Yuanyuan Ma＊, Lei Yang Xianghan Zheng

1 College of Information Science and Engineering, Shandong Agricultural University, Taian 271018, China

2 College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China

Abstract: This paper derives a non-stationary multiple-input multiple-output(MIMO) from the one-ring scattering model. The proposed channel model characterizes vehicular radio propagation channels with considerations of moving base and mobile stations, which makes the angle of arrivals (AOAs) along with the angle of departures (AODs) time-variant.We introduce the methodology of including the time-variant impacts when characterizing non-stationary radio propagation channels through the geometrical channel modelling approach. We analyze the statistical properties of the proposed channel model including the local time-variant autocorrelation function(ACF) and the space cross-correlation functions (CCFs). We show that the model developed in this paper for non-stationary scenarios includes the existing one-ring wide-sense stationary channel model as its special case.

Keywords: vehicular communications; mobile fading channels; non-stationary channels;multiple antennas; geometrical channel modeling; one-ring scattering model

I. INTRODUCTION

Recently, vehicular communications have gained extensive attentions as the large demands for highway intelligence, road safety,collision avoidance, and trafficc efficciency [1],[2]. Through vehicle-to-anything technologies,a vehicle is entitled to exchange information with other elements in the networks e.g.,vehicles and road infrastructure. The trafficc accidents and jams reduce signi ficantly by the exchanged warning messages and road information sent to the drivers through vehicular ad hoc networks (VANETs) [3], [4].

With regard to VANETs, the underlying propagation environments of vehicular communications have signi ficant impacts on the reliability, accuracy and efficciency of the transferred information. Therefore, the deep understanding of vehicular radio propagation channels is essential for investigating and designing vehicular communication systems [5].Moreover, the research on VANETs mainly relies on simulations because of high cost of constructing the real testbed. It is well motivated to mathematically represent the channel characterizations for vehicular communications. For high mobility vehicular communication environments, the main challenge comes from accurately characterizing the fast variations of mobile radio channels when the transmitters and receivers move in relative high speeds.

For this purpose, the authors in [6], [7]characterize vehicular propagation channels through measurements campaigns. The channel parameters or statistical properties are concreted from the experimental results, which are very accurate. However, the channel models developed by this approach are restricted to speci fic environments and invalid in case of environmental changes.

Alternatively, physical channel models have been developed for various vehicular propagation environments through the geometry-based channel modeling approach.The scatterer locations at the base and mobile stations are modeled by a speci fic geometric pattern, e.g., one ring and two rings. The narrowband MIMO geometrical channel models were proposed in [8] for vehicle-to-vehicle(V2V) communications, which were further extended in [9], [10] to address the frequency selectivity. The channel characterizations are mathematically represented by the main physical channel parameters, e.g., antenna spacings, AODs, and AOAs. The fundamental knowledge of various radio propagation channels can easily be obtained by changing the channel parameters.

Nevertheless, the temporal variations of AODs and AOAs due to the movement of the BS and MS have not been taken into account in all aforementioned geometrical channel models. The underlying channels turn out to be wide-sense stationary. However, the pre-condition of having constant AODs and AOAs is restricted to very short time intervals within a few tens of the wavelength that the BS (or MS) moves [11], [12]. As shown by the empirical investigations [13], if communication vehicles move at relatively high speeds,the AODs and AOAs turn out to be time-variant, which results in a non-stationary propagation scenario.

The authors in [14]– [17] have contributed on modeling nonstationary propagation channels for vehicular communications. Under the one-ring scattering environments, [14] have developed a non-stationary channel model for the single antenna scenario, in which the base station (BS) is fixed and only the mobile station (MS) moves. The AOA and the direction along which the MS moves have been included in the developed channel model. The impacts of other physical channel parameters on the statistical properties are unknown. Papers[15] and [16] have characterized non-stationary Rayleigh and Double Rayleigh channels for mobile-to-mobile communications, respectively. Authors in [14]–[16] have focused on nonstationary single antenna scenarios, meaning only one antenna is installed at the base and mobile station. A MIMO channel model has been developed based on the T-junction scattering pattern in [17] for V2V propagation channels. Both AODs and AOAs are time dependent due to vehicles’ movements, which indicates the underlying channel non-stationary. The T-junction scattering environment is not a typical driving environment that vehicles experience the most, which limits the use of the developed channel model. Moreover, the assumption of the double-bouncing scattering in [17] brings more restrictions to the applicable situations. The two communicating vehicles need to be located in two separate streets and move in a small range close to the junction.

