# A General 3D Nonstationary Vehicle-to-Vehicle Channel Model Allowing 3D Arbitrary Trajectory and 3D-Shaped Antenna Array

Authors: Qiuming Zhu; Weidong Li; Ying Yang; Dazhuan Xu; Weizhi Zhong; Xiaomin Chen
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

Most of the existing channel model for multiple-input multiple-output (MIMO) vehicle-to-vehicle (V2V) communications only considered that the terminals were equipped with linear antenna arrays and moved with fixed velocities. Nevertheless, under the realistic environment, those models are not practical since the velocities and trajectories of mobile transmitter (MT) and mobile receiver (MR) could be time-variant and unpredictable due to the complex traffic conditions. This paper develops a general 3D nonstationary V2V channel model, which is based on the traditional geometry-based stochastic models (GBSMs) and the twin-cluster approach. In contrast to the traditional models, this new model is characterized by 3D scattering environments, 3D antenna arrays, and 3D arbitrary trajectories of both terminals and scatterers. The calculating methods of channel parameters are also provided. In addition, the statistical properties, i.e., spatial-temporal correlation function (STCF) and Doppler power spectrum density (DPSD), are derived in detail. Simulation results have demonstrated that the output statistical properties of the proposed model agree well with the theoretical and measured results, which verifies the effectiveness of theoretical derivations and channel model as well.


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## Body

## 1. Introduction

Vehicle-to-vehicle (V2V) communications can improve the safety of life and property by collecting and exchanging environment information under complex traffic [1]. Meanwhile, multiple-input multiple-output (MIMO) technologies can expand the capacity and improve the communication efficiency. It is promising to apply MIMO technologies to V2V communications. To provide reasonable references for developing, analyzing, and testing V2V communication systems, a general, accurate, and easy-to-use channel model is required. Moreover, the movements, insufficient antenna space, and lack of rich scattering should be considered in channel modeling and statistical property analysis [2–7].The wide-sense stationary (WSS) assumption was adopted in the traditional geometry-based stochastic models (GBSMs) [5, 8]. Nevertheless, the authors in [9, 10] found that the WSS was only suitable for short distances between transmitters and receivers based on the measured data. Several GBSMs for nonstationary V2V channels are proposed in the literature [11–25]. Among them, 2D nonstationary V2V GBSMs with fixed clusters [11, 12] or moving scatterers [13] were studied and the statistical properties, i.e., the temporal correlation function (TCF), spatial correlation function (SCF), and Doppler power spectrum density (DPSD), were also analyzed. The authors in [14] proposed a 2D V2V channel model with random movement scatterers. The aforementioned 2D channel models were only considered on the horizontal plane. Under the realistic scenarios, the scatterers, e.g., vehicles, pedestrian, and infrastructures, may distribute or move in the 3D space. Meanwhile, the authors in [7] have proved that the 3D channel model is more accurate than the 2D ones for evaluating the system performance. By extending the scatterers distributed on the surface of 3D regular shapes, i.e., a hemisphere [15], two cylinders [16, 17], two spheres [18], an elliptic-cylinder [19], a rectangular tunnel [20], several 3D nonstationary V2V channel models were proposed recently. In [21], the authors proposed a 3D irregular-shaped GBSM for nonstationary V2V channels. Some 3D cluster-based nonstationary V2V channel models can also been addressed in [22–25]. However, the output phases related to the Doppler frequencies in [11–25] were inaccurate compared with the theoretical results [26, 27]. The authors in [28, 29] modified the phase item to overcome this shortcoming, but it is only suitable for 2D scattering environments. A modified 3D channel model with accurate Doppler frequency can be addressed in [30].Note that most of the existing V2V channel models [11–26] only took the fixed velocities of the mobile transmitter (MT) and mobile receiver (MR) into account. However, under realistic traffic environments, the velocities of MT and MR could be time-variant. Although the time-variant velocities were considered in [29, 31–33], their trajectories were 2D. Meanwhile, the models in the literature [11–33] only considered 2D or even 1D antenna arrays for simplicity. With the development of antenna technologies, 3D-shaped antenna arrays begin to be used in MIMO communication systems. Very recently, a general 3D nonstationary GBSM between base station and MT with 3D arbitrary trajectories and 3D antenna arrays was proposed in [34]. This idea was adopted by [35] to model the 3D V2V channels allowing MT and MR with 3D arbitrary trajectories. However, the details of channel parameter computation and theoretical analyses of SCF, TCF, and DPSD were lacked. This paper aims at filling these gaps. The major contributions and novelties are summarized as follows:(1)
Combining the GBSM and twin-cluster approach [36], a general 3D V2V channel model was proposed. The 3D antenna array and 3D arbitrary trajectory of each terminal are allowed under the 3D scattering environment, which guarantees the proposed model is more general and has the nonstationary aspect.(2)
The upgraded computation procedure of channel parameters for the proposed model, such as the number of paths, path delays, and path powers, are developed and analyzed in detail.(3)
The theoretical closed-form expressions of statistical properties, i.e., the SCFs, TCFs, and DPSDs are investigated and verified by the simulation method and measurement data.This paper is structured as follows. Section2 presents a 3D general V2V channel model characterized by nonstationary aspect and generalization. The channel parameter updating algorithms are given in Section 3. In Section 4, the theoretical SCFs, TCFs, and DPSDs of the proposed model are derived in detail. Section 5 shows and compares the simulated results with the derived ones and measured data. Finally, the conclusions are drawn in Section 6.


