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298 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 1, JANUARY 2014

Novel 3D Geometry-Based Stochastic Models forNon-Isotropic MIMO Vehicle-to-Vehicle Channels

Yi Yuan, Cheng-Xiang Wang, Senior Member, IEEE, Xiang Cheng, Senior Member, IEEE,Bo Ai, Senior Member, IEEE, and David I. Laurenson, Member, IEEE

Abstract—This paper proposes a novel three-dimensional (3D)theoretical regular-shaped geometry-based stochastic model (RS-GBSM) and the corresponding sum-of-sinusoids (SoS) simulationmodel for non-isotropic multiple-input multiple-output (MIMO)vehicle-to-vehicle (V2V) Ricean fading channels. The proposedRS-GBSM, combining line-of-sight (LoS) components, a two-sphere model, and an elliptic-cylinder model, has the abilityto study the impact of the vehicular traffic density (VTD) onchannel statistics, and jointly considers the azimuth and elevationangles by using the von Mises Fisher distribution. Moreover,a novel parameter computation method is proposed for jointlycalculating the azimuth and elevation angles in the SoS channelsimulator. Based on the proposed 3D theoretical RS-GBSM andits SoS simulation model, statistical properties are derived andthoroughly investigated. The impact of the elevation angle inthe 3D model on key statistical properties is investigated bycomparing with those of the corresponding two-dimensional(2D) model. It is demonstrated that the 3D model is moreaccurate to characterize real V2V channels, in particular for picocell scenarios. Finally, close agreement is achieved between thetheoretical model, SoS simulation model, and simulation results,demonstrating the utility of the proposed models.

Index Terms—MIMO vehicle-to-vehicle channels, 3D RS-GBSM, non-isotropic scattering, vehicular traffic density, sta-tistical properties.

I. INTRODUCTION

IN recent years, vehicle-to-vehicle (V2V) communications[1] have been encountered in many new applications, such

Manuscript received March 10, 2013; revised August 2, 2013; acceptedOctober 16, 2013. The associate editor coordinating the review of this paperand approving it for publication was M. J. Hossain.

This paper was presented in part at IEEE VTC’10-Fall, Ottawa, Canada,Sep. 2010.

This work was supported by the Opening Project of the Key Laboratory ofCognitive Radio and Information Processing (Guilin University of ElectronicTechnology), Ministry of Education (No.: 2013KF01), the National NaturalScience Foundation of China (Grant No. 61222105 & 61101079), BeijingMunicipal Natural Science Foundation (Grant No. 4112048), and the Sci-ence Foundation for the Youth Scholar of Ministry of Education of China(Grant No. 20110001120129).

Y. Yuan and C.-X. Wang are with the Joint Research Institute for Sig-nal and Image Processing, School of Engineering & Physical Sciences,Heriot-Watt University, Edinburgh, EH14 4AS, U.K. (e-mail: {yy120, cheng-xiang.wang}@hw.ac.uk). C.-X.Wang is also with the School of InformationScience and Engineering, Shandong University, Jinan 250100, China. C.-X. Wang is the corresponding author.

X. Cheng is with the Research Institute for Modern Communications,School of Electronics Engineering & Computer Science, Peking University,Beijing, 100871, China (e-mail: [email protected]).

B. Ai is with the State Key Laboratory of Rail Traffic Controland Safety, Beijing Jiaotong University, Beijing, 100044, China (e-mail:[email protected]).

D. I. Laurenson is with the Joint Research Institute for Signal and ImageProcessing, Institute for Digital Communications, University of Edinburgh,Edinburgh, EH9 3JL, U.K. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2013.120313.130434

as wireless mobile ad hoc peer-to-peer networks [2], [3]cooperative systems [4], [5], and intelligent transportationsystems. In V2V communication systems, both the transmitter(Tx) and receiver (Rx) are in motion and equipped withlow elevation antennas. This is different from conventionalfixed-to-mobile (F2M) cellular radio systems, where onlyone terminal moves. Moreover, multiple-input multiple-output(MIMO) technologies, where multiple antennas are deployedat both the Tx and Rx [6], have widely been adopted inadvanced F2M cellular systems and have also been receivingmore and more attention in V2V systems [7].

In order to evaluate the performance of a V2V commu-nication system, accurate channel models are indispensable.Existing channel models for F2M communications systemscannot be used directly for the design of V2V systems. V2Vchannel models available in the literature [8]–[23] can beclassified as geometry-based deterministic models (GBDMs)[8] and stochastic models, which can further be categorizedas non-geometry-based stochastic models (NGSMs) [9] andgeometry-based stochastic models (GBSMs) [10]–[23]. Fur-thermore, GBSMs can be classified as regular-shaped GBSMs(RS-GBSMs) [11]–[20] and irregular-shaped GBSMs (IS-GBSMs) [21]–[24], depending on whether effective scatterersare located on regular shapes, e.g., one-ring, two-ring, ellipses,or irregular shapes.

RS-GBSMs [11]–[20] have widely been used to mimic V2Vchannels due to their convenience for theoretical analysis ofchannel statistics. To preserve the mathematical tractability,RS-GBSMs assume that all the effective scatterers are locatedon regular shapes. Akki and Haber were the first to pro-pose a two-dimensional (2D) RS-GBSM [11] and investigatecorresponding statistical properties for narrowband isotropicscattering single-input single-output (SISO) V2V Rayleighchannels [12]. In [13], the authors proposed a 2D two-ring RS-GBSM with both single- and double-bounced raysfor narrowband non-isotropic scattering MIMO V2V Riceanchannels. In [14], the authors proposed an adaptive RS-GBSMconsisting of two rings and one ellipse also with both single-and double-bounced rays for narrowband non-isotropic MIMOV2V Ricean channels. As 2D models assume that waves travelonly in the horizontal plan, they neglect signal variations inthe vertical plane and are valid only when the Tx and Rxare sufficiently separated. In reality, waves do travel in threedimensions. Therefore, a three-dimensional (3D) two-cylinderRS-GBSM was developed for narrowband non-isotropic scat-tering MIMO V2V channels in [16]. It was further extendedto a wideband one in [17]. Other 3D V2V channel modelsinclude a 3D two-sphere RS-GBSM for narrowband non-

1536-1276/14$31.00 c© 2014 IEEE

YUAN et al.: NOVEL 3D GEOMETRY-BASED STOCHASTIC MODELS FOR NON-ISOTROPIC MIMO VEHICLE-TO-VEHICLE CHANNELS 299

isotropic SISO V2V channels [18] and a 3D two-concentric-quasi-sphere RS-GBSM for wideband non-isotropic MIMOV2V channels [19].

The aforementioned 3D RS-GBSMs [16]–[19] all assumedthat the azimuth angle and elevation angle are completelyindependent and thus analyzed them separately. Moreover,although the measurement campaigns in [9] demonstrated thatthe vehicular traffic density (VTD) significantly affects theV2V channel statistical properties, the impact of the VTD onchannel statistics was not considered in the existing 3D RS-GBSMs [16]–[19].

To fill the above research gaps, the first part of thispaper proposes a novel theoretical 3D RS-GBSM, which isthe combination of line-of-sight (LoS) components, a two-sphere model, and an elliptic-cylinder model [15], for non-isotropic MIMO V2V channels. The proposed 3D RS-GBSMis sufficiently generic and adaptive to model various V2Vchannels in different scenarios. It is the first 3D RS-GBSMthat has the ability to study the impact of the VTD on channelstatistics, and jointly considers the azimuth and elevationangles by applying the von Mises-Fisher (VMF) distribution asthe scatterer distribution. As the 3D theoretical RS-GBSM as-sumes infinite numbers of effective scatterers, which results inthe infinite complexity, it cannot be implemented in practice.However, a theoretical model can be used as a starting pointto design a realizable simulation model that considers limitednumbers of scatterers and has a reasonable complexity. Hence,the second part of this paper concentrates on developing acorresponding 3D MIMO V2V sum-of-sinusoids (SoS) basedsimulation model with a novel parameter computation method.Note that the proposed models have already considered theeffect of diffuse scattering [23] by using double-bounced rays.Also, the impact of vehicles as obstacles on the LoS obstruc-tion, as studied in measurements [24] and [25], can be capturedin our models by adjusting relevant model parameters, e.g., theRicean factor.