Given both the MS and BS in motion, this paper models non-stationary mobile radio channels, which serves vehicular communications. Multiple antenna scenarios are concentrated in this paper, which differs with[14]. The scattering conditions at the MS are characterized by the well-known one-ring geometry model [18], [19], i.e., all scatterers are placed on a circle. However, we remove the assumption made in [18], [19] that the ring’s radius is much smaller than the distance between the mobile and base stations. All the considerations, i.e., non-stationary scenarios,moving BS and MS, and removed assumption,bring a totally new form to the mathematical channel model. We present the methodology of introducing the time-variant in fluence when modeling non-stationary propagation environments through the geometrical channel modeling approach. The in fluence of time-dependent AODs and AOAs are investigated.The complex channel gain is formulated as a function with more physical channel parameters included. Thus, the derived mathematic model enables to characterize non-stationary fading behaviors of given propagation scenarios more precisely. The statistical properties of the derived model are analyzed with emphasis not only on the local ACF but also on the space CCF. It is essential to investigate the space CCF since both theory and simulations indicate that antenna spacings have strong impacts on the vehicular communication system performance. We also demonstrate that our proposed non-stationary one-ring channel model reduces to the wide-sense stationary one existing in [19] when considering the special case of constant AODs and AOAs.

Fig. 1. Geometrical one-ring scattering model with scatterer surrounding the MS.

Fig. 2. Geometrical one-ring scattering model with moving BS and MS; time-variant AODs

The remaining contents of the paper are organized into five sections. We first review the one-ring geometry scattering model in Section II. Then, the time-dependent AODs and AOAs are studied in this section. Section III presents the mathematical expressions of the geometrical non-stationary one-ring channel model.Section IV addresses the statistical properties including the time ACF and space CCF. Section V shows the numerical results. Last but not least, Section VI gives the concluding remarks.

II. ONE-RING SCATTERING MODEL

This section concisely reviews the one-ring MIMO scattering model, which has been employed in [18], [19] for modeling stationary scenarios. It is depicted in figure1 that at the BS,MTtransmit antennas are installed with the orientation denoted by the angleαT. The antenna spacing isδT. The MS is surrounded withNscatterers, which are situated over a circle with radiusR. At the MS side,there areMRantennas separated by the antenna spacingδR. These antennas are tilted at the angleαR.

At reference timet=0, the MS is located at the (0,0) of thexy-plane with a distance D away from the BS. The initial AOA, represented by, is the orientation of the plane-wave bounced by the nth scatterer before arriving at the original MS positionOR. The BS is originally situated onx-axis. The orientation of the plane-wave originating from the BS and re flected by the nth scatterer is depicted by the angle, namely AOD.Furthermore, as shown in figure2, the MS moves at a fixed velocityvRalong with the direction speci fied by the motion angleβR.Suppose that at timet, the MS reaches the new positionO'R. The values of the AOAs change against timet. For convenience,we represent the time-dependent AOAs bywhich speci fies the propagation direction of the plane-wave reaching the new MS’s locationthrough the nth scatterer. The BS moves at a speedvTwith the angle of motionβT. At timet, the BS moves to the pointO'T.Similarly, the AOD also depends on time due to the moving BS. We useto represent the angle of departure from the BS located atand bounced by thenth scatterer.The time-dependent AOAand AODcan be calculated by

where atan2(y,x) is the four-quadrant inverse tangent function,denotes the distance that the MS moves during the intervalrepresents the distance that the BS moves in this time slot. The symbolxnandynare the coordinates of thenth scatterer.

According to the geometrical relationship,we find that the original AODin eq.(3d)is related with the AOAby the following equation

As presented in eq.(2b), the time-variant AODsdepend on the constant AODs. Therefore, the time-variant AODscan be finally expressed by a function related withinstead ofif we submit eq. (4) into eq.(3d). To keep simplicity, we present the time-variant AODs by the general termin all formulae of the paper where the time-variant AODs appear.

III. NON-STATIONARY MIMO CHANNEL MODEL UNDER ONE-RING SCATTERING ENVIRONMENTS

3.1 Channel impulse response

This subsection is devoted to the derivation of the mathematical expression for the channel impulse response. Based on the plane-wave propagation principle, the channel impulse response between the antennaand the antennais calculated by

whereλis the wavelength.

As the BS and the MS move, the positions of the antennas vary against time. At timet,the new location of thekth receive antenna is denoted in figure 2 by. Thelth transmit antenna reaches. The time-dependent termin eq.(7) represents the total propagation distance of thenth plan-wave from. The total propagation distanceis composed by two part: the plane-wave travelling distance from thenh scatterer todenoted byand the distance of the plane-wave emitted from thelth transmit antennareaching the obstacle, denoted by

If we apply the law of cosine to the triangleand

Similarly, applying the law of sine (cosine)to the geometry at the BS, the distancein eq. (7) can be written as

Submitting the equations (6)-(11) into (5),we obtain the mathematic expression of the channel impulse response

3.2 Special case with constant AOAs and AODs

In this subsection, we discuss a wide-sense scenario that has been widely investigated in literature [18], [19], where the BS is fixed and the MS moves. The AOAs and AODs are constant.