## 2. 3D Nonstationary GBSM for V2V Channels

Figure1 shows a typical 3D V2V communication system between the MT and MR, which is characterized by 3D arbitrary trajectories and 3D antenna arrays. The MT and MR are configured with S and U antennas, respectively. In the figure, the coordinate systems at the MT and MR with the centers of corresponding terminals are named as xyz and x˜y˜z˜, respectively. Suppose that the travel directions of the MT and MR at the initial time correspond to x-axis’s direction, respectively. Due to the movement and rotation of two coordinate systems, the antenna element of MT or MR is 3D and time-variant and can be expressed by dsMTt=ds,xMTt,ds,yMTt,ds,zMTtT or duMRt=du,x˜MRt,du,y˜MRt,du,z˜MRtT. Figure 1 shows that between the MT and MR, there are many propagation paths and subpaths (or rays). The 3D location of the first cluster SnMT or the last cluster SnMR affects the corresponding elevation angles, such as elevation angle of departure (EAoD) θn,mMTt and elevation angle of arrival (EAoA) θn,mMRt, and azimuth angles, such as the azimuth angle of departure (AAoD) ϕn,mMTt and the azimuth angle of arrival (AAoA) ϕn,mMRt. In addition, by adopting twin-cluster approach, the rest clusters can be viewed as a virtual link [36]. In the figure, vit and vSnit, i∈MT,MR, represent time-variant velocities of i and Sni, respectively. Table 1 shows the detailed definitions of channel parameters.Figure 1
Typical 3D V2V communication system.Table 1
Definitions of channel parameters.
N
t
Valid path number
P
n
t
,
τ
n
t
Thenth path power and delay
M
The number of subpaths
d
s
MT
t
,
d
u
MR
t
Location vectors of transmitter antenna elements and receiver antenna element u, respectively
L
i
t
,
L
S
n
i
t
Location vectors ofi and Sni, i∈MT,MR, respectively
v
i
t
,
v
S
n
i
t
Velocity vectors ofi and Sni, i∈MT,MR, respectively
v
i
t
,
v
S
n
i
t
Speeds ofi and Sni, i∈MT,MR, respectively
ϕ
v
i
t
,
ϕ
v
S
n
i
t
Traveling azimuth angles ofi and Sni, i∈MT,MR, respectively
θ
v
i
t
,
θ
v
S
n
i
t
Traveling elevation angles ofi and Sni, i∈MT,MR, respectively
s
˜
n
,
m
i
t
Spherical unit vector ofi along ray m of cluster n, i∈MT,MR
ϕ
n
,
m
MT
t
,
θ
n
,
m
MT
t
Azimuth and elevation departure angles along raym of cluster n, respectively
ϕ
n
,
m
MR
t
,
θ
n
,
m
MR
t
Azimuth and elevation arrival angles along raym of cluster n, respectivelyUnder the V2V communication scenario of Figure1, the V2V MIMO channel can be modeled as [30](1)Ht,τ=h1,1t,τh1,2t,τ⋯h1,St,τh2,1t,τhu,st,τ⋯h2,St,τ⋮⋮⋱⋮hU,1t,τhU,2t,τ⋯hU,St,τ,where hu,st,τ denotes the complex channel impulse response (CIR) between the transmitting antenna ss=1,2,…,S and receiving antenna uu=1,2,…,U. In this paper, we modify the model of hu,st,τ in [30] and express it as(2)hu,st,τ≜ΠT0t∑n=1NtPnth˜u,s,ntδτ−τnt,where ΠT0t is a rectangular window function:(3)ΠT0t≜1,0≤t≤T0,0,otherwise,and it is introduced to limit the length of CIR, and thus, the large-scale variations can be negligible within the time interval; Nt represents the valid path number and is related to the path delay τnt, path power Pnt, and normalized coefficient h˜u,s,nt, which is expressed as(4)h˜u,s,nt=limM⟶∞1M∑m=1MejΦn,mDt+Φu,s,n,mLt+Φn,mI,where M denotes the subpath number, Φn,mI stands for the random initial phase distributing over 0,2π uniformly, and Φn,mDt means the phase somehow affected by variant Doppler frequency. Here, we model Φn,mDt as(5)Φn,mDt=k∫0tvMT,SnMTt′⋅s˜n,mMTt′+vSnMR,MRt′⋅s˜n,mMRt′dt′,where k=2πfc/c denotes the wave number with fc and c representing the carrier frequency and light speed, respectively. vi,Snit, i∈MT,MR, denotes the relative velocity between i and Sni. The departure or arrival angle unit vector of the mth subpath within the nth path s˜n,mit is defined as(6)sn,mit=cosθn,mitcosϕn,mitcosθn,mitsinϕn,mitsinθn,mit.In (4), Φu,s,n,mLt denotes the offset phase caused by the movements of terminals:(7)Φu,s,n,mLt=k⋅s˜n,mMTtT⋅RvMTt⋅dsMT,t0+k⋅s˜n,mMTtT⋅RvMTt⋅duMT,t0,where ⋅T means transpose operation and dsMT,t0 represents sth antenna position at initial time instant. Similarly, duMR,t0 represents the corresponding antenna position at initial corresponding time. Note that Rvit is a rotation matrix due to the 3D arbitrary trajectory, and it can be written as(8)Rvit=cosθvitcosϕvit−sinϕvit−sinθvitcosϕvitcosθvitsinϕvitcosϕvit−sinθvitsinϕvitsinθvit0cosθvit.


