# Three-Dimensional Vehicle-to-Vehicle Channel Modeling with Multiple Moving Scatterers

Authors: Derong Du; Xiaoping Zeng; Xin Jian; Lijuan Miao; Haobo Wang
Journal: Mobile Information Systems
 (2017)

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## Abstract

Connected vehicles have received much attention in recent years due to their significant societal benefit and commercial value. However, a suitable channel model for vehicle-to-vehicle (V2V) communications is difficult to build due to the dynamic communication environment. In this paper, a three-dimensional (3D) geometrical propagation model that includes line-of-sight (LoS), single bounced (SB), and multiple bounced (MB) rays is proposed. Each of multiple scatterers in the model is moving with a random velocity in a random direction. Based on the geometrical propagation model, a generalized 3D reference model for narrowband multiple-input-multiple-output (MIMO) V2V multipath fading channels is developed. The corresponding space-time correlation functions (ST-CFs), time correlation functions (T-CFs), and space correlation functions (S-CFs) are analytically investigated and numerically simulated in terms of various factors. Several notable ST-CFs for V2V and fixed-to-mobile (F2M) communications become the special cases of ST-CFs of the proposed model by adjusting the corresponding channel parameters. Finally, the theoretical results of the space-Doppler power spectral density (SD-PSD) are compared with the available measured data. The close agreements between the theoretical and measured SD-PSD curves confirm the utility and generality of the proposed model.


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## Body

## 1. Introduction

Connected vehicles have the potential to improve the safety and efficiency of the automobile transportation [1], and they are expected to be a pillar of a smart society and to revolutionize the way people move. Different from the conventional fixed-to-mobile (F2M) cellular systems, vehicle-to-vehicle (V2V) is a kind of the mobile-to-mobile (M2M) communication which allows both the transmitter and receiver to be in motion [2]. Suitable channel models and channel characterizations are absolutely essential for successful design of V2V systems, where the quality of wireless links between vehicles can vary greatly and rapidly from one environment to another as one or both ends move [3].Many M2M channel models have been proposed in various ways, some of which are summarized in [4], and these models have important reference values in modeling the V2V channel. The models in M2M communication environment can be traced back from two-dimensional (2D) [5–7] and three-dimensional (3D) [8–11] fixed scattering models to 2D [12–17] and 3D [18–20] moving scattering models. The M2M channel models with the assumption of stationary scatterers have been proposed in [5–11]. However, moving scatterers are unavoidable in M2M communications. Moving foliage, walking pedestrians, and passing vehicles are only a few examples of scatterers in motion, which can be observed in most of the real-world radio propagation environments [14, 21]. The impact of moving scatterers on channel characteristics has been studied in [22–25] for 2D F2M cellular and fixed-to-fixed (F2F) communications with line-of-sight (LoS) and single bounced (SB) rays.Recently, modeling of M2M/V2V channels in the presence of moving scatterers has been discussed in [12–20]. A nonstationary multiple-input-multiple-output (MIMO) V2V channel model based on the geometrical street model was derived, and the impact of fixed and moving clusters of scatterers on the channel statistics was studied in [12, 15]. A single-input-single-output (SISO) V2V channel model was derived assuming a typical propagation scenario in which the local scatterers without specific constraints of positions moved with random velocities in random directions in [13, 14]. The impact of mobile and stationary scattering clusters on the Doppler spectrum was investigated for wideband V2V communication channels in an urban canyon oncoming environment in [16, 17]. All previously reported models in [12–17] are 2D propagation models in which the rays are LoS, SB, or double bounced (DB). However, this 2D assumption does not seem to be appropriate for many communication scenarios, for example, urban V2V communications in which the transmitter and receiver antenna arrays are often located in close proximity to or lower than the surrounding scatterers.A 3D geometrical propagation model that included both stationary and moving scatterers around the transmitter and receiver was proposed in [18], and this work was extended to wideband channels in [19]. The models in [18, 19] took into account that the rays in V2V channels could be both SB and DB; however, they imposed some constraints on the position of the local scatterers and assumed that the transmitter, receiver, and scatterers were in motion with constant velocities in 2D space. Therefore, the models in [18, 19] cannot fully capture the 3D spatial information. A preliminary investigation of the impact of multiple moving scatterers on the Doppler spectrum in 3D V2V communication scenarios was presented in [20]. However, the work of [20] did not investigate and take into account the space-time correlation functions (ST-CFs), scatterer velocity distributions, angle distributions, and so on.It is unavoidable in V2V communications that the rays from the transmitter to receiver are multiple bounced (MB) by moving scatterers, especially in environments with high-density scatterers, for example, urban area. However, to the best of the authors’ knowledge, the model and statistical properties of 3D V2V or M2M channels in the presence of multiple moving scatterers have been investigated rarely so far. This paper strives to alleviate the current lack of analytical studies by investigating the model and statistical properties of a narrowband 3D MIMO V2V channel in which the local multiple scatterers are moving with random velocities in random directions. The proposed reference model constructs the channel impulse response as a combination of LoS and MB components, and SB is considered as a special case of MB. Different from the assumption of all non-LoS (NLoS) path gains having the same size in [13, 14] and meanwhile compared with models in [18, 19], SB, DB, and other MB rays have more flexible and precise power weights in the proposed model. From the reference model, the corresponding ST-CFs, time correlation functions (T-CFs), and space correlation functions (S-CFs) are analytically investigated and numerically simulated in terms of various factors such as the maximum bounces, scattering forms, scatterer velocity distributions, and spacing between adjacent antenna elements. Finally, the theoretical space-Doppler power spectral density (SD-PSD) results with SD-PSDs in [18] and measured data in [26, 27] are compared. The close agreements between the analytically and empirically obtained SD-PSDs confirm the utility and generality of the proposed model and show the importance of including multiple moving scatterers in propagation models. The contributions and novelties of this paper are summarized as follows.(i)
We propose a generalized geometrical model and a generalized reference model that include LoS, SB, and MB rays between the transmitter and receiver for 3D narrowband MIMO V2V communications. The proposed model can be adapted to a wide variety of scenarios, for example, F2F, F2M, and M2M with certain bounces by adjusting model parameters.(ii)
To the best of the authors’ knowledge, the impact of rays’ different maximum bounces caused by the multiple moving scatterers on the ST-CFs and T-CFs is deeply investigated for the first time in V2V or M2M communication environments.(iii)
The ST-CFs obtained from the proposed reference model are relatively generalized and can be reduced to several existing ST-CFs and T-CFs, for example, those in [13, 14, 23, 28, 29].The remainder of the paper is organized as follows. Section2 describes the geometrical propagation model and presents a 3D reference model for narrowband MIMO V2V channels in the presence of multiple moving scatterers. Section 3 derives the channel ST-CFs, T-CFs, S-CFs, and SD-PSDs for different parametric sets. Numerical results and comparison between the theoretical results and measured data are provided in Section 4. Finally, Section 5 provides some concluding remarks.


## 2. 3D Geometrical Propagation Model and Reference Model

### 2.1. Geometrical Propagation Model

This paper considers the MIMO V2V communication links between transmitterTX and receiver RX, as shown in Figure 1. The radio propagation environment is characterized by 3D multiple moving scattering with either LoS or NLoS conditions between TX and RX. The MB waves emitted from the qth antenna element of TX at an angle of departure (AOD) reach the pth antenna element of RX at an angle of arrival (AOA) after being multiple scattered by the local moving scatterers.Figure 1
Geometrical propagation model with LoS and MB rays for the 3D MIMO V2V communication scenario. Empty circles represent the antenna elements, and solid squares represent the moving scatterers.For ease of reference, in this geometrical propagation model, the main parameters are summarized in Table1, and the main assumptions are summarized as follows:(i)
The received signal power consists of LoS, SB, and MB scattering components with the corresponding power weights.(ii)
BothTX and RX are equipped with uniform linear arrays consisting of omnidirectional antenna elements.(iii)
TX and RX move with constant velocities in 3D space described by the fixed azimuth angles and elevation angles.(iv)
Each of the local scatterers on the links betweenTX and RX for the multiple bounces is in motion with a random velocity in a random direction in 3D space.(v)
The geometrical propagation model does not impose specific constraints on the position of the local moving scatterers like [13, 14]. Owing to high path loss, we neglect the energy contribution of remote scatterers.Table 1
Definition of the parameters used in the geometrical model.
Parameters
Definition
Attributes
S
i
Theith moving scatterer for the multiple bounces
Symbol
D
The distance betweenTX and RX
Deterministic
q, p
The antenna element identifier ofTX and RX, respectively
Deterministic
Q, P
The number of antenna elements ofTX and RX, respectively
Deterministic
d
T, dR
The spacing between two adjacent antenna elements ofTX and RX, respectively
Deterministic
α
q, βq
The azimuth angle and elevation angle ofTX’s antenna array, respectively
Deterministic
α
p, βp
The azimuth angle and elevation angle ofRX’s antenna array, respectively
Deterministic
α
p
q
T, βpqT
The azimuth angle and elevation angle of AOD, respectively
Statistical
α
p
q
R, βpqR
The azimuth angle and elevation angle of AOA, respectively
Statistical
v
T, vR
The velocities ofTX and RX, respectively
Deterministic
v
p
q
i
The velocity ofSi
Statistical
α
T, βT
The azimuth angle and elevation angle ofvT, respectively
Deterministic
α
R, βR
The azimuth angle and elevation angle ofvR, respectively
Deterministic
α
p
q
i, βpqi
The azimuth angle and elevation angle ofvpqi, respectively
Statistical


### 2.2. Reference Model

We can observe from Figure1 that the complex faded envelope of the link between the qth antenna element of TX and the pth antenna element of RX can be written as a superposition of LoS and MB components; that is,(1)hpqt=hpqLoSt+hpqMBt,(2)hpqLoSt=ρpqexp⁡j2πfpq,ρt+2πfpq,ρq+fpq,ρp+θpq,ρ,(3)hpqMBt=∑m=1Mpmhpqmt,(4)hpqmt=limNm→∞⁡∑nm=1Nmcpq,nmexp⁡j2πfpq,nmt+2πfpq,nq+fpq,np+θpq,nm,where ρpq, fpq,ρ, θpq,ρ, fpq,ρq, and fpq,ρp denote the path gain, Doppler shift, phase shift, frequency shift for qth antenna element of TX, and frequency shift for pth antenna element of RX of hpqLoSt, respectively; m denotes the amount of bounces, M is the maximum of m, pm and Nm denote the power weight and path number of hpqmt, respectively; cpq,nm, fpq,nm, θpq,nm, fpq,nq, and fpq,np denote the path gain, Doppler shift, phase shift, frequency shift for qth antenna element of TX, and frequency shift for pth antenna element of RX of the nmth path in hpqmt. The LoS component hpqLoSt can be described by a complex sinusoid, and (2) is an extension of the (3.17) in [30]. hpqmt denotes the NLoS component with m bounced (mB) rays, and it is an extension of the (3.12) in [30]. Note that MB component hpqMBt of the channel impulse response consists of M clusters of rays reflected m∈1,2,3,…,M times from moving scatterers.The channel gainsρpq,cpq,nm, Doppler shifts and frequency shifts fpq,ρ,fpq,ρq,fpq,ρp,fpq,nm,fpq,nq,fpq,np, and phases (θpq,ρ, θpq,nm) in this model can be calculated as follows.