IV. STATISTICAL CHARACTERIZATION

4.1 Time-variant ACF

The following equation de fines the temporal ACF of the channel model

wherecalculates the expectation andtakes the complex conjugation. After submitting eq.(12) into eq.(14) and carrying out mathematical calculations, we finally obtain the expression of the ACF as below

where sinc {·} denotes the sinc function given by sinc (x)=sin(x)x. In the equations above,are the derivatives of the Doppler frequencieswith respect to timet. We have

As observed from eq. (15), the ACF of the underlying channel model is a function of time separation τ as expected. Besides, the ACF also depends on time t. For the special case discussed in SubsectionIII-B, where the BS is fixed and the AOAsare constant, i.e.,, the time-dependent ACF reduces to

which is the ACF for the wide-sense stationary case [19]. In this scenario, the ACF becomes independent on timetand close to the shape of the 0th Bessel function.

4.2 Time-variant CCF

The space cross correlation function of the channel model is de fined as

After submitting eq. (12) into eq.(18), we obtain the time-variant space CCF given by

The three-dimensional (3D) time-variant CCF in eq. (19) will reduce to a two-dimensional (2D) CCF for the situation described in SubsectionIII-B, which is time-invariant and only related with the antenna spacings at both stations. The expression of 2D CCF is aligned with the 2D CCF presented in [19] for the wide-sense MIMO one-ring model if we follow the assumptionR?Dmade in [19].

We can derive the 2D spatial correlation function at the BS (or MS) from the 3D CCF in eq. (19) by settingk'=k(orl'=l). Analytical expressions are presented as below

V. NUMERICAL RESULTS

This section illustrates the analytical findings through several numerical examples.

For this purpose, we assume a propagation environment that the MS is surrounded withN=20 scatterers placed on a circle with radiusR=200m. The distanceDbetween two stations is 2000m. The MS moves at speedalong with the motion angleβR=60°. The BS moves at the velocityin the direction ofβT=120°.

The maximum Doppler frequencyfTmaxis set to 91Hz andfRmax=100Hz . The BS hasMT=2 transmit antennas with the tilt angleαT=30°,whereas the MS hasMR=2 receive antennas with the tilt angleαR=60°.The initial AOAsare determined by certain pa-rameter calculation methods. Here, we adopt the extended method of exact Doppler spread[20] to calculate the original AOAs and we have

Figure 3 depicts the time-variant ACFof the proposed non-stationary model.

We can see from this figure that att=0, the ACF follows the shape of the 0th order Bessel function. However, as time t proceeds, the ACF differs more and more. This is clearly shown through figure4, where 3 snapshots of theat timet=0s,t=1s, andt=1.5s were captured.

Figure 5 illustrates the special case having constant AOAs, which has been discussed in Subsection III-B. The time-dependent ACF in eq. (15) follows the 0th order Bessel function and independent on timet, which indicates that the channel becomes wide-sense stationary.

The space correlation property at the MS side is presented in figure 6. The figure shows that the CF at the MS is nonstationary and changes against time t. This can be explained by the time-variant AOAs. If we setthe AOAs become constant. The channel is again wide-sense stationary. Therefore, the CF is independent on time (see figure 7). The space CF at the BS is also time-dependent and illustrated in figure 8.

Fig. 3. Time-dependent ACF of the proposed channel model with the time-variant AOAs

Fig. 4. Snapshots of the time-dependent ACF of the developed non-stationary channel model at different time t .

Fig. 5. ACF of the developed channel model with the constant AOAs

Fig. 6. Time-dependent CF of the developed channel model with the time-variant AOAs

Fig. 7. CF of the developed channel model with the constant AOAs

Fig. 8. Time-dependent CF of the developed channel model with the time-variant AODs

VI. CONCLUSION

This paper was devoted to derive a non-stationary MIMO channel model for vehicular communications. We modeled the scatterers around the MS by a ring pattern. Different with traditional one-ring scattering environments, we considered that both the base and mobile stations are in motion. We removed the assumption made in the traditional onering channel model that ring’s radius is much smaller than the distance between the two stations. All the considerations brought a totally new form for the mathematical channel model.We presented the methodology of including the time-variant in fluence when characterizing non-stationary channels by the geometrical channel modeling approach. We also studied the local ACF and 3D space CCF of the model. It can be seen from theory and numerical results that the statistical properties of non-stationary channel models change against time. We demonstrated that our non-stationary channel model includes the existing widesense stationary one-ring channel model as its special case.

ACKNOWLEDGMENT

This work was supported by Shandong Agricultural University Funding of First-class Disciplines and Shandong Agricultural University Key Cultivation Discipline Funding for NSFC Proposers. The work was also supported by Grant of Beihang University Beidou Technology Transformation and Industrialization (BARI1709) and Open Project of National Engineering Research Center for Information Technology in Agriculture (No.KF2015W003).