## 3. Time-Variant Channel Parameters

### 3.1. Path Number

The valid path number has been proved to be time-variant under V2V scenarios by the measured data in [37], due to the moving clusters and two terminals. In other words, the old disappearing clusters and new appearing ones are random. We model these with a Markov process [38]. λG is used to denote the birth rate, and λR denotes the death rate. The probability of each path remaining from t to t+Δt is calculated as [19](9)Prt;Δt=e−λR⋅PcvSMTt+vSMRtΔt⋅e−λRvMRt−vMTtΔt,where Pc means the moving percentage and vSit, i∈MT,MR, represents clusters average speed. Nnewt is introduced to describe newly generated path number and calculated as(10)ENnewt=λGλR1−Prt;Δt.Combining (9) with (10), the averaged path number can be expressed as(11)ENt=Nt−ΔtPrt;Δt+ENnewt=λGλR.


### 3.2. Delays and Powers

For thenth valid path, the total delay at time instant t consists of the first and last bounce delays and virtual link delay, and it is a function of total distance as(12)τnt=LMTt−LSnMTt+LMRt−LSnMRtc+τ˜nt,where Lit and LSnit, i∈MT,MR, mean the instantaneous locations of i and Sni and can be updated from the values of previous time instant by(13)Lit=Lit−Δt+vit−ΔtΔt,LSnit=LSnit−Δt+vSnit−ΔtΔt,where τ˜nt denotes the equivalent delay of virtual link and it can be updated by the first-order filtering method in [26] as follows:(14)τ˜nt=τ˜nt−Δte−Δt/τdec+1−e−Δt/τdecX,where X∼ULMTt−LMRt/c,τmax, in which τmax denotes the maximum delay, and τdec denotes the decorrelation speed of time-variant delays. Based on the measurement-based method, we can get the averaged power as(15)Pn′t=e−τnt1−rDS/rDSσDS×10−ξn/10,where ξn, rDS, and σDS mean the shadowing degree, delay factor, and delay spread, respectively, which are all related with the environments. Finally, all the normalized power can be denoted as(16)Pnt=Pn′t∑n=1NtPn′t.


### 3.3. Time-Variant Angles

Measurements in [39] revealed that the azimuth and elevation angles at two terminals are not independent and cannot be depicted by two independent distributions such as Gaussian or Laplacian. In this paper, we take the 3D von Mises–Fisher (VMF) distribution to describe these two joint angles [40]. The probability density function (PDF) of VMF distribution can be expressed as [39](17)pϕ,θ=κcosθeκcosθcosθ¯cosϕ−ϕ¯+sinθsinθ¯4πsinhκ,−π≤ϕ≤π,−π2≤θ≤π2,where ϕ¯ and θ¯ represent the mean values of azimuth and elevation angles, respectively, and κ is the shape factor and can be extracted from the measurement data.As long asκ is determined, the time-variant angle of departure (AoD) and angle of arrival (AoA) can be obtained by ϕ¯t and θ¯t, respectively, which can be expressed as follows:(18)θ¯nit=arcsinLzSnit−LzitLSnit−Lit,ϕ¯nit=arccosLxSnit−LxitLSnit−Litcosθ¯nit,LySnit−Lyit≥0−arccosLxSnit−LxitLSnit−Litcosθ¯nit,LySnit−Lyit<0,where i∈MT,MR. Note that κ almost unchanged during the short simulation time period. Figure 2 gives an example of VMF distributed angles of AoA (or AoD) on the unit sphere with different parameters. It also reveals that with the increase in shape factor κ, the shape of distribution becomes more concentrated.Figure 2
The VMF distributed angles on the unit sphere.


## 3.1. Path Number

The valid path number has been proved to be time-variant under V2V scenarios by the measured data in [37], due to the moving clusters and two terminals. In other words, the old disappearing clusters and new appearing ones are random. We model these with a Markov process [38]. λG is used to denote the birth rate, and λR denotes the death rate. The probability of each path remaining from t to t+Δt is calculated as [19](9)Prt;Δt=e−λR⋅PcvSMTt+vSMRtΔt⋅e−λRvMRt−vMTtΔt,where Pc means the moving percentage and vSit, i∈MT,MR, represents clusters average speed. Nnewt is introduced to describe newly generated path number and calculated as(10)ENnewt=λGλR1−Prt;Δt.Combining (9) with (10), the averaged path number can be expressed as(11)ENt=Nt−ΔtPrt;Δt+ENnewt=λGλR.


## 3.2. Delays and Powers

For thenth valid path, the total delay at time instant t consists of the first and last bounce delays and virtual link delay, and it is a function of total distance as(12)τnt=LMTt−LSnMTt+LMRt−LSnMRtc+τ˜nt,where Lit and LSnit, i∈MT,MR, mean the instantaneous locations of i and Sni and can be updated from the values of previous time instant by(13)Lit=Lit−Δt+vit−ΔtΔt,LSnit=LSnit−Δt+vSnit−ΔtΔt,where τ˜nt denotes the equivalent delay of virtual link and it can be updated by the first-order filtering method in [26] as follows:(14)τ˜nt=τ˜nt−Δte−Δt/τdec+1−e−Δt/τdecX,where X∼ULMTt−LMRt/c,τmax, in which τmax denotes the maximum delay, and τdec denotes the decorrelation speed of time-variant delays. Based on the measurement-based method, we can get the averaged power as(15)Pn′t=e−τnt1−rDS/rDSσDS×10−ξn/10,where ξn, rDS, and σDS mean the shadowing degree, delay factor, and delay spread, respectively, which are all related with the environments. Finally, all the normalized power can be denoted as(16)Pnt=Pn′t∑n=1NtPn′t.