#### 2.2.1. Channel Gains

The central limit theorem states thathpqmt equals a complex valued Gaussian random process with zero mean and variance 2σm2=Var⁡hpqmt=limNm→∞⁡∑nmNmEcpq,nm2. pm in (3) denotes the power weight of the mth clusters of rays, and ∑m=1Mpm=1. The channel gain of hpqt is normalized, i.e., 2∑m=1Mpmσm2+ρpq2=1, and the Rice factor can be denoted as K=ρpq2/2∑m=1Mpmσm2. These parameters have to be either set during simulations or estimated from measurements.


#### 2.2.2. Doppler Shifts and Frequency Shifts

The frequency shiftsfpq,ρq, fpq,ρp, fpq,nq, and fpq,np depend on the difference of the propagation distance (TPD) changes between hpqt and h11t. On the other hand, the Doppler shifts fpq,ρ and fpq,nm depend on the geometrical relation between directions of movement of TX, RX, and multiple moving scatterers and the directions of AOD and AOA. For maxdT,dR≪D, both the AOD and AOA of LoS rays are approximately equal to zero. Appendix A shows that fpq,ρq, fpq,ρp, fpq,nq, fpq,np, and fpq,ρ are, respectively,(5)fpq,ρq=q-1dTf0ccos⁡αqcos⁡βq,(6)fpq,ρp=p-1dRf0ccos⁡αpcos⁡βp,(7)fpq,nq=q-1dTf0ccos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT,(8)fpq,np=p-1dRf0ccos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR,(9)fpq,ρ=f0cvTcos⁡αTcos⁡βT+vRcos⁡αRcos⁡βR,where f0 is the carrier frequency and c denotes the speed of light.(10)fpq,nm=fpqT+fpqR+fpq,nmAOD+fpq,nmAOA,where fpqT and fpqR are caused by the movement of TX and RX, respectively. fpq,nmAOD and fpq,nmAOA are caused by the movement of m scatterers relative to the directions of AOD and AOA, respectively. Using the similar mathematical manipulations in Appendix A, the respective components in (10) are(11)fpqT=vTf0ccos⁡αT-αpqTcos⁡βTcos⁡βpqT+sin⁡βTsin⁡βpqT,fpqR=vRf0ccos⁡αR-αpqRcos⁡βRcos⁡βpqR+sin⁡βRsin⁡βpqR.According to [31], the maximum Doppler shift for MB link over m moving scatterers with velocities vi≪c is(12)fmmax=f0cν1+2∑i=2m-1νi+νm.Therefore, the remainder components of fpq,nm are [20](13)fpq,nmAOD≈f0cνpq1PpqAOD,1+2∑i=2m-1νpqiPpqAOD,i+νpqmPpqAOD,m,fpq,nmAOA≈f0cνpq1PpqAOA,1+2∑i=2m-1νpqiPpqAOA,i+νpqmPpqAOA,m,where PpqAOD,i and PpqAOA,ii=1,2,3,…,m are(14)PpqAOD,i=cos⁡αpqi-αpqTcos⁡βpqicos⁡βpqT+sin⁡βpqisin⁡βpqT,PpqAOA,i=cos⁡αpqi-αpqRcos⁡βpqicos⁡βpqR+sin⁡βpqisin⁡βpqR.Note that ifβT=βR=βpqT=βpqR=βpqi=0 and m=1, (10) equals (7) in [13] and (5) in [14] regardless of the plus or minus signs. The differences among these plus or minus signs are caused by the different forms of angles’ expression.


#### 2.2.3. Phases

The phase shiftθpq,ρ can be assumed to be constant [30]. The phase shift θpq,nm consists of the phase change caused by the interaction of the transmitted signal with the scatterers and the phase change caused by the TPD between the first and the last scatterers. Without loss of generality, we can assume that the phases θpq,nm  m=1,2,3,…,M are independent random variables. Here, it is assumed that they are uniformly distributed on the interval 0,2π and independent of any other random variable.


## 2.1. Geometrical Propagation Model

This paper considers the MIMO V2V communication links between transmitterTX and receiver RX, as shown in Figure 1. The radio propagation environment is characterized by 3D multiple moving scattering with either LoS or NLoS conditions between TX and RX. The MB waves emitted from the qth antenna element of TX at an angle of departure (AOD) reach the pth antenna element of RX at an angle of arrival (AOA) after being multiple scattered by the local moving scatterers.Figure 1
Geometrical propagation model with LoS and MB rays for the 3D MIMO V2V communication scenario. Empty circles represent the antenna elements, and solid squares represent the moving scatterers.For ease of reference, in this geometrical propagation model, the main parameters are summarized in Table1, and the main assumptions are summarized as follows:(i)
The received signal power consists of LoS, SB, and MB scattering components with the corresponding power weights.(ii)
BothTX and RX are equipped with uniform linear arrays consisting of omnidirectional antenna elements.(iii)
TX and RX move with constant velocities in 3D space described by the fixed azimuth angles and elevation angles.(iv)
Each of the local scatterers on the links betweenTX and RX for the multiple bounces is in motion with a random velocity in a random direction in 3D space.(v)
The geometrical propagation model does not impose specific constraints on the position of the local moving scatterers like [13, 14]. Owing to high path loss, we neglect the energy contribution of remote scatterers.Table 1
Definition of the parameters used in the geometrical model.
Parameters
Definition
Attributes
S
i
Theith moving scatterer for the multiple bounces
Symbol
D
The distance betweenTX and RX
Deterministic
q, p
The antenna element identifier ofTX and RX, respectively
Deterministic
Q, P
The number of antenna elements ofTX and RX, respectively
Deterministic
d
T, dR
The spacing between two adjacent antenna elements ofTX and RX, respectively
Deterministic
α
q, βq
The azimuth angle and elevation angle ofTX’s antenna array, respectively
Deterministic
α
p, βp
The azimuth angle and elevation angle ofRX’s antenna array, respectively
Deterministic
α
p
q
T, βpqT
The azimuth angle and elevation angle of AOD, respectively
Statistical
α
p
q
R, βpqR
The azimuth angle and elevation angle of AOA, respectively
Statistical
v
T, vR
The velocities ofTX and RX, respectively
Deterministic
v
p
q
i
The velocity ofSi
Statistical
α
T, βT
The azimuth angle and elevation angle ofvT, respectively
Deterministic
α
R, βR
The azimuth angle and elevation angle ofvR, respectively
Deterministic
α
p
q
i, βpqi
The azimuth angle and elevation angle ofvpqi, respectively
Statistical


## 2.2. Reference Model

We can observe from Figure1 that the complex faded envelope of the link between the qth antenna element of TX and the pth antenna element of RX can be written as a superposition of LoS and MB components; that is,(1)hpqt=hpqLoSt+hpqMBt,(2)hpqLoSt=ρpqexp⁡j2πfpq,ρt+2πfpq,ρq+fpq,ρp+θpq,ρ,(3)hpqMBt=∑m=1Mpmhpqmt,(4)hpqmt=limNm→∞⁡∑nm=1Nmcpq,nmexp⁡j2πfpq,nmt+2πfpq,nq+fpq,np+θpq,nm,where ρpq, fpq,ρ, θpq,ρ, fpq,ρq, and fpq,ρp denote the path gain, Doppler shift, phase shift, frequency shift for qth antenna element of TX, and frequency shift for pth antenna element of RX of hpqLoSt, respectively; m denotes the amount of bounces, M is the maximum of m, pm and Nm denote the power weight and path number of hpqmt, respectively; cpq,nm, fpq,nm, θpq,nm, fpq,nq, and fpq,np denote the path gain, Doppler shift, phase shift, frequency shift for qth antenna element of TX, and frequency shift for pth antenna element of RX of the nmth path in hpqmt. The LoS component hpqLoSt can be described by a complex sinusoid, and (2) is an extension of the (3.17) in [30]. hpqmt denotes the NLoS component with m bounced (mB) rays, and it is an extension of the (3.12) in [30]. Note that MB component hpqMBt of the channel impulse response consists of M clusters of rays reflected m∈1,2,3,…,M times from moving scatterers.The channel gainsρpq,cpq,nm, Doppler shifts and frequency shifts fpq,ρ,fpq,ρq,fpq,ρp,fpq,nm,fpq,nq,fpq,np, and phases (θpq,ρ, θpq,nm) in this model can be calculated as follows.


### 2.2.1. Channel Gains

The central limit theorem states thathpqmt equals a complex valued Gaussian random process with zero mean and variance 2σm2=Var⁡hpqmt=limNm→∞⁡∑nmNmEcpq,nm2. pm in (3) denotes the power weight of the mth clusters of rays, and ∑m=1Mpm=1. The channel gain of hpqt is normalized, i.e., 2∑m=1Mpmσm2+ρpq2=1, and the Rice factor can be denoted as K=ρpq2/2∑m=1Mpmσm2. These parameters have to be either set during simulations or estimated from measurements.


### 2.2.2. Doppler Shifts and Frequency Shifts

The frequency shiftsfpq,ρq, fpq,ρp, fpq,nq, and fpq,np depend on the difference of the propagation distance (TPD) changes between hpqt and h11t. On the other hand, the Doppler shifts fpq,ρ and fpq,nm depend on the geometrical relation between directions of movement of TX, RX, and multiple moving scatterers and the directions of AOD and AOA. For maxdT,dR≪D, both the AOD and AOA of LoS rays are approximately equal to zero. Appendix A shows that fpq,ρq, fpq,ρp, fpq,nq, fpq,np, and fpq,ρ are, respectively,(5)fpq,ρq=q-1dTf0ccos⁡αqcos⁡βq,(6)fpq,ρp=p-1dRf0ccos⁡αpcos⁡βp,(7)fpq,nq=q-1dTf0ccos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT,(8)fpq,np=p-1dRf0ccos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR,(9)fpq,ρ=f0cvTcos⁡αTcos⁡βT+vRcos⁡αRcos⁡βR,where f0 is the carrier frequency and c denotes the speed of light.(10)fpq,nm=fpqT+fpqR+fpq,nmAOD+fpq,nmAOA,where fpqT and fpqR are caused by the movement of TX and RX, respectively. fpq,nmAOD and fpq,nmAOA are caused by the movement of m scatterers relative to the directions of AOD and AOA, respectively. Using the similar mathematical manipulations in Appendix A, the respective components in (10) are(11)fpqT=vTf0ccos⁡αT-αpqTcos⁡βTcos⁡βpqT+sin⁡βTsin⁡βpqT,fpqR=vRf0ccos⁡αR-αpqRcos⁡βRcos⁡βpqR+sin⁡βRsin⁡βpqR.According to [31], the maximum Doppler shift for MB link over m moving scatterers with velocities vi≪c is(12)fmmax=f0cν1+2∑i=2m-1νi+νm.Therefore, the remainder components of fpq,nm are [20](13)fpq,nmAOD≈f0cνpq1PpqAOD,1+2∑i=2m-1νpqiPpqAOD,i+νpqmPpqAOD,m,fpq,nmAOA≈f0cνpq1PpqAOA,1+2∑i=2m-1νpqiPpqAOA,i+νpqmPpqAOA,m,where PpqAOD,i and PpqAOA,ii=1,2,3,…,m are(14)PpqAOD,i=cos⁡αpqi-αpqTcos⁡βpqicos⁡βpqT+sin⁡βpqisin⁡βpqT,PpqAOA,i=cos⁡αpqi-αpqRcos⁡βpqicos⁡βpqR+sin⁡βpqisin⁡βpqR.Note that ifβT=βR=βpqT=βpqR=βpqi=0 and m=1, (10) equals (7) in [13] and (5) in [14] regardless of the plus or minus signs. The differences among these plus or minus signs are caused by the different forms of angles’ expression.


