Rubing Jia, Yiran Li, Xiang Cheng,＊, Bo Ai

1 State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China

2 State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China

3 School of electronic and information engineering of Beijing Jiaotong University, Beijing Jiaotong University, Beijing 100044, China

Abstract: A more general narrowband regular-shaped geometry-based statistical model(RS-GBSM) combined with the line of sight(LoS) and single bounce (SB) rays for unmanned aerial vehicle (UAV) multiple-input multiple-output (MIMO) channel is proposed in this paper. The channel characteristics,including space-time correlation function(STCF), Doppler power spectral density(DPSD), level crossing rate (LCR) and average fade duration (AFD), are derived based on the single sphere reference model for a non-isotropic environment. The corresponding sum-of-sinusoids (SoS) simulation models including both the deterministic model and statistical model with finite scatterers are also proposed for practicable implementation. The simulation results illustrate that the simulation models well reproduce the channel characteristics of the single sphere reference model with sufficient simulation scatterers. And the statistical model has a better approximation of the reference model in comparison with the deterministic one when the simulation trials of the stochastic model are sufficient. The effects of the parameters such as flight height, moving direction and Rice factor on the characteristics are also studied.

Keywords: UAV-MIMO; geometry-based model; channel characteristics; simulation model

I. INTRODUCTION

In recent years, due to the low cost and high flexibility, the unmanned aerial vehicles(UAVs) have been widely used in weather monitoring, forest fire detection, traffic control, communication relay [1]. The UAV communication systems have gradually become a research hotspot, and it is necessary to establish an accurate and reliable UAV channel model.

The research of the UAV channel model is still lacking, and the existing models can be roughly divided into two categories， deterministic model and statistical model. The Ray-tracing model [2], [3] and finite-difference time domain method (FDTD) [4] model are two kinds of common UAV deterministic models. Deterministic models usually have high precision, but require a large amount of actual data and long operational time to describe a specific propagation environment.Therefore, the universality of the deterministic models is limited. Statistical model contains non-geometrical statistical model (NGSM) and geometry-based statistical model (GBSM). [5],[6] and [7] proposed a wideband tapped delay line (TDL) model, which is a common kind of the NGSM and composed of the line of sight(LoS) component, a ground reflection, and up to seven intermittent multi-path components(MPCs). However the models in [5]–[8] are based on the measurement and have some degree of complexity. The regular-shaped geometry-based statistical models (RS-GBSMs) can do us a favor to study the channel characteristics and provide a reference for the channel measurements in the absence of a large number of measured data for the UAV.

RS-GBSMs can greatly reduce the complexity of the channel model by assuming that the scatterers are uniformly distributed in the regular geometrical shapes such as circle,cylinder and ellipse. The RS-GBSMs are often used in cellular communication and vehicle-to-vehicle communication scenes, showing high accuracy and generality [9]–[13]. As a new scenario for communication systems,UAV moves in the three-dimensional (3D)space. The flight parameters such as height and moving directions have important effects on the statistical characteristics of channel,and thus there are more requirements on the new RS-GBSMs for UAV. [14]–[16] proposed two air to ground (A2G) models which assumed that the scatterers around the ground station were uniformly distributed within the truncated ellipsoidal-shaped scattering region,which is more complicated than the common regular shapes such as cylinders and spheres.Besides, the distribution of both the azimuth and elevation angles is completely derived from geometrical relations in [14]–[16]. In contrast, cylinder and sphere models are more realistic for the reason that the angles of arrival and departure are modeled by the specific statistical mathematical forms such as cosine and von Mises distributions. These common statistical distributions make it easier to derive the closed-form expressions for the channel characteristics and have shown good fit to the previously measured data. Then second order statistics including level crossing rate (LCR)and average fade duration (AFD) were studied in [14]. Nevertheless, the model in [14]is merely for the single-input single-output(SISO) channel, while the UAV-MIMO channel is obviously more suitable for practicable application. [17] proposed a GBSM for A2G channel, but the architecture of the model had fairly complex six elements. A series of cylinder UAV-MIMO models including both the narrowband and wideband models with reduced complexity was proposed in [18]–[21], but they all assume that the elevation and azimuth angles are independent. A 3D double-cylinder RS-GBSM for non-isotropic UAV-MIMO channel were proposed in [22],and the second order statistics, including LCR and AFD, were derived under the 3D propagation environment. In order to consider the joint effect of the azimuth and elevation angles, a 3D single-sphere RS-GBSM based on the von Mises Fisher (VMF) distribution was proposed in [23] for the UAV-MIMO channels and the space-time correlation function(STCF) under a 3D moving environment was derived. However, [23] was based on urban scenarios with the assumption that the line of sight component was blocked by obstacles.According to the actual measurements of the UAV in some environments such as overwater[5] and sub-urban [6], the LoS component is dominant and cannot be ignored, and the more general models remain to be proposed.