Overall, the major contributions and novelties of this paperare summarized as follows:

1) Based on the novel 3D theoretical RS-GBSM, com-prehensive statistical properties are derived and thor-oughly investigated, i.e., amplitude and phase probabil-ity density functions (PDFs), space-time (ST) correlationfunction (CF), Doppler power spectral density (PSD),envelope level crossing rate (LCR), and average fade du-ration (AFD). Meanwhile, some inaccurate expressionsin [15] are corrected.

2) The impacts of the VTD and elevation angle on afore-mentioned channel statistical properties are investigatedby comparing with those of the corresponding 2Dmodel.

3) The corresponding SoS simulation model is proposedby considering a finite number of scatterers at the Txand Rx.

4) A novel parameter computation method, namely themethod of equal volume (MEV), is proposed to cal-culate the azimuth and elevation angles of proposedSoS simulation model. It is the first method for 3DMIMO channel models jointly computing the azimuthand elevation angles.

3( )nR�

T�

RO

3( )nR�

TO

pT

'pT

qT

'qT

x

z 'z

'y y

T� R� R�

3( )ns

1( )ns2( )ns

3pn�

3n q�

pq�

R�

T� �

3( )nT�

3( )nT�

LoSR� �

3( )nT

3( )nR

LoSR�

TR 2D f�

RR

2a

Fig. 1. The proposed 3D MIMO V2V RS-GBSM combining a two-spheremodel and an elliptic-cylinder model (only showing the detailed geometryof LoS components and single-bounced rays in the elliptic-cylinder model).

1( )nT�

RO TO

'pT

qT

'qT

x

z 'z

'y y

T� R�

1( )ns 2( )ns

1pn�

1( )nT�

R� T� �

1n q�

2( )nR�

pT

1( )nR�

2( )nT�

1( )nR� �

T R

1 2n n�

2pn�

2n q�

2n

1n

2( )nR�

2( )nT�

Fig. 2. The detailed geometry of the single- and double-bounced rays inthe two-sphere model of the proposed 3D RS-GBSM.

5) The statistical properties of our SoS simulation modelare verified by comparing with those of the referencemodel and simulated results. The results show that thesimulation model is an excellent approximation of thereference model according to their statistical properties.

The remainder of this paper is structured as follows. Sec-tion II introduces a novel 3D theoretical RS-GBSM for non-isotropic narrowband MIMO V2V Ricean channels. In Sec-tion III, the corresponding 3D simulation model is developedwith parameters calculated by the MEV. Simulation resultsand analysis are unveiled in Section IV. Finally, we drawconclusions in Section V.

II. A NOVEL 3D MIMO V2V THEORETICAL RS-GBSM

A. Description of the 3D MIMO V2V theoretical RS-GBSM

Let us consider a narrowband MIMO V2V communicationsystem with MT transmit and MR receive omnidirectionalantenna elements. The radio propagation environment is char-acterized by 3D effective scattering with LoS and non-LoS(NLoS) components between the Tx and Rx. Different fromphysical scatterers, an effective scatterer may include severalphysical scatterers which are unresolvable in delay and angledomains. Figs. 1 and 2 illustrate the proposed 3D RS-GBSM,which is the combination of LoS components, a single-and double-bounced two-sphere model, and a single-bouncedelliptic-cylinder model. To consider the impact of the VTDon channel statistics, we need to distinguish between themoving vehicles around the Tx and Rx and the stationaryroadside environments (e.g., buildings, trees, parked cars, etc.).Therefore, we use a two-sphere model to mimic the moving


TABLE IDEFINITION OF PARAMETERS IN FIGS. 1 AND 2.

D distance between the centers of the Tx and Rx spheresRT , RR radius of the Tx and Rx spheres, respectivelya, f semi-major axis and half spacing between two foci of the elliptic-cylinder, respectively

δT , δR antenna element spacing at the Tx and Rx, respectivelyθT , θR orientation of the Tx and Rx antenna array in the x-y plane, respectivelyϕT , ϕR elevation of the Tx and Rx antenna array relative to the x-y plane, respectivelyυT , υR velocities of the Tx and Rx, respectively.γT , γR moving directions of the Tx and Rx in the x-y plane, respectivelyα(ni)T azimuth angle of departure (AAoD) of the waves that impinge on the effective

(i = 1, 2, 3) scatterers s(ni)

α(ni)R azimuth angle of arrival (AAoA) of the waves traveling from the effective


β(ni)T elevation angle of departure (EAoD) of the waves that impinge on the effective


β(ni)

R elevation angle of arrival (EAoA) of the waves traveling from the effective(i = 1, 2, 3) scatterers s(ni)

αLoSR , βLoS

R AAoA and EAoA of the LoS paths, respectivelyεpq , εpni , εn1n2 ,

εniq , ξ, ξn3T (R), distances d (Tp, Tq), d

(Tp, s

(ni))

, d(s(n1), s(n2)

), d

(s(ni), Tq

), d (Tp, OR),

ξn1 , ξn2 d(OT (OR), s

(n3))

, d(s(n1), OR

), d

(OT , s

(n2))

, respectively(i = 1, 2, 3)

vehicles and an elliptic-cylinder model to depict the stationaryroadside environments. It is worth mentioning that in orderto significantly reduce the complexity of the 3D theoreticalRS-GBSM, only the double-bounced rays via scatterers onthe two-sphere model are considered because other double-bounced rays (via one scatterer on a sphere and the otherone on the elliptic-cylinder) show similar channel statistics[14]. For readability purposes, Fig. 1 only shows the geometryof LoS components, and the single-bounced elliptic-cylindermodel. The detailed geometry of the single- and double-bounced two-sphere model is given in Fig. 2. Note that in bothFigs. 1 and 2, we adopted uniform linear antenna arrays withMT =MR = 2 as an example. The proposed RS-GBSM canbe extended with arbitrary numbers of antenna elements. Bymodeling effective scatterers, we assume that the two-spheremodel defines two spheres of effective scatterers, one aroundthe Tx and the other around the Rx. Suppose there are N1

effective scatterers around the Tx lying on a sphere of radiusRT and the n1th (n1 = 1, ..., N1) effective scatterer is denotedby s(n1). Similarly, assume there are N2 effective scatterersaround the Rx lying on a sphere of radius RR and the n2th(n2 = 1, ..., N2) effective scatterer is denoted by s(n2). Forthe elliptic-cylinder model, N3 effective scatterers lie on anelliptic-cylinder with the Tx and Rx located at the foci andthe n3th (n3 = 1, ..., N3) effective scatterer is denoted bys(n3). The parameters in Figs. 1 and 2 are defined in Table I.Note that the reasonable assumptions D � max{RT , RR}and min{RT , RR, a− f} � max{δT , δR} are applied in thistheoretical model [14].