### 2.2.3. Phases

The phase shiftθpq,ρ can be assumed to be constant [30]. The phase shift θpq,nm consists of the phase change caused by the interaction of the transmitted signal with the scatterers and the phase change caused by the TPD between the first and the last scatterers. Without loss of generality, we can assume that the phases θpq,nm  m=1,2,3,…,M are independent random variables. Here, it is assumed that they are uniformly distributed on the interval 0,2π and independent of any other random variable.


## 2.2.1. Channel Gains

The central limit theorem states thathpqmt equals a complex valued Gaussian random process with zero mean and variance 2σm2=Var⁡hpqmt=limNm→∞⁡∑nmNmEcpq,nm2. pm in (3) denotes the power weight of the mth clusters of rays, and ∑m=1Mpm=1. The channel gain of hpqt is normalized, i.e., 2∑m=1Mpmσm2+ρpq2=1, and the Rice factor can be denoted as K=ρpq2/2∑m=1Mpmσm2. These parameters have to be either set during simulations or estimated from measurements.


## 2.2.2. Doppler Shifts and Frequency Shifts

The frequency shiftsfpq,ρq, fpq,ρp, fpq,nq, and fpq,np depend on the difference of the propagation distance (TPD) changes between hpqt and h11t. On the other hand, the Doppler shifts fpq,ρ and fpq,nm depend on the geometrical relation between directions of movement of TX, RX, and multiple moving scatterers and the directions of AOD and AOA. For maxdT,dR≪D, both the AOD and AOA of LoS rays are approximately equal to zero. Appendix A shows that fpq,ρq, fpq,ρp, fpq,nq, fpq,np, and fpq,ρ are, respectively,(5)fpq,ρq=q-1dTf0ccos⁡αqcos⁡βq,(6)fpq,ρp=p-1dRf0ccos⁡αpcos⁡βp,(7)fpq,nq=q-1dTf0ccos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT,(8)fpq,np=p-1dRf0ccos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR,(9)fpq,ρ=f0cvTcos⁡αTcos⁡βT+vRcos⁡αRcos⁡βR,where f0 is the carrier frequency and c denotes the speed of light.(10)fpq,nm=fpqT+fpqR+fpq,nmAOD+fpq,nmAOA,where fpqT and fpqR are caused by the movement of TX and RX, respectively. fpq,nmAOD and fpq,nmAOA are caused by the movement of m scatterers relative to the directions of AOD and AOA, respectively. Using the similar mathematical manipulations in Appendix A, the respective components in (10) are(11)fpqT=vTf0ccos⁡αT-αpqTcos⁡βTcos⁡βpqT+sin⁡βTsin⁡βpqT,fpqR=vRf0ccos⁡αR-αpqRcos⁡βRcos⁡βpqR+sin⁡βRsin⁡βpqR.According to [31], the maximum Doppler shift for MB link over m moving scatterers with velocities vi≪c is(12)fmmax=f0cν1+2∑i=2m-1νi+νm.Therefore, the remainder components of fpq,nm are [20](13)fpq,nmAOD≈f0cνpq1PpqAOD,1+2∑i=2m-1νpqiPpqAOD,i+νpqmPpqAOD,m,fpq,nmAOA≈f0cνpq1PpqAOA,1+2∑i=2m-1νpqiPpqAOA,i+νpqmPpqAOA,m,where PpqAOD,i and PpqAOA,ii=1,2,3,…,m are(14)PpqAOD,i=cos⁡αpqi-αpqTcos⁡βpqicos⁡βpqT+sin⁡βpqisin⁡βpqT,PpqAOA,i=cos⁡αpqi-αpqRcos⁡βpqicos⁡βpqR+sin⁡βpqisin⁡βpqR.Note that ifβT=βR=βpqT=βpqR=βpqi=0 and m=1, (10) equals (7) in [13] and (5) in [14] regardless of the plus or minus signs. The differences among these plus or minus signs are caused by the different forms of angles’ expression.


## 2.2.3. Phases

The phase shiftθpq,ρ can be assumed to be constant [30]. The phase shift θpq,nm consists of the phase change caused by the interaction of the transmitted signal with the scatterers and the phase change caused by the TPD between the first and the last scatterers. Without loss of generality, we can assume that the phases θpq,nm  m=1,2,3,…,M are independent random variables. Here, it is assumed that they are uniformly distributed on the interval 0,2π and independent of any other random variable.


## 3. Space-Time Correlation Function and Space-Doppler Power Spectral Density

Using the reference model described in Section2, we can now derive the key temporal and spatial characteristics of MIMO V2V narrowband multipath fading channels with the local multiple moving scatters.


### 3.1. Space-Time Correlation Function

The normalized ST-CF between two complex faded envelopeshpqt and hp-q-t is defined as(15)Rp-q-,pqdT,dR,τ=Ehp-q-t∗hpqt+τ,where •∗ denotes the complex conjugate operation, E• is the statistical expectation operator, p,p-∈1,2,3,…,P, and q,q-∈1,2,3,…,Q. The normalized T-CF can be obtained, if p=p- and q=q- in (15). The normalized space correlation function (S-CF) can be obtained by setting τ to zero in (15).Sincehpq1t,hpq2t,…,hpqMt, and hpqLoSt are independent of each other, (15) can be simplified to(16)Rp-q-,pqdT,dR,τ=Rp-q-,pqMBdT,dR,τ+Rp-q-,pqLoSdT,dR,τ=∑m=1MpmRp-q-,pqmdT,dR,τ+Rp-q-,pqLoSdT,dR,τ,where Rp-q-,pqmdT,dR,τ and Rp-q-,pqLoSdT,dR,τ denote the normalized ST-CFs of the mB and LoS components, respectively, and they are defined as(17)Rp-q-,pqmdT,dR,τ=Ehp-q-mt∗hpqmt+τ,(18)Rp-q-,pqLoSdT,dR,τ=Ehp-q-LoSt∗hpqLoSt+τ.


#### 3.1.1. ST-CF of LoS Component

By substituting (2) into (18), the expression for the ST-CF of the LoS component can be written as (19)Rp-q-,pqLoSdT,dR,τ=ρpqρp-q-Eexpj2πtfpq,ρ-fp-q-,ρ+2πfpq,ρτ+2πfpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-+θpq,ρ-θp-q-,ρ.Formax⁡dT,dR≪D, we assume ρpq=ρp-q- and θpq,ρ=θp-q-,ρ. Then, (19) can be written as(20)Rp-q-,pqLoSdT,dR,τ=ρpq2Eexpj2πfpq,ρτ+Δfρ=ρpq2exp⁡j2πfpq,ρτ+Δfρ,where Δfρ is defined as(21)Δfρ=fpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-.By substituting (5) and (6) into (21), Δfρ can be written as(22)Δfρ=f0cq-q-dTcos⁡αqcos⁡βq+p-p-dRcos⁡αpcos⁡βp.


#### 3.1.2. ST-CF ofmB Component

By substituting (4) into (17), the expression for the ST-CF of the mB component can be written as(23)Rp-q-,pqmdT,dR,τ=limNn→∞⁡limNm→∞⁡∑nn=1Nncp-q-,nn∑nm=1Nmcpq,nm×Eexp⁡j2πfpq,nm-fp-q-,nnt+2πfpq,nmτ+2πfpq,nq+fpq,np-fp-q-,nq--fp-q-,np-+θpq,nm-θp-q-,nn.Under the assumption thatθpq,nmand θp-q-,nn are uniformly distributed on the interval 0,2π and independent of each other, Eexp⁡jθpq,nm-θp-q-,nn equals 1. It is assumed that all the path gains of the mB component have the same size; that is,(24)cp-q-,nn=cpq,nm=σm2Nm.Then, (23) can be written as(25)Rp-q-,pqmdT,dR,τ=2σm2limNm→∞⁡∑nm=1NmEexp⁡j2πfpq,nmτ+Δfnm,where Δfnm is defined as(26)Δfnm=fpq,nq+fpq,np-fp-q-,nq--fp-q-,np-.It is assumed that AOD and AOA ofhpqmt and hp-q-mt are independent and identically distributed (i.i.d.). By substituting (7) and (8) into (26), Δfnm can be written as(27)Δfnm=f0cq-q-dTcos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT+p-p-dRcos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR.Since the number of local scatterers in the reference model is infinite, the parametersαpqT, βpqT, αpqR, βpqR, αpqi, and βpqi can be seen as continuous random variables with corresponding probability density functions (PDFs). Then, the ST-CF of the mB component (25) can be written as(28)Rp-q-,pqmdT,dR,τ=2σm2×∫⋯∫∫⋯∫∫⋯∫∫∫∫∫exp⁡j2πfpq,nmτ+Δfnm×∏i=1mpνpqi∏i=1mpαpqi,βpqi·pαpqT,βpqT,αpqR,βpqRdνpq1⋯dνpqmdαpq1⋯dαpqmdβpq1⋯dβpqmdαpqTdβpqTdαpqRdβpqR.Now, the complete expression of (16) can be obtained by substituting (20) and (28) into (16). pαpqT,βpqT,αpqR,βpqR describes the joint distribution of AOD and AOA, and it can be optionally used to present some propagation channel models. As a result, the ST-CF in (28) can provide a suitable platform to study the statistical properties of some different channel models, such as the random scattering model [13, 14], Jakes model [28], one-ring model [23], and two-ring model [29] as described in Section 3.1.3. Therefore, the ST-CF in (28) is a generalized and parametric expression. However, due to the complex nature of pαpqT,βpqT,αpqR,βpqR, it is assumed that AOD and AOA are independent [13, 14, 32, 33], and azimuth angles and elevation angles in (28) are also independent [18, 19, 32]. The parameters in (28) such as the velocities of the multiple moving scatterers and random angles can be calculated as follows.(i) Scatterer Velocity Distributions. The Gaussian, Laplace, exponential, and uniform distributions can be used to describe the velocity of moving scatterers [13]. In fact, the scatterer velocity νpqi is always positive or equal to zero. We use the uniform distribution in (29) and half-Gaussian distribution in (30) to describe the velocity of multiple moving scatterers.(29)pνpqi=1vmaxi,0≤νpqi≤vmaxi,where vmaxi is the maximum of νpqi.(30)pνpqi=2πσpq,iexp⁡-νpqi22σpq,i2,νpqi≥0,where σpq,i is the standard deviation of νpqi.(ii) Angle Distributions. To characterize the statistical angles in Table 1, we use the uniform distribution in (31) in the isotropic scattering environment and use the von Mises distribution in (32) and the cosine distribution in (33) in the nonisotropic scattering environment. In addition, the interval of azimuth angles αpqT, αpqR, and αpqi is -π,π, and the interval of elevation angles βpqT, βpqR, and βpqi is -π/2,π/2.(31)pγ=1γ2-γ1,γ1≤γ≤γ2,(32)pα=exp⁡kcos⁡α-α-2πI0k,α1≤α≤α1+2π,where I0• is the zeroth-order modified Bessel function of the first kind, α- is the mean angle, and k controls the spread of angles around the mean. The von Mises distribution PDF with α-=0 is used to describe the azimuth angles.(33)pβ=π4βmcos⁡π2ββm,-βm≤β≤βm,where βm is the maximum of β. The cosine distribution PDF is used to describe the elevation angles.


