (TF-CF), and TF-dependent delay and Doppler profiles of the proposed channel model. Such expressions are valid regardless of the underlying geometry of the propagation area. Furthermore, a closed-form solution of the 4D TF-CF is presented for the particular case of the geometrical two-ring scattering model. The obtained results provide new theoretical insights into the correlation and spectral properties of small-scale nonstationary V2V double-Rayleigh fading channels.

#### 1. Introduction

Terrestrial vehicle-to-vehicle (V2V) communication systems are emerging as an enabling technology for a variety of new wireless applications and services, such as information relaying for mobile cellular networks [1] and peer-to-peer data transmission for vehicular communications [2]. Some of the most important applications of these systems target the prevention of vehicular accidents and the optimization of traffic flow. Such applications have caught the attention of the automotive industry and different government bodies around the world, who have become major promoters of the V2V communications technology [3].

One of the main challenges in the design of V2V communication systems is to develop a robust air interface that supports delay sensitive applications under the constraints of a rapidly changing propagation environment and a dynamic network topology. To successfully design and optimize such an air interface, a realistic reference model of the time-frequency (TF) dispersive V2V fading channel is required. This is of primary importance, since the performance of the wireless communication systems is highly influenced by the propagation environment. In addition to measurement-based models, proper analytic channel models are needed that provide insights into the physics of V2V radio reception and, at the same time, that lend themselves to mathematical and numerical system performance investigations.

Important advances in the analytical characterization of fixed-to-mobile (F2M) multipath fading channels were prompted by the emergence of the mobile cellular communication systems in the late 1970s. However, the modeling of fading channels for V2V communications not only demands more exhaustive research work, but also requires a shift of paradigm, because some of the assumptions that are often invoked for the characterization of F2M channels are not valid when the terminals at both ends of the radio link are able to move at high speeds. For example, most of the existing statistical models for F2M channels have been formulated assuming the fulfillment of the wide-sense stationary uncorrelated scattering (WSSUS) condition introduced by Bello in [4] (e.g., see [5–7]). This assumption facilitates the mathematical analysis of TF-dispersive channels, as it implies that the channel’s statistics are simultaneously wide-sense stationary (WSS) in the time and the frequency domains. Nevertheless, recent empirical investigations carried out in vehicular communication environments suggest that the WSSUS condition is not valid for V2V channels [8]. While the nonstationary features of multipath wireless channels have been a subject of analysis since the early days of the mobile radio communications (e.g., see [4, 9]), the modeling of such nonstationarities has predominantly been addressed from a large-scale propagation perspective. Measured data obtained independently in [10–12] demonstrates that the nonstationary characteristics of V2V channels are also meaningful at a small-scale level.

Empirical investigations have further shown that the signal fading induced by V2V channels is more severe than the one produced by F2M channels [13], which is typically modeled by Rayleigh or Rice distributions. The exacerbation of signal fading is not surprising if one considers that, in a V2V communications system, the mobile terminals are located at ground level. This scenario increases the chances of receiving echoes of the transmitted signal that interact with multiple interfering objects (IOs) on their way to the receiver antenna. As a result of such multiple interactions, the received multipath signal is subject to a form of cascaded fading that is modeled by worse-than-Rayleigh distributions [13, 14] (e.g., the double-Rayleigh [15] and double-Rice distributions [16]).

Notable contributions to the analytical characterization of non-WSSUS V2V channels have recently been made in [17–19] on the basis of the geometry-based statistical modeling approach. The geometrical channel models proposed in these papers assume the propagation of spherical waves to account for the nonstationarities of V2V channels stemming from small-scale propagation. In the spherical wave propagation (SWP) framework, the angle of departure (AOD) and angle of arrival (AOA) of the received multipath signals are determined by the instantaneous spatial position of the transmitting and receiving mobile stations (MSs). The angular statistics of the resulting channel models are therefore time-dependent. This feature is particularly convenient for the characterization of nonstationary small-scale channels but renders the mathematical analysis of the channel’s statistics a cumbersome task.