The 3D MIMO V2V channel is described by an MT ×MR matrix of complex fading envelopes, i.e., H (t) =[hpq (t)]MT×MR

. The subscripts p and q denote the MIMOantenna elements. Therefore, the received complex fadingenvelope between the pth (p = 1, ...,MT ) Tx and the qth

(q = 1, ...,MR) Rx at the carrier frequency fc is a superposi-tion of the LoS, single- and double-bounced components, andcan be expressed as

hpq (t) = hLoSpq (t) +

I∑i=1

hSBipq (t) + hDBpq (t) (1)

where

hLoSpq (t) =

√K

K + 1e−j2πfcτpq

× ej2πfTmax t cos(αLoST −γT ) cosβLoS

T (2a)

× ej2πfRmax t cos(αLoSR −γR) cos βLoS

R

hSBipq (t) =

√ηSBi

K + 1lim

Ni→∞

Ni∑ni=1

1√Niej(ψni

−2πfcτpq,ni)

× ej2πfTmax t cos

(α

(ni)

T −γT)cosβ

(ni)

T (2b)

× ej2πfRmax t cos

(α

(ni)

R −γR)cos β

(ni)

R

hDBpq (t) =

√ηDBK + 1

× limN1,N2→∞

N1,N2∑n1,n2=1

1√N1N2

ej(ψn1,n2−2πfcτpq,n1,n2)

× ej2πfTmax t cos

(α

(n1)

T −γT)cos β

(n1)

T (2c)

× ej2πfRmax t cos

(α

(n2)

R −γR)cosβ

(n2)

R

with αLoST ≈ βLoST ≈ βLoSR ≈ 0, αLoSR ≈ π, τpq = εpq/c,τpq,ni = (εpni + εniq)/c, and τpq,n1,n2 = (εpn1 + εn1n2 +εn2q)/c. Here, c is the speed of light, K designates theRicean factor, and I = 3 which means there are three


subcomponents for single-bounced rays, i.e., SB1 from the Txsphere, SB2 from the Rx sphere, and SB3 from the elliptic-cylinder. Power-related parameters ηSBi and ηDB specify theamount of powers that the single- and double-bounced rayscontribute to the total scattered power 1/(K + 1). Note thatthese power-related parameters satisfy

∑Ii=1 ηSBi +ηDB = 1.

The phases ψni and ψn1,n2 are independent and identicallydistributed (i.i.d.) random variables with uniform distributionsover [−π, π), fTmax and fRmax are the maximum Dopplerfrequencies with respect to the Tx and Rx, respectively. Notethat we have corrected inaccurate expressions (2a) and (2c) in[15], corresponding to (2a) and (2c) in this paper, respectively.

Based on the law of cosines in appropriate triangles andsmall angle approximations (i.e., sinx ≈ x and cosx ≈ 1 forsmall x), we have

εpq ≈ ξ − δR2ξ

[δT2

sinϕT sinϕR −Q cosϕR cos θR

](3a)

εpn1 ≈ RT − δT2

[sinβ

(n1)T sinϕT (3b)

+ cosβ(n1)T cosϕT cos(θT − α

(n1)T )

]

εn1q ≈ ξn1 −δR2ξn1

[RT sinβ

(n1)T sinϕR (3c)

− Qn1 cosϕR cos(α(n1)R − θR)

]

εpn2 ≈ ξn2 −δT2ξn2

[RR sinβ

(n2)R sinϕT (3d)

+ Qn2 cosϕT cos(α(n2)T − θT )

]

εn2q ≈ RR − δR2

[sinβ

(n2)R sinϕR (3e)

+ cosβ(n2)R cosϕR cos(θR − α

(n2)R )

]

εn1n2 ≈{[D −RT cosα

(n1)T −RR cos(α

(n1)R − α

(n2)R )

]2

+[RT cosβ

(n1)T −RR cosβ

(n2)R

]2}1/2

(3f)

εpn3 ≈ ξ(n3)T − δT

2ξ(n3)T

[ξ(n3)R sinβ

(n3)R sinϕT

+ Qn3 cosϕT cos(α(n3)T − θT )

](3g)

εn3q ≈ ξ(n3)R − δR

[sinβ

(n3)R sinϕR

+ cosβ(n3)R cosϕR cos(α

(n3)R − θR)

](3h)

where ξ ≈ Q ≈ D − δT2 cosϕT cos θT , ξn1 =√

Q2n1

+R2T sin2 β

(n1)T , Qn1 ≈ D−RT cosβ

(n1)T ×cosα

(n1)T ,

ξn2 =

√Q2n2

+R2R sin2 β

(n2)R , Qn2 ≈ D +

RR cosβ(n2)R cosα

(n2)R , ξ

(n3)R =

2a−Qn3

cosβ(n3)

R

, ξ(n3)T =√

Q2n3

+ (ξ(n3)R )2 sin2 β

(n3)R , and Qn3 =

a2+f2+2af cosα(n3)

R

a+f cosα(n3)

R

.

−π −π/2 0 π/2 π0

0.1

0.2

0.3

0.4

α

2D V

MF

PD

F

(b)

−π−π/2

0π/2

π

ππ/2

0−π/2

−π0

0.1

0.2

0.3

0.4

β

(a)

α

3D V

MF

PD

F

k=0.6

k=1.3

k=3.6

k=3.6

Fig. 3. (a) The 3D VMF PDF (α0 = 0◦ , β0 = 31.6◦ , k = 3.6) and (b)2D VMF PDF (α0 = 0◦, β0 = 31.6◦ , β = 0◦, k = 0.6, 1.3, 3.6).

Note that the azimuth/elevation angle of departure(AAoD/EAoD), (i.e., α

(ni)T , β

(ni)T ), and azimuth/elevation

angle of arrival (AAoA/EAoA), (i.e., α(ni)R , β(ni)

R ), are in-dependent for double-bounced rays, while are correlatedfor single-bounced rays. According to geometric algorithms,for the single-bounced rays resulting from the two-spheremodel, we can derive the relationship between the AoDsand AoAs as α

(n1)R ≈ π − RT

D sinα(n1)T , β

(n1)R ≈

arccos(D−RT cos β

(1)T cosα

(1)T

ξn1

), and α

(n2)T ≈ RR

D sinα(n2)R ,

β(n2)T ≈ arccos

(D+RR cosβ(2)R cosα

(2)R

ξn2

). For the single-

bounced rays resulting from elliptic-cylinder model, the

angular relationship α(n3)T = arcsin

( b2 sinα(n3)

R

a2+f2+2af cosα(n3)

R

)and β

(n3)T = arccos

[ a2+f2+2af cosα(n3)

R(a+f cosα

(n3)

R

)ξ(n3)

T

]hold with b =√

a2 − f2 denoting the semi-minor axis of the elliptic-cylinder. The undefined β

(n1)R , β(n2)

T , and β(n3)T in Line 12

of the left column on Page 3 in [15] have been given here.For the theoretical RS-GBSM, as the number of scatterers

tends to infinity, the discrete AAoD α(ni)T , EAoD β

(ni)T , AAoA

α(ni)R , and EAoA β

(ni)R can be replaced by continuous random

variables α(i)T , β(i)

T , α(i)R , and β

(i)R , respectively. In [26], the

assumption of 3D scattering has been validated. To jointlyconsider the impact of the azimuth and elevation angles onchannel statistics, we use the VMF PDF to characterize thedistribution of effective scatterers, which is defined as [27]

f (α, β) =k cosβ

4π sinh k× ek[cosβ0 cosβ cos(α−α0)+sin β0 sin β]

(4)

where α, β ∈ [−π, π), α0 ∈ [−π, π) and β0 ∈ [−π, π)account for the mean values of the azimuth angle α andelevation angle β, respectively, and k (k ≥ 0) is a real-valuedparameter that controls the concentration of the distributionrelative to the mean direction identified by α0 and β0.