#### 3.1.3. Special Cases ofmB’s ST-CF

IfβT=βR=βpqT=βpqR=βpqi=0, that is, the scattering environment is 2D, some different special cases can be derived from the general expression of the mB’s ST-CF in (28).In the 2D scattering environment, (28) can be written as (B.2) in Appendix B. If p=p-, q=q-, and m=1, (B.2) equals (11) in [13] and (6) in [14] regardless of the plus or minus signs under the assumption that the angles αT and αR are independent of each other in [13, 14]. In isotropic scattering environments, some other special cases with closed-form expressions can be derived as follows.AppendixB shows that an approximate ST-CF of (B.2) can be written as(34)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR∫⋯∫J02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ∏i=1mpνpqidνpq1⋯dνpqm,where J0• denotes the zeroth-order Bessel function of the first kind and k0=2πf0/c is the wave number.Note that ifm approaches to the infinity, (34) can be written as(35)limm→∞⁡Rp-q-,pqmdT,dR,τ=2σm2J0q-q-k0dTJ0p-p-k0dR,τ=0,0τ>0.In the NLoS communication environment with very high-density scatterers, (35) implies that the ST-CF approaches to zero at the nonzero time difference τ and has nothing to do with velocities of scatterers, TX and RX. The large-scale antenna arrays are very suitable to be used in this environment because the antenna element spacing can be reduced to a smaller value.Ifνpqi is constant, (34) can be presented as the following closed-form expression:(36)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRJ02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ.The T-CF of the classical F2M scenario with fixed scatterers is obtained, ifvT=vpqi=0i=1,2,3,…,m, q=q-, and p=p- in (36). In this case, the ST-CF in (36) is 2σm2J0k0vRτ, which is known as the Jakes model [28]. If vT=0,m=1, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vpq1τJ0k0vRτ, which equals the T-CF of F2M single-ring channel model in the presence of moving scatterers reported in (11) of [23]. If vpqi=0i=1,2,3,…,m, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vTτJ0k0vRτ, which equals the T-CFs of the classical M2M two-ring channel model in the presence of fixed scatterers reported in (46) of [29]. If vpqi=τ=0i=1,2,3,…,m, q=p=2, and q-=p-=1, the ST-CF in (36) results in 2σm2J0k0dTJ0k0dR, which equals the S-CF in (46) of [29].The velocities of moving scatterers such as moving foliage, walking pedestrians, and passing vehicles generally are random variables. Ifνpqi is described by the uniform distribution in (29), (34) can be presented as the following closed-form expression:(37)Rp-q-,pqmdT,dR,τ=σm22m-1J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR2J02k0vmax1τ+πJ12k0vmax1τH02k0vmax1τ-πJ02k0vmax1τ×H12k0vmax1τ2J02k0vmaxmτ+πJ12k0vmaxmτH02k0vmaxmτ-πJ02k0vmaxmτH12k0vmaxmτ×∏i=2m-12J04k0vmaxiτ+πJ14k0vmaxiτ×H04k0vmaxiτ-πJ04k0vmaxiτH14k0vmaxiτ,where J1• denotes the first-order Bessel function of the first kind, H0• denotes the zeroth-order Struve function, and H1• denotes the first-order Struve function.Ifνpqi is described by the half-Gaussian distribution in (30), (34) can be presented as the following closed-form expression:(38)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRI0k02σpq,12τ2×I0k02σpq,m2τ2exp⁡-k02σpq,12τ2exp⁡-k02σpq,m2τ2×∏i=2m-1I04k02σpq,i2τ2exp⁡-4k02σpq,i2τ2.


### 3.2. Space-Doppler Power Spectral Density

The SD-PSD can be obtained by taking the Fourier transform of the ST-CF in (16) with respect to time difference τ. From (16), it follows that the SD-PSD is a summation of the SD-PSDs of the MB and LoS components; that is,(39)Sp-q-,pqdT,dR,ω=FτRp-q-,pqdT,dR,τ=∑m=1MpmSp-q-,pqmdT,dR,ω+Sp-q-,pqLoSdT,dR,ω.The SD-PSD of the LoS component can be written as(40)Sp-q-,pqLoSdT,dR,ω=2πρpq2exp⁡j2πΔfρδω-2πfpq,ρ,where δ• is the Dirac delta function.Since the ST-CF of themB component is the multiple integral as (28), the closed-form expression of SD-PSD cannot be derived. In Section 4.2, we show the SD-PSD of the 3D V2V channel in the presence of multiple moving scatterers by means of numerical integrations.


## 3.1. Space-Time Correlation Function

The normalized ST-CF between two complex faded envelopeshpqt and hp-q-t is defined as(15)Rp-q-,pqdT,dR,τ=Ehp-q-t∗hpqt+τ,where •∗ denotes the complex conjugate operation, E• is the statistical expectation operator, p,p-∈1,2,3,…,P, and q,q-∈1,2,3,…,Q. The normalized T-CF can be obtained, if p=p- and q=q- in (15). The normalized space correlation function (S-CF) can be obtained by setting τ to zero in (15).Sincehpq1t,hpq2t,…,hpqMt, and hpqLoSt are independent of each other, (15) can be simplified to(16)Rp-q-,pqdT,dR,τ=Rp-q-,pqMBdT,dR,τ+Rp-q-,pqLoSdT,dR,τ=∑m=1MpmRp-q-,pqmdT,dR,τ+Rp-q-,pqLoSdT,dR,τ,where Rp-q-,pqmdT,dR,τ and Rp-q-,pqLoSdT,dR,τ denote the normalized ST-CFs of the mB and LoS components, respectively, and they are defined as(17)Rp-q-,pqmdT,dR,τ=Ehp-q-mt∗hpqmt+τ,(18)Rp-q-,pqLoSdT,dR,τ=Ehp-q-LoSt∗hpqLoSt+τ.


### 3.1.1. ST-CF of LoS Component

By substituting (2) into (18), the expression for the ST-CF of the LoS component can be written as (19)Rp-q-,pqLoSdT,dR,τ=ρpqρp-q-Eexpj2πtfpq,ρ-fp-q-,ρ+2πfpq,ρτ+2πfpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-+θpq,ρ-θp-q-,ρ.Formax⁡dT,dR≪D, we assume ρpq=ρp-q- and θpq,ρ=θp-q-,ρ. Then, (19) can be written as(20)Rp-q-,pqLoSdT,dR,τ=ρpq2Eexpj2πfpq,ρτ+Δfρ=ρpq2exp⁡j2πfpq,ρτ+Δfρ,where Δfρ is defined as(21)Δfρ=fpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-.By substituting (5) and (6) into (21), Δfρ can be written as(22)Δfρ=f0cq-q-dTcos⁡αqcos⁡βq+p-p-dRcos⁡αpcos⁡βp.


### 3.1.2. ST-CF ofmB Component

By substituting (4) into (17), the expression for the ST-CF of the mB component can be written as(23)Rp-q-,pqmdT,dR,τ=limNn→∞⁡limNm→∞⁡∑nn=1Nncp-q-,nn∑nm=1Nmcpq,nm×Eexp⁡j2πfpq,nm-fp-q-,nnt+2πfpq,nmτ+2πfpq,nq+fpq,np-fp-q-,nq--fp-q-,np-+θpq,nm-θp-q-,nn.Under the assumption thatθpq,nmand θp-q-,nn are uniformly distributed on the interval 0,2π and independent of each other, Eexp⁡jθpq,nm-θp-q-,nn equals 1. It is assumed that all the path gains of the mB component have the same size; that is,(24)cp-q-,nn=cpq,nm=σm2Nm.Then, (23) can be written as(25)Rp-q-,pqmdT,dR,τ=2σm2limNm→∞⁡∑nm=1NmEexp⁡j2πfpq,nmτ+Δfnm,where Δfnm is defined as(26)Δfnm=fpq,nq+fpq,np-fp-q-,nq--fp-q-,np-.It is assumed that AOD and AOA ofhpqmt and hp-q-mt are independent and identically distributed (i.i.d.). By substituting (7) and (8) into (26), Δfnm can be written as(27)Δfnm=f0cq-q-dTcos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT+p-p-dRcos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR.Since the number of local scatterers in the reference model is infinite, the parametersαpqT, βpqT, αpqR, βpqR, αpqi, and βpqi can be seen as continuous random variables with corresponding probability density functions (PDFs). Then, the ST-CF of the mB component (25) can be written as(28)Rp-q-,pqmdT,dR,τ=2σm2×∫⋯∫∫⋯∫∫⋯∫∫∫∫∫exp⁡j2πfpq,nmτ+Δfnm×∏i=1mpνpqi∏i=1mpαpqi,βpqi·pαpqT,βpqT,αpqR,βpqRdνpq1⋯dνpqmdαpq1⋯dαpqmdβpq1⋯dβpqmdαpqTdβpqTdαpqRdβpqR.Now, the complete expression of (16) can be obtained by substituting (20) and (28) into (16). pαpqT,βpqT,αpqR,βpqR describes the joint distribution of AOD and AOA, and it can be optionally used to present some propagation channel models. As a result, the ST-CF in (28) can provide a suitable platform to study the statistical properties of some different channel models, such as the random scattering model [13, 14], Jakes model [28], one-ring model [23], and two-ring model [29] as described in Section 3.1.3. Therefore, the ST-CF in (28) is a generalized and parametric expression. However, due to the complex nature of pαpqT,βpqT,αpqR,βpqR, it is assumed that AOD and AOA are independent [13, 14, 32, 33], and azimuth angles and elevation angles in (28) are also independent [18, 19, 32]. The parameters in (28) such as the velocities of the multiple moving scatterers and random angles can be calculated as follows.(i) Scatterer Velocity Distributions. The Gaussian, Laplace, exponential, and uniform distributions can be used to describe the velocity of moving scatterers [13]. In fact, the scatterer velocity νpqi is always positive or equal to zero. We use the uniform distribution in (29) and half-Gaussian distribution in (30) to describe the velocity of multiple moving scatterers.(29)pνpqi=1vmaxi,0≤νpqi≤vmaxi,where vmaxi is the maximum of νpqi.(30)pνpqi=2πσpq,iexp⁡-νpqi22σpq,i2,νpqi≥0,where σpq,i is the standard deviation of νpqi.(ii) Angle Distributions. To characterize the statistical angles in Table 1, we use the uniform distribution in (31) in the isotropic scattering environment and use the von Mises distribution in (32) and the cosine distribution in (33) in the nonisotropic scattering environment. In addition, the interval of azimuth angles αpqT, αpqR, and αpqi is -π,π, and the interval of elevation angles βpqT, βpqR, and βpqi is -π/2,π/2.(31)pγ=1γ2-γ1,γ1≤γ≤γ2,(32)pα=exp⁡kcos⁡α-α-2πI0k,α1≤α≤α1+2π,where I0• is the zeroth-order modified Bessel function of the first kind, α- is the mean angle, and k controls the spread of angles around the mean. The von Mises distribution PDF with α-=0 is used to describe the azimuth angles.(33)pβ=π4βmcos⁡π2ββm,-βm≤β≤βm,where βm is the maximum of β. The cosine distribution PDF is used to describe the elevation angles.