## 3.3. Time-Variant Angles

Measurements in [39] revealed that the azimuth and elevation angles at two terminals are not independent and cannot be depicted by two independent distributions such as Gaussian or Laplacian. In this paper, we take the 3D von Mises–Fisher (VMF) distribution to describe these two joint angles [40]. The probability density function (PDF) of VMF distribution can be expressed as [39](17)pϕ,θ=κcosθeκcosθcosθ¯cosϕ−ϕ¯+sinθsinθ¯4πsinhκ,−π≤ϕ≤π,−π2≤θ≤π2,where ϕ¯ and θ¯ represent the mean values of azimuth and elevation angles, respectively, and κ is the shape factor and can be extracted from the measurement data.As long asκ is determined, the time-variant angle of departure (AoD) and angle of arrival (AoA) can be obtained by ϕ¯t and θ¯t, respectively, which can be expressed as follows:(18)θ¯nit=arcsinLzSnit−LzitLSnit−Lit,ϕ¯nit=arccosLxSnit−LxitLSnit−Litcosθ¯nit,LySnit−Lyit≥0−arccosLxSnit−LxitLSnit−Litcosθ¯nit,LySnit−Lyit<0,where i∈MT,MR. Note that κ almost unchanged during the short simulation time period. Figure 2 gives an example of VMF distributed angles of AoA (or AoD) on the unit sphere with different parameters. It also reveals that with the increase in shape factor κ, the shape of distribution becomes more concentrated.Figure 2
The VMF distributed angles on the unit sphere.


## 4. Time-Variant Statistical Properties

The normalized spatial-temporal correlation function (STCF) between two different subchannels can be defined by [16, 19, 25](19)R˜u1,s1,nu2,s2,nt;Δt,Δd=Eh˜u1,s1,nth˜u2,s2,n∗t+ΔtEh˜u1,s1,nt2Eh˜u2,s2,nt+Δt2,where E⋅ represents the expectation function, ∗ is complex conjugate, and Δt means time lag. Moreover, Δd=ΔdMT,ΔdMR denotes the space lag and consists of ΔdMT and ΔdMR, which are the corresponding space lags at two terminals, respectively. By substituting our proposed channel model (4) into (19), the expression of STCF can be obtained as(20)R˜u1,s1,nu2,s2,nt;Δt,Δd=limM⟶∞1M∑m=1MejΦu2,s2,n,mDt+Δt+Φu2,s2,n,mLt+Δt+Φn,mIe−jΦu1,s1,n,mDt+Φu1,s1,n,mLt+Φn,mI=∫​∫​∫​∫​θnMT,ϕnMT,θnMR,ϕnMRpϕnMTt+Δt,θnMTt+ΔtpϕnMRt+Δt,θnMRt+ΔtpϕnMRt,θnMRt⋅pϕnMTt,θnMTtejΦu2,s2,nDt+Δt−Φu1,s1,nDt+Φu2,s2,nLt+Δt−Φu1,s1,nLtdθnMTdϕnMTdθnMRdϕnMR.