In this paper, the authors proposed a narrowband single sphere UAV-MIMO channel model and two new corresponding SoS based simulation models.

Because the reference model assumes that the scatterers are infinite and cannot be applied in practice, it is especially important to propose the corresponding simulation models.In [10], deterministic and statistical sum-of-sinusoids (SoS) simulation models based on two-ring model for MIMO mobile-to-mobile(M2M) fading channels were proposed. [24]proposed both the deterministic and statistical simulation models for non-isotropic scattering M2M scenarios. As for UAV-MIMO channels,there are very few simulation models. [25]proposed deterministic and statistical simulation models based on the narrowband single-cylinder UAV-MIMO reference model, but only the time correlation function and Doppler power spectral density (DPSD) were verified by comparing with the corresponding statistical properties of reference model.

In this paper, we present a narrowband 3D UAV-MIMO single-sphere channel model.The main contributions are summarized as follows.

1) Taking the numerous UAV scenarios into account, we propose a more general UAV-MIMO model combined with line of sight and single bounce rays. The joint effects of the azimuth and elevation angles are considered by applying the VMF distribution.

2) The expressions of STCF, DPSD, LCR and AFD are derived based on the UAV reference model, and the effects of flight parameters are studied.

3) We propose the deterministic and statistical simulation models with finite scatterers based on the reference model by adopting a novel approach to design the angles which have joint probability density function (PDF).

The rest of the paper is organized as follows. In Section II, we propose a single-sphere UAV-MIMO reference model combined with LoS and single bounce (SB) rays. In Section III, the channel statistical properties expressions of the reference model including STCF,DPSD, LCR and AFD are derived. In Section IV, the deterministic and statistical simulation models based on the reference model are proposed, and the corresponding channel characteristics are also derived. In Section V,numerical simulations are discussed to verify the simulation models and the effects of the flight parameters on characteristics are also analysed. Finally, conclusions are drawn in Section VI.

Fig. 1. The IoT network scenario.

II. GEOMETRY-BASED UAV-MIMO CHANNEL REFERENCE MODEL

In this section, a 3D narrowband RS-GBSM for A2G UAV-MIMO communication channel is proposed and the multipath component of the reference model consists of line of sight component and single bounce rays. The geometry relationships are shown in figure 1,Tx and Rx represent the locations of the UAV and the ground user respectively. Considering that the UAV is quite high from the ground,we suppose that there is no scatterer around the UAV. The scatters around the ground station are uniformly distributed on the spherical surface with radius R, the j th scatterer of the sphere is denoted by, j =(1,2,…,NR).

The uniform linear antenna arrays are used in both transmitter and receiver, with nTtransmitting antenna elements and nRreceiving ones. For easy analysis, we assume nT= nR=2 and it can be expanded to arbitrary number of antenna elements later. Assuming that both the UAV and the ground users are in a state of motion, the UAV can move in both the horizontal direction and vertical direction, while the ground users only have movement in the horizontal direction. The elevation angle between Tx and Rx is definedwhere HTand HRrepresent the heights of transmitter and receiver respectively, D is the horizontal distance between the Tx and Rx. The elevation angle formula can be approximately simplified to β0=arctan(HT/D) as a result of the fact that HT? HR. δTand δRdenote the spacing between two adjacent antenna elements and we usually assume {δR,δT}?R?D. More parameters used in the model are defined in Table I.

The expression of channel impulse response(CIR) can be obtained by superimposing the LoS and SB ray,

where j2= ?1, λ is the is the carrier wavelength, the random phase shift ψR(j)is uniformly distributed over [?π,π), ?plis the power of the sub-channel p?l. K represents the Rice factor, which is the ratio of the receiving power of the LoS component to that of the non-direct component.

Applying the sine law to the triangles and small angle approximation (i.e., sin x ≈x and cos x ≈ 1 for small x), we can derive the following approximations for the AAoD and EAoD. Through the calculation and simplification of geometric relations, we can get the distances

Table I. Parameters used in the UAV model.

III. CHANNEL CHARACTERISTICS OF THE 3D UAV-MIMO REFERENCE CHANNEL MODEL

In this section, considering the actual UAV communication scenes, several attractive statistic channel characteristics of the proposed 3D UAV-MIMO reference model for a non-isotropic scattering environment are derived, including the STCF, DPSD, LCR and AFD.

3.1 The STCF and DPSD

Assuming that the channel is wide-sense stationary uncorrelated scattering (WSSUS), the STCF of arbitrary two sub-channels p1→l1,p2→l2can be written as

where h*(?) represents complex conjugate operations and E(?) stands for mathematical expectation.