### 3.1.3. Special Cases ofmB’s ST-CF

IfβT=βR=βpqT=βpqR=βpqi=0, that is, the scattering environment is 2D, some different special cases can be derived from the general expression of the mB’s ST-CF in (28).In the 2D scattering environment, (28) can be written as (B.2) in Appendix B. If p=p-, q=q-, and m=1, (B.2) equals (11) in [13] and (6) in [14] regardless of the plus or minus signs under the assumption that the angles αT and αR are independent of each other in [13, 14]. In isotropic scattering environments, some other special cases with closed-form expressions can be derived as follows.AppendixB shows that an approximate ST-CF of (B.2) can be written as(34)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR∫⋯∫J02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ∏i=1mpνpqidνpq1⋯dνpqm,where J0• denotes the zeroth-order Bessel function of the first kind and k0=2πf0/c is the wave number.Note that ifm approaches to the infinity, (34) can be written as(35)limm→∞⁡Rp-q-,pqmdT,dR,τ=2σm2J0q-q-k0dTJ0p-p-k0dR,τ=0,0τ>0.In the NLoS communication environment with very high-density scatterers, (35) implies that the ST-CF approaches to zero at the nonzero time difference τ and has nothing to do with velocities of scatterers, TX and RX. The large-scale antenna arrays are very suitable to be used in this environment because the antenna element spacing can be reduced to a smaller value.Ifνpqi is constant, (34) can be presented as the following closed-form expression:(36)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRJ02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ.The T-CF of the classical F2M scenario with fixed scatterers is obtained, ifvT=vpqi=0i=1,2,3,…,m, q=q-, and p=p- in (36). In this case, the ST-CF in (36) is 2σm2J0k0vRτ, which is known as the Jakes model [28]. If vT=0,m=1, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vpq1τJ0k0vRτ, which equals the T-CF of F2M single-ring channel model in the presence of moving scatterers reported in (11) of [23]. If vpqi=0i=1,2,3,…,m, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vTτJ0k0vRτ, which equals the T-CFs of the classical M2M two-ring channel model in the presence of fixed scatterers reported in (46) of [29]. If vpqi=τ=0i=1,2,3,…,m, q=p=2, and q-=p-=1, the ST-CF in (36) results in 2σm2J0k0dTJ0k0dR, which equals the S-CF in (46) of [29].The velocities of moving scatterers such as moving foliage, walking pedestrians, and passing vehicles generally are random variables. Ifνpqi is described by the uniform distribution in (29), (34) can be presented as the following closed-form expression:(37)Rp-q-,pqmdT,dR,τ=σm22m-1J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR2J02k0vmax1τ+πJ12k0vmax1τH02k0vmax1τ-πJ02k0vmax1τ×H12k0vmax1τ2J02k0vmaxmτ+πJ12k0vmaxmτH02k0vmaxmτ-πJ02k0vmaxmτH12k0vmaxmτ×∏i=2m-12J04k0vmaxiτ+πJ14k0vmaxiτ×H04k0vmaxiτ-πJ04k0vmaxiτH14k0vmaxiτ,where J1• denotes the first-order Bessel function of the first kind, H0• denotes the zeroth-order Struve function, and H1• denotes the first-order Struve function.Ifνpqi is described by the half-Gaussian distribution in (30), (34) can be presented as the following closed-form expression:(38)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRI0k02σpq,12τ2×I0k02σpq,m2τ2exp⁡-k02σpq,12τ2exp⁡-k02σpq,m2τ2×∏i=2m-1I04k02σpq,i2τ2exp⁡-4k02σpq,i2τ2.


## 3.1.1. ST-CF of LoS Component

By substituting (2) into (18), the expression for the ST-CF of the LoS component can be written as (19)Rp-q-,pqLoSdT,dR,τ=ρpqρp-q-Eexpj2πtfpq,ρ-fp-q-,ρ+2πfpq,ρτ+2πfpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-+θpq,ρ-θp-q-,ρ.Formax⁡dT,dR≪D, we assume ρpq=ρp-q- and θpq,ρ=θp-q-,ρ. Then, (19) can be written as(20)Rp-q-,pqLoSdT,dR,τ=ρpq2Eexpj2πfpq,ρτ+Δfρ=ρpq2exp⁡j2πfpq,ρτ+Δfρ,where Δfρ is defined as(21)Δfρ=fpq,ρq+fpq,ρp-fp-q-,ρq--fp-q-,ρp-.By substituting (5) and (6) into (21), Δfρ can be written as(22)Δfρ=f0cq-q-dTcos⁡αqcos⁡βq+p-p-dRcos⁡αpcos⁡βp.


## 3.1.2. ST-CF ofmB Component

By substituting (4) into (17), the expression for the ST-CF of the mB component can be written as(23)Rp-q-,pqmdT,dR,τ=limNn→∞⁡limNm→∞⁡∑nn=1Nncp-q-,nn∑nm=1Nmcpq,nm×Eexp⁡j2πfpq,nm-fp-q-,nnt+2πfpq,nmτ+2πfpq,nq+fpq,np-fp-q-,nq--fp-q-,np-+θpq,nm-θp-q-,nn.Under the assumption thatθpq,nmand θp-q-,nn are uniformly distributed on the interval 0,2π and independent of each other, Eexp⁡jθpq,nm-θp-q-,nn equals 1. It is assumed that all the path gains of the mB component have the same size; that is,(24)cp-q-,nn=cpq,nm=σm2Nm.Then, (23) can be written as(25)Rp-q-,pqmdT,dR,τ=2σm2limNm→∞⁡∑nm=1NmEexp⁡j2πfpq,nmτ+Δfnm,where Δfnm is defined as(26)Δfnm=fpq,nq+fpq,np-fp-q-,nq--fp-q-,np-.It is assumed that AOD and AOA ofhpqmt and hp-q-mt are independent and identically distributed (i.i.d.). By substituting (7) and (8) into (26), Δfnm can be written as(27)Δfnm=f0cq-q-dTcos⁡αq-αpqTcos⁡βqcos⁡βpqT+sin⁡βqsin⁡βpqT+p-p-dRcos⁡αp-αpqRcos⁡βpcos⁡βpqR+sin⁡βpsin⁡βpqR.Since the number of local scatterers in the reference model is infinite, the parametersαpqT, βpqT, αpqR, βpqR, αpqi, and βpqi can be seen as continuous random variables with corresponding probability density functions (PDFs). Then, the ST-CF of the mB component (25) can be written as(28)Rp-q-,pqmdT,dR,τ=2σm2×∫⋯∫∫⋯∫∫⋯∫∫∫∫∫exp⁡j2πfpq,nmτ+Δfnm×∏i=1mpνpqi∏i=1mpαpqi,βpqi·pαpqT,βpqT,αpqR,βpqRdνpq1⋯dνpqmdαpq1⋯dαpqmdβpq1⋯dβpqmdαpqTdβpqTdαpqRdβpqR.Now, the complete expression of (16) can be obtained by substituting (20) and (28) into (16). pαpqT,βpqT,αpqR,βpqR describes the joint distribution of AOD and AOA, and it can be optionally used to present some propagation channel models. As a result, the ST-CF in (28) can provide a suitable platform to study the statistical properties of some different channel models, such as the random scattering model [13, 14], Jakes model [28], one-ring model [23], and two-ring model [29] as described in Section 3.1.3. Therefore, the ST-CF in (28) is a generalized and parametric expression. However, due to the complex nature of pαpqT,βpqT,αpqR,βpqR, it is assumed that AOD and AOA are independent [13, 14, 32, 33], and azimuth angles and elevation angles in (28) are also independent [18, 19, 32]. The parameters in (28) such as the velocities of the multiple moving scatterers and random angles can be calculated as follows.(i) Scatterer Velocity Distributions. The Gaussian, Laplace, exponential, and uniform distributions can be used to describe the velocity of moving scatterers [13]. In fact, the scatterer velocity νpqi is always positive or equal to zero. We use the uniform distribution in (29) and half-Gaussian distribution in (30) to describe the velocity of multiple moving scatterers.(29)pνpqi=1vmaxi,0≤νpqi≤vmaxi,where vmaxi is the maximum of νpqi.(30)pνpqi=2πσpq,iexp⁡-νpqi22σpq,i2,νpqi≥0,where σpq,i is the standard deviation of νpqi.(ii) Angle Distributions. To characterize the statistical angles in Table 1, we use the uniform distribution in (31) in the isotropic scattering environment and use the von Mises distribution in (32) and the cosine distribution in (33) in the nonisotropic scattering environment. In addition, the interval of azimuth angles αpqT, αpqR, and αpqi is -π,π, and the interval of elevation angles βpqT, βpqR, and βpqi is -π/2,π/2.(31)pγ=1γ2-γ1,γ1≤γ≤γ2,(32)pα=exp⁡kcos⁡α-α-2πI0k,α1≤α≤α1+2π,where I0• is the zeroth-order modified Bessel function of the first kind, α- is the mean angle, and k controls the spread of angles around the mean. The von Mises distribution PDF with α-=0 is used to describe the azimuth angles.(33)pβ=π4βmcos⁡π2ββm,-βm≤β≤βm,where βm is the maximum of β. The cosine distribution PDF is used to describe the elevation angles.