To demonstrate the VMF distribution, we set the meanangles α0 = 0◦ and β0 = 31.6◦ as an example, and plotthe corresponding PDF in both 3D and 2D figures in Figs. 3(a) and (b), respectively. Fig. 3 (a) shows the 3D VMF PDFwith k = 3.6. For the purpose of comparison, in Fig. 3 (b), weplot the 2D VMF PDF only for azimuth angle α with β = 0◦


and different k = 0.6, 1.3, 3.6. Fig. 3 (b) tells that the largerthe value of k, the VMF PDF is more concentrated towardsthe mean direction. For k → 0 the distribution is isotropic,while for k → ∞ the distribution becomes extremely non-isotropic. For the high VTD scenario with many movingvehicles around the Tx and Rx, k is small and the scattererdistribution approaches isotropic. Note that when the elevationangle β = β0 = 0◦, the VMF PDF reduces to von Mises PDF,which has widely been applied as a scatterer distribution in 2Dpropagation environments [28]. In this paper, for the anglesof interest, i.e., the AAoD α

(1)T and EAoD β

(1)T for the Tx

sphere, the AAoA α(2)R and EAoA β

(2)R for the Rx sphere,

and the AAoA α(3)R and EAoA β

(3)R for the elliptic-cylinder,

the parameters (α0, β0, and k) of the VMF PDF in (4) can bereplaced by (α(1)

T0 , β(1)T0 , and k(1)), (α(2)

R0, β(2)R0 , and k(2)), and

(α(3)R0, β(3)

R0 , and k(3)), respectively.It is important to emphasize that the proposed model is

adaptable to a wide variety of V2V propagation environmentsby adjusting important parameters, which are the Ricean factorK , energy-related parameters ηSBi and ηDB , and environmentparameters k(i). In general, for a low VTD, the value of K islarge since the LoS component can bear a significant amountof power. In addition, the received scattered power is mainlyfrom waves reflected by the stationary roadside environmentsdescribed by the scatterers located on the elliptic-cylinder.The moving vehicles represented by the scatterers locatedon the two spheres are sparse and thus more likely to besingle-bounced, rather than double-bounced. This indicatesthat ηSB3 > max{ηSB1 , ηSB2} > ηDB . For a high VTD,the value of K is smaller than that in the low VTD scenario.Also, due to dense moving vehicles, the double-bounced raysof the two-sphere model bear more energy than single-bouncedrays of the two-sphere and elliptic-cylinder models, i.e.,ηDB > max{ηSB1 , ηSB2 , ηSB3}. Therefore, the considerationof the VTD can be well characterized by utilizing a combinedtwo-sphere model and elliptic-cylinder model with the LoScomponent.

B. Statistical properties of the 3D MIMO V2V RS-GBSM

For the proposed 3D MIMO V2V theoretical RS-GBSM,statistical properties will be derived in this section, i.e.,amplitude and phase PDFs, ST CF, Doppler PSD, envelopeLCR, and AFD.

1) Amplitude and phase PDFs: Based on the proposed 3Dtheoretical RS-GBSM, the amplitude and phase processes canbe expressed as ζ(t) = |hpq(t)| and ϑ(t) = arg {hpq(t)},respectively. According to the similar procedure in [29], theamplitude PDF of the 3D V2V reference model can be derivedas

pζ(z) =z

σ20

e− z2+K2

02σ2

0 I0(zK0

σ20

) (5)

where z presents the amplitude variable, K0 =√

KK+1 and

I0(·) is the zeroth-order modified Bessel function of the firstkind.

In addition, the phase PDF of the reference model can be

derived as

pϑ(θ) =e− K2

02σ2

0

2π

{1 +

K0

σ0

√π

2cos (θ − θK) e

K20cos2(θ−θK )

2σ20

×[1 + erf

(K0 cos(θ − θK)

σ0√2

)]}(6)

where θK = arg{hLoSpq (t)

}. Due to the page limit, detailed

derivations are omitted here.2) ST CF: Under the wide-sense stationary (WSS) condi-

tion, the normalized ST CF between any two complex fadingenvelopes hpq (t) and hp′q′ (t) is defined as [15]

ρhpqhp′q′ (τ) =E[hpq(t)h

∗p′q′(t− τ)

]√E[|hpq(t)|2]E [

|hp′q′(t)|2] (7)

= E[hpq (t)h

∗p′q′ (t− τ)

](K + 1)

where (·)∗ denotes the complex conjugate operation and E[·]designates the statistical expectation operator. Substituting (1)into (7) and applying the corresponding VMF distribution, wecan obtain the ST CF of the LoS, single-, and double-bouncedcomponents as follows:

(a) In the case of the LoS component,

ρhLoSpq hLoS

p′q′(τ) = Ke

j2πλ ALoS+j2πτ(fTmax cos γT−fRmax cos γR)

(8)

where ALoS = 2D cosϕR cos θR.(b) In terms of the single-bounced components SBi (i =

1, 2, 3) resulting from the Tx sphere, Rx sphere, and elliptic-cylinder, respectively,

ρhSBipq h

SBip′q′

(τ) = ηSBi

∫ π

−π

∫ π

−π

[e−

j2πλ A(i)

(9)

×ej2πτ(fTmaxB(i)+fRmaxC

(i))f(α(i)T/R, β

(i)T/R)

]d(α

(i)T/R, β

(i)T/R)

with A(1) = δT[sinβ

(1)T sinϕT + cosβ

(1)T cosϕT cos

(θT −

α(1)T

)]+ δRξn1

[RT sinβ

(1)T sinϕR−Qn1 cosϕR cos

(θR−α(1)

R

)],

B(i) = cos(α(i)T − γT

)cos

(β(i)T

), C(i) = cos

(α(i)R − γR

) ×cos

(β(i)R

), A(2) = δR

[sinβ

(2)R sinϕR + cosβ

(2)R cosϕR ×

cos(θR − α

(2)R

)]+ δT

ξn2

[RR sinβ

(2)R sinϕT + Qn2 cosϕT ×

cos(θT − α

(2)T

)], A(3) = δT

ξ(n3)

T

[ξ(n3)R sinβ

(3)R sinϕT +Qn3 ×

cosϕT cos(θT − α

(3)T

)]+ δR

[sinβ

(3)R sinϕR + cosβ

(3)R ×

cosϕR cos(θR − α

(3)R

)], where the expressions of α(i)

R , β(i)R ,

Qni , ξn1 , ξn2 , and ξn3

T (R) are given in Section II. A. Note thatthe subscripts T and R are applied to i = 1 and i = 2, 3,respectively.

(c) In terms of the double-bounced component resultingfrom the Tx and Rx spheres,

ρhDBpq hDB

p′q′(τ) = ρTpp′(τ)ρ

Rqq′ (τ) = ηDB

∫ π

−π

∫ π

−π

∫ π

−π

∫ π

−π[e−

j2πλ ADB · ej2πτ(fTmaxB

DB+fRmaxCDB) (10)

× f(α(1)T , β

(1)T ) · f(α(2)

R , β(2)R )

]d(α

(1)T , β

(1)T )d(α

(2)R , β

(2)R )


where ADB = δT[sinβ

(1)T sinϕT +cosβ

(1)T cosϕT cos

(θT −

α(1)T

)]+δR

[sinβ

(2)R sinϕR+cosβ

(2)R cosϕR cos

(θR−α(2)

R

)],

BDB = cos(α(1)T − γT

)cosβ

(1)T , and CDB = cos

(α(2)R −

γR)cosβ

(2)R .

The normalized theoretical ST CF can be expressed as thesummation of (8) – (10), i.e.,

ρhpqhp′q′ (τ) = ρhLoSpq hLoS

p′q′(τ) +

I∑i=1

ρhSBipq h

SBip′q′

(τ)

+ ρhDBpq hDB

p′q′(τ) . (11)

3) Doppler PSD: Applying the Fourier transformto the ST CF, we can obtain the correspondingDoppler PSD as Shpqhp′q′ (fD) = F

{ρhpqhp′q′ (τ)

}=∫∞

−∞ ρhpqhp′q′ (τ)e−j2πfDτdτ , where fD is the Doppler

frequency. Substituting (11) into the above equation, theDoppler PSD can be expressed as

Shpqhp′q′ (fD) = F{ρhLoS

pq hLoSp′q′

(τ)}+

I∑i=1

F

{ρhSBipq h

SBip′q′

(τ)

}

+ F{ρTpp′ (τ)

} F{ρRqq′ (τ)

}(12)

where denotes the convolution and F{·} indicates theFourier transform.