### 4.1. Time-Variant SCFs

The SCFs is reduced from STCF when the time lagΔt equals to zero. Then, it can be expressed as(21)R˜u1s1,nu2s2,nt;Δd=∫​∫​∫​∫​θnMT,ϕnMT,θnMR,ϕnMRpϕnMTt,θnMTtpϕnMRt,θnMRt⋅ejΦu2,s2,nLt−Φu1,s1,nLtdθnMTdϕnMTdθnMRdϕnMR.Since the clustersSnMT and SnMR are independent, we can rewrite (21) as(22)R˜u1,s1,nu2,s2,nt;Δd=R˜nMTt;ΔdMT⋅R˜nMRt;ΔdMR,where R˜nit;Δdi, i∈MT,MR, denotes the normalized SCF at the MT or MR. By substituting (7) into (21), the following equation can be obtained:(23)R˜nit;Δdi=∫​−ππ∫​−ππejks˜nitT⋅RVit⋅Δdi,t0⋅pϕn,mit,θn,mitϕnit,θnitdϕnidθni.It is shown in (23) that not only the angle of horizontal plane ϕnit but also the one of vertical plane θnit affects the result of (23). Thus, traditional 2D models cannot obtain accurate correlation functions in 3D scattering environments. By setting θnit=θ¯nit+ζ and ϕnit=ϕ¯nit+υ and substituting them into (23), it yields(24)R˜nit;Δdi=∫​∫​ζ,υκcosθ¯nit+ζAiBi4πsinhκdζdυ,where(25)Ai=eκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit,(26)Bi=ejks˜ϕ¯nit+υ,θ¯nit+ζRVitΔdi,t0.The measurement data demonstrate that the angle spread is usually narrow under V2V communication scenarios. With this assumption, we havecosζ≈cosυ≈1, sinζ≈ζ, sinυ≈υ, κcosυ≈κ1−υ2/2, and κcosζ≈κ1−ζ2/2. Consequently, we can rewrite (25) and (26) as(27)Ai≈eκ1−ζ2/2−2cos2θ¯nitυ/22,(28)Bi≈ejks˜ϕ¯nit,θ¯nit+Citζ+EitυRVitΔdi,t0,where(29)Cit=−sinθ¯nitcosϕ¯nit,−sinθ¯nitsinϕ¯nit,cosθ¯nit,(30)Dit=−sinϕ¯nit,cosϕ¯nit,0.By substituting (27)–(30) into (24) and using Euler formula, we can obtain the following equation:(31)R˜nit;Δdi≈ejks˜ϕ¯nit,θ¯nitRVitΔdi,t0+κcosθ¯nitκ1−m/22πm/2Im/2−1κ⋅∫−ΔϕiΔϕie−Eitυ2coskDitRVitΔdi,t0υdυ⋅∫−ΔθiΔθie−κζ2/2coskCitRVitΔdi,t0ζdζ,where Eit=κcos2θ¯nit/2 and Δϕi and Δθi are the angle spreads of AAoA and EAoA. With the help of integration formula,(32)∫−cce−ax2cosbxdx=jπe−b2/4aerfib−2jac/2a−erfib+2jac/2a2a.The approximate closed-form result of SCF is as follows:(33)R˜nit;Δdi≈−ImerfiFitImerfiGit8sinhκ⋅ejks˜ϕ¯nit,θ¯nitRVitΔdi,t0⋅e−kRVitΔdi,t02Dit/2κ2+Cit/2κ2+κ,where erfi⋅ is the imaginary error function.(34)Fit=kDitRVitΔdi,t0−jκcos2θ¯nitΔϕi2κcos2θ¯nit,(35)Git=kCitRVitΔdi,t0−jκΔθi2κ,(36)R˜u1,s1,nu1,s1,nt;Δt=∫​∫​∫​∫​θnMT,ϕnMT,θnMR,ϕnMRpϕnMTt+Δt,θnMTt+ΔtpϕnMRt+Δt,θnMRt+ΔtpϕnMRt,θnMRt⋅pϕnMTt,θnMTtejΦu1,s1,nDt−Φu1,s1,nDt+Δt+Φu1,s1,nLt−Φu1,s1,nLt+ΔtdθnMTdϕnMTdθnMRdϕnMR,(37)R˜nit;Δt=∬ϕni,θnipϕnit,θnitpϕnit+Δt,θnit+Δte−jk∫tt+Δtvi,Snit′⋅sn,mit′dt′dϕnidθni,(38)R˜nit;Δt=κ4πsinhκ∫−ΔθΔθ∫−ΔϕΔϕcosθ¯nit+ζeκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit⋅cosθ¯nit+Δt+ζeκcosθ¯nit+Δt+ζcosθ¯nit+Δtcosυ+sinθ¯nit+Δt+ζsinθ¯nit+Δt⋅e−jk∫tt+Δtvi,Snitcosϕvi,Snit−ϕ¯nit+υcosθ¯nit+ζcosθvi,Snit+sinθ¯nit+ζsinθvi,Snitdtdυdζ,(39)R˜nit;Δt=κ4πsinhκ∫−ΔθΔθ∫−ΔϕΔϕcosθ¯nit+ζeκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit⋅cosθ¯nit+Δt+ζeκcosθ¯nit+Δt+ζcosθ¯nit+Δtcosυ+sinθ¯nit+Δt+ζsinθ¯nit+Δt⋅e−jkvi,SnitcosAit;Δt;υ,ζDit;Δt+cosBit;Δt;υ,ζEit;Δt+sinCit;Δt;ζFit;Δtdυdζ.Finally, by substituting (33)–(35) into (22), the final result of SCF in our proposed model can be obtained. Due to time-variant communication environments, SCF is also time-dependent.


### 4.2. Time-Variant TCFs

The TCF is reduced from STCF whenΔd is set to be zero. Then, TCF can be derived as (36). It also equals to the product of TCFs at the MT and MR R˜nit;Δt, i∈MT,MR, when the clusters SnMT and SnMR are independent. For simplicity, we assumed that the antenna array is placed at the origin of the coordinate system. By substituting (5) and (7) into (36), it can be obtained as (37).By settingθnit=θ¯nit+ζ and ϕnit=ϕ¯nit+υ and substituting them into (37), it yields (38), where ϕvi,Snit and θvi,Snit mean the azimuth and elevation angles of the relative velocity between i and cluster Sni, respectively. It is reasonable to assume that the elevation AoDs and azimuth AoDs change linearly during the short time interval as ϕ¯nit=k1t+b1,θ¯nit=k2t+b2, where k1=ϕ¯nit+Δtmax−ϕ¯nit/Δtmax, k2=θ¯nit+Δtmax−θ¯nit/Δtmax, b1=ϕ¯nit−k1t, and b2=θ¯nit−k2t. By integrating the Doppler frequency, we can rewrite (38) as (39), where(40)Ait;Δt;υ,ζ=t+Δt2−k1+k2+ϕvi,Snit−b1+b2−υ+ζ,Bit;Δt;υ,ζ=t+Δt2−k1−k2+ϕvi,Snit−b1−b2−υ−ζ,Cit;Δt;ζ=k2t+Δt2+b2+ζ,Dit;Δt=cosθvi,SnitsinΔt−k1+k2/2−k1+k2,Eit;Δt=cosθvi,SnitsinΔtk1+k2/2k1+k2,Fit;Δt=2sinθvi,Snitsink2Δt/2k2,(41)R˜nit;Δt≈κeκ4πsinhκcosθ¯nitcosθ¯nit+Δte−jkGit;Δt⋅∫−ΔθΔθe−κζ2/2coskIit;Δtζdζ⋅∫−ΔϕΔϕe−Jit;Δtυ2coskHit;Δtυdυ.Measurement data have revealed that small angle spreads exist under some scenarios. In other words,κ is usually big or ζ and υ are small. Holding this condition, we have cosυ−ζ≈1, cosυ+ζ≈1, sinυ−ζ≈υ−ζ, sinυ+ζ≈υ+ζ, cosζ≈1, cosυ≈1, sinζ≈ζ, and sinυ≈υ. Thus, we can approximate (39) as (41), where(42)Git;Δt=vi,Snit⋅Dit;ΔtcosAit;Δt+vi,Snit⋅Eit;ΔtcosBit;Δt+vi,Snit⋅Fit;ΔtsinCit;Δt,Hit;Δt=vi,Snit⋅Dit;ΔtsinAit;Δt+vi,Snit⋅Eit;ΔtsinBit;Δt,Iit;Δt=−vi,Snit⋅Dit;ΔtsinAit;Δt+vi,Snit⋅Eit;ΔtsinBit;Δt+vi,Snit⋅Fit;ΔtcosCit;Δt,Jit;Δt=κcos2θ¯nit+cos2θ¯nit+Δt4,(43)R˜nit;Δt=−κ8sinhκcosθ¯nitcosθ¯nit+Δteκ−k2Ii2t,Δt/2κ−k2Hi2t,Δt/4Jit,Δt−jkGit,Δt⋅ImerfikHit,Δt−2jJit,ΔtΔϕ/2Jit,ΔtJit,ΔtImerfikIit,Δt−jκΔθ/2κ2κ.where Ait;Δt, Bit;Δt, and Cit;Δt are the value of Ait;Δt;υ,ζ, Bit;Δt;υ,ζ, and Cit;Δt;ζ when υ=ζ=0, respectively. Using the integration formula in (32), the closed-form expression of (41) can be expressed as (43). On this basis, we can obtain the final TCF expression of the proposed model.