## 3.1.3. Special Cases ofmB’s ST-CF

IfβT=βR=βpqT=βpqR=βpqi=0, that is, the scattering environment is 2D, some different special cases can be derived from the general expression of the mB’s ST-CF in (28).In the 2D scattering environment, (28) can be written as (B.2) in Appendix B. If p=p-, q=q-, and m=1, (B.2) equals (11) in [13] and (6) in [14] regardless of the plus or minus signs under the assumption that the angles αT and αR are independent of each other in [13, 14]. In isotropic scattering environments, some other special cases with closed-form expressions can be derived as follows.AppendixB shows that an approximate ST-CF of (B.2) can be written as(34)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR∫⋯∫J02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ∏i=1mpνpqidνpq1⋯dνpqm,where J0• denotes the zeroth-order Bessel function of the first kind and k0=2πf0/c is the wave number.Note that ifm approaches to the infinity, (34) can be written as(35)limm→∞⁡Rp-q-,pqmdT,dR,τ=2σm2J0q-q-k0dTJ0p-p-k0dR,τ=0,0τ>0.In the NLoS communication environment with very high-density scatterers, (35) implies that the ST-CF approaches to zero at the nonzero time difference τ and has nothing to do with velocities of scatterers, TX and RX. The large-scale antenna arrays are very suitable to be used in this environment because the antenna element spacing can be reduced to a smaller value.Ifνpqi is constant, (34) can be presented as the following closed-form expression:(36)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRJ02k0νpq1τ×J02k0νpqmτ∏i=2m-1J04k0νpqiτ.The T-CF of the classical F2M scenario with fixed scatterers is obtained, ifvT=vpqi=0i=1,2,3,…,m, q=q-, and p=p- in (36). In this case, the ST-CF in (36) is 2σm2J0k0vRτ, which is known as the Jakes model [28]. If vT=0,m=1, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vpq1τJ0k0vRτ, which equals the T-CF of F2M single-ring channel model in the presence of moving scatterers reported in (11) of [23]. If vpqi=0i=1,2,3,…,m, q=q-, and p=p-, the ST-CF in (36) results in 2σm2J0k0vTτJ0k0vRτ, which equals the T-CFs of the classical M2M two-ring channel model in the presence of fixed scatterers reported in (46) of [29]. If vpqi=τ=0i=1,2,3,…,m, q=p=2, and q-=p-=1, the ST-CF in (36) results in 2σm2J0k0dTJ0k0dR, which equals the S-CF in (46) of [29].The velocities of moving scatterers such as moving foliage, walking pedestrians, and passing vehicles generally are random variables. Ifνpqi is described by the uniform distribution in (29), (34) can be presented as the following closed-form expression:(37)Rp-q-,pqmdT,dR,τ=σm22m-1J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dR2J02k0vmax1τ+πJ12k0vmax1τH02k0vmax1τ-πJ02k0vmax1τ×H12k0vmax1τ2J02k0vmaxmτ+πJ12k0vmaxmτH02k0vmaxmτ-πJ02k0vmaxmτH12k0vmaxmτ×∏i=2m-12J04k0vmaxiτ+πJ14k0vmaxiτ×H04k0vmaxiτ-πJ04k0vmaxiτH14k0vmaxiτ,where J1• denotes the first-order Bessel function of the first kind, H0• denotes the zeroth-order Struve function, and H1• denotes the first-order Struve function.Ifνpqi is described by the half-Gaussian distribution in (30), (34) can be presented as the following closed-form expression:(38)Rp-q-,pqmdT,dR,τ=2σm2J0k0vTτ+q-q-dT×J0k0vRτ+p-p-dRI0k02σpq,12τ2×I0k02σpq,m2τ2exp⁡-k02σpq,12τ2exp⁡-k02σpq,m2τ2×∏i=2m-1I04k02σpq,i2τ2exp⁡-4k02σpq,i2τ2.


## 3.2. Space-Doppler Power Spectral Density

The SD-PSD can be obtained by taking the Fourier transform of the ST-CF in (16) with respect to time difference τ. From (16), it follows that the SD-PSD is a summation of the SD-PSDs of the MB and LoS components; that is,(39)Sp-q-,pqdT,dR,ω=FτRp-q-,pqdT,dR,τ=∑m=1MpmSp-q-,pqmdT,dR,ω+Sp-q-,pqLoSdT,dR,ω.The SD-PSD of the LoS component can be written as(40)Sp-q-,pqLoSdT,dR,ω=2πρpq2exp⁡j2πΔfρδω-2πfpq,ρ,where δ• is the Dirac delta function.Since the ST-CF of themB component is the multiple integral as (28), the closed-form expression of SD-PSD cannot be derived. In Section 4.2, we show the SD-PSD of the 3D V2V channel in the presence of multiple moving scatterers by means of numerical integrations.


## 4. Numerical Results and Validation

This section demonstrates the normalized correlation functions and validates the Doppler power spectral density (D-PSD) described in Section3 through the MATLAB numerical simulations. Unless indicated otherwise, the values of the numerical simulation parameters are summarized in Table 2.Table 2
Parameters used in the numerical simulation.
Parameters
Figures2, 3, and 6
Figures4, 5, and 7
Figures8 and 9
Figure10
Figure11
f
0 (GHz)
2.435
2.435
2.435
29.5
2.435
K
2
2
2
2.4
2.41
k
5.4
5.4
3
—
3
q
,
p
2,2
1,1
2,2
1,1
1,1
q
-
,
p
-
1,1
1,1
1,1
1,1
1,1
d
T
,
d
R
λ
1
/
2,1
/
2
0,0
variable
0,0
0,0
α
q
,
β
q (rad)
π
/
2
,
π
/
6
π
/
2
,
π
/
6
π
/
2
,
π
/
6
π
/
2
,
π
/
6
0,0
α
p
,
β
p (rad)
π
/
2
,
π
/
6
π
/
2
,
π
/
6
π
/
2
,
π
/
6
π
/
2
,
π
/
6
0,0
v
T
,
v
R (m/s)
25,25
25,25
25,25
0,0
12,12
α
T
,
β
T
r
a
d
π
/
2
,
0
π
/
2
,
0
π
/
2
,
0
0,0
π
/
2
,
0
α
R
,
β
R
r
a
d
π
/
2
,
0
π
/
2
,
0
π
/
2
,
0
0,0
π
/
2
,
0


### 4.1. Numerical Results

In this section, the numerical curves of ST-CFs, T-CFs, and S-CFs influenced by some important contributory factors are presented. As in the typical urban environments, the power weight of themth cluster of rays has been set to p1=p2=1/2 for M=2, p1=p2=p3=1/3 for M=3, p1=p2=1/3, p3=p4=1/6 for M=4, p1=1/3, p2=1/4, p3=p4=1/6, and p5=1/12 for M=5.


#### 4.1.1. ST-CFs and T-CFs for Different Scattering Scenarios and Bounces

Figures2–5 demonstrate the ST-CFs and T-CFs for the different maximum bounces M in the 3D isotropic and nonisotropic scattering scenarios. The scatterer velocity is uniformly distributed with an average speed of 25 m/s that may be the velocity of passing vehicles. The other parameters used to obtain curves in Figures 2–5 are summarized in (Table 2, Cols. 2 and 3). As shown in Figures 2 and 4, the larger M is, the faster the ST-CFs and T-CFs decrease in the isotropic scattering scenario. However, the descent rates of ST-CFs and T-CFs increase slowly when M is larger than 3. The similar conclusions can be obtained from Figures 3 and 5; however, the ST-CF curves in Figure 3 have oscillations which may be caused by the nonisotropic scattering. However, this conclusion differs from [27] which showed that the triple- or higher-order bounced rays had statistical properties very similar to those of the double-bounced rays and could be approximated as double-bounced rays. The discrepancy may be caused by the different communication environments considered by us and [27]. Specifically, channel-sounding experimental campaign in [27] was conducted along surface streets around the Georgia Tech campus and on the Interstate highways in the Midtown Atlanta metropolitan area which may have lower dense scatterers than our communication environments. The ST-CF curves in Figure 3 converge relatively slower for the larger M that means there are more local scatterers in motion with random velocities and random directions. As can be observed in Figures 4 and 5, the T-CFs in both figures are very similar that indicates that the scattering forms have no significant effect on the T-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Furthermore, in Figures 2–5, it can be observed that a tendency to a limit value curve of ST-CFs and T-CFs can be inferred if M approaches infinity, and the similar inference also can be obtained in (35) for the 2D scattering environment.Figure 2
ST-CFs in (16) in the isotropic scattering scenario for different maximum bounces.Figure 3
ST-CFs in (16) in the nonisotropic scattering scenario for different maximum bounces.Figure 4
T-CFs in (16) with p=p- and q=q- in the isotropic scattering scenario for different maximum bounces.Figure 5
T-CFs in (16) with p=p- and q=q- in the nonisotropic scattering scenario for different maximum bounces.


#### 4.1.2. ST-CFs and T-CFs for Different Scattering Scenarios and Scatterer Velocity Distributions

Figures6 and 7 demonstrate the ST-CFs and T-CFs for the different scatterer velocity distributions in the 3D isotropic and nonisotropic scattering scenarios. We use two uniform distributions in the isotropic scattering environment and use the von Mises distribution and the cosine distribution in the nonisotropic scattering environment to describe all the statistical azimuth and elevation angles in Table 1, respectively. The average speed of the uniform and half-Gaussian distributions is 25 m/s, and the standard deviation of the half-Gaussian distribution is 31 m/s. The other parameters used to obtain curves in Figures 6 and 7 are summarized in (Table 2, Cols. 2 and 3). As shown in Figure 6, the ST-CF for the uniform distributed velocity of scatterers in the isotropic scattering environment decreases fastest, and both the nonisotropic scattering and the half-Gaussian distribution can result in swift oscillations of the fading curves. The comparison results indicate that the scatterer velocity distributions have more significant effect than the scattering forms on the ST-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Figure 7 indicates that, not like scattering forms, the scatterer velocity distributions have significant effect on the T-CFs. Generally, the realistic propagation environment is nonisotropic scattering, and the scatterer velocity can be modeled by the half-Gaussian or uniform distribution. Therefore, the results of Figures 6 and 7 may be helpful in analysis of the system performance and modeling a specific propagation channel for V2V communications.Figure 6
ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.Figure 7
T-CFs in (16) with p=p- and q=q- in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.


#### 4.1.3. ST-CFs and S-CFs for Different Scattering Scenarios and Spacing between Adjacent Antenna Elements

Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for the different spacing between adjacent antenna elements of TX and RX in isotropic and nonisotropic scattering scenarios. The set of the scatterer velocity in Figure 8 is the same as Figures 2–5, and the other parameters used to obtain curves in Figures 8 and 9 are summarized in (Table 2, Col. 4). Figures 8 and 9 show that spacing between adjacent antenna elements has significant effect on the 3D ST-CFs and S-CFs. Furthermore, the 3D ST-CFs in isotropic and nonisotropic scattering scenarios decrease to the bottom when dR=dT tends to 1.2λ and 0.7λ, respectively. In addition, the 3D S-CF in nonisotropic scattering scenarios is explicitly less than that in isotropic scattering scenarios for most spacing between adjacent antenna elements as shown in Figure 9. Therefore, the large-scale antenna arrays employing the spatial multiplexing are more suitable to be used in the nonisotropic scattering scenarios owing to the high antenna space utilization. The spacing between adjacent antenna elements of antenna arrays is suggested to be more than λ in the V2V communications, especially in the environments with low-density scatterers.Figure 8
3D ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5. λ is the carrier wavelength.Figure 9
3D S-CFs in (16) with τ=0 in the isotropic and nonisotropic scattering scenarios. λ is the carrier wavelength.