4) Envelope LCR and AFD: The LCR at a specified level r,L(r), is defined as the rate at which the signal envelope crosseslevel r in the positive/negative going direction. Using thetraditional PDF-based method [30], we derive the expressionof the LCR for V2V channels as

L(r) =2r√K + 1

π3/2

√b2b0

− b21b20

× e−K−(K+1)r2

×∫ π/2

0

cosh(2√K(K + 1) · r cos θ

)(13)

×[e−(χ sin θ)2 +

√πχ sin θ · erf(χ sin θ)

]dθ

where cosh(·) is the hyperbolic cosine function, erf(·) is the

error function, and χ =

√Kb21

(b0b2−b21). Finally, parameters b0,

b1, and b2 are defined as

b0�= E

[hInpq (t)

2]= E

[hQupq (t)

2]

(14)

b1�= E

[hInpq (t)h

Qupq (t)

]= E

[hQupq (t)h

Inpq (t)

](15)

b2�= E

[hInpq (t)

2]= E

[hQupq (t)

2]

(16)

where hInpq (t) and hQupq (t) denote the in-phase and quadra-ture components of the complex fading envelope hpq(t),and hInpq (t) and hQupq (t) denote the first derivative of hInpq (t)and hQupq (t), respectively. By substituting (1) into (14), theparameter b0 becomes

b0 =

I∑i=1

bSBi0 + bDB0 =

1

2(K + 1)(17)

where

bSBi0 =

ηSBi

2(K + 1)

∫ π

−π

∫ π

−πf(α

(i)T , β

(i)T )d(α

(i)T , β

(i)T )

=ηSBi

2(K + 1)(18a)

bDB0 =ηDB

2(K + 1)

∫ π

−π

∫ π

−πf(α

(1)T , β

(1)T )d(α

(1)T , β

(1)T )

×∫ π

−π

∫ π

−πf(α

(2)R , β

(2)R )d(α

(2)R , β

(2)R ) =

ηDB2(K + 1)

. (18b)

Similarly, by substituting (1) into (15) and (16), the parametersb1 and b2 become

bm =

I∑i=1

bSBim + bDBm , (19)

where m ∈ {1, 2} and

bSBim =

ηSBi

2(K + 1)(2π)m

∫ π

−π

∫ π

−πf(α

(i)R , β

(i)R )

×[fTmax cos

(α(i)T − γT

)cosβ

(i)T

]m(20a)

×[fRmax cos

(α(i)R − γR

)cosβ

(i)R

]md(α

(i)R , β

(i)R )

bDBm =ηDB

2(K + 1)(2π)m

∫ π

−π

∫ π

−πf(α

(1)T , β

(1)T )

×[fTmax cos

(α(1)T − γT

)cosβ

(1)T

]md(α

(1)T , β

(1)T )

×∫ π

−π

∫ π

−πf(α

(2)R , β

(2)R ) (20b)

×[fRmax cos

(α(2)R − γR

)cosβ

(2)R

]md(α

(2)R , β

(2)R ).

The AFD, T (r), is defined as the average time over whichthe signal envelope, |hpq(t)|, remains below a certain level r.In the proposed 3D RS-GBSM, the AFD can be written as[31]

T (r) =1−Q

(√2K,

√2(K + 1)r2

)L(r)

(21)

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

III. THE 3D SOS SIMULATION MODEL FOR MIMO V2VCHANNELS

Based on the proposed 3D theoretical RS-GBSM describedin Section II, the corresponding SoS simulation model canbe further developed by using finite numbers of scatterers orsinusoids N1, N2, and N3. According to (1) – (2c), the SoSsimulation model for the link Tp → Tq can be expressed as

hpq (t) = hLoSpq (t) +I∑i=1

hSBipq (t) + hDBpq (t) (22)

where

hLoSpq (t) =

√K

K + 1e−j2πfcτpq

× ej2πfTmax t cos(αLoST −γT ) cosβLoS

T (23a)

× ej2πfRmax t cos(αLoSR −γR) cos βLoS

R


hSBipq (t) =

√ηSBi

K + 1

Ni∑ni=1

1√Niej(ψni

−2πfcτpq,ni)

× ej2πfTmax t cos

(α

(ni)

T −γT)cos β

(ni)

T (23b)

× ej2πfRmax t cos

(α

(ni)

R −γR)cosβ

(ni)

R

hDBpq (t) =

√ηDBK + 1

N1,N2∑n1,n2=1

1√N1N2

ej(ψn1,n2−2πfcτpq,n1,n2)

× ej2πfTmax t cos

(α

(n1)

T −γT)cosβ

(n1)

T (23c)

× ej2πfRmax t cos

(α

(n2)

R −γR)cos β

(n2)

R .

It is clear that the unknown simulation model parametersto be determined are only the discrete AoDs and AoAs,while the remaining parameters are identical to those ofthe theoretical model. Our task is thus to determine thediscrete AAoDs (α(n1)

T , α(n2)T , α(n3)

T ), EAoDs (β(n1)T , β(n2)

T ,β(n3)T ), AAoAs (α(n1)

R , α(n2)R , α(n3)

R ), and EAoAs (β(n1)R ,

β(n2)R , β(n3)

R ) for the simulation model. Furthermore, thereare actually correlations between AoDs and AoAs for thesingle-bounce case. Therefore, we only need to determine the

discrete sets of{α(n1)T , β

(n1)T

}N1

n1=1,{α(n2)R , β

(n2)R

}N2

n2=1, and{

α(n3)R , β

(n3)R

}N3

n3=1. In [32], different parameter computation

methods have been introduced. In general, there are threewidely adopted methods, i.e., Extended Method of ExactDoppler Spread (EMEDS), Modified Method of Equal Areas(MMEA), and Lp-Norm method (LPNM). The EMEDS isespecially recommended for isotropic scattering. However, allthe above methods are only valid for 2D horizontal models. Tojointly calculate the azimuth and elevation angles, we proposea novel parameter computation method that can be applied toour 3D channel models. The method is named as MEV, whichis developed from MMEA [33].

A. MEV for parameterization of the proposed SoS simulationmodel

As we mentioned before, the VMF distribution is adoptedin order to jointly consider the impact of the azimuth andelevation angles on channel statistics. Furthermore, the cu-mulative distribution function (CDF) of α and β, i.e., thedouble integral of the 3D VMF PDF, denotes the volumeof Fig. 3 (a). The idea of MEV is designed to selectthe set of

{α(ni), β(ni)

}Ni

ni=1in such a manner that the

volume of the VMF PDF f(α, β) in different ranges of{α(ni−1), β(ni−1)

}� {α, β} < {

α(ni), β(ni)}

are equal to

each other with the initial condition∫ α(1)

−π∫ β(1)

−π f(α, β)dαdβ =1−1/4Ni

. The application of the MEV to the 3D V2V channelmodel requires the joint computation of the discrete model pa-

rameters, i.e.,{α(n1)T , β

(n1)T

}N1

n1=1,{α(n2)R , β

(n2)R

}N2

n2=1, and{

α(n3)R , β

(n3)R

}N3

n3=1. In the following, we will derive the MEV

that has the ability to meet the two accuracy-efficiency designcriteria [33] for 3D scattering MIMO V2V channels withthe joint VMF distribution. Using the design of the AAoDs

{α(n1)T

}N1

n1=1and EAoDs

{β(n1)T

}N1

n1=1as an example, the

MEV includes the following three steps:Step 1: Define a pair of random variables, i.e., αT

(n1) ∈[α(1)T0 − π, α

(1)T0 + π

)and βT

(n1) ∈[β(1)T0 − π, β

(1)T0 + π

).

They follow the VMF distribution having the same α(1)T0 , β(1)

T0 ,and k1.

Step 2: Temporarily design the proper set of{αT

(n1)}N1

n1=1

and{βT

(n1)}N1

n1=1, as αT

(n1), βT(n1)

:= F−1α/β

(n1−1/4N1

),

where F−1α/β(·) denotes the inverse function of the VMF CDF

derived from VMF PDF for αT(n1) and βT

(n1).

Step 3: Obtain the desired set of{α(n1)T

}N1

n1=1and{

β(n1)T

}N1

n1=1by mapping

{αT

(n1)}N1

n1=1and

{βT

(n1)}N1

n1=1into the range of [−π, π), respectively.