### 4.3. Time-Variant DPSDs

The DPSD is the Fourier transform of TCFR˜u,s,nt;Δt. For the nonstationary aspect of our proposed model, the DPSD can be calculated with short-time Fourier transform as(44)Snf;t=∫−∞∞R˜u,s,nt;Δte−j2πfΔtνt−ΔtdΔt,where window function νt−Δt lasting time is shorter than the stationary interval (about several milliseconds [10]).


## 4.1. Time-Variant SCFs

The SCFs is reduced from STCF when the time lagΔt equals to zero. Then, it can be expressed as(21)R˜u1s1,nu2s2,nt;Δd=∫​∫​∫​∫​θnMT,ϕnMT,θnMR,ϕnMRpϕnMTt,θnMTtpϕnMRt,θnMRt⋅ejΦu2,s2,nLt−Φu1,s1,nLtdθnMTdϕnMTdθnMRdϕnMR.Since the clustersSnMT and SnMR are independent, we can rewrite (21) as(22)R˜u1,s1,nu2,s2,nt;Δd=R˜nMTt;ΔdMT⋅R˜nMRt;ΔdMR,where R˜nit;Δdi, i∈MT,MR, denotes the normalized SCF at the MT or MR. By substituting (7) into (21), the following equation can be obtained:(23)R˜nit;Δdi=∫​−ππ∫​−ππejks˜nitT⋅RVit⋅Δdi,t0⋅pϕn,mit,θn,mitϕnit,θnitdϕnidθni.It is shown in (23) that not only the angle of horizontal plane ϕnit but also the one of vertical plane θnit affects the result of (23). Thus, traditional 2D models cannot obtain accurate correlation functions in 3D scattering environments. By setting θnit=θ¯nit+ζ and ϕnit=ϕ¯nit+υ and substituting them into (23), it yields(24)R˜nit;Δdi=∫​∫​ζ,υκcosθ¯nit+ζAiBi4πsinhκdζdυ,where(25)Ai=eκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit,(26)Bi=ejks˜ϕ¯nit+υ,θ¯nit+ζRVitΔdi,t0.The measurement data demonstrate that the angle spread is usually narrow under V2V communication scenarios. With this assumption, we havecosζ≈cosυ≈1, sinζ≈ζ, sinυ≈υ, κcosυ≈κ1−υ2/2, and κcosζ≈κ1−ζ2/2. Consequently, we can rewrite (25) and (26) as(27)Ai≈eκ1−ζ2/2−2cos2θ¯nitυ/22,(28)Bi≈ejks˜ϕ¯nit,θ¯nit+Citζ+EitυRVitΔdi,t0,where(29)Cit=−sinθ¯nitcosϕ¯nit,−sinθ¯nitsinϕ¯nit,cosθ¯nit,(30)Dit=−sinϕ¯nit,cosϕ¯nit,0.By substituting (27)–(30) into (24) and using Euler formula, we can obtain the following equation:(31)R˜nit;Δdi≈ejks˜ϕ¯nit,θ¯nitRVitΔdi,t0+κcosθ¯nitκ1−m/22πm/2Im/2−1κ⋅∫−ΔϕiΔϕie−Eitυ2coskDitRVitΔdi,t0υdυ⋅∫−ΔθiΔθie−κζ2/2coskCitRVitΔdi,t0ζdζ,where Eit=κcos2θ¯nit/2 and Δϕi and Δθi are the angle spreads of AAoA and EAoA. With the help of integration formula,(32)∫−cce−ax2cosbxdx=jπe−b2/4aerfib−2jac/2a−erfib+2jac/2a2a.The approximate closed-form result of SCF is as follows:(33)R˜nit;Δdi≈−ImerfiFitImerfiGit8sinhκ⋅ejks˜ϕ¯nit,θ¯nitRVitΔdi,t0⋅e−kRVitΔdi,t02Dit/2κ2+Cit/2κ2+κ,where erfi⋅ is the imaginary error function.(34)Fit=kDitRVitΔdi,t0−jκcos2θ¯nitΔϕi2κcos2θ¯nit,(35)Git=kCitRVitΔdi,t0−jκΔθi2κ,(36)R˜u1,s1,nu1,s1,nt;Δt=∫​∫​∫​∫​θnMT,ϕnMT,θnMR,ϕnMRpϕnMTt+Δt,θnMTt+ΔtpϕnMRt+Δt,θnMRt+ΔtpϕnMRt,θnMRt⋅pϕnMTt,θnMTtejΦu1,s1,nDt−Φu1,s1,nDt+Δt+Φu1,s1,nLt−Φu1,s1,nLt+ΔtdθnMTdϕnMTdθnMRdϕnMR,(37)R˜nit;Δt=∬ϕni,θnipϕnit,θnitpϕnit+Δt,θnit+Δte−jk∫tt+Δtvi,Snit′⋅sn,mit′dt′dϕnidθni,(38)R˜nit;Δt=κ4πsinhκ∫−ΔθΔθ∫−ΔϕΔϕcosθ¯nit+ζeκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit⋅cosθ¯nit+Δt+ζeκcosθ¯nit+Δt+ζcosθ¯nit+Δtcosυ+sinθ¯nit+Δt+ζsinθ¯nit+Δt⋅e−jk∫tt+Δtvi,Snitcosϕvi,Snit−ϕ¯nit+υcosθ¯nit+ζcosθvi,Snit+sinθ¯nit+ζsinθvi,Snitdtdυdζ,(39)R˜nit;Δt=κ4πsinhκ∫−ΔθΔθ∫−ΔϕΔϕcosθ¯nit+ζeκcosθ¯nit+ζcosθ¯nitcosυ+sinθ¯nit+ζsinθ¯nit⋅cosθ¯nit+Δt+ζeκcosθ¯nit+Δt+ζcosθ¯nit+Δtcosυ+sinθ¯nit+Δt+ζsinθ¯nit+Δt⋅e−jkvi,SnitcosAit;Δt;υ,ζDit;Δt+cosBit;Δt;υ,ζEit;Δt+sinCit;Δt;ζFit;Δtdυdζ.Finally, by substituting (33)–(35) into (22), the final result of SCF in our proposed model can be obtained. Due to time-variant communication environments, SCF is also time-dependent.