### 4.2. Comparison with Measurements

The parametric nature of the proposed channel model makes it adaptable to a variety of propagation environments. D-PSD is one of the most important and unique channel characteristics for V2V communication channels. To illustrate the validity of the proposed model, we compare the modeled SD-PSDs of the 3D V2V channel with multiple moving scatterers with the measured F2F D-PSD in [26], the theoretical V2V SD-PSD in [18], and the measured V2V SD-PSD in [27].Figure10 shows the comparison between the modeled SD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario and the measured D-PSDs of Figure  4 in [26]. The outdoor measurement experiments in [26] have shown that, at millimeter wavelengths, fading caused by foliage movement and the motion of nearby vehicles is a significantly deleterious effect in systems with static subscribers. Figure  4 in [26] shows the D-PSD generated by foliage movements and the D-PSD caused by passing vehicles, both in a fixed wireless channel at a carrier frequency of 29.5 GHz. We have used the half-Gaussian distribution with an average speed of 0.4 m/s and the standard deviation of 0.5 m/s to model the velocity of moving foliage scatterers, while the uniform distribution with an average speed of 6 m/s has been used to model the velocity of passing vehicles scatterers. The power weight of the mth cluster of rays scattered by the moving foliage or passing vehicles has been set to p1=9/16, p2=5/16, p3=1/16, and p4=p5=1/32. The other parameters used to obtain the modeled SD-PSD curves in Figure 10 are summarized in (Table 2, Col. 5). As can be observed in Figure 10, 40 dB fading bandwidths for the effect of the foliage and vehicles movements are 500 Hz and 1 kHz approximately, respectively. Fade depth due to foliage is less than the variation due to passing vehicles. Close agreements between the modeled and measured results are shown in Figure 10. These close matches can be seen not only for relatively slow moving scatterers but also for relatively fast moving scatterers as well.Figure 10
Comparison between the modeled SD-PSDs withM=5 and the measured D-PSDs of Figure  4 in [26].Figure11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CF presented in (16) in the nonisotropic scattering scenario, the theoretical SD-PSD of Figure  8 in [18], and the measured SD-PSD of Figure  11 in [27]. The theoretical SD-PSD reproduced here is based on the two-cylinder model with moving and stationary scatterers for dR=dT=0. The channel measurements in [27] for the measured SD-PSD were collected at 2.435 GHz in the urban street surface environment, and the spacing between two adjacent antenna elements of TX and RX was set to zero. We have used the half-Gaussian distribution with average speeds of 2.394 m/s, 0.798 m/s, and 0.4 m/s and the standard deviation of 3 m/s, 1 m/s, and 0.5 m/s to model the velocity of the first, the second, and the remaining moving scatterers on the communication links, respectively. The power weight of the mth cluster of rays has been set to p1=8/15, p2=1/3, p3=1/15, p4=1/30, and p5=1/30 for M=5. The other parameters used to obtain the modeled SD-PSD curve in Figure 11 are summarized in (Table 2, Col. 6). We note that, different from the measured SD-PSDs in [26], the measured SD-PSD curve in [27] has oscillations. The oscillating pattern in measurements for the case dR=dT=0 may appear because it is not a true SISO scenario where only one transmit and one receive antenna are active [27]. As can be observed in Figure 11, the modeled SD-PSD matches better with measured SD-PSD than the theoretical SD-PSD in [18]. This may be because of the introduction of the multiple bounced rays into the propagation model or the constraints on the position of local scatterers in the two-cylinder model proposed in [18].Figure 11
Comparison among the modeled SD-PSD withM=5, the theoretical SD-PSD of Figure  8 in [18], and the measured SD-PSD of Figure  11 in [27].Figure12 demonstrates the CDFs of the relative deviations between modeled/theoretical SD-PSDs and corresponding measured values in Figures 10 and 11. The mean relative deviation of the modeled SD-PSDs for foliage in Figure 10, modeled SD-PSDs for vehicles in Figure 10, modeled SD-PSDs in Figure 11, and theoretical SD-PSDs in Figure 11 is 2.56%, 2.16%, 3.9%, and 8.32%, respectively. As shown in Figure 12, more than 80% of the relative deviations of modeled SD-PSDs in Figures 10 and 11 are less than 5%, and the relative deviations of theoretical SD-PSDs in Figure 11 are explicitly larger than those of modeled SD-PSDs. In addition, from Figures 10 and 11, we note that the modeled SD-PSD matches better with measured SD-PSDs at the lower and higher Doppler frequencies than the middle Doppler frequencies. This may be because TX and RX were not strictly fixed, and not all the scatterers were in motion in measurements. The approximate calculation of the frequency shift in (A.4) and Doppler shift in (10) may be another factor which causes the mismatch at the middle Doppler frequencies, and more investigations concerning the accuracy of the proposed model will be addressed in future works. However, as shown in Figures 10–12, the acceptable matches confirm the utility and generality of the proposed model and show the need for including multiple moving scatterers in propagation model.Figure 12
The relative deviation cumulative distribution functions (CDFs) of the modeled and theoretical SD-PSDs in Figures10 and 11.


## 4.1. Numerical Results

In this section, the numerical curves of ST-CFs, T-CFs, and S-CFs influenced by some important contributory factors are presented. As in the typical urban environments, the power weight of themth cluster of rays has been set to p1=p2=1/2 for M=2, p1=p2=p3=1/3 for M=3, p1=p2=1/3, p3=p4=1/6 for M=4, p1=1/3, p2=1/4, p3=p4=1/6, and p5=1/12 for M=5.


### 4.1.1. ST-CFs and T-CFs for Different Scattering Scenarios and Bounces

Figures2–5 demonstrate the ST-CFs and T-CFs for the different maximum bounces M in the 3D isotropic and nonisotropic scattering scenarios. The scatterer velocity is uniformly distributed with an average speed of 25 m/s that may be the velocity of passing vehicles. The other parameters used to obtain curves in Figures 2–5 are summarized in (Table 2, Cols. 2 and 3). As shown in Figures 2 and 4, the larger M is, the faster the ST-CFs and T-CFs decrease in the isotropic scattering scenario. However, the descent rates of ST-CFs and T-CFs increase slowly when M is larger than 3. The similar conclusions can be obtained from Figures 3 and 5; however, the ST-CF curves in Figure 3 have oscillations which may be caused by the nonisotropic scattering. However, this conclusion differs from [27] which showed that the triple- or higher-order bounced rays had statistical properties very similar to those of the double-bounced rays and could be approximated as double-bounced rays. The discrepancy may be caused by the different communication environments considered by us and [27]. Specifically, channel-sounding experimental campaign in [27] was conducted along surface streets around the Georgia Tech campus and on the Interstate highways in the Midtown Atlanta metropolitan area which may have lower dense scatterers than our communication environments. The ST-CF curves in Figure 3 converge relatively slower for the larger M that means there are more local scatterers in motion with random velocities and random directions. As can be observed in Figures 4 and 5, the T-CFs in both figures are very similar that indicates that the scattering forms have no significant effect on the T-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Furthermore, in Figures 2–5, it can be observed that a tendency to a limit value curve of ST-CFs and T-CFs can be inferred if M approaches infinity, and the similar inference also can be obtained in (35) for the 2D scattering environment.Figure 2
ST-CFs in (16) in the isotropic scattering scenario for different maximum bounces.Figure 3
ST-CFs in (16) in the nonisotropic scattering scenario for different maximum bounces.Figure 4
T-CFs in (16) with p=p- and q=q- in the isotropic scattering scenario for different maximum bounces.Figure 5
T-CFs in (16) with p=p- and q=q- in the nonisotropic scattering scenario for different maximum bounces.


### 4.1.2. ST-CFs and T-CFs for Different Scattering Scenarios and Scatterer Velocity Distributions

Figures6 and 7 demonstrate the ST-CFs and T-CFs for the different scatterer velocity distributions in the 3D isotropic and nonisotropic scattering scenarios. We use two uniform distributions in the isotropic scattering environment and use the von Mises distribution and the cosine distribution in the nonisotropic scattering environment to describe all the statistical azimuth and elevation angles in Table 1, respectively. The average speed of the uniform and half-Gaussian distributions is 25 m/s, and the standard deviation of the half-Gaussian distribution is 31 m/s. The other parameters used to obtain curves in Figures 6 and 7 are summarized in (Table 2, Cols. 2 and 3). As shown in Figure 6, the ST-CF for the uniform distributed velocity of scatterers in the isotropic scattering environment decreases fastest, and both the nonisotropic scattering and the half-Gaussian distribution can result in swift oscillations of the fading curves. The comparison results indicate that the scatterer velocity distributions have more significant effect than the scattering forms on the ST-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Figure 7 indicates that, not like scattering forms, the scatterer velocity distributions have significant effect on the T-CFs. Generally, the realistic propagation environment is nonisotropic scattering, and the scatterer velocity can be modeled by the half-Gaussian or uniform distribution. Therefore, the results of Figures 6 and 7 may be helpful in analysis of the system performance and modeling a specific propagation channel for V2V communications.Figure 6
ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.Figure 7
T-CFs in (16) with p=p- and q=q- in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.


### 4.1.3. ST-CFs and S-CFs for Different Scattering Scenarios and Spacing between Adjacent Antenna Elements

Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for the different spacing between adjacent antenna elements of TX and RX in isotropic and nonisotropic scattering scenarios. The set of the scatterer velocity in Figure 8 is the same as Figures 2–5, and the other parameters used to obtain curves in Figures 8 and 9 are summarized in (Table 2, Col. 4). Figures 8 and 9 show that spacing between adjacent antenna elements has significant effect on the 3D ST-CFs and S-CFs. Furthermore, the 3D ST-CFs in isotropic and nonisotropic scattering scenarios decrease to the bottom when dR=dT tends to 1.2λ and 0.7λ, respectively. In addition, the 3D S-CF in nonisotropic scattering scenarios is explicitly less than that in isotropic scattering scenarios for most spacing between adjacent antenna elements as shown in Figure 9. Therefore, the large-scale antenna arrays employing the spatial multiplexing are more suitable to be used in the nonisotropic scattering scenarios owing to the high antenna space utilization. The spacing between adjacent antenna elements of antenna arrays is suggested to be more than λ in the V2V communications, especially in the environments with low-density scatterers.Figure 8
3D ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5. λ is the carrier wavelength.Figure 9
3D S-CFs in (16) with τ=0 in the isotropic and nonisotropic scattering scenarios. λ is the carrier wavelength.


