Wireless Communications and Mobile Computing

Volume 2018, Article ID 6396173, 9 pages

https://doi.org/10.1155/2018/6396173

## Modelling and Analysis of Nonstationary Vehicle-to-Infrastructure Channels with Time-Variant Angles of Arrival

Correspondence should be addressed to Matthias Pätzold; on.aiu@dlozteap.saihttam

Received 17 October 2017; Revised 16 January 2018; Accepted 4 February 2018; Published 15 March 2018

Academic Editor: Enrico M. Vitucci

Copyright © 2018 Matthias Pätzold and Carlos A. Gutierrez. 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

In mobile radio channel modelling, it is generally assumed that the angles of arrival (AOAs) are independent of time. This assumption does not in general agree with real-world channels in which the AOAs vary with the position of a moving receiver. In this paper, we first present a mathematical model for the time-variant AOAs. This model serves as the basis for the development of two nonstationary multipath fading channels models for vehicle-to-infrastructure communications. The statistical properties of both channel models are analysed with emphasis on the time-dependent autocorrelation function (ACF), time-dependent mean Doppler shift, time-dependent Doppler spread, and the Wigner-Ville spectrum. It is shown that these characteristic quantities are greatly influenced by time-variant AOAs. The presented analytical framework provides a new view on the channel characteristics that goes well beyond ultra-short observation intervals over which the channel can be considered as wide-sense stationary.

#### 1. Introduction

In a typical downlink scenario, where plane waves travel from a base station (BS) to a mobile station (MS) via a large number of fixed scattering objects, the angles of arrival (AOAs) of the received signals are changing along the moving route of the MS. Only for very short observation intervals in which the MS travels a few tens of the wavelengths [1], the temporal variation of the AOAs can be neglected justifying the wide-sense stationary assumption of multipath fading channels. The lengths of the stationary intervals during which the mobile radio channel can be considered as wide-sense stationary or quasi-stationary have been investigated (e.g., in [2–4] and the references therein). By pushing the observation interval beyond the stationary interval, the received signal captures nonstationary effects that call for new channel modelling approaches using time-frequency analysis techniques [5]. One of the effects that come with long observation intervals is that the AOAs and thus the Doppler frequencies are changing with time along the MS’s moving route.

Attempts to include the temporal variations of the AOAs in mobile radio channel models have been made in [6–8]. In [6], a nonstationary multiple-input multiple-output (MIMO) vehicle-to-vehicle (V2V) channel model has been derived by assuming that the AOAs and AODs are piecewise constant. In [7], a proposal has been made for the extension of the IMT-Advanced channel model [9] by replacing the time-invariant model parameters, such as the propagation delays, AOAs, and the angels of departure (AODs) by time-variant parameters. In [8], a nonstationary one-ring model has been introduced in which the time-variant AOAs have been modelled by stochastic processes rather than random variables.

This paper is an extended version of our conference paper [10]. It expands on the recent results by studying the impact of time-variant AOAs on the statistical properties of multipath fading channels. It is shown that the multipath fading channel becomes non-wide-sense stationary if the AOAs change with time. Two new nonstationary channel models with time-variant AOAs are derived. The first one has an instantaneous channel phase that is related to the instantaneous Doppler frequency via the phase-frequency relationship [11], while the second one is based on a sum-of-cisoids (SOC) model in which the time-independent Doppler frequencies are replaced by time-dependent Doppler frequencies. The latter approach is simple, straightforward, and intuitive but results in a less accurate nonstationary channel model. The statistical properties of both channel models are investigated with emphasis on the time-dependent autocorrelation function (ACF), time-dependent mean Doppler shift, time-dependent Doppler spread, and the Wigner-Ville spectrum. Our analysis shows that our first proposed nonstationary channel model is consistent with respect to the mean Doppler shift and the Doppler spread, while this consistency property is not fulfilled by the SOC model with time-variant Doppler frequencies. The two proposed nonstationary channel models provide a trade-off between accuracy and complexity concerning the mathematical expressions.

One of the main differences between [6–8] and our paper is that the AOAs are modelled in different ways. For example, in [6], the AOAs are modelled as piecewise constant functions, that is, these parameters are considered as constant apart from a finite number of jumps, while in our paper the AOAs are modelled in our paper as continuous time-variant functions. Another difference is that the models in [6–8] have been developed for different propagation scenarios. The V2V channel model in [6] has been developed to simulate propagation scenarios which are typical for T-junctions. The BS-to-MS channel model in [7] covers basically the same scenarios as the IMT-Advanced channel model [9], while the model in [8] is restricted to scenarios that can be generated by the one-ring model under the assumption of isotropic scattering. This contrasts with our nonstationary generic model which is not restricted to any specific propagation scenario. The drawback of the models in [6–8] is that they are inconsistent with respect to the mean Doppler shift and the Doppler spread. Our preferred model avoids this drawback by using an integral relationship between the instantaneous channel phases and the corresponding instantaneous Doppler frequencies.

The organization of this paper is as follows. Section 2 presents the derivation of two nonstationary multipath fading channel models with time-variant AOAs. Their statistical properties will be analysed in Section 3. The numerical key results of our study are visualized in Section 4. Section 5 provides guidelines for various extensions of the model. Finally, Section 6 draws the conclusion and suggests possible future research topics in relation to the issues addressed in this paper.

#### 2. Derivation of the Nonstationary Multipath Channel Models

##### 2.1. Time-Variant AOAs

We consider a downlink non-line-of-sight (NLOS) propagation scenario in which a fixed BS operates as transmitter, and an MS acts as receiver. It is supposed that the BS and the MS are equipped with omnidirectional antennas. The BS antenna is elevated and unobstructed by any object, whereas the MS antenna is surrounded by a large number of fixed scattering objects called henceforth scatterers . The coordinate system has been chosen such that the MS is located at the origin of the -plane at . Furthermore, it is assumed that the MS moves with constant velocity in the direction determined by the angle of motion as indicated in Figure 1. For reasons of clarity, this figure highlights only the location of the scatterer from which the MS receives the th multipath component (plane wave) in the form of , where denotes the path gain which is supposed to be constant, and is the associated channel phase that will be studied in Section 2.3. The corresponding AOA is defined as the angle between the propagation direction of the th incident plane wave and the -axis, that is,for , where denotes the four-quadrant inverse tangent function. It should be mentioned that the four-quadrant inverse tangent function returns the angle of the vector with the positive -axis in the range . This function contrasts with the inverse tangent function , whose results are limited to the interval . In (1), the symbols and denote the coordinates of the scatterer ; and and indicate the position of the MS at time . According to (1), the AOA is a nonlinear function of time , which can be turned into a linear function by developing in a Taylor series around and retaining only the first two terms. This results in the following model for the time-variant AOA:whereIn (4), denotes the distance from the scatterer to the origin of the -plane, that is, , as can be deduced from the geometrical model in Figure 1. In Section 4, it is shown that the two-term Taylor series expansion of in (2) is sufficiently accurate for small observation intervals .