Consequently, the jointly calculated AAoDs and EAoDs,{α(n1)T , β

(n1)T

}N1

n1=1are obtained. Similarly, AAoAs{

α(n2)R

}N2

n2=1and

{α(n3)R

}N3

n3=1and EAoAs

{β(n2)R

}N2

n2=1

and{β(n3)R

}N3

n3=1can be obtained by following the same

procedure.

B. Statistical properties of the proposed SoS simulation model

Based on our 3D MIMO V2V theoretical RS-GBSM andits statistical properties, it is achievable to derive the corre-sponding statistical properties for the SoS simulation model.As the detailed derivations have been explained in SectionII.B, those of the corresponding simulation model with similarderivations are only briefly explained. Applying the discretemodel parameters to (5), (6), (11), (12), (13), and (21),we have the corresponding statistical properties for the SoSsimulation model as follows:

1) Amplitude and Phase PDFs: The amplitude and phaseprocesses of the SoS simulation model can be expressed asζ(t) =

∣∣∣hpq(t)∣∣∣ and ϑ(t) = arg{hpq(t)

}, respectively. Still

using the similar procedure in [29], the amplitude PDF of theSoS simulation model can be derived as

pζ(z) = 4π2z

∫ ∞

0

[N1∏n1=1

J0 (2π |GSB1 |x) (24)

×N2∏n2=1

J0 (2π |GSB2 |x)×N3∏n3=1

J0 (2π |GSB3 |x)

×N1,N2∏n1,n2=1

J0 (2π |GDB |x)]J0 (2πzx)J0 (2πK0x)xdx

where GSBi =√

ηSBi

Ni(K+1) (i = 1, 2, 3), and GDB =√ηDB

N1N2(K+1) .

In addition, the phase PDF of the SoS simulation model


can be derived as

pϑ(θ) = 2π

∫ ∞

0

∫ ∞

0

[N1∏n1=1

J0 (2π |GSB1 |x)

×N2∏n2=1

J0 (2π |GSB2 |x)×N3∏n3=1

J0 (2π |GSB3 |x)

×N1,N2∏n1,n2=1

J0 (2π |GDB |x)]

(25)

× J0

(2πx

√z2 +K2

0 − 2zK0 cos (θ − θK)

)xzdxdz.

2) ST CF: As we should represent the spatial components,here we rewrite the ST CF as

ρhpqhp′q′ (δT , δR, τ) = ρhLoSpq hLoS

p′q′(δT , δR, τ) (26)

+

I∑i=1

ρhSBipq h

SBip′q′

(δT , δR, τ) + ρhDBpq hDB

p′q′(δT , δR, τ) .

(a) In the case of the LoS component,

ρhLoSpq hLoS

p′q′(δT , δR, τ) = Ke

j2πλ ALoS

(27)

× ej2πτ(fTmax cos γT−fRmax cos γR).

Please note that the LoS ST CF of the SoS simulation modelis identical to that of the 3D theoretical RS-GBSM.

(b) In terms of the single-bounced components SBi (i =1, 2, 3) resulting from the Tx sphere, Rx sphere, and elliptic-cylinder, respectively,

ρhSBipq h

SBip′q′

(δT , δR, τ) =ηSBi

Ni

Ni∑ni=1

ej2πλ A(i)

(28)

× ej2πτ(fTmaxB(i)+fRmaxC

(i)).

(c) In terms of the double-bounced component resultingfrom the Tx and Rx spheres,

ρhDBpq hDB

p′q′(δT , δR, τ) = ρTpp′(δT , τ)ρ

Rqq′ (δR, τ)

= ηDB × 1

N1

N1∑n1

ej2πλ ADBT × ej2πτfTmaxB

DB

× 1

N2

N2∑n2

ej2πλ ADBR × ej2πτfRmaxC

DB

(29)

where ADBT = δT[sinβ

(1)T sinϕT +

cosβ(1)T cosϕT cos

(θT − α

(1)T

)], ADBR =

δR[sinβ

(2)R sinϕR + cosβ

(2)R cosϕR × cos

(θR − α

(2)R

)], and

ALoS , A(i), B(i), C(i), BDB , and CDB have been given inSection II. B.

3) Doppler PSD: The Doppler PSD of the SoS simulationmodel can be expressed as

Shpqhp′q′ (fD)= F{ρhLoS

pq hLoSp′q′

(τ)}+

I∑i=1

F

{ρhSBipq h

SBip′q′

(τ)

}

+ F{ρTpp′ (τ)

} F{ρRqq′ (τ)

}. (30)

4) Envelope LCR and AFD: Similarly, according to (13),the envelope LCR of the SoS simulation model, L(r), can bederived as

L(r) =2r√K + 1

π3/2

√b2

b0− b21

b20

× e−K−(K+1)r2 ×∫ π/2

0

cosh(2√K(K + 1) · r cos θ

)×[e−(χ sin θ)2 +

√πχ sin θ · erf(χ sin θ)

]dθ (31)

with χ =

√Kb21

b0 b2−b21, b0 = 1

2(K+1) , and bm =∑Ii=1 b

SBim +

bDBm (m = 1, 2), where

bSBim =

ηSBi

2(K + 1)(2π)m

× 1

Ni

Ni∑ni=1

[fTmax cos

(α(ni)T − γT

)cosβ

(ni)T (32a)

× fRmax cos(α(ni)R − γR

)cosβ

(ni)R

]m

bDBm =ηDBK + 1

(2π)m (32b)

× 1

N1

N1∑n1=1

[fTmax cos

(α(n1)T − γT

)cosβ

(n1)T

]m

× 1

N2

N2∑n2=1

[fRmax cos

(α(n2)R − γR

)cosβ

(n2)R

]m.

Similarly, according to (21), the envelope AFD of the SoSsimulation model, T (r), can be expressed as

T (r) =1−Q

(√2K,

√2(K + 1)r2

)L(r)

. (33)

IV. SIMULATION RESULTS AND ANALYSIS

In this section, we investigate both the 3D and 2D modelsin detail for each statistical property. Based on measuredscenarios in [9], the following main parameters were chosenfor our simulations: fc = 5.9 GHz, D = 300 m, fTmax =fRmax = 570 Hz, a = 180 m, RT = RR = 15 m,γT = γR = 0◦, ϕT = ϕR = 45◦, θT = θR = 45◦,α(1)T0 = 21.7◦, β(1)

T0 = 6.7◦, α(2)R0 = 147.8◦, β(2)

R0 = 17.2◦,α(3)R0 = 171.6◦, and β

(3)R0 = 31.6◦. Considering the con-

straints of the Ricean factor and power-related parametersin [15], we have k(1) = 9.6, k(2) = 3.6, k(3) = 11.5,K = 3.786, ηSB1 = 0.335, ηSB2 = 0.203, ηSB3 = 0.411,and ηDB = 0.051 for low VTD scenario. For high VTDscenario, we have k(1) = 0.6, k(2) = 1.3, k(3) = 11.5,K = 0.156, ηSB1 = 0.126, ηSB2 = 0.126, ηSB3 = 0.063, andηDB = 0.685. Please note that k(3) = 11.5 for both low VTDand high VTD scenarios are applied. Table II summarizeskey parameters adopted by low and high VTD scenarios. Theenvironment-related parameters k(1), k(2), and k(3) are relatedto the distribution of scatterers (normally, the smaller values ofk(1) and k(2) the more dense moving vehicles/scatterers, i.e.,the higher VTD). In both high and low VTDs, k(3) is largeas the scatterers reflected from static roadsides are normally


TABLE IIKEY PARAMETERS OF DIFFERENT VTD SCENARIOS.