When the number of effective scatterers tends to be infinite NR→∞, the discrete expression of the arrival and departure anglescan be replaced by four continuous random variables αTR, βTR, αRR,βRR. Similar to the channel impulse response function, we can obtain the STCF by superimposing the two components,

In order to jointly consider the effects of the azimuth and elevation angles, the distribution of effective scatterers is represented by the VMF distribution. The PDF of the VMF is defined as [24]

where αRR∈[?π ,π),βRR∈[?π /2,π /2). The αu∈[?π ,π) and βu∈[?π /2,π/2) are the parameters of the VMF PDF, which represent the statistical means of αRRand βRRrespectively.The non-negative real coefficient kRcontrols the distribution concentration of the scatterers, when kRis larger, the scatterers are more concentrated around the direction of the mean angle.

The DPSD is defined as the Fourier transform of the time correlation function.

where fDis the Doppler frequency. (9) can also be written as a sum of LoS and SB ray

where,

3.2 The LCR and AFD

The LCR of the specified level r is defined as the number of times the signal envelope crosses the level r at a positive (negative) slope.Using traditional PDF-based methods [27],general expressions of LCR for Ricean UAV channels can be obtained.

where cosh(?) is the hyperbolic cosine, erf(?) is the error function, andCombined with the reference model proposed above, we can derive the parameters bm,m=(0,1,2) as the follows,

The AFD T(r) is defined as the average time over which the signal envelope remains below a certain level r [28]. The expression of the AFD can be written as

where, Q(?) is the Marcum Q function.

IV. NEW SIMULATION MODELS FOR 3D UAV-MIMO CHANNELS

Because the reference model assumes that the scatterers are infinite, which prevents the practical implementation, the simulation models with finite numbers of scaterers is needed[11]. The SoS simulation model is a kind of common simulation models using several sine waves to fit the channel, which can effectively reduce the computation. In this section, we propose both the deterministic and statistical simulation models for the non-isotropic scattering environment based on the UAV reference model above. The channel characteristics of the simulation model are also derived to reproduce those of reference model.

4.1 New deterministic simulation model

The deterministic simulation model defines only the phases shift ψR(j)as random variables and the statistical channel characterises of the simulation model can be obtained through one simulation trial. The CIR of the deterministic simulation model can be get by the superposition of LOS and SB ray,

Different from other models such as cylinder-based models, whose azimuth and elevation angles are independent, the AAoA and EAoA of the sphere-based models have joint PDF. It is impossible to achieve the corresponding angles set by using the inverse function of the distribution function directly.In this particular case we have adopted a novel approach combined with the traditional modified method of equal area (MMSE) method.That is to say, the AAoA is segmented into several subintervals firstly. The main steps are as follows

1) Divide the angle equally into Nαparts firstly.Because the AAoAis uniformly distributed over [?π,π), we can define the random variable as+2π ?(j? 1/2)/Nα, j =1,2,...,NαThen we have Naintervals [?π,?π+2π/Nα),[? π+ 2π / Nα,? π+4π / Nα)… ,[π ?2π /Nα, π).

2) Integrate the VMF PDF at each interval above, and the cumulative distribution function(CDF) sequence of the Naintervals is represented by CDF(j), j=1,2,...,Nα,which can be obtained

3) Combined with the traditional MMSE method [11], the EAoAcan be defined aswhere fβRR|αRR(αRR,βRR) is the conditional VMF PSD given that the AAoAis known.So we can get the expression of the EAoA thatwhere(?) is the inverse function of fβRR|αRR(αRR,βRR). The formula can be further expressed as

4.2 New statistical simulation model

Fig. 2. The effect of the Ricean factor and simulation scatterers on the normalized time correlation function.

Fig. 3. Comparison of the normalized space correlation of the transmitter derived from the reference model and two simulation models with different VMF PSD parametersn kR.

It is fast and easy to implement deterministic simulation model, however, because their scatterers are placed in a specific position in the simulation, they cannot reflect the actual channel. To better simulate the fading process,the deterministic simulation model can be modified into a statistical simulation model by defining the AoAs as random variables. The statistical channel characterises of the simulation model vary in each simulation trial. By averaging the sufficient number of simulation trials, they will converge to those of the reference model [10].

There are still three main steps and the first two steps are similar to the deterministic model.

1) Divide the horizontal angle equally into Nαparts firstly. Similar to the deterministic simulation model, we can define the random variable AAoA as

2) Calculate the VMF PSD integral of each interval and we can get the CDF sequence of the Nαintegrals that can be obtained as

3) Different from the deterministic simulation model, random variable σβis introduced in this step so thatvaries with different simulation trials. Theis defined as

where σβis uniformly distributed over[?1 /2,1/2).

In this way, we finally get the deterministic and statistical simulation models. Similar to the derivation of the reference model channel characteristics, we extract the STCF, PSD,LCR and AFD of both the deterministic and statistical simulation model. The specific derivation process is no longer given as the length limit and the comparison between the reference model and the two simulation models will be given in the next section.