## 4.2. Time-Variant TCFs

The TCF is reduced from STCF whenΔd is set to be zero. Then, TCF can be derived as (36). It also equals to the product of TCFs at the MT and MR R˜nit;Δt, i∈MT,MR, when the clusters SnMT and SnMR are independent. For simplicity, we assumed that the antenna array is placed at the origin of the coordinate system. By substituting (5) and (7) into (36), it can be obtained as (37).By settingθnit=θ¯nit+ζ and ϕnit=ϕ¯nit+υ and substituting them into (37), it yields (38), where ϕvi,Snit and θvi,Snit mean the azimuth and elevation angles of the relative velocity between i and cluster Sni, respectively. It is reasonable to assume that the elevation AoDs and azimuth AoDs change linearly during the short time interval as ϕ¯nit=k1t+b1,θ¯nit=k2t+b2, where k1=ϕ¯nit+Δtmax−ϕ¯nit/Δtmax, k2=θ¯nit+Δtmax−θ¯nit/Δtmax, b1=ϕ¯nit−k1t, and b2=θ¯nit−k2t. By integrating the Doppler frequency, we can rewrite (38) as (39), where(40)Ait;Δt;υ,ζ=t+Δt2−k1+k2+ϕvi,Snit−b1+b2−υ+ζ,Bit;Δt;υ,ζ=t+Δt2−k1−k2+ϕvi,Snit−b1−b2−υ−ζ,Cit;Δt;ζ=k2t+Δt2+b2+ζ,Dit;Δt=cosθvi,SnitsinΔt−k1+k2/2−k1+k2,Eit;Δt=cosθvi,SnitsinΔtk1+k2/2k1+k2,Fit;Δt=2sinθvi,Snitsink2Δt/2k2,(41)R˜nit;Δt≈κeκ4πsinhκcosθ¯nitcosθ¯nit+Δte−jkGit;Δt⋅∫−ΔθΔθe−κζ2/2coskIit;Δtζdζ⋅∫−ΔϕΔϕe−Jit;Δtυ2coskHit;Δtυdυ.Measurement data have revealed that small angle spreads exist under some scenarios. In other words,κ is usually big or ζ and υ are small. Holding this condition, we have cosυ−ζ≈1, cosυ+ζ≈1, sinυ−ζ≈υ−ζ, sinυ+ζ≈υ+ζ, cosζ≈1, cosυ≈1, sinζ≈ζ, and sinυ≈υ. Thus, we can approximate (39) as (41), where(42)Git;Δt=vi,Snit⋅Dit;ΔtcosAit;Δt+vi,Snit⋅Eit;ΔtcosBit;Δt+vi,Snit⋅Fit;ΔtsinCit;Δt,Hit;Δt=vi,Snit⋅Dit;ΔtsinAit;Δt+vi,Snit⋅Eit;ΔtsinBit;Δt,Iit;Δt=−vi,Snit⋅Dit;ΔtsinAit;Δt+vi,Snit⋅Eit;ΔtsinBit;Δt+vi,Snit⋅Fit;ΔtcosCit;Δt,Jit;Δt=κcos2θ¯nit+cos2θ¯nit+Δt4,(43)R˜nit;Δt=−κ8sinhκcosθ¯nitcosθ¯nit+Δteκ−k2Ii2t,Δt/2κ−k2Hi2t,Δt/4Jit,Δt−jkGit,Δt⋅ImerfikHit,Δt−2jJit,ΔtΔϕ/2Jit,ΔtJit,ΔtImerfikIit,Δt−jκΔθ/2κ2κ.where Ait;Δt, Bit;Δt, and Cit;Δt are the value of Ait;Δt;υ,ζ, Bit;Δt;υ,ζ, and Cit;Δt;ζ when υ=ζ=0, respectively. Using the integration formula in (32), the closed-form expression of (41) can be expressed as (43). On this basis, we can obtain the final TCF expression of the proposed model.