K ηSB1 ηSB2 ηSB3 ηDB k(1) k(2) k(3)

Low VTD 3.786 0.335 0.203 0.411 0.051 9.6 3.6 11.5High VTD 0.156 0.126 0.126 0.063 0.685 0.6 1.3 11.5

concentrated. Also, Ricean factor K is small in higher VTD,as the LoS component does not have dominant power. Thereason is that dense vehicles (i.e., more vehicles/obstaclesbetween Tx and Rx) on the road result in less likelihood ofstrong LoS components. For the SoS simulation model, wemust first choose adequate values for the numbers of discretescatterers N1, N2, and N3. Based on our own simulationexperiences and suggested by [32], a reasonable values forNi can be 40, which can be considered as a good trade-off between realization complexity and accuracy. Certainly,if we simulate rigorous channels, e.g., very high VTD, thenumber of effective scatterers can be increased to improvethe performance of the channel simulator. In addition, whenβ(n1)T = β

(n2)R = β

(n3)R = 0◦, the proposed 3D model will

be reduced to a 2D two-ring and elliptic model. The impactof elevation angle is evaluated in this section by comparingbetween the 3D and 2D models in terms of their statisticalproperties.

A. Amplitude and phase PDFs

Figs. 4 and 5 show the amplitude and phase PDFs, respec-tively, for the 3D reference model, 3D simulation model withN1 = N2 = N3 = 40, and 3D simulation results for bothlow and high VTD scenarios. Note that the simulation resultswere obtained from the channel coefficients generated by theproposed channel simulator. It is clear that both amplitudeand phase PDFs of the simulation model, i.e., (24) and (25),respectively, are completely determined by the number ofscatterers Ni, the gains GSBi and GDB , and LoS amplitudeK0, whereas other model parameters have no influence atall. In addition, Figs. 4 and 5 demonstrate that the choiceof N1 = N2 = N3 = 40 is sufficient to obtain an excellentagreement between the simulation model and reference modelin both low and high VTD scenarios.

B. Temporal autocorrelation function

We investigate the temporal autocorrelation function (ACF),which can be derived from the ST CF (26) by setting dT =dR = 0. Therefore, the temporal ACF can be expressed as

ρhpqhp′q′ (τ) = ρhpqhp′q′ (0, 0, τ) . (34)

Fig. 6 presents the absolute values of the temporal ACFsfor the 3D reference model, 3D simulation model with N1 =N2 = N3 = 40, and 3D simulation result for both lowVTD and high VTD scenarios. The temporal ACFs of the 2Dsimulation model by setting β(n1)

T = β(n2)R = β

(n3)R = 0◦ are

also plotted in Fig. 6. It is clear that no matter what the VTDis, the ACFs of the 2D model always show higher correlationthan that of the 3D model. This means that the 2D modeloverestimates the temporal ACFs. From Fig. 6, we observethat both the 3D simulation model and 3D simulation resultclosely match the 3D reference model. Moreover, the VTD

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Amplitude

Am

plitu

de P

DF

3D Reference model3D Simulation model3D Simulation resultLow VTD

High VTD

Fig. 4. The amplitude PDFs for the 3D reference model, 3D simulationmodel, and 3D simulation result (δT = δR = 0, β(1)

T0 = 6.7◦, β(2)R0 =

17.2◦ , β(3)R0 = 31.6◦).

−π −π/2 0 π/2 π0

0.2

0.4

0.6

0.8

1

Phase

Pha

se P

DF

3D Reference model3D Simulation model3D Simulation result

Low VTD

High VTD

Fig. 5. The phase PDFs for the 3D reference model, 3D simulation model,and 3D simulation result (δT = δR = 0, β(1)

T0 = 6.7◦, β(2)R0 = 17.2◦,

β(3)R0 = 31.6◦).

significantly affects the temporal ACF. In low VTD scenario,the temporal ACF is always higher than that in high VTDscenario.

C. Spatial cross-correlation function

The spatial cross CF (CCF) can be derived from the ST CFby setting τ = 0. Therefore, the spatial CCF can be expressedas

ρhpqhp′q′ (δT , δR) = ρhpqhp′q′ (δT , δR, 0) . (35)

In simulations, the basic parameters are the same as beforeexcept for δT = 0.5λ. Fig. 7 presents the absolute values ofthe spatial CCFs for the 3D reference model, 3D simulationmodel, 3D simulation result, and 2D simulation model forthe low VTD and high VTD scenarios. Both 3D and 2D


0 0.002 0.004 0.006 0.008 0.010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time separation, τ (s)

Abs

olut

e va

lues

of t

he te

mpo

ral A

CF

3D Reference model3D Simulation model3D Simulation result2D Simulation model

High VTD

Low VTD

Fig. 6. The absolute values of the temporal ACFs for the 3D referencemodel, 3D simulation model, 3D simulation result, and 2D simulationmodel (δT = δR = 0, 3D model: β(1)

T0 = 6.7◦, β(2)R0 = 17.2◦, β(3)

R0 =

31.6◦, 2D model: β(n1)T = β

(n2)R = β

(n3)R = β

(1)T0 = β

(2)R0 = β

(3)R0 = 0◦).

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised antenna spacing, dR

/λ

Abo

solu

te v

alue

s of

the

spat

ial C

CF

3D Reference model3D Simulation model3D Simulation result2D Simulation model

High VTD

Low VTD

Fig. 7. The absolute values of the spatial CCFs for the 3D referencemodel, 3D simulation model, 3D simulation result, and 2D simulationmodel (τ = 0, 3D model: β

(1)T0 = 6.7◦, β(2)

R0 = 17.2◦ , β(3)R0 = 31.6◦,

2D model: β(n1)T = β

(n2)R = β

(n3)R = β

(1)T0 = β

(2)R0 = β

(3)R0 = 0◦).

simulation models have the number of effective scatterersN1 = N2 = N3 = 40. Again, from Fig. 7, it is clear thathigher VTD leads to lower spatial correlation properties. Thisis because the higher the VTD, the more spatial diversity theV2V channel has. Compared with the 3D models in Fig. 7,2D simulation model overestimates the spatial correlations. Inother words, the 2D model underestimates the spatial diversitygain. The reason is that the 2D model cannot capture thespatial diversity gain in the vertical plane. Moreover, in Figs. 6and 7 we have shown that 3D simulation results match those ofthe 3D simulation model very well, indicating the correctnessof our derivations. For clarity purposes, we only present 2Dand 3D simulation models in the rest of the figures.

D. Doppler PSD

As the Doppler PSD is derived from the Fourier transformof corresponding temporal ACF, Fig. 8 shows the DopplerPSD of the proposed 3D model compared with 2D one atdifferent VTDs. Comparing the Doppler PSDs with differentVTDs in Fig. 8, it shows that the higher the VTD, the

−1000 −500 0 500 1000−70

−60

−50

−40

−30

−20

−10

0

Doppler frequency, fD (Hz)

Nor

mal

ized

Dop

pler

PS

D (

dB)

3D Simulation model2D Simulation model

Low VTD

High VTD

Fig. 8. The normalized Doppler PSDs for the 3D and 2D simulation models(δT = δR = 0, 3D model: β(1)

T0 = 6.7◦ , β(2)R0 = 17.2◦, β(3)

R0 = 31.6◦, 2D

model: β(n1)T = β

(n2)R = β

(n3)R = β

(1)T0 = β

(2)R0 = β

(3)R0 = 0◦).

more evenly distributed the Doppler PSD is. The underlyingphysical reason is that in the high VTD scenario, the receivedpower comes from all directions reflected by moving vehicles.However, in the low VTD scenario, the received power comesmainly from specific directions identified by main stationaryroadside scatterers and LoS components. Fig. 8 also tells thatcompared with the 3D model, the 2D model underestimatesthe Doppler PSD in both low VTD and high VTD scenarios.

E. Envelope LCR and AFD

Figs. 9 and 10 depict the envelope LCRs and AFDs fordifferent VTD scenarios (low and high), respectively. Again,the VTD significantly affects the envelope LCR and AFD forV2V channels. Fig. 9 shows that the LCRs are smaller whenthe VTD is lower. Fig. 10 illustrates that the AFD tends tobe larger with lower VTD. However, the elevation angles donot influence the LCR and AFD remarkably. If we used thesame elevation parameters (i.e., β(1)

T0 = 6.7◦, β(2)R0 = 17.2◦,

and β(3)R0 = 31.6◦) as before, the LCR and AFD are barely

discernible. The difference is noticeable when we increase theelevation parameters to β

(1)T0 = β

(2)R0 = β

(3)R0 = 60◦ in Figs. 9

and 10. For the envelope LCR in Fig. 9, the 2D model showshigher LCR than the 3D model. For the envelope AFD, the2D model exhibits smaller AFD than the 3D model. Overall,the elevation angle has minor impact on the envelope LCRand AFD.