To facilitate the modeling and analysis of non-WSSUS V2V channels, we recently proposed in [20–22] a novel framework that builds instead on the principles of plane wave propagation (PWP). Our proposal is well suited for the analysis of V2V radio reception over small local areas spanning a few tens of wavelengths, where a plane wave approximation of the more realistic spherical electromagnetic waves can be applied. For such propagation scenarios, the angular statistics of the V2V channel can be modeled by time-invariant distributions, which are more mathematically tractable than their time-varying counterparts. The focus of [20, 21] was on the characterization of nonstationary Rayleigh fading channels for single-input single-output (SISO) and multiple-input multiple-output (MIMO) V2V communication systems, respectively. In [21], the MSs are assumed to move at constant speeds and on linear trajectories, whereas the effects of acceleration and nonlinear motion are investigated in [20] following a parallel approach to the TF analysis techniques employed in [23]. On the other hand, in [22], we apply our modeling framework to the characterization of non-WSSUS SISO V2V channels that experience double-Rayleigh fading.

In this paper, we complete our preliminary work presented in [22] by providing a detailed description and a thorough statistical analysis of the proposed geometrical model for non-WSSUS V2V double-Rayleigh fading channels. The scope and depth of the work in [22] are expanded here as follows:(i)Important statistical quantities of the proposed channel model, such as the autocorrelation functions in the time and frequency domains, as well as the TF-dependent delay and Doppler profiles, were not investigated in [22]. An in-depth analysis of these statistical quantities is presented here.(ii)Details on the derivations of the four-dimensional (4D) TF correlation function (4D TF-CF) of the proposed channel model were not presented in [22]. An outline of the derivations is given here in the Appendix.(iii)In [22], our discussion of the proposed channel model’s stationary (nonstationary) characteristics was constrained to a single paragraph due to space limitations. In this paper, we complement our discussion with important additional remarks.(iv)Finally, new numerical examples are presented in this paper to illustrate our findings regarding the autocorrelation, spectral, and stationary (nonstationary) characteristics of the proposed geometry-based statistical model (GBSM) for non-WSSUS V2V double-Rayleigh fading channels.

The remainder of the paper is organized as follows. Our proposal for the geometrical modeling of nonstationary V2V double-Rayleigh fading channels is presented in Section 2. In Section 3, we derive general expressions for the envelope and phase distributions, the 4D TF-CF, and the TF-dependent delay and Doppler profiles of the proposed channel model. It is worth pointing out that such expressions are valid regardless of the underlying geometry of the propagation area. In Section 4, we compute a closed-form solution of the 4D TF-CF by considering the particular case of the geometrical two-ring scattering model. Numerical examples illustrating our theoretical findings are presented in Section 4. Finally, our conclusions are given in Section 5.

*Notation*. The complex conjugate, the argument, and the absolute value operations are denoted by , , and , respectively. Vectors are written in bold face. The transpose operation is denoted by , stands for the Euclidean norm, and the scalar product between two vectors and is represented as . The operator designates the statistical expectation. The set of positive real numbers is denoted by , and the operator indicates set cardinality.

#### 2. The Proposed Geometrical Model for Non-WSSUS V2V Channels

##### 2.1. Geometrical Modeling of the Propagation Scenario

The aim of this paper is to model the nonstationarities of TF-dispersive V2V channels stemming from small-scale propagation. We are particularly interested in characterizing the nonstationarities arising from the time-varying nature of the propagation delays. For that purpose, we consider a SISO V2V communication system and assume that the transmitted signal reaches the receiver antenna through a double interaction with nonmoving IOs randomly located in the propagation environment. Specifically, we assume that the transmitted signal interacts first with a set of IOs () that are located in the surroundings of the transmitting MS (). Then, the scattered signals that result from such an interaction impinge on a second set of IOs () that lie in the vicinity of the receiving MS (). Thereby, a total of double-scattered waves are produced, which combine with one another at the receiver antenna. Figure 1 shows an illustration of the propagation scenario under consideration at the time instant when the MSs start to communicate with each other.