V. NUMERICAL SIMULATION AND ANALYSIS

In this Section, we give the simulation results of the reference model, deterministic simulation model and statistical simulation model.The STCF, DPSD, LCR and AFD are presented to verify the simulation models and analyse the effect of various parameters. Because the channel characteristics of the statistical simulation model change with each simulation trial, to avoid the judgment of the reliability and practicability of the model, we average the multiple simulation results and the number of the simulation times is 50.

Combining with the actual UAV communication scenes, the simulation parameters are set as follows, D=500m, R=10m, θT= π/3,θR=2π /3, β0= π/4, γT= π/6, ?T= π/12,γR= π/12, vT=100 m /s , vR=10 m /s, the VMF PDF parameters αu= π/6, βu= π/6,k=0.5, the Ricean factor K=5, the number of simulation scatters Nα= Nβ=50.

We can obtain the the normalized time-CFs in figure 2 by changing the Ricean factor K={3,5,10} and the number of scatterers Nα= Nβ={10,30,50}. Figure 2 shows that the time correlation increases as K grows. It can be easily explained that, when K increases,the LoS component becomes more dominant and the influence of the change of scattering environment becomes smaller, which results in the time stability increase of the UAV channel. What’s more, it can be seen that, when the number of simulation scatterers increases,the simulation model can better reproduce the statistical characteristics of the reference model, which also proves the validity of the two simulation models. Sufficient number of simulation scatterers is needed for a good approximation. When the number of simulation scatterers is the same, the statistical model is more approximate to the reference model than the deterministic channel model with sufficient simulation trials.

Fig. 4. Comparison of the normalized space correlation of the receiver derived from the reference model and two simulation models.

Fig. 5. The Doppler power spectra characteristic for the UAV-MIMO channel model. The curves are obtained by varying the parameters of moving direction γT,?T.

Figure 3 and 4 show the space correlation of the transmitter and receiver respectively. Changing the VMF PDF parameter kR={0.5,3,5}, we can observe that the spatial correlation increases with the growth of kR.Because the parameter kRcontrols the distribution concentration of the scatterers, when kRis larger, the scatterers are more concentrated around the direction of the mean angle and thus the space correlation increases. By comparing the figure 3 and figure 4, we can see that the spatial correlation of the receiver is significantly smaller than that of the transmit-ter, and the change of the number of simulation scatterers has a smaller effect on the transmitter. That is because the model proposed in this paper assumes that all the scatterers are around the ground station, while there is no scatterer around the UAV.

Fig. 6. The envelope level crossing rate of the reference and simulation models with different horizonal moving directions γT.

Fig. 7. The envelope average fade duration of the reference and simulation models with different altitude.

Through taking the Fourier transform of the time correlation function, we can obtain the DPSD in figure 5. Varying the moving direction of the UAV, we can see that the moving direction of UAV has a great influence on the Doppler power spectrum. {γT=π/ 6， ?T=π /3}is the set of statistical means of the PDF of AAoA and EAoA, which represents the concentrated direction of the scatterers. The Doppler energy is more concentrated when moving towards the direction where the scatterers are concentrated. When the UAV moves in other directions, the doppler power spectrum expands obviously.

Figure 6 shows the level crossing rate with the horizontal moving direction of the UAV changed, γT= {π/ 12， π/ 6， π/3}. The simulation results show that when the UAV is moving towards sacatterers γT=π /6, the level crossing rate is larger than those in other directions. Just as the explanation of the DPSD above, the environment with more scatterers change the channel seriously. From figure 7,as the β0increases β0= {π/ 12， π/ 6， π/3},the average fade duration gradually increases.In the actual UAV communication scenario,UAV becomes higher from the ground as the elevation angle between Tx and Rx increases.The height difference results in the increase of time stability for the UAV channel.

VI. CONCLUSIONS

In this paper, we have proposed a narrowband single sphere UAV-MIMO channel model and two new corresponding SoS based simulation models. The influence of the azimuth and elevation angles has been considered by using the VMF distribution. Considering the specification of the UAV channel, the the closed form expressions of the channel statistical properties including STCF, DPSD, LCR and AFD for a non-isotropic environment have been derived. Both the deterministic and statistical simulation models based on the reference model with finite scatters have been presented,and the channel characteristics have also been derived to verify the simulation models. The simulation results have shown that the sim-ulation models have a better approximation of the reference model with more simulation scatterers and the statistical simulation model is better than the deterministic model with sufficient simulation trials. The altitude and moving direction of the UAV, the Ricean factor and distribution concentration of the scatterers have significant impacts on the channel statistical properties, which indicates that the flight parameters of UAV and the scattering environment both influence the channel jointly.

ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of China under Grant 61622101 and Grant 61571020,and National Science and Technology Major Project under Grant 2018ZX03001031.