V. CONCLUSIONS

In this paper, we have proposed a novel 3D theoreticalRS-GBSM and corresponding SoS simulation model for non-isotropic scattering MIMO V2V fading channels. The pro-posed models have the ability to investigate the impact of theVTD and elevation angle on channel statistics. Furthermore,a novel parameter computation method, named as MEV,has been developed for jointly calculating the azimuth andelevation angles. Based on proposed models, comprehensivestatistical properties have been derived and thoroughly inves-tigated. The simulation results have validated the utility of theproposed model. The impact of the elevation angle on channel


−25 −20 −15 −10 −5 0 510

−4

10−3

10−2

10−1

100

101

Normalized envelope Level, r (dB)

Nor

mal

ized

env

elop

e LC

R, L

(R)/

f Tm

ax


Low VTD

High VTD

Fig. 9. The normalized envelope LCRs for the 3D and 2D simulationmodels (3D model: β(1)

T0 = β(2)R0 = β

(3)R0 = 60◦, 2D model: β(n1)

T = β(n2)R

= β(n3)R = β

(1)T0 = β

(2)R0 = β

(3)R0 = 0◦).

−25 −20 −15 −10 −5 0 510

−4

10−3

10−2

10−1

100

101

102

Normalized envelope Level, r (dB)

Nor

mal

ized

env

elop

e A

FD

, T(R

)*f T

max


High VTD

Low VTD

Fig. 10. The normalized envelope AFDs for the 3D and 2D simulationmodels (3D model: β

(1)T0 = β

(2)R0 = β

(3)R0 = 60◦ , 2D model: β

(n1)T =

β(n2)R = β

(n3)R = β

(1)T0 = β

(2)R0 = β

(3)R0 = 0◦).

statistical properties has been investigated and analyzed, i.e.,the difference between the 3D and 2D models. By comparingthese results, we can see that the VTD has a great impact onall channel statistical properties, whereas the elevation anglehas significant impact only on ST CF and Doppler PSD. Inaddition, our simulations and analysis have clearly addressedthat the low VTD condition always shows better channelperformance than the high VTD case. Compared with theexisting less complex 2D RS-GBSMs and 3D RS-GBSMs,the proposed 3D MIMO V2V RS-GBSMs are more practicalto mimic a real V2V communication environment. Our re-search work can be considered as a theoretical guidance forestablishing more purposeful V2V measurement campaigns inthe future.

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B. Jiao, “An improved parameter computation method for a MIMOV2V Rayleigh fading channel simulator under non-isotropic scatteringenvironments,”” IEEE Commun. Lett., vol. 17, no. 2, pp. 265–268, Feb.2013.

Yi Yuan received the BSc degree with distinctionin Electronic Engineering from Tianjin University ofTechnology, Tianjin, China, in 2006 and MSc degreewith distinction in Mobile Communications fromHeriot-Watt University, Edinburgh, U.K., in 2009.He has been a joint Ph.D. student at Heriot-WattUniversity and The University of Edinburgh since2010. His main research interests include mobile-to-mobile communications and advanced wirelessMIMO channel modeling and simulation.

Cheng-Xiang Wang (S’01-M’05-SM’08) receivedthe BSc and MEng degrees in Communicationand Information Systems from Shandong University,China, in 1997 and 2000, respectively, and the Ph.D.degree in Wireless Communications from AalborgUniversity, Denmark, in 2004.

He has been with Heriot-Watt University, Edin-burgh, U.K., since 2005 and was promoted to aProfessor in 2011. He is also an Honorary Fel-low of the University of Edinburgh, U.K., and aChair/Guest Professor of Shandong University and

Southeast University, China. He was a Research Fellow at the Universityof Agder, Grimstad, Norway, from 2001–2005, a Visiting Researcher atSiemens AG-Mobile Phones, Munich, Germany, in 2004, and a ResearchAssistant at Technical University of Hamburg-Harburg, Hamburg, Germany,from 2000–2001. His current research interests include wireless channelmodeling and simulation, green communications, cognitive radio networks,vehicular communication networks, Large MIMO, cooperative MIMO, and5G wireless communications. He has edited one book and published one bookchapter and over 180 papers in refereed journals and conference proceedings.

Prof. Wang served or is currently serving as an editor for eight inter-national journals, including IEEE TRANSACTIONS ON VEHICULAR TECH-NOLOGY (2011–) and IEEE TRANSACTIONS ON WIRELESS COMMUNICA-TIONS (2007–2009). He was the leading Guest Editor for IEEE JOURNALON SELECTED AREAS IN COMMUNICATIONS, Special Issue on VehicularCommunications and Networks. He served or is serving as a TPC member,TPC Chair, and General Chair for over 70 international conferences. Hereceived the Best Paper Awards from IEEE Globecom 2010, IEEE ICCT2011, ITST 2012, and IEEE VTC 2013-Fall. He is a Fellow of the IET, aFellow of the HEA, and a member of EPSRC Peer Review College.

Xiang Cheng (S’05-M’10-SM’13) received the PhDdegree from Heriot-Watt University and the Univer-sity of Edinburgh, Edinburgh, U.K., in 2009, wherehe received the Postgraduate Research Thesis Prize.

He has been with Peking University, Bejing,China, since 2010, first as a Lecturer, and then as anAssociate Professor since 2012. His current researchinterests include mobile propagation channel model-ing and simulation, next generation mobile cellularsystems, intelligent transport systems, and hardwareprototype development and practical experiments.

He has published more than 80 research papers in journals and conferenceproceedings. He received the Best Paper Award from the IEEE InternationalConference on ITS Telecommunications (ITST 2012) and the IEEE Inter-national Conference on Communications in China (ICCC 2013). Dr. Chengreceived the “2009 Chinese National Award for Outstanding Overseas Ph.D.Student” for his academic excellence and outstanding performance. He hasserved as Symposium Co-Chair and a Member of the Technical ProgramCommittee for several international conferences.

Bo Ai (M’01-SM’09) is now working in BeijingJiaotong University as a professor and PhD advisor.He is vice director of State Key Lab. of Rail TrafficControl and Safety. He is also a vice director ofModern Communications Research Institute. He hasauthored five books and published over 140 scien-tific research papers in his research area. He has hold10 national invention patents and one US patent. Hehas been the research team leader for 16 nationalprojects and has won some scientific research prizessuch as the First Grade of Technology Advancement

Award of Shaanxi Province.He is serving as an associate editor for IEEE TRANSACTIONS ON

CONSUMER ELECTRONICS and an editorial member of Wireless PersonalCommunications. He has been notified by Council of Canadian Academies(CCA) that, based on Scopus database, Prof. Ai Bo has been listed as one ofthe Top 1% authors in his field all over the world. Prof. Ai Bo has also beenFeature Interviewed by IET Electronics Letters. His research interests includeradio wave propagation and wireless channel modeling, power amplifierpredistortion, LTE-R system and Cyber-physical System. He is an IET Fellow.

David I. Laurenson (M’90) is currently a SeniorLecturer at The University of Edinburgh, Scotland.His interests lie in mobile communications: at thelink layer this includes measurements, analysis andmodeling of channels, whilst at the network layerthis includes provision of mobility management andQuality of Service support. His research extendsto practical implementation of wireless networksto other research fields, such as prediction of firespread using wireless sensor networks, deploymentof communication networks for distributed control

of power distribution networks, and sensor networks for environmentalmonitoring and structural analysis.

He is the UK representative for URSI Commission C, and is an associateeditor for the Journal of Electrical and Computer Engineering, and theEURASIP Journal on Wireless Communications. He is a member of the IEEEand the IET.

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