## 4.1.1. ST-CFs and T-CFs for Different Scattering Scenarios and Bounces

Figures2–5 demonstrate the ST-CFs and T-CFs for the different maximum bounces M in the 3D isotropic and nonisotropic scattering scenarios. The scatterer velocity is uniformly distributed with an average speed of 25 m/s that may be the velocity of passing vehicles. The other parameters used to obtain curves in Figures 2–5 are summarized in (Table 2, Cols. 2 and 3). As shown in Figures 2 and 4, the larger M is, the faster the ST-CFs and T-CFs decrease in the isotropic scattering scenario. However, the descent rates of ST-CFs and T-CFs increase slowly when M is larger than 3. The similar conclusions can be obtained from Figures 3 and 5; however, the ST-CF curves in Figure 3 have oscillations which may be caused by the nonisotropic scattering. However, this conclusion differs from [27] which showed that the triple- or higher-order bounced rays had statistical properties very similar to those of the double-bounced rays and could be approximated as double-bounced rays. The discrepancy may be caused by the different communication environments considered by us and [27]. Specifically, channel-sounding experimental campaign in [27] was conducted along surface streets around the Georgia Tech campus and on the Interstate highways in the Midtown Atlanta metropolitan area which may have lower dense scatterers than our communication environments. The ST-CF curves in Figure 3 converge relatively slower for the larger M that means there are more local scatterers in motion with random velocities and random directions. As can be observed in Figures 4 and 5, the T-CFs in both figures are very similar that indicates that the scattering forms have no significant effect on the T-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Furthermore, in Figures 2–5, it can be observed that a tendency to a limit value curve of ST-CFs and T-CFs can be inferred if M approaches infinity, and the similar inference also can be obtained in (35) for the 2D scattering environment.Figure 2
ST-CFs in (16) in the isotropic scattering scenario for different maximum bounces.Figure 3
ST-CFs in (16) in the nonisotropic scattering scenario for different maximum bounces.Figure 4
T-CFs in (16) with p=p- and q=q- in the isotropic scattering scenario for different maximum bounces.Figure 5
T-CFs in (16) with p=p- and q=q- in the nonisotropic scattering scenario for different maximum bounces.


## 4.1.2. ST-CFs and T-CFs for Different Scattering Scenarios and Scatterer Velocity Distributions

Figures6 and 7 demonstrate the ST-CFs and T-CFs for the different scatterer velocity distributions in the 3D isotropic and nonisotropic scattering scenarios. We use two uniform distributions in the isotropic scattering environment and use the von Mises distribution and the cosine distribution in the nonisotropic scattering environment to describe all the statistical azimuth and elevation angles in Table 1, respectively. The average speed of the uniform and half-Gaussian distributions is 25 m/s, and the standard deviation of the half-Gaussian distribution is 31 m/s. The other parameters used to obtain curves in Figures 6 and 7 are summarized in (Table 2, Cols. 2 and 3). As shown in Figure 6, the ST-CF for the uniform distributed velocity of scatterers in the isotropic scattering environment decreases fastest, and both the nonisotropic scattering and the half-Gaussian distribution can result in swift oscillations of the fading curves. The comparison results indicate that the scatterer velocity distributions have more significant effect than the scattering forms on the ST-CFs of the 3D V2V channel in the presence of multiple moving scatterers. Figure 7 indicates that, not like scattering forms, the scatterer velocity distributions have significant effect on the T-CFs. Generally, the realistic propagation environment is nonisotropic scattering, and the scatterer velocity can be modeled by the half-Gaussian or uniform distribution. Therefore, the results of Figures 6 and 7 may be helpful in analysis of the system performance and modeling a specific propagation channel for V2V communications.Figure 6
ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.Figure 7
T-CFs in (16) with p=p- and q=q- in the isotropic and nonisotropic scattering scenarios with M=5 for different scatterer velocity distributions.


## 4.1.3. ST-CFs and S-CFs for Different Scattering Scenarios and Spacing between Adjacent Antenna Elements

Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for the different spacing between adjacent antenna elements of TX and RX in isotropic and nonisotropic scattering scenarios. The set of the scatterer velocity in Figure 8 is the same as Figures 2–5, and the other parameters used to obtain curves in Figures 8 and 9 are summarized in (Table 2, Col. 4). Figures 8 and 9 show that spacing between adjacent antenna elements has significant effect on the 3D ST-CFs and S-CFs. Furthermore, the 3D ST-CFs in isotropic and nonisotropic scattering scenarios decrease to the bottom when dR=dT tends to 1.2λ and 0.7λ, respectively. In addition, the 3D S-CF in nonisotropic scattering scenarios is explicitly less than that in isotropic scattering scenarios for most spacing between adjacent antenna elements as shown in Figure 9. Therefore, the large-scale antenna arrays employing the spatial multiplexing are more suitable to be used in the nonisotropic scattering scenarios owing to the high antenna space utilization. The spacing between adjacent antenna elements of antenna arrays is suggested to be more than λ in the V2V communications, especially in the environments with low-density scatterers.Figure 8
3D ST-CFs in (16) in the isotropic and nonisotropic scattering scenarios with M=5. λ is the carrier wavelength.Figure 9
3D S-CFs in (16) with τ=0 in the isotropic and nonisotropic scattering scenarios. λ is the carrier wavelength.


## 4.2. Comparison with Measurements

The parametric nature of the proposed channel model makes it adaptable to a variety of propagation environments. D-PSD is one of the most important and unique channel characteristics for V2V communication channels. To illustrate the validity of the proposed model, we compare the modeled SD-PSDs of the 3D V2V channel with multiple moving scatterers with the measured F2F D-PSD in [26], the theoretical V2V SD-PSD in [18], and the measured V2V SD-PSD in [27].Figure10 shows the comparison between the modeled SD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario and the measured D-PSDs of Figure  4 in [26]. The outdoor measurement experiments in [26] have shown that, at millimeter wavelengths, fading caused by foliage movement and the motion of nearby vehicles is a significantly deleterious effect in systems with static subscribers. Figure  4 in [26] shows the D-PSD generated by foliage movements and the D-PSD caused by passing vehicles, both in a fixed wireless channel at a carrier frequency of 29.5 GHz. We have used the half-Gaussian distribution with an average speed of 0.4 m/s and the standard deviation of 0.5 m/s to model the velocity of moving foliage scatterers, while the uniform distribution with an average speed of 6 m/s has been used to model the velocity of passing vehicles scatterers. The power weight of the mth cluster of rays scattered by the moving foliage or passing vehicles has been set to p1=9/16, p2=5/16, p3=1/16, and p4=p5=1/32. The other parameters used to obtain the modeled SD-PSD curves in Figure 10 are summarized in (Table 2, Col. 5). As can be observed in Figure 10, 40 dB fading bandwidths for the effect of the foliage and vehicles movements are 500 Hz and 1 kHz approximately, respectively. Fade depth due to foliage is less than the variation due to passing vehicles. Close agreements between the modeled and measured results are shown in Figure 10. These close matches can be seen not only for relatively slow moving scatterers but also for relatively fast moving scatterers as well.Figure 10
Comparison between the modeled SD-PSDs withM=5 and the measured D-PSDs of Figure  4 in [26].Figure11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CF presented in (16) in the nonisotropic scattering scenario, the theoretical SD-PSD of Figure  8 in [18], and the measured SD-PSD of Figure  11 in [27]. The theoretical SD-PSD reproduced here is based on the two-cylinder model with moving and stationary scatterers for dR=dT=0. The channel measurements in [27] for the measured SD-PSD were collected at 2.435 GHz in the urban street surface environment, and the spacing between two adjacent antenna elements of TX and RX was set to zero. We have used the half-Gaussian distribution with average speeds of 2.394 m/s, 0.798 m/s, and 0.4 m/s and the standard deviation of 3 m/s, 1 m/s, and 0.5 m/s to model the velocity of the first, the second, and the remaining moving scatterers on the communication links, respectively. The power weight of the mth cluster of rays has been set to p1=8/15, p2=1/3, p3=1/15, p4=1/30, and p5=1/30 for M=5. The other parameters used to obtain the modeled SD-PSD curve in Figure 11 are summarized in (Table 2, Col. 6). We note that, different from the measured SD-PSDs in [26], the measured SD-PSD curve in [27] has oscillations. The oscillating pattern in measurements for the case dR=dT=0 may appear because it is not a true SISO scenario where only one transmit and one receive antenna are active [27]. As can be observed in Figure 11, the modeled SD-PSD matches better with measured SD-PSD than the theoretical SD-PSD in [18]. This may be because of the introduction of the multiple bounced rays into the propagation model or the constraints on the position of local scatterers in the two-cylinder model proposed in [18].Figure 11
Comparison among the modeled SD-PSD withM=5, the theoretical SD-PSD of Figure  8 in [18], and the measured SD-PSD of Figure  11 in [27].Figure12 demonstrates the CDFs of the relative deviations between modeled/theoretical SD-PSDs and corresponding measured values in Figures 10 and 11. The mean relative deviation of the modeled SD-PSDs for foliage in Figure 10, modeled SD-PSDs for vehicles in Figure 10, modeled SD-PSDs in Figure 11, and theoretical SD-PSDs in Figure 11 is 2.56%, 2.16%, 3.9%, and 8.32%, respectively. As shown in Figure 12, more than 80% of the relative deviations of modeled SD-PSDs in Figures 10 and 11 are less than 5%, and the relative deviations of theoretical SD-PSDs in Figure 11 are explicitly larger than those of modeled SD-PSDs. In addition, from Figures 10 and 11, we note that the modeled SD-PSD matches better with measured SD-PSDs at the lower and higher Doppler frequencies than the middle Doppler frequencies. This may be because TX and RX were not strictly fixed, and not all the scatterers were in motion in measurements. The approximate calculation of the frequency shift in (A.4) and Doppler shift in (10) may be another factor which causes the mismatch at the middle Doppler frequencies, and more investigations concerning the accuracy of the proposed model will be addressed in future works. However, as shown in Figures 10–12, the acceptable matches confirm the utility and generality of the proposed model and show the need for including multiple moving scatterers in propagation model.Figure 12
The relative deviation cumulative distribution functions (CDFs) of the modeled and theoretical SD-PSDs in Figures10 and 11.


## 5. Conclusion

Without specific constraints on the position of the local moving scatterers, a 3D geometrical V2V propagation model that includes LoS, single bounced, and multiple bounced links between the transmitter and receiver was proposed. Based on the geometrical propagation model, a 3D reference model for narrowband MIMO V2V multipath fading channels was developed. From the reference model, the corresponding mathematical expressions and numerical results of ST-CFs, T-CFs, S-CFs, and SD-PSDs were studied for different parametric sets. It has been shown that the maximum bounces, scattering forms, scatterer velocity distributions, and spacing between adjacent antenna elements have significant effect on the ST-CFs, and only the maximum bounces and scatterer velocity distributions have significant effect on the T-CFs. Finally, the modeled SD-PSD results were compared with the measured data and the other literature’s theoretical result. The close agreements between the analytically and empirically obtained channel statistics confirmed the utility and generality of the proposed model.



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*Source: 7231417-2017-07-10.xml*