## 4.3. Time-Variant DPSDs

The DPSD is the Fourier transform of TCFR˜u,s,nt;Δt. For the nonstationary aspect of our proposed model, the DPSD can be calculated with short-time Fourier transform as(44)Snf;t=∫−∞∞R˜u,s,nt;Δte−j2πfΔtνt−ΔtdΔt,where window function νt−Δt lasting time is shorter than the stationary interval (about several milliseconds [10]).


## 5. Simulation Results and Validation

Firstly, in order to evaluate the accuracy of our derived SCF and TCF, we calculate the maximum difference between the approximated and numerical integral results overϕ¯ni∈−180°,180° and θ¯ni∈−10°,10°. Figure 3 shows the results of maximum absolute error with different κ, normalized space lags, and time lags. As we can see that the maximum absolute error of SCF is less than 0.025 when κ is more than 50 and d/λ is less than 3, and it increases as κ decreases or d/λ increases. Meanwhile, the absolute error of TCF is less than 0.02 when κ is more than 50 and the time lag is less than 0.05 s, and it increases as κ decreases or the time lag increases. Overall, the maximum error is acceptable, and thus, the derived expressions can be used to calculate the SCF and TCF efficiently.Figure 3
Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs.
(a)
(b)Secondly, in order to verify the generality of the proposed channel model. Three V2V communication scenarios with different trajectories are selected and given in Figure4. In the figure, the MT travels straightly with the fixed velocity, and the MR travels in three different velocities with the same departure and arrival points. Path I has fixed speed and travel direction, and Path II allows speed variation with 2D direction movement. Furthermore, Path III has 3D variations in both vertical and horizontal directions. For demonstration purpose, the distribution of clusters is uniform, and we adopt VMF distribution to depict AoAs and AoDs and set κ as 3.95 [39]. With the referred data in [41], we assume that the speeds of cluster are Gaussian-distributed with 1 km/h mean value and 0.1 variance. Furthermore, the carrier frequency fc=2.4GHz, ϕvSnMT, ϕvSnMR, θvSnMT and θvSnMR are all uniformly distributed. The former pair distributes over 0 and 2π, but the later one only distributes over −π/36 and π/36.Figure 4
Three typical trajectories of the MR.It should be mentioned that the V2V channel models in [11–26, 28, 30] only considered fixed velocities of two terminals like Path I, while the models in [31–33] allowed for 2D curve trajectories like Path II. The absolute values of the theoretical SCFs of different models including the model considering Path I in [30] and the one focusing on Path II in [32] are simulated and compared in Figure 5. The good agreement of SCFs between our proposed and other models indicates the generalization and compatibility of our proposed model. In particular, the models in [30, 32] can be viewed as two special applications in straight or curved trajectories.Figure 5
Comparison of absolute SCFs for (a) Path I and (b) Path II.
(a)
(b)Thirdly, in order to verify the correctness of statistical properties of our proposed model under 3D trajectories like Path III, the theoretical, approximated, and simulated values of SCFs and TCFs are shown in Figure6. The x-axis of Figure 6(a) represents normalized space between antennas, which is similar at two terminals in the communication system. By substituting u1=s1=1, u2=s2=2, and n=1 into (21) and (33), we can get the theoretical and approximated values of SCFs. In addition, the measured results in [42] are also shown in Figure 6(a). The good agreement of SCFs shows the correctness of both the theoretical model and derivations. Similarly, by substituting u=s=1 and n=1 into (36) and (43), we obtain and show the theoretical and approximated values of TCFs in Figure 6(b). The good agreements between theoretical, simulated, and approximated results reveal that our simulation and derivation are correct.Figure 6
Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants.
(a)
(b)Finally, substituting TCFs into (44), we can get the theoretical and simulated values of DPSDs for Path III, which are also shown in Figure 7. It clearly shows that moving clusters and two terminals have an influence on the drifting of DPSDs. Moreover, good agreements between the simulated and theoretical results indicate that our proposed model is correct.Figure 7
(a) Theoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III.
(a)
(b)


## 6. Conclusion

This paper has proposed a general channel model characterized by 3D scattering environments, 3D movements, and 3D-shaped antenna arrays of two terminals for V2V communication system. The upgraded algorithms for time-evolving parameters such as the number of paths, path delays, path powers, and angles have been developed and analyzed in detail. Meanwhile, the approximated expressions of statistical properties including SCF, TCF, and DPSD have been derived and verified by simulations. Simulated and analyzed results show that different moving trajectories significantly affect the V2V channel characteristic. Meanwhile, the generalization and compatibility of our proposed model also demonstrate that some previous models with 2D or even 1D movements have special applications. The new model can be used to develop, analyze, and test realistic V2V communication systems in the future.



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*Source: 8708762-2019-10